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2305.07540 | Content-based jewellery item retrieval using the local region-based
histograms | Jewellery item retrieval is regularly used to find what people want on online
marketplaces using a sample query reference image. Considering recent
developments, due to the simultaneous nature of various jewelry items, various
jewelry goods' occlusion in images or visual streams, as well as shape
deformation, content-based jewellery item retrieval (CBJIR) still has
limitations whenever it pertains to visual searching in the actual world. This
article proposed a content-based jewellery item retrieval method using the
local region-based histograms in HSV color space. Using five local regions, our
novel jewellery classification module extracts the specific feature vectors
from the query image. The jewellery classification module is also applied to
the jewellery database to extract feature vectors. Finally, the similarity
score is matched between the database and query features vectors to retrieve
the jewellery items from the database. The proposed method performance is
tested on publicly available jewellery item retrieval datasets, i.e. ringFIR
and Fashion Product Images dataset. The experimental results demonstrate the
dominance of the proposed method over the baseline methods for retrieving
desired jewellery products. | Amin Muhammad Shoib, Summaira Jabeen, Changbo Wang, Tassawar Ali | 2023-05-12T15:06:17Z | http://arxiv.org/abs/2305.07540v1 | # Content-based jewellery item retrieval using the local region-based histograms
###### Abstract
Jewellery item retrieval is regularly used to find what people want on online marketplaces using a sample query reference image. Considering recent developments, due to the simultaneous nature of various jewelry items, various jewelry goods' occlusion in images or visual streams, as well as shape deformation, content-based jewellery item retrieval (CBJIR) still has limitations whenever it pertains to visual searching in the actual world. This article proposed a content-based jewellery item retrieval method using the local region-based histograms in HSV color space. Using five local regions, our novel jewellery classification module extracts the specific feature vectors from the query image. The jewellery classification module is also applied to the jewellery database to extract feature vectors. Finally, the similarity score is matched between the database and query features vectors to retrieve the jewellery items from the database. The proposed method performance is tested on publicly available jewellery item retrieval datasets, i.e. ringFIR and Fashion Product Images dataset. The experimental results demonstrate the dominance of the proposed method over the baseline methods for retrieving desired jewellery products.
Keywords:Fashion item retrieval jewellery retrieval region extractor histograms similarity matching
## 1 Introduction
Content-based fashion item retrieval (CBFIR) methods retrieve similar products from a huge cluster of fashion item images. This lessens the dependency of the CBFIR system on text, enabling a better precise and immediate search for the wanted fashion item [1; 2]. CBFIR has many applications in the fashion industry, like fashion item retrievals [3], fashion product parsing [4], fashion outfit recommendations [5], fashion analysis [6], and many more. Many unimodal and multimodal deep learning techniques are used recently for the retrieval of desired fashion products from the huge collection of data streams [7; 8; 9; 10; 11]. Whenever a customer submits a keyword, source image, or image with some additional text, drawing, or visual stream from their everyday events, the CBFIR may locate fashion items or goods using identical or comparable attributes for e-commerce purchases [12]. Many e-commerce websites provide clothing retrieval services for users by keyword query [13; 14]. But keyword-based retrieval can hardly
meet the requirement of exact recovery of a particular clothing style. Content-based fashion product retrieval can fill the gap to a certain extent, and some existing websites and applications support content-based fashion item retrieval. But the results of low-level feature-based image retrieval still have semantic gaps with user requirements, and still, there is ample space for improvement in this area.
In fashion item retrieval systems, desired fashion products from large databases are retrieved using text or reference images. Traditional text-based retrieval systems employ the real text on website information, which simplifies indexing and keyword extraction. Keyword-based or text-based fashion item retrieval techniques have a simpler architecture design when compared to the CBFIR systems [15; 16]. Prior text-based retrieval systems relied heavily on manual labeling of images, where the experts annotate the portions of images using the specified keywords. Such manual labelling is then utilized to retrieve fashionable goods or products. The manually performed annotation of the content of images is subjective and time-consuming. Different annotators add different descriptions to the same clothing item. In a similar way based on the situation at hand, a single individual may think about the same sight many times at various periods.
To get the desired jewelry goods, content-based jewelry item retrieval (CBJIR) is often employed in online shopping platforms and searches on sites such as Taobao, Jingdong, Google, Baidu, and many more. Individuals adhere photographing their daily surroundings and purchasing their favorite goods online [17; 18]. Consumers may speedily find the selected jewelery products online with the help of the CBJIR methodology. However, research on complex fashion products such as jewellery items has little momentum due to the simultaneous nature of multiple jewellery products, different jewellery products' occlusion in images or visual streams, shape deformation, and the unavailability of appropriate datasets. According to industry studies, the scale of the local and worldwide fashion marketplaces is continually growing. Although the technology for obtaining images of fashion goods has progressed substantially over the last two decades, retrieving jewelry items in cross-domain situations still needs improvement.
To overcome the above challenges, we proposed a content-based jewellery item retrieval system that uses the local region-based histograms in HSV color space to retrieve jewellery items better. Our novel Jewellery Classification Module (JCM) plays an integral part in extracting the specific features from the queried and jewellery databases. The JCM extracts the specific feature vectors from the query image using five local regions explained in section 3. The JCM is also applied to the jewellery databases to remove the feature vectors from the vast collection of items. Finally, the similarity score between the database and the query features is matched to retrieve the most relevant jewellery items from the database.
The rest of the article is arranged as follows; section 2 reviews some recent related work of content-based fashion image retrieval systems. The complete description of the proposed Content-based jewellery item retrieval using the local region-based histograms is presented in section 3. The implementation of the proposed method and comparative analysis with the baseline methods are discussed in section 4. Finally, the conclusion of the article is presented in section 5.
## 2 Related Work
In recent decades, fashion research has advanced significantly, with the use of reference photos for fashion retrieval being among the most effective methodologies and a research hotspot. Every day, thousands of images and their associated electronic information are being added to online shopping web pages, worsening the fashion retrieval problem. Utilizing similarity measurement and feature extraction viewpoints, researchers have contributed to improving the precision of Fashion Item Retrieval (FIR) in current years [19; 20]. Several artificial intelligence frameworks were developed in recent decades to improve FIR performance while maximizing computational resource utilization. FIR frameworks are categorized into two main parts, i.e. text-based fashion item retrieval systems and content-based fashion item retrieval systems.
**Text-based fashion items retrieval (TBFIR) systems:** The TBFIR systems retrieve the desired products for the consumers based on provided text or keyword queries. Text/Keyword-based fashion item retrieval systems have a simpler architecture than content-based retrieval systems. The approaches proposed for TBFIR systems range from a "simple frequency-of-occurrence-based scheme to an ontology-based scheme [21]." TBFIR is more effective as compared to content-based FIR models in handling semantic inquiries.
Previous unimodal-text retrieval systems relied heavily on manual picture annotations, where the user annotated the content of fashion items using keywords. Such manual comments are then utilized to retrieve the clothing items from the databases [1]. The manually performed annotation of the content of the image is time-consuming and a subjective process. Different annotation experts add distinct explanations to the
same clothing item. In a similar vein based on the situation at hand, a single individual may think about the same sight countless times during various periods [22]. As a result, annotations made manually can be used in particular domains such as virtual museums, online libraries, individual recordings, and many more.
In text-based retrieval models, automatic image indexing could be an approach to this problem. Additionally, there are numerous autonomous indexing strategies, the most prominent of which is "count the frequency of occurrence of words." The actual distance between the words (used to describe the image) determines the weighting of the words. Moreover, reactions from customers to image results may be utilized to enhance keywords. The weighting technique for image keywords similarly heavily depends upon the domain. Many aesthetic features of fashion products are difficult to convey in terms. Figure 1 presents the general outcome of text-based fashion item retrieval methods using text or keywords as a query.
**Content-based fashion item retrieval (CBFIR) systems:** The CBFIR systems are used to retrieve the desired products for the consumer based on provided reference images[23]. In content-based retrieval systems, feature extraction strategies are critical for the retrieval of desired clothing items from the huge collection of images [24; 25; 26]. Where the similarity and feature vectors (FV) are two essential mechanisms. In the CB
Figure 1: General outcome of Text-Based Fashion Item Retrieval methods using text/keywords query.
Figure 2: General outcome of Content-Based Fashion Item Retrieval methods using image query.
FIR system, the features are extracted to represent the unique instance of queried and databased images. The extracted features are used for calculating database and reference image similarities. Utilizing similarity measurement and feature extraction approaches, researchers have contributed to increasing the precision of unimodal FIR in recent years. Several artificial intelligence frameworks have already been suggested in recent decades to improve FIR achievement while making the best use of computing power [27]. As compared to TBFIR systems, using a reference image as an input in CBFIR enables customers to convert rich information regarding their preferred fashion product or item.
Likewise, using a reference image as an input query to the retrieval system has unified features as compared to putting text/keywords as an input query. But if some consumers desire some additional attributes regarding the fashion item along with the queried reference image, then the content-based fashion item retrieval systems are not able to retrieve the desired product with these additional attributes. Figure 2 presents the general outcome of Content-based fashion item retrieval methods using reference images as a query.
## 3 Proposed Method
Content-based Jewellery item retrieval (CBJIR) method quantifies the content of images. Mostly, consumers use jewelry item retrieval systems to search for their desired jewellery products or items on online marketplaces using a sample query reference image. Considering recent developments, due to the simultaneous nature of various jewelry items, jewelry goods' occlusion in images or visual streams, as well as shape deformation, CBJIR still has limitations whenever it pertains to visual searching in the actual world. To overcome these limitations, in this research we proposed a region-based jewellery classification module using local histograms to extract the specific feature vectors from the queried image to retrieve desired jewellery items. The general structure diagram of the proposed content-based jewellery item retrieval method is presented in Figure 3. We provide the cropped jewellery item image to the jewellery classification module to retrieve selected jewellery products. In the preprocessing stage, we first convert the RGB space of the query image into the 3D HSV color space. RGB values are used to quantify the simple values of an image and cannot impressionist the human-style perception of colors. We designed a specific 3D HSV color space to illustrate human-style color perception better. In the jewellery classification module, the image descriptor optimally selects the number of bins for 3D HSV color space histograms. The quantization of pixel intensities of a query image is elaborated with these histogram bins. We optimally select the number of bins for 3D HSV color space histograms. To yield the maximum dimensions of feature vectors, we choose 10 bins for the hue channel, 3 bins for the value channel, and 14 bins for the saturation channel.
Figure 3: Flow diagram of proposed content-based jewellery item retrieval method.
We proposed a novel region extractor in the jewellery classification module to compute the local region-based histogram for 3D HSV color space. We add the local region-based histogram rather than the global histogram of the entire queried image to calculate the locality while extracting feature vectors. To calculate the locality of the queried image and database images, we partitioned the images into the top left and right, bottom left and right, and central regions. The Jewellery classification module used region-based localities to extract the optimum feature vectors from the query image to enhance the retrieval outcome of jewellery items. Figure 4 represents the division of \(rTl,rTr,rBl,rBr\), and \(ellipseC\) region of queried image. We can calculate these regions by using four indexes of the input image, i.e. startX, endX, startY, and endY. The \(rTl,rTr,rBl,rBr\), and \(ellipseC\) are calculated using Equations 1 and 2, simultaneously.
\[\begin{split}& rTl=(\theta,cX,\theta,cY)\\ & rTr=(cX,w,\theta,cY)\\ & rBl=(\theta,cX,cY,h)\\ & rBr=(cX,w,cY,h)\\ & region=[(rTl),(rTr),(rBr),(rBl)]\\ \end{split} \tag{1}\]
Where \(rTl,rTr,rBr\), and \(rBl\) contain the input image's starting and ending index of \((X,Y)\)-coordinate. The value of \(\theta\) is equal to zero here to indicate the starting point of \((X,Y)\)-coordinate, \(w\) is the width and \(h\) is the height of an image, and \((cX,cY)\) represents the center \((X,Y)\)-coordinate.
In this jewellery classification module, we calculate the center region of the queried image with the perspective of an ellipse. The construction of the ellipse is represented in Equation 2.
\[\begin{split}& r=(int(w*0.7)/2,int(h*0.7)/2)\\ & ellipseC=(r,(cX,cY),(axesX,axesY),0,0,360,255,-1)\\ \end{split} \tag{2}\]
Where \((axesX,axesY)\) represents the length of the ellipse and \(r\) is the radius of ellipse. To optically extract feature vectors of jewellery items, we use a radius of the ellipse equal to 70 percent of the height and width of the queried image. 0 and 360 are starting and ending angles of an ellipse. 255 represents the color of an ellipse, and -1 represents the border size.
We loop over all five local regions of the queried image and construct a mask for each region separately to extract features from it. We update the feature vector list by popping over the rTl, rTr, rBl, rBr, and ellipse region of the image. The local histograms of the queried image are calculated using the explained number of bins for HSV space. Similarly, feature vectors of all the database images are calculated using the exact mechanism through the jewellery classification module. Chi-square distance calculates the similarities and dissimilarities between region-based histogram bins. The similarity matching between query and database features is calculated using chi-square distance using Equation 3.
\[X^{2}(xbin,ybin)=\frac{1}{2}\sum_{a=1}^{k}\frac{(xbin_{a}-ybin_{a})^{2}}{xbin _{a}+ybin_{a}} \tag{3}\]
Where xbin and ybin calculate similarities between the database and query feature bins. After similarity matching, the closest jewellery retrieval results from the database are retrieved. The proposed method is tested on the two well-known jewellery image retrieval datasets, which details are provided in the section 4.
## 4 Experiments and Results
We performed various sorts of experiments to assess the efficiency of the proposed methodology. On the Ring-FIR [28] and Fashion Product Images [29] datasets, we have contrasted the retrieval accuracy of the suggested approach to that of other mentioned retrieval strategies. The following demonstration shows the complete
Figure 4: Local region-based division for 3D HSV color space, where \(rTl=\) top-left region, \(rTr=\) top-right region, \(rBl=\) bottom-left region, \(rBr=\) bottom-right region, and \(ellipseC\) = central region.
detail of experimental configuration, setup, and results on the RingFIR and Fashion Product Images dataset. The performance of the content-based jewellery item retrieval using the local region-based histogram methods experiments are evaluated using the Top-k accuracy. Top-k accuracy measures the retrieval accuracy of relevant retrieved images from the database. Our experiments compute the top 1, 5, 10, 15, and 20 retrieval accuracy on the RingFIR and Fashion Product Images dataset.
### Datasets
**RingFIR:** The RingFIR [28] is a diversified collection of earrings from various jewellery catalogues. This dataset consists of approximately 2,651 high-quality different images of golden earrings. This collection of different images is structurally categorized into 46 classes according to their patterns, design, and structures. The RingFIR is one of the most suitable datasets for our proposed method for retrieval of desired jewellery items from the database. Figure 5 presents the random sample images of the RingFIR dataset.
**Fashion Product Images dataset:** The Fashion Product Images dataset [29] contains 44,441 number of images along with a 3-class hierarchy. Additionally, the dataset contains a masterCategory and subCategory levels with 4 and 21 classes simultaneously. This dataset contains mixed jewellery items like necklaces, earrings, bracelets, rings, etc. To assess the performance of the proposed Content-based jewellery item retrieval using the local region-based histograms method we extract all the jewellery item images from the vast collection of Fashion Product Images dataset. A total of 1,081 different jewellery item images are available in this dataset which we used for the retrieval task of the proposed method. Figure 6 presents the random sample images of the RingFIR dataset.
### Experimental setup and results:
The proposed method experiments on the referred above datasets are conducted on Intel(R) Core(TM) i7-9750H CPU, 32GB-RAM system with NVIDIA GeForce GTX 1660Ti GPU. Anaconda is used for the development environment with Pillow, Flask, TensorFlow, Keras, and other libraries. In the proposed method, the jewellery classification module is used to extract the specific feature vectors from the queried image, which are then used for similarity matching of extracted database features. The region and region-based feature extractors are vital in extracting specific jewellery item features from the query image and jewellery databases.
The following demonstrations show the detailed results and effectiveness of this work. The performance of the proposed content-based jewellery item retrieval using the local region-based histograms method is first evaluated on the RingFIR dataset, where we used a 90:10 percent ratio for training and testing of images for the retrieval of jewellery items. For better evaluation of results, we perform experiments on test split of the RingFIR dataset and some experiments on the cropped jewellery items images from out-of-database sources. Figure 7 shows the retrieved images results of the test split on the RingFIR dataset. Query images are taken from the test split of the RingFIR dataset which is presented on the left side of the figure and the right side shows the top 1, 2, 3, 4, and 5 retrieved images results. Similarly, Figure 8 shows the retrieved images of out-of-database queried photos, where cropped query images of jewellery items are taken from the miscellaneous web sources presented on the left side of the figure. The right side presents the top 1, 2, 3, 4, and 5 retrieved image results.
Figure 5: RingFIR dataset: A) random sample examples of different classes, B) Images distribution in different classes [28]
Figure 6: Random sample examples of different Fashion Product Images dataset [29].
Figure 8: RingFIR retrieved results from the test split, where the green box presents successful retrieval and the red box presents unsuccessful retrieval.
Figure 7: RingFIR retrieved results from the test split, where the green box presents successful retrieval and the red box presents unsuccessful retrieval.
The performance of the content-based jewellery item retrieval using the local region-based histograms method's experiments is evaluated using the Top-k accuracy. Where, we compute the top 1, 5, 10, 15, and 20 retrieval accuracy on the RingFIR dataset. The quantitative retrieval results of the baseline methods and the proposed method on the test split of the RingFIR dataset are presented in Table 1. The experimental results clearly show the improvement of the proposed method over the baseline methods. Our proposed method obtains 32.67, 59.31, 74.24, 78.18, and 90.18 top 1, 5, 10, 15, and 20 respectively. The results clearly show improvement in top 1, 5, 10, and 20 retrieval accuracy over the baseline methods except for the top-15 accuracy of the DSSN [34] method. Additionally, the quantitative retrieval results of the baseline methods and the proposed method from the out-of-database reference image on the RingFIR dataset are presented in Table 2. The experimental results clearly show the improvement of the proposed method over the baseline methods. Our proposed method obtains 22.67, 56.26, 72.19, 75.51, and 85.25 top 1, 5, 10, 15, and 20 respectively. The results clearly show improvement in the top 5, 10, and 20 retrieval accuracy over the baseline methods except for the top 1 and 15 accuracy of the DSSN [34] method.
Additionally, the proposed method is also evaluated on the Fashion Product Images dataset, where we similarly used a 90:10 percent ratio for training and testing images for the retrieval of jewellery items. Firstly, we perform experiments on the test split of the Fashion Product Images dataset and then perform additional experiments on the cropped jewellery items images from out-of-database sources. Figure 9 shows the retrieved images results of the test split on the Fashion Product Images dataset. Query images are taken from the test split of the Fashion Product Images dataset which is presented on the left side of the figure and the right side shows the top 1, 2, 3, 4, and 5 retrieved images results. Similarly, Figure 10 shows the retrieved images results of the out-of-database queried images on the Fashion Product Images dataset, where cropped query images of jewellery items are taken from the miscellaneous web sources presented on the left side of the figure and the right side presents the top 1, 2, 3, 4, and 5 retrieved images results.
Figure 10: Fashion Product Images dataset retrieved results from the test split, where the green box presents successful retrieval, and the red box presents unsuccessful retrieval.
Figure 9: Fashion Product Images dataset retrieved results from the test split, where the green box presents successful retrieval and the red box presents unsuccessful retrieval.
## 5 Conclusion
A content-based jewellery item retrieval method using the local region-based histograms in HSV color space is proposed to achieve better accuracy. The core part of the proposed method is the jewellery classification module, which extracts the localities of jewellery items from the queried image based on five regions. The retrieval accuracy of the proposed method also improves by applying this jewellery classification module to the jewellery databases to extract the feature vectors optimally. The experimental results on the RingFIR and the Fashion Product Images dataset show the dominance of the proposed method over the baseline methods. This CBJIR method has significantly impacted the fashion industry to retrieve the desired jewellery items for their outfits. Still, there needs to be more databases available publicly to make the retrieval predictions better. In the future, a vast collection of jewellery item databases will be required to improve the retrieval accuracy of the CBJIR methods.
## Declarations
### Conflict of interest
The authors declare that they have no conflict of interest. The authors also declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper
|
2310.03590 | Dielectronic satellite emission from a solid-density Mg plasma:
relationship to models of ionisation potential depression | We report on experiments where solid-density Mg plasmas are created by
heating with the focused output of the Linac Coherent Light Source x-ray
free-electron-laser. We study the K-shell emission from the Helium and
Lithium-like ions using Bragg crystal spectroscopy. Observation of the
dielectronic satellites in Lithium-like ions confirms that the M-shell
electrons appear bound for these high charge states. An analysis of the
intensity of these satellites indicates that when modelled with an
atomic-kinetics code, the ionisation potential depression model employed needs
to produce depressions for these ions which lie between those predicted by the
well known Stewart-Pyatt and Ecker-Kroll models. These results are largely
consistent with recent Density Functional Theory calculations. | G. Pérez-Callejo, T. Gawne, T. R. Preston, P. Hollebon, O. S. Humphries, H. -K. Chung, G. L. Dakovski, J. Krzywinski, M. P. Minitti, T. Burian, J. Chalupský, V. Hájková, L. Juha, V. Vozda, U. Zastrau, S. M. Vinko, S. J. Rose, J. S. Wark | 2023-10-05T15:14:17Z | http://arxiv.org/abs/2310.03590v2 | Dielectronic satellite emission from a solid-density Mg plasma: relationship to models of ionisation potential depression
###### Abstract
We report on experiments where solid-density Mg plasmas are created by heating with the focused output of the Linac Coherent Light Source x-ray free-electron-laser. We study the K-shell emission from the Helium and Lithium-like ions using Bragg crystal spectroscopy. Observation of the dielectronic satellites in Lithium-like ions confirms that the M-shell electrons appear bound for these high charge states. An analysis of the intensity of these satellites indicates that when modelled with an atomic-kinetics code, the ionisation potential depression model employed needs to produce depressions for these ions which lie between those predicted by the well known Stewart-Pyatt and Ecker-Kroll models. These results are largely consistent with recent Density Functional Theory calculations.
## I Introduction
The focused output of hard x-ray free-electron-lasers (FELs), such as the Linac Coherent Light Source (LCLS), with peak spectral brightnesses many orders of magnitude greater than those of any synchrotron, provides a means to create hot (many hundreds of eV) plasmas at exactly solid density. Each pulse created by the FEL can, when it is operating in self-amplified spontaneous emission (SAE) mode, contain of order a few mJ of energy, which can be focused to micron-scale spots with Be lenses or Kirkpatrick-Baez mirrors [1]. The short duration of the pulses (typically sub-100 fs) ensures that the x-ray energy is deposited in the solid target in the focal plane on a timescale short compared with its disassembly time. The combination of energy, spot size, and pulse duration noted above corresponds to intensities on target of order at least \(10^{17}\) Wcm\({}^{-2}\).
If the photon energy of the incoming FEL radiation exceeds that of the K-edge of the atoms in the cold target, and the subsequent ions created, the photoionization by the FEL results in copious K-shell core holes being created, which are subsequently filled by radiative decay from the upper levels or via the Auger process. Both the photoionized and Auger electrons are ejected into the continuum, rapidly thermalising via further collisional ionisation and electron-electron scattering, heating the electrons in the system to many tens or even hundreds of eV. The filling of the core holes created by the photoionisation typically occurs on a femtosecond timescale, short compared with the FEL pulse duration (which is typically several tens of femtoseconds), and certainly much shorter than the target disassembly time, effectively ensuring that the resultant K-shell emission comes from the target when it is still at solid density. Thus K-shell spectroscopy of the solid density plasma produced provides detailed information on the charge states produced.
There have now been several studies of the x-ray spectra produced from solid targets in the manner described above [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13], which have allowed a wealth of information to be gleaned concerning various properties and processes occurring in these dense plasmas, such as opacity [9], collisional rates [7; 12], and the phenomenon of saturable absorption in the x-ray regime [8]. Of particular relevance here are those studies which have looked at ionization potential depression (IPD) [3; 6; 11]. The lowering of the energy of the start of the continuum, or IPD, is a fundamental process that occurs in dense plasmas as a result of the electrostatic interactions between the atom or ion and the surrounding charged particles [14; 15; 16; 17], leading to a reduction in the atomic binding energies. A knowledge of where the continuum lies for ions has a direct impact on many important plasma processes, such as the ionization and ion charge state distribution, the equation of
state, and the opacity and transport properties.
In the work cited above, the IPD was ascertained experimentally for a particular charge state by noting the photon energy of the FEL for which copious K\({}_{\alpha}\) radiation for the relevant ion was produced, this being interpreted as the photon energy needed to create a K-shell core hole by photoionization into the continuum. This method was applied to elements of relatively low \(Z\) (12-14), and relied on the fact that for the electron temperatures produced - typically less than 200 eV - the atomic energies and plasma temperatures are such that the vast majority of the resultant \(K\)-shell emission cannot be produced via thermal processes, but only due to photoionization, and thus recorded x-ray spectra, although time-integrated over each individual pulse, are effectively gated by the pulse duration. The results were compared with predictions from simple semi-classical models often employed in atomic-kinetics calculations, for example the Stewart-Pyatt (SP) [16] and Ecker-Kroll (EK) models [17], whose mathematical expressions are discussed later in the text.
In the first of these x-ray heating experiments, studying Al, it was found that the EK model gave better agreement width the data [3]. As a result of matching the K-shell binding energy, no M-shell electrons were found to be bound. In that particular experiment the authors note that the spectra from some of the highest charge states (up to Helium-like) were complicated by the presence of K\({}_{\beta}\) radiation from the cold solid, and in this original work no firm conclusion about the degree of IPD for them was made. However, it was subsequently noted that were the EK model to be applied for the higher charge states as well, the M-shell would also not be bound [5]. This latter conclusion is at odds with the work of Hoarty and co-workers [18], who observed the presence of radiation from the He-\(\beta\) transition of Al in experiments using optical lasers for heating and compression, and for which the spectra could not be reconciled with the EK model. At the same time, the SP model was also found to be inadequate for modelling both experiments.
The difficulty of finding a single, semi-classical IPD model that captures the pertinent physics that such an approach purports to describe is perhaps not surprising, given that the plasmas under study are dense quantum systems. Indeed, the authors of ref. [3] note in their original work that the EK and SP models are _"ultimately both unlikely to capture fully the complex physics of atomic systems embedded within dense plasma environments over wide ranges of plasma conditions and charge states."_ The veracity of the above statement has been given further credence by recent extensive computational studies based on Density Functional Theory (DFT) [19]. These calculations reveal that one of the main underlying issues in such dense systems is that a binary distinction between electrons that are free, and in the so-called continuum, and those that are bound to an atom or ion, cannot definitively be made - the problem is inherently quantum mechanical in nature.
Nevertheless, owing to the manner in which the majority of standard atomic-kinetics-based calculations are constructed, their use requires that such a distinction between bound and free electrons is made and thus, at least for the foreseeable future, simple IPD models are likely to continue to be adopted. As a result, within the work encompassed by the DFT calculations cited above, the use of the technique of the inverse participation ratio was employed to give a measure of the degree of boundness of the Kohn-Sham wavefunctions [19], and hence deduce where the energy of the continuum ought best to be placed, were one forced to make such an artificial division between bound and free. Within the above constraints and caveats, it was found that for solid density Al and Mg plasmas, for the highest charge states the most appropriate energies at which to consider the electrons to be free would correspond to a position lying somewhere between those predicted by the SP and EK models, although the EK model would still give a better fit for the lower charge states. In particular, it was noted that for Li and He-like ions, it would appear that the M-shell should be treated as being bound, and radiation from the M-shell states should be experimentally observable. In fact, the authors of [19] report the direct observation of He- and Li-like Mg K\({}_{\beta}\) emission.
It is in the above context that we present here an analysis of the emission spectrum from the Helium and Lithium-like ions in an FEL-generated solid-density, optically-thin, Mg plasma. We specifically investigate the intensity of the dielectronic satellites, where the spectator electron lies in the \(n=3\) principal quantum number, i.e. the M-shell. Studies of the intensity of such satellites, be they due to L or M-shell spectator electrons, has a long history, and has previously been shown to be of use in determining plasma conditions in both astrophysical [20; 21; 22; 23; 24] and laboratory based plasmas [25; 26; 27; 28; 29; 30; 31; 32; 33].
In analysing the intensity of these M-shell satellites, our main finding is that the Li-like states for which the \(n=3\) level is occupied by one electron are in good thermal contact with the equivalent He-like state without the \(n=3\) electron, and as a result the intensity of these satellites is sensitive to the IPD. Furthermore, for best agreement with the experimental data given the parameters of the FEL, we need to place the IPD for these charge states between the EK and SP limits, as advised by the DFT studies, and to this degree the data is consistent with them.
## II Experimental set-up
The experiment was performed at the Soft X-Ray (SXR) end station [34] of the LCLS, using a set-up which has been thoroughly discussed in previous publications [9; 10; 35; 13].
Targets consisted of 54 -nm-thick Mg foils. These were irradiated using 100 fs x-ray pulses with a photon energy of 1540 eV. The nominal pulse energy was 1.7 mJ, which is reduced to 0.51 mJ after transmission through
the beamline optics [9; 10; 35]. The size of the focal spot on target was measured _ex situ_ using imprint measurements on PbI\({}_{2}\)[36], obtaining an effective focal spot area of 8.5 um\({}^{2}\), which corresponds to a maximum irradiance of \(\sim 10^{17}\) W cm\({}^{-2}\).
The targets were irradiated at a 45 \({}^{\circ}\) angle with respect to their normal, and their x-ray emission was collected at an angle of 20 \({}^{\circ}\) to that normal by means of a flat-crystal Bragg spectrometer. We employed a Beryl (10\(\bar{1}\)0) crystal, whose lattice spacing (2\(d=\)15.96 A) corresponds to a diffraction angle of \(\sim 35\) \({}^{\circ}\) for the He\({}_{\alpha}\) line of Mg. The diffracted X rays were then recorded with a Princeton Instruments Charge Coupled Device (CCD) camera. A schematic drawing of the experimental set-up is shown in Fig. 1a.
Figure 1b shows a typical example of the K-shell emission spectra from the solid Mg plasma that was obtained in the experiment. Owing to the time-integrated nature of the measured spectra, emission from the different ionization species present in the plasma as it heats up is present in the data. The lines are labelled according to the number of bound electrons in the core of the emitting ion (the label 'He' corresponds to two bound electrons, 'Li' to three, and so on). As mentioned in the introduction, in this work we will focus on the emission from He-like Mg, and the associated Li-like satellites which lie in the energy range of \(1340-1370\) eV.
## III Modelling
### Atomic kinetics
The plasma evolution and resultant spectra were modelled using a combination of two codes: the atomic kinetics non-LTE (Local Thermodynamic Equilibrium) code SCFLY [10], and a separate, stand-alone, LTE Saha-Boltzmann code. As we shall explain in more detail below, we use SCFLY to determine the overall evolution of the plasma in terms of superconfigurations, and also to confirm that the system is very close to LTE. We then use a Saha-Boltzmann approach, with more detailed atomic physics (but now assuming LTE) to model the spectra of the satellites for comparison with the experimental results.
SCFLY is based upon the commonly used FLYCHK code [37], adapted to treat x-ray laser problems. SCFLY is based on a superconfiguration approach - i.e. it provides the populations for states defined solely by the number of electrons with specific principal quantum numbers. Thus the ground state of a lithium-like ion, with two electrons in the K-shell, and one in the L-shell, is denoted (210), whereas in its first excited state, with the L-shell electron excited to the M-shell, it would be denoted (201). It takes as its input the x-ray laser intensity as a function of time, and solves for the evolution of ground and excited-state superconfiguration populations. The electrons in the continuum are assumed to obey classical statistics, and to instantaneously thermalise to a temperature dictated by their overall energy content. In contrast with the electrons, we assume that on the timescale of typical FEL pulses the ions remain at room tempera
Figure 1: (1a) Schematic drawing of the experimental set-up. (1b) Experimental data of Mg K-shell emission spectrum showing a whole set of emission lines from different ionization stages. The lines have been labelled according to the number of electrons of the emitting ion.
ture throughout the calculation, given that the timescale for electron-ion equilibriation, in terms of their temperatures, is several picoseconds [38; 39].
Within both the SCFLY and Saha-Boltzmann solver we take into account the degree of IPD by considering two widely used models, namely the EK model and the SP model. The levels of continuum lowering predicted by these models in the high-density limit, where the SP model is effectively equivalent to the Ion-Sphere model [16], are given by [5]
\[\Delta I_{EK}=C_{EK}\cdot\frac{(Z+1)e^{2}}{4\pi\varepsilon_{0}}\cdot\left[ \frac{4\pi\left(n_{e}+n_{i}\right)}{3}\right]^{1/3} \tag{1}\]
and
\[\Delta I_{SP}=C_{SP}\cdot\frac{3}{2}\cdot\frac{(Z+1)e^{2}}{4\pi\varepsilon_{0} }\cdot\left(\frac{4\pi n_{e}}{3\cdot(Z+1)}\right)^{1/3}, \tag{2}\]
where, for each model, \(C_{EK}=C_{SP}=1\), \(Z\) is the charge of the ion (0 for the neutral atom), \(e\) is the electron charge, \(n_{e}\) and \(n_{i}\) are the electron and ion number density respectively and \(\varepsilon_{0}\) is the electric permittivity of vacuum. Note that in order to explore models that lie between SP and EK we will also in what follows show results for which these two constants \(C_{EK}\) and \(C_{SP}\) differ from unity.
As noted in the introduction, the main result we will be studying in this work is the intensity of the lithium-like dielectronic satellites. Within SCFLY, the transition between superconfigurations that corresponds to satellites associated with an upper state containing an L-shell electron is the (120)-(210) transition, and for M-shell satellites is the (111)-(201) transition. At the more detailed level, taking into account the various configurations and fine-structure effects, these two superconfigurational transitions encompass the gamut of satellites that are well known to accompany the helium-like resonance line, and which, for the L-shell satellites, are usually labelled according to the notation proposed by Gabriel [20]. It is these detailed transitions that are modelled by the Saha-Boltzmann code, under its LTE assumption, and which we justify in the next section.
To model the plasma kinetics, we first calculated the temperature evolution of the plasma with SCFLY. It is even at this early stage that the choice of an IPD model affects our results, since the electron temperature of the plasma is dependent on the level of ionization and therefore on the level of continuum lowering. We show this effect in Fig. 2, where the temperature evolution for the peak laser irradiance (\(\sim 3\times 10^{17}\,\mathrm{W\ cm^{-2}}\)) is shown for the SP and EK IPD models. In these simulations, and all that follow, we assume that the incident FEL pulse is gaussian in time with a FWHM of \(100\,\mathrm{fs}\), which peaks at a time of \(100\,\mathrm{fs}\). In the simulations shown here, which assume constant ion density, the cooling of the plasma after the FEL pulse is only due to radiation while in practice, disassembly of the target will take place on a timescale of order picoseconds.
The main difference between these IPD models can be seen in Figs. 3a and 3b, where we show the ionization energy of the M-shell for He- and Li-like ions as a function of time. The SP model predicts the M-shell to be always bound, with a binding energy around \(\sim 100\,\mathrm{eV}\), which is of the same order as the electron energy. This means that a large proportion of the free electrons are able to collisionally ionize these states, losing part of their thermal energy. In contrast, in the EK model the M-shell becomes completely free when the plasma heats up, so collisional ionization loses are somewhat reduced. Also shown in Fig. 3 are the results of scaled SP and EK models, such that the M-shell remains bound (in the bound-free picture of atomic kinetics codes), but with a lower ionization energy, which increases the rate of collisional ionization, thus increasing the energy losses and reducing the overall plasma temperature.
### Spectrum
Having used SCFLY to determine the temperature evolution of the plasma, the more detailed atomic kinetics were solved using an iterative Saha-Boltzmann LTE code which treats Ne-like to Be-like ions using configuration averaged levels, while explicitly including the configurations and fine structure levels of Li-like to H-like ions. The energy of the different atomic states and the transition probabilities were obtained from the Los Alamos National Laboratory (LANL) atomic codes [40].
Figure 2: Temperature evolution of solid density Mg, as predicted by SCFLY for different IPD models, upon irradiation with an FEL pulse with irradiance \(3\times 10^{17}\,\mathrm{W\ cm^{-2}}\) at \(1540\,\mathrm{eV}\). The plot is shown up to \(10\,\mathrm{ps}\) for illustrative purposes, but it should be noted that disassembly of the target will take place on a timescale of order picoseconds.
The particular region of the spectrum in which we are interested is around the Helium-like resonance line, along with its associated satellites, that is, the region between 1340 and 1370 eV shown in Fig. (b)b. Since the targets used in this experiment were 54 nm-thick, the peak optical depth of the plasma in the spectral range of the He\({}_{\alpha}\) emission (determined by the resonance line) is \(\tau<0.2\)[9]. All of the radiation in this region, between 1340 and 1370 eV, is produced by radiative transitions from the three superconfigurations (110)-(200), (120)-(210), and (111)-(201).
As noted in the introduction, for the solid-density plasmas produced here, the electron temperatures are such that very few K-shell holes are produced thermally for the He and Li ionisation stages, compared with the number that are produced due to photoionisation by the incident FEL radiation. Indeed, during the FEL pulse, the photoionisation production of such inner core holes exceeds that due to thermal collisional processes by about two orders of magnitude. However, and importantly for our analysis here, the electron density of the system is so high, and thus electron collisional processes so fast, that the superconfigurations (and also configurations and fine structure levels within them) are extremely close to LTE, apart from the deviation in the population of the K-shell induced by the photoionisation process. However, importantly, the photoionization due to the FEL does not significantly disturb the other aspects of LTE relationships within the system. This is best illustrated by example. Consider two superconfigurations that do not have K-shell holes: (210) and (211). Photoionisation by the FEL would, from these superconfigurations, produce (110) and (111) respectively, and indeed towards the peak of the pulse the fractional populations of ions with such K-shell holes is of order a few percent. However, the collisional processes are so fast that the ratio of (110) to (111) remains almost identical to that which would pertain in the circumstances of no FEL, but the same electron temperature (i.e. that given by the Saha-Boltzmann equation at this temperature). Note also, that given that the (111) superconfiguration can undergo Auger decay, collisional processes are also much more important than this effect.
As an example of this we plot the ratio of the populations of the (110) superconfiguration to that of the (111) superconfiguration, as predicted by SCFLY, firstly with the FEL on and running the simulation in full non-LTE mode, and then secondly, switching the FEL off (so no photoionisation occurs), but assuming LTE, following the time-dependent temperature from the first non-LTE simulation. As can be seen in Fig. (a)a, the ratio between the populations of the superconfigurations is almost identical at each point in time, and whether the FEL is on or off. This is despite the fact that the absolute value of the
Figure 3: Time-evolution of the level of the continuum edge with respect to the M-shell of He-like (3a) and Li-like (3b) ions, for the different IPD models mentioned in the text. In this picture, when the continuum energy is positive, it means that the M-shell is _bound_ and when it is negative, the M-shell becomes _free_. Note how, while for the usual SP and EK models the M-shell is respectively bound or free for both ion species (when the temperature is sufficiently high), for the case of 88% EK and 130% SP, the M-shell is barely bound for both charge states and much closer to the continuum edge.
populations of the superconfigurations is quite different between the non-LTE calculation including the FEL, and the one that assumes LTE: this can be seen in Fig. 4b, where we plot the number density of ions in the (110) and (111) superconfigurations as a function of time for the non-LTE and LTE case. Note that the populations are about two orders of magnitude higher whilst the FEL is on, due to photoionisation of the K-shell, in the non-LTE case.
As LTE between superconfigurations has been shown to be maintained in the presence of an FEL drive, it is therefore reasonable to use the time-dependent temperature from SCFLY as a basis of the model, assuming LTE between configurations, and the fine structre levels within them in order to construct a detailed spectrum. In this limit, as the plasma is optically thin, the intensity emitted at a given time from a given detailed transition depends only on its population and spontaneous emission rate. We use the Saha-Boltzmann solver, with the time-dependent temperature provided by SCFLY, to determine the time-dependent populations of all of the relevant levels, and thus the detailed emission. For the Helium-like line and its satellites we included the resonance (\(1s2p\,\,^{1}P_{1}\to 1s^{2}\,\,^{1}S_{0}\)) and intercombination (\(1s2p\,\,^{3}P_{1}\to 1s^{2}\,\,^{1}S_{0}\)) lines from the He\({}_{\alpha}\) complex and all Li-like L- and M-shell satellite transitions with energies above \(1330\,\mathrm{eV}\) and with spontaneous emission rates \(A\) above \(0.08\times 10^{13}\,\mathrm{s}^{-1}\). This corresponds to 13 L-shell satellite lines and 32 M-shell satellite lines, which are shown in Table 1. Note that, although the spectral region of interest for this work lies between 1340 and 1370 eV, we set the lower limit of the energy of the lines considered to be 10 eV below the region of interest, to include the emission from the wings of the lines.
The intensity of each transition is distributed across its lineshape. Given that, for the duration of the emission, the ions are static, the contribution of the Doppler effect to the broadening of the line can be considered negligible. By studying the mechanisms affecting line broadening in the experimental conditions, using the code ALICE [41], we observed that, under these conditions, the lineshape is mostly determined by the rate of collisional ionization and recombination, and thus, the lineshape is well modelled by a Lorentzian curve. The experimental width of the isolated He\({}_{\alpha}\) resonance line was determined by fitting the high-energy side of the line, where no satellite emission is present. The obtained FWHM of the Lorentzian was \(\Delta E=4.4\pm 0.5\,\mathrm{eV}\). We observed that the shape of the total spectrum was not strongly dependent on the width of the individual satellite lines -given the numerous satellite transitions, the shape of the individual features is lost. For this reason, and given that the rate of collisional recombination to the M-shell for a He-like ion is the same as the rate of collisional ionization of an M-shell electron for a Li-like ion, for simplicity we considered all the individual transitions to have the same Lorentzian width as the He\({}_{\alpha}\) line. The lineshapes are then modelled
Figure 4: (4a) Population ratio between the 110 and 111 states obtained from SCFLY compared with those obtained by assuming LTE for the SP IPD model as a function of time. (4b) Fractional population of the He-like 110 states (red) and the Li-like 111 satellite states (blue) as a function of time as predicted by SCFLY (solid lines) and pure LTE (dashed lines).
as
\[I(E)=\frac{I_{\rm total}}{\pi}\frac{\Delta E/2}{(E-E_{0})^{2}+(\Delta E/2)^{2}}, \tag{3}\]
where \(I_{\rm total}\) is the total intensity of a transition defined by the populations and spontaneous rate, and \(E_{0}\) is the transition energy.
### Effect of the focal spot
In order to take into account the spatial distribution of the focussed FEL x-rays, for each pulse we model 25 intensity bins spanning six orders of magnitude, as detailed by Ciricosta _et al._[10]. The intensity of the \(n\)-th bin was obtained as
\[I_{n}(\omega,t)=I_{0}(\omega,t)\cdot e^{-2(n\cdot 0.1)^{2}}. \tag{4}\]
It is worth mentioning that not all the bins contribute to the He\({}_{\alpha}\) emission, since for \(n\gtrsim 11\) the temperature is not sufficiently high to ionize the the plasma up to neither Li nor He-like state. This spatial distribution was measured experimentally by the imprint method, as described in more detail in [10]. The full spectrum was then obtained as a sum of the time-integrated spectra calculated for each intensity bin, weighted by the fluence scan of the laser spot.
## IV Results
Figure 5 shows both the experimental data, and simulated spectra, for the Helium-like emission and associated Li-like satellites. We show in red the simulated spectra of the He\({}_{\alpha}\) region for both the SP (Fig. 5a) and the EK (Fig. 5b) IPD models. The brown line corresponds to the contribution from the He\({}_{\alpha}\) emission, while the green and black solid lines correspond to the collective emission from the Li-like satellites with an L- and M-shell spectator electron respectively (the dotted green and black lines correspond to each individual satellite transition). The red band corresponds to the total spectrum, with the width of the band corresponding to the uncertainty introduced by the error in the fit to the width of the Lorentzian. When comparing with the experimental data, shown in blue, it can be seen that the SP model overestimates the intensity of the M-shell satellites, thus predicting a He\({}_{\alpha}\) feature \(\sim 2\,\)eV wider than the experimental result. The opposite happens with the EK model, where there is almost no M-shell satellite contribution, and therefore the line appears narrower than observed.
It is worth noting that the brown line labeled He-\(\alpha\) in the figures includes the contribution from the resonance line \(1s2p\ ^{1}P_{1}\to 1s^{2}\ {}^{1}S_{0}\) (labeled \(w\) following Gabriel's notation Gabriel (1980)), centered at \(E_{w}=1352.5\,\)eV; and the intercombination line \(1s2p\ ^{3}P_{1}\to 1s^{2}\ {}^{1}S_{0}\ (y)\), centered
Figure 5: Shape of the He\({}_{\alpha}\) emission obtained by the (5a) Steward-Pyatt and (5b) Ecker-Kröll models (red lines) compared with the experimental data (blue line) and the associated \(1\sigma\) uncertainty (grey area). Below the main line, the total contribution from the satellite lines with an M or an L-shell spectator electron, as well as that of the He\({}_{\alpha}\) emission are indicated with the green, black and brown solid lines respectively. The green and black dotted lines correspond to the emission from individual fine structure states.
at \(E_{y}=1343.1\,\mathrm{eV}\). The contribution of the intercombination line, however, can barely be resolved, since its intensity is \(\sim 0.2\%\) that of the resonance line. Whilst the intercombination line is a well-known intense feature of spectra from plasmas produced by irradiation with optical lasers, it is almost completely absent here. This is due to the very high electron densities present in these experiments, which contrast with the critical electron densities in optical experiments. With optically produced laser plasmas, which are far from LTE, significant population can build up in the \(1s2p\)\({}^{3}P_{1}\) state, giving rise to a large intercombination line intensity, despite the very low tran
\begin{table}
\begin{tabular}{c c c c c} Type & Transition & Energy (eV) & A-rate (\(10^{13}\,\mathrm{s}^{-1}\)) & Gabriel’s notation \\ \hline He-like & \(1s2p\)\({}^{3}P_{1}\to 1s^{2}\)\({}^{1}S_{0}\) & 1343.1 & \(1.21\times 10^{-3}\) & y \\ He-like & \(1s2p\)\({}^{1}P_{1}\to 1s^{2}\)\({}^{1}S_{0}\) & 1352.5 & 2.0469 & w \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}D_{3/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{3/2}\) & 1331.3 & 0.0499 & l \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}D_{5/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{3/2}\) & 1331.3 & 0.9596 & j \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}D_{3/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{1/2}\) & 1331.8 & 0.9270 & k \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}P_{1/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{3/2}\) & 1333.4 & 0.9147 & c \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}P_{3/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{3/2}\) & 1333.8 & 2.6167 & a \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}P_{1/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{1/2}\) & 1333.8 & 2.0883 & d \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}P_{3/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{1/2}\) & 1334.3 & 0.3683 & b \\ L-shell satellite & \(1s2s2p\)\({}^{2}P_{1/2}\to 1s^{2}2s^{1}\)\({}^{2}S_{1/2}\) & 1335.3 & 1.7592 & r \\ L-shell satellite & \(1s2s2p\)\({}^{2}P_{3/2}\to 1s^{2}2s^{1}\)\({}^{2}S_{1/2}\) & 1335.6 & 1.8214 & q \\ L-shell satellite & \(1s2s2p\)\({}^{2}P_{1/2}\to 1s^{2}2s^{1}\)\({}^{2}S_{1/2}\) & 1339.9 & 0.2026 & t \\ L-shell satellite & \(1s2s2p\)\({}^{2}P_{3/2}\to 1s^{2}2s^{1}\)\({}^{2}S_{1/2}\) & 1340.4 & 0.1406 & s \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}S_{1/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{3/2}\) & 1343.0 & 0.7842 & m \\ L-shell satellite & \(1s2p^{2}\)\({}^{2}S_{1/2}\to 1s^{2}2p^{1}\)\({}^{2}P_{1/2}\) & 1343.2 & 0.2636 & n \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{3})P^{2}P_{3/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{3/2}\) & 1337.4 & 0.0807 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{3})P^{2}P_{1/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{1/2}\) & 1337.5 & 0.0854 & \\ M-shell satellite & \(1s^{1}2s^{1}3d^{1}\)\(({}^{3})P^{2}P_{5/2}\to 1s^{3}3p^{2}\)\({}^{2}P_{3/2}\) & 1341.9 & 0.2530 & \\ M-shell satellite & \(1s^{1}2s^{1}3p^{1}\)\(({}^{3})P^{2}D_{5/2}\to 1s^{3}3p^{1}\)\({}^{2}P_{3/2}\) & 1341.9 & 0.1543 & \\ M-shell satellite & \(1s^{1}2s^{1}3d^{1}\)\(({}^{3})P^{2}D_{3/2}\to 1s^{3}3p^{2}\)\({}^{2}P_{1/2}\) & 1342.0 & 0.2760 & \\ M-shell satellite & \(1s^{1}2p^{1}3d^{1}\)\(({}^{3})P^{2}P_{5/2}\to 1s^{3}3d^{1}\)\({}^{2}D_{3/2}\) & 1342.6 & 0.3080 & \\ M-shell satellite & \(1s^{1}2p^{1}3d^{1}\)\(({}^{3})P^{2}F_{7/2}\to 1s^{3}3d^{1}\)\({}^{2}D_{5/2}\) & 1343.1 & 0.3324 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{3})P^{2}S_{1/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{3/2}\) & 1343.8 & 0.3058 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{3})P^{2}S_{1/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{1/2}\) & 1344.0 & 0.0910 & \\ M-shell satellite & \(1s^{1}2p^{1}3s^{1}\)\(({}^{3})P^{2}P_{1/2}\to 1s^{2}3s^{1}\)\({}^{2}S_{1/2}\) & 1344.9 & 0.2500 & \\ M-shell satellite & \(1s^{1}2p^{1}3s^{1}\)\(({}^{3})P^{2}P_{3/2}\to 1s^{2}3s^{1}\)\({}^{2}S_{1/2}\) & 1345.4 & 0.1850 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{1})P^{2}D_{5/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{3/2}\) & 1347.4 & 1.8410 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{1})P^{2}D_{3/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{3/2}\) & 1347.4 & 0.2166 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{1})P^{2}D_{3/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{1/2}\) & 1347.5 & 1.5855 & \\ M-shell satellite & \(1s^{1}2p^{1}3s^{1}\)\(({}^{1})P^{2}P_{1/2}\to 1s^{2}3s^{1}\)\({}^{2}S_{1/2}\) & 1348.1 & 1.8705 & \\ M-shell satellite & \(1s^{1}2p^{1}3s^{1}\)\(({}^{1})P^{2}P_{3/2}\to 1s^{2}3s^{1}\)\({}^{2}S_{1/2}\) & 1348.1 & 1.8226 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{1})P^{2}P_{1/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{3/2}\) & 1348.4 & 0.4731 & \\ M-shell satellite & \(1s^{1}2p^{1}3p^{1}\)\(({}^{1})P^{2}P_{1/2}\to 1s^{2}3p^{1}\)\({}^{2}P_{1/2}\) & 1348.6 & 1.4666 & \\ M-shell satellite & \(1s^{1}2p^{1}3d^{1}\)\(({}^{1})P^{2}P_{3/2}\to 1s^{2}3p^{
sition rate to the lower \(1s^{2}\)\({}^{1}S_{0}\) level. However, here the collisional effects cause a much lower LTE-dictated population of \(1s2p\)\({}^{3}P_{1}\), leading to almost complete absence of the transition.
From these figures, it seems clear that the intensity of the M-shell satellites seen experimentally cannot be fitted by either the full SP or EK models, at least for the FEL intensity used. However, if we consider the intensity of these satellites to indicate in some way where the IPD should actually lie, it would be somewhere between the two. We thus reran the SCFLY and Saha-Boltzmann solver for the two IPD models, but changed the value of \(C\) within them from unity, while keeping the functional form of the model the same (see Eqns. 1 and 2). The best fits were obtained for \(C_{EK}=0.88\) and \(C_{SP}=1.30\), cases for which both models predict the M-shell to remain just _bound_ for the whole duration of the emission (see Figs. 3a and 3b respectively).
The resulting spectra are shown in Fig. 6, where 6a shows the total spectra for both cases and Fig. 6b corresponds to the individual contribution of each line for the 130% SP IPD model, following the same color convention as for Fig. 5. It can be seen that the emission from the M-shell satellites is reponsible for the shape of the low-energy wing of the line (where it creates a shoulder-like feature that we are indicating with an arrow), whereas the He\({}_{\alpha}\) emission is the only contribution on the high-energy end.
In the context of the recent DFT calculations by Gawne _et al._[19], it is encouraging that to obtain the best fit to the satellite spectra we need to invoke an IPD value that lies between the SP and EK limits, as that was precisely the conclusion of that work. Furthermore, the required scaling of both the EK and SP models to match the experimental data is in excellent agreement with the required scaling to match the DFT results (Gawne _et al._ found an IPD value corresponding to 133% that predicted by the SP model and 89% the prediction of the EK model). However, before stating definitively that this is the case, care should be taken to note that the simulated intensity of the satellites compared with the resonance line will depend on accurate modelling of the temperature of the system, as the satellite intensity is determined by the ratio of the ionisation energy of their upper levels to the temperature. This in turn, as well as any inherent limitations of the model, entails accurately knowing the experimental x-ray FEL intensity incident upon the target: a figure that is generally quoted to be known within about 30% [10]. It is therefore important to also investigate the sensitivity of the results to the FEL intensity on target.
To this end we ran several simulations with FEL intensities that differed from those measured experimentally by up to a factor of two lower and up to 50% higher than the experimental value. As described previously in subsection III.3, these simulations were integrated over the
Figure 6: (6a) Comparison of the shapes of the He\({}_{\alpha}\) emission obtained for both 130% of the SP model and 88% of the EK model (green-dotted and red-dashed regions respectively) compared with the experimental data. The width of the error regions corresponds to the uncertainty in the line widths. (6b) Line profile obtained using the SP IPD model scaled to 130% (red) compared to the experimental data (blue line and grey area). The contribution from individual lines is shown following the same colour convention as Fig. 5. The black arrow marks the shoulder-like feature caused by the M-shell satellite emission.
shape of the focal spot, in order to be directly comparable with the current results. The results are shown in Fig. (a)a for the full EK model and in Fig. (b)b for the full SP model. In both figures, the current result for an IPD level of \(130\%\) of the SP model is included for comparison. As expected, in the spectral region of the M-shell satellites, the predicted line width differs very little in the case of the EK model, as the M-shell electrons are not bound in any event. The main changes to the spectrum appear in the region of the L-shell satellites, which become more intense with respect to the He\({}_{\alpha}\) emission as the laser intensity is decreased.
However, in the case of using the full SP model, a reduction in the laser intensity does lead to an increase in satellite intensity predicting an even wider line. On the other hand, an increase in the intensity slightly reduces the satellite contribution from both M- and L-shell satellites. We find that increasing the laser intensity by \(50\%\) results in a line width that is still \(\sim 1\,\mathrm{eV}\) wider than the experimental result, while the emission around \(1340\)-\(1345\,\mathrm{eV}\) starts deviating from the data as well. Furthermore, looking in more detail we see the shoulder asymmetry mentioned before in the lower energy side to the main He\({}_{\alpha}\) peak (\(\sim 1348\,\mathrm{eV}\)) appearing for the scaled-IPD models. This feature is not present in the full SP simulations, but does appear in the data. We thus conclude that given we believe we know the incident FEL flux to within about \(30\%\), the simulations still indicate that the satellite intensities are more consistent with an IPD model that lies between the EK and SP limits once the He and Li-like ion stages are reached.
## V Conclusions
In this work we have investigated the sensitivity of the intensity of M and L-shell satellites to the He\({}_{\alpha}\) emission from a solid-density plasma to the IPD model used. By adjusting the IPD level in an atomic-kinetics code, in conjunction with an LTE Saha-Boltzmann solver, we find that both the SP and EK IPD models fail to reproduce the experimental spectra, but obtain best agreement with the experimental data by employing a degree of IPD that lies between these two extremes, (\(88\%\) of EK and \(130\%\) of SP). These values are in good agreement with recent results obtained using first principles simulations, and indicate that the M-shell of Li-like and He-like Mg under these conditions lies very close to the continuum edge. Whilst the intensity of the M-shell satellites does depend on the intensity of the FEL, our knowledge of the incident intensity, and the observation of asymmetry in the He\({}_{\alpha}\) peak, does lead credence to the conclusion that the best fit IPD lies between the EK and SP limits.
As outlined in the introduction, it should be borne in mind that for such dense quantum systems the distinction between bound and free states is somewhat artificial, yet in the mode in which current atomic-kinetics calculations are performed, we are often forced to make this division. The fact that the experimental spectra can be reasonably reproduced within this simple framework, us
Figure 7: Spectra obtained for the EK ((a)a) and SP ((b)b) IPD models for the nominal laser intensity (yellow squares), and the results obtained by modifying the laser intensity to \(150\%\) (red circles) and \(50\%\) the nominal value (brown diamonds). The results obtained by scaling the IPD models, as presented in Fig. 6, are also shown for comparison (green crosses).
ing IPD values guided by first-principle simulations, gives confidence that they are still of use in the modelling of the spectra of hot dense plasmas.
## Acknowledgements
T.G., J.S.W. and S.M.V. acknowledge support from AWE via the Oxford Centre for High Energy Density Science (OxCHEDS). S.M.V. acknowledges support from the Royal Society. J.S.W. and S.M.V. acknowledge support from the UK EPSRC under grants EP/P015794/1 and EP/W010097/1. G.P.-C. acknowledges support from Spanish Ministry of Science and Innovation under Research Grant No. PID2019-108764RB-I00. S.M.V. is a Royal Society University Research Fellow. T.B., J.Ch., V.H., L.J., and V.V. appreciate financial support from the Czech Ministry of Education (LG15013 and CZ.02.1.01/0.0/0.0/16-013/0001552--ERDF) and Czech Science Foundation (grant No. 20-08452S)
Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515.
|
2304.13634 | HausaNLP at SemEval-2023 Task 12: Leveraging African Low Resource
TweetData for Sentiment Analysis | We present the findings of SemEval-2023 Task 12, a shared task on sentiment
analysis for low-resource African languages using Twitter dataset. The task
featured three subtasks; subtask A is monolingual sentiment classification with
12 tracks which are all monolingual languages, subtask B is multilingual
sentiment classification using the tracks in subtask A and subtask C is a
zero-shot sentiment classification. We present the results and findings of
subtask A, subtask B and subtask C. We also release the code on github. Our
goal is to leverage low-resource tweet data using pre-trained Afro-xlmr-large,
AfriBERTa-Large, Bert-base-arabic-camelbert-da-sentiment (Arabic-camelbert),
Multilingual-BERT (mBERT) and BERT models for sentiment analysis of 14 African
languages. The datasets for these subtasks consists of a gold standard
multi-class labeled Twitter datasets from these languages. Our results
demonstrate that Afro-xlmr-large model performed better compared to the other
models in most of the languages datasets. Similarly, Nigerian languages: Hausa,
Igbo, and Yoruba achieved better performance compared to other languages and
this can be attributed to the higher volume of data present in the languages. | Saheed Abdullahi Salahudeen, Falalu Ibrahim Lawan, Ahmad Mustapha Wali, Amina Abubakar Imam, Aliyu Rabiu Shuaibu, Aliyu Yusuf, Nur Bala Rabiu, Musa Bello, Shamsuddeen Umaru Adamu, Saminu Mohammad Aliyu, Murja Sani Gadanya, Sanah Abdullahi Muaz, Mahmoud Said Ahmad, Abdulkadir Abdullahi, Abdulmalik Yusuf Jamoh | 2023-04-26T15:47:50Z | http://arxiv.org/abs/2304.13634v1 | # HausaNLP at SemEval-2023 Task 12: Leveraging African Low Resource TweetData for Sentiment Analysis
###### Abstract
We present the findings of SemEval-2023 Task 12, a shared task on sentiment analysis for low-resource African languages using Twitter dataset. The task featured three subtasks; subtask A is monolingual sentiment classification with 12 tracks which are all monolingual languages, subtask B is multilingual sentiment classification using the tracks in subtask A and subtask C is a zero-shot sentiment classification. We present the results and findings of subtask A, subtask B and subtask C. We also release the code on github. Our goal is to leverage low-resource tweet data using pre-trained Afro-xlmr-large, AfriBERTa-Large, Bert-base-arabic-camelbert-da-sentiment (Arabic-camelbert), Multilingual-BERT (mBERT) and BERT models for sentiment analysis of 14 African languages. The datasets for these subtasks consists of a gold standard multi-class labeled Twitter datasets from these languages. Our results demonstrate that Afro-xlmr-large model performed better compared to the other models in most of the languages datasets. Similarly, Nigerian languages: Hausa, Igbo, and Yoruba achieved better performance compared to other languages and this can be attributed to the higher volume of data present in the languages.
## 1 Introduction
Social media offers an opinionated platform of data content on many topics of interest, such as product reviews, feedback on purchases or services, political interest, etc., that is dynamically created by users in different languages. Sentiment analysis in many languages is required due to the need for effective classification of these contents. However, the majority of research on sentiment analysis has been done in high-resource languages [1, 16, 17, 18, 19, 20] while several low-resource African languages receive little attention in Natural Language Processing (NLP) application due to insufficient of publicly available data. Although, recently, some considerable efforts are made for the development of sentiment analysis for some low-resource African languages [18, 21, 22, 23, 24], nonetheless, sentiment analysis for low-resourced African languages still is a misrepresented research area.
The AfriSenti1 shared task 12 [25] aims at building sentiment analysis for 14 low-resource African languages using a Twitter dataset which include Hausa, Yoruba, Igbo, Nigerian Pidgin from Nigeria, Amharic, Tigrinya, and Oromo from Ethiopia, Swahili from Kenya and Tanzania, Algerian Arabic dialect from Algeria, Kinyarwanda from Rwanda, Twi from Ghana, Mozambique Portuguese from Mozambique and Moroccan Arabic/Darija from Morocco. The task featured the following subtasks: Subtask A (Monolingual Sentiment Classification): Given a training document, analyze the sentiment classification of 12 individual African languages. Subtask B (Multilingual Sentiment Classification): Given a training document, analyze the sentiment classification of multiple languages using the languages in task A. Subtask C (Zero-Shot Sentiment Classification): Given evaluation data only, analyze the sentiment classification of only
two African languages.
In this paper, we demonstrate our approach to tackling the concerns raised in SemEval Task 12 as well as the obstacles posed by low-resource languages [23]. We used the Bert, AfriBERTa_large and Afro-xlmr-large model to facilitate the classification of tweets using a monolingual, multilingual and zero shot approach. The rest of the paper is organized as follows. Section 2 is the related works. Section 3 describes the proposed approach. Experimentation and evaluation are discussed in section 4, while section 5 draws some conclusions and discusses some directions for future work.
## 2 Related Work
Sentiment Analysis is the process of generating positive or negative sentiment from data using computational techniques, and to some extent is capable of predicting and classifying tempers such as excitement, anger, and sadness[19]. Sentiment and emotion are considered to be related, [1]therefore, the process of assigning polarity to text and emotion among others has become a prevalent task in NLP. [10]. Hence, sentiment analysis is to generate opinions based on a given input provided by users [13], the current campaign in NLP is that sentiment analysis is extensively adopted for opinion mining to collect information about users from different fields of a particular aspect.
[1] authors use a machine learning approach to combine English and Hausa features to measure classification performance and create a more precise sentiment classification process. [20, 1] demonstrated that training is feasible with less than 1GB of text to build a competitive multilingual language model. Their results show that "smalldata" technique which uses languages that are similar to one another may occasionally be more effective than combined training on big datasets with high-resource languages.
Although sentiment analysis has been extensively used in many high-resource languages like English and French just to mention a few, little attention is paid to African low-resource languages. [23] presented the first extensive human-annotated Twitter sentiment dataset for the Hausa, Igbo, Nigerian-Pidgin, and Yoruba languages--the four most widely spoken in Nigeria--consisting of roughly 30,000 annotated tweets per language (and 14,000 for Nigerian-Pidgin) and a sizable portion of code-mixed tweets. For these low-resource languages, they suggested text collecting, filtering, processing, and labeling techniques that let us build datasets. They used a variety of pre-trained models and transfer methods. They discovered that the most effective methods are language-specific models and language-adaptive fine-tuning.
## 3 Shared Task Description
The AfriSenti-SemEval Shared Task 12 is based on a collection of Twitter datasets in 14 African languages for sentiment classification. Participants are provided with a training dataset and are required to make a prediction using multiclass sentiment classification. It consists of three sub-tasks: Monolingual sentiment classification, multilingual sentiment classification, and zero-shot sentiment classification. In this paper, we concentrate on all the three subtasks with a total of 14 languages.
### Subtask A: Monolingual Sentiment Classification
For this subtask, we used a single language for sentiment classification. Given training data in a target language, we determine the polarity of a tweet in the target language (positive, negative, or neutral). If a tweet conveys both a positive and negative sentiment, whichever is the stronger sentiment should be chosen. This subtask consists of 12 African languages: Hausa, Yoruba, Igbo, Nigerian-Pidgin, Amharic, Algerian Arabic, Moroccan Arabic/Darija, Swahili, Kinyarwanda, Twi, Mozambique Portuguese, Xitsonga (Mozam-bique dialect). For this subtask, the dataset is split into 70% training and 30% validation. We select an individual language, fine-tune the models with the provided training data, and fine-tune several hyper-parameters to obtain the optimal performances using only 2 models: Afro-xlmr-large and Bert-base-arabic-camelbert-da-sentiment. Afro-xlmr-large is used in all the languages with the exception of Darija and Algerian Arabic and this is because Afro-xlmr-large model was not trained using the two languages, therefore, we used the Bert-base-arabic-camelbert-da-sentiment for these 2 languages.
### Subtask B: Multilingual Sentiment Classification
Given combined training data from Subtask-A (Track 1 to 12), we determine the polarity of a tweet in the target language (positive, negative, or neutral). For this subtask, the multilingual dataset is split into three parts: 90% training, 5% each for validation and test set. The three-part split allows for a more robust evaluation of the model, avoids overfitting, and produces a better estimate of the model's generalization performance. We implemented only 1 model for this subtask: Afroxlmr-large-ner-masakhaner-1.0-2.0 and this is because the model was trained using almost all the African languages.
### Subtask C: Zero Shot Classification
Zero-shot learning is a type of machine learning that allows a model to perform a task on a new data point without having been trained on any data points from that class. As given in Subtask C, working with languages that have limited resources. Zero-shot learning can be used to classify the sentiment of text in two non-labelled African languages, Tigrinya and Oromo. The implementation of zero-shot learning for sentiment analysis is to use a multilingual language model like AfroXLM-R, which has been pre-trained on a large corpus of text from multiple African languages. The pre-trained language model, would learn to understand the underlying patterns in language across the different languages and use this knowledge to classify text in the 2 new languages.
However, limitations such as the reliance on language similarity and the assumption that the learned representations can transferable across languages might cause it to underperform due to the fact that the target language can be significantly different from the languages in the pre-training corpus Wang and Jiang (2021)
### Dataset Description
The AfriSenti dataset is a collection of Twitter datasets for sentiment analysis of African languages. The dataset used for the AfriSentiSemEval 2023 shared task 12 consists of 14 languages: Hausa, Yoruba, Igbo, NigerianPidgin, Amharic, Algerian Arabic, Moroccan Arabic/Darija, Swahili, Kinarywanda, Twi, MozambiqueCan Portuguese, Setswana, isiZulu, Tigrinya, Xitsonga, and Oromo. The datasets are gold standard with multi-class labels (positive, negative, and neutral). Each tweet is annotated by three annotators following the annotation guidelines in Mohammad (2016) as shown in Table 1 and Figure 1. Table 1 shows the distribution of the languages datasets, sentiment labels, and sizes.
## 4 Proposed Approach
In this section, we describe our proposed approach for the SemEval shared task i.e leveraging low-resource tweet data for sentiment analysis of African languages. Our goal is to identify the sentiment classification of low-resource African languages using AfriBERTa-large, BERT-base-multilingual-cased and Afro-xlmr-large models. We have also shared the code on github 2.
Footnote 2: [https://github.com/ahmadmwali/SemEval-AfriSenti](https://github.com/ahmadmwali/SemEval-AfriSenti)
### Models Description
#### 4.1.1 Afro-xlmr-large
Afro-xlmr-large Alabi et al. (2022) was created by Masked Language Modelling (MLM) adaptation of XLM-R-large model on 17 African languages (Afrikaans, Amharic, Hausa, Igbo, Malagasy, Chichewa, Oromo, Nigerian Pidgin, Kinarywanda, Kirundi, Shona, Somali, Sesotho, Swahili, isiXhosa, Yoruba and isiZulu), covering the major African language families and 3 high resource languages (Arabic, French, and English). Afro-xlmr-large is used to facilitate sentiment classification of tweets in the low-resource African languages included in the AfriSenti shared task 12. The motivation behind using Afro-xlmr-large is that it is a multilingual model that is adapted to cover a wide range of African languages, including low-resource ones. The multilingual model was created to help overcome the challenge of insufficient data in low-resource languages. However, one potential weakness of Afro-xlmr-large is that it may not perform as well in sentiment analysis tasks for languages that are not included in the model and the model usually requires large amounts of computational resources which is a limiting factor.
#### 4.1.2 Bert
Bert language model (Bidirectional Encoder Representations from Transformers) was introduced by researchers from Google in 2018 and has become a popular and effective approach in natural
language processing tasks such as sentiment analysis, named entity recognition and question- answering (Devlin et al., 2018). Bert has achieved state-of-the-art performance in various benchmarks, and its pre-training on large corpora of text has shown to be effective in capturing contextual information of words. However, one limitation of Bert is its computational cost and memory requirements. The model architecture is complex, and training on large amounts of data can be computationally expensive and time-consuming. Another limitation is its vulnerability to adversarial attacks, where small perturbations in the input text can lead to significant changes in the output prediction. Bert has shown to be effective in capturing the nuances and complexities of language, especially the impact of the model architecture on the input text.
\begin{table}
\begin{tabular}{l c|c|c|c|c|c|c} \hline
**Subtask A: Monolingual** & **Pos** & **Pos\%** & **Neg** & **Neg\%** & **Neu** & **Neu \%** & **Total** \\ \hline Amharic(am) & 1332 & 22.26 & 1548 & 25.87 & 3104 & 51.88 & 05984 \\ Algerian Arabic(dz) & 417 & 25.26 & 892 & 54.03 & 342 & 20.72 & 01651 \\ Hausa(ha) & 4687 & 33.08 & 4573 & 32.27 & 4912 & 34.66 & 14172 \\ Igbo(ig) & 3084 & 30.26 & 2600 & 25.52 & 4508 & 44.24 & 10192 \\ Kinyarwanda(kr) & 899 & 27.23 & 1146 & 34.71 & 1257 & 38.07 & 03302 \\ Darija(ma) & 1758 & 31.49 & 1664 & 29.81 & 2161 & 38.71 & 05584 \\ Naija(pcm) & 1808 & 35.31 & 3241 & 63.29 & 72 & 1.41 & 05121 \\ Mozambiquen Portuguese(pt) & 681 & 22.24 & 782 & 25.54 & 1600 & 52.24 & 03063 \\ Swahili(sw) & 1072 & 59.23 & 547 & 30.23 & 191 & 10.56 & 01810 \\ Xitsonga(ts) & 384 & 47.77 & 284 & 35.33 & 136 & 16.92 & 00804 \\ Twit(twi) & 1644 & 47.23 & 1315 & 37.78 & 522 & 15.00 & 03481 \\ Yóubá(yo) & 3542 & 41.57 & 1872 & 21.97 & 3108 & 36.48 & 08522 \\ \hline
**Subtask B: MultiLingual** & & & & & & & \\ \hline Multilingual & 20783 & 32.63 & 20108 & 31.57 & 22794 & 35.79 & 63685 \\ \hline
**Subtask C: Zero Shot** & & & & & & & \\ \hline Tigrinya (Ti) & & & & & & & \\ Oromo (Or) & & & & & & & \\ \hline \end{tabular}
\end{table}
Table 1: **Distribution of tweets across Languages**. Showing the balance distribution of various tweets across the 14 languages for Monolingual Subtask A, Multilingual Subtask B and Zero Shot Subtask C
Figure 1: **Distribution of Tweets Across Languages**. We show the graphical comparison of the various tweets across the 12 Languages for Monolingual Subtask A
cially in languages with rich morphology and syntax. Inoue et al. (2021) used Bert-based models trained on Arabic text data and achieved state-of-the-art performance on Arabic sentiment analysis benchmarks. Similarly, AfroXLMR-large Alabi et al. (2022) and MasakhaNER-1.0/2.0 Adelani et al. (2021) used Bert-based models for named entity recognition and achieved high accuracy on African language datasets. We experiment with multiple pre-trained BERT based models to competitively select the best across the datasets:
Bert-base-arabic-camelbert-da-sentiment: Bert-base-arabic-camelbert-da-sentiment Inoue et al. (2021) is a collection of BERT models pre-trained on Arabic texts with different sizes and variants. Arabic Sentiment Tweet Dataset (ASTD) Nabil et al. (2015), an Arabic SpeechAct and Sentiment Corpus of Tweets ArSAS Elmadany et al. (2018) and SemEval datasets are used for fine-tuning the model.
Afroxlmr-large-ner-masakhaner-1.0-2.0:masakhaner/afroxlmr-large-ner-masakhaner-1.0-2.0 Adelani et al. (2021) is a Named Entity Recognition (NER) model for 21 African languages. Specifically, this model is a Davlan/afroxlmr-large model that was fine-tuned on an aggregation of African language datasets obtained from two versions of MasakhaNER dataset i.e. MasakhaNER 1.0 and MasakhaNER 2.0. One major advantage of using this model is that it has been trained on a wide range of African languages and has been fine-tuned on datasets specific to those languages. This means that it is well- suited for analysing sentiment in African languages, which can be challenging for other models as Bert is trained on Arabic language and Afroxmlr-large does not cover 5 other languages.
Multilingual Bert (mBERT):mBERT Devlin et al. (2018) is a multilingual version of BERT pretrained on top 104 large language dataset from Wikipedia. It uses Masked Language Modeling (MLM).
AfriBERTa-Large: The AfriBERTa-Large Ogueji et al. (2021) is trained on mBERT using 11 African languages namely, Afaan Oromoo (also called Oromo), Amharic, Gahuza (a mixed language containing Kinyarwanda and Kirundi), Hausa, Igbo, Nigerian Pidgin, Somali, Swahili, Tigrinya and Yoruba. It outperformed mBERT and XLM-R on several languages and is very competitive overall.
Table 3 reports the results of the Large Language Models (LLMs) using Weighted F1 metric. Extensive experiments were carried out before selecting the best across the datasets and subtasks. Afro-XLMR-Large Performed best in most of the languages. Also Notably is Bert-base-arabic-camelbert-da-sentiment performance in arabic based datasets.
## 5 Experiment and Evaluation
This section describes the experimental and evaluation settings for our proposed approach for the SemEval- 2023 Task 12.
### Experimental Settings
The 3 subtasks use different training and testing size. We fine-tune the models with the provided training data and tune several hyper-parameters to obtain the optimal performances as shown in Table 2. Subtask C uses the same hyper parameters as Subtask A.
All subtasks will be evaluated using standard evaluation metric of weighted F1.
## 6 Results and Discussion
Table 3 represents the performance of the 14 African languages evaluated by the weighted F1 metric.
For the monolingual subtask, the best dataset performance across the Pre-trained models is the Hausa language with an Average of 74.50%, followed by Igbo and Yoruba with 71.32% and 64.55.% respectively. In terms of model performance, each of the pre-trained model performed best in at least one of the monolingual dataset. However, Afro-xlmr-Large ranked highest in best performances in 7 monolingual datasets, followed by AfriBERTa-Large and Arabic-Camelbert with 3 and 2 best performances respectively. BERT and mBERT with least best performances of 1 each.
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Hyper Parameters** & **Subtask A** & **Subtask B** \\ \hline Max-Length & 128 & 150 \\ Batch Size & 16 & 32 \\ Epoch & 5 & 10 \\ Optimizers & AdamW & AdamW \\ Learning Rate & 1e-5 & 2e-5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Subtasks Hyper-Parameter Set
It is noteworthy that Arabic-Camelbert pre-trained model performed best in Darija and Moroccan Arabic due to the fact that it was originally trained in Arabic datasets. Likewise for Afro-xlmr-Large and AfriBERTa-Large which were largely trained in several Afro-centric datasets.
For multilingual subtask, Afro-xlmr-Large also performed best. This is much expected since the dataset is composed of all the 12 languages in the Monolingual subtask. The impressive 69.50% performance is also attributed to the large volume of dataset. AfriBERTa-Large performance was also impressive with 69.30% just 0.20% difference. This performance shows that there's a potential for building a cross-lingual model for a more advanced NLP system.
For zero shot subtask, with two datsets, AfriBERTa-Large and Afro-xlmr-Large share the top spots with each emerging best in 1 of the tracks. AfriBERTa-Large dominated the Tigrinya zero shot track with an impressive 62.50% performance. While Afro-xlmr-Large still claim another excellent performance on Oromo dataset track with 46.20% as the best score.
Lastly, to average the performance across all the three subtasks, Afro-xlmr-Large came top with overall top score of 62.26%. It was trailed behind by AfriBERTa-Large with 60.53%. mBERT, Arabic-CamelBERT and BERT coming distant third with 59.53.00%, fourth with 56.35% and fifth with 53.12% respectively.
### Ablation Study
We further perform an ablation study to attribute the reason for the performance across the datasets and subtasks. For Monolingual, the Xitsonga language achieved the least best performance with a weighted F1 of 50.3%. While Hausa achieved the best of the bests performance with 81.00%.
As shown in Table 1, the Xitsonga language has a smaller volume of data of just 804 training size and thus the lower performance. While Hausa has 14172 and achieved a superior performance across all the five pre-trained models. Therefore, we are of the opinion that there is a correlation between performance and volume of data for a given language. There is better performance when implemented in languages with larger datasets than in smaller dataset.
\begin{table}
\begin{tabular}{r|c c c c c c} \hline \hline \multicolumn{2}{c|}{**DATASETS**} & \multicolumn{4}{c}{**PERFORMANCE ON WEIGHTED F1 METRIC**} \\ \multicolumn{2}{c|}{**Subtask A: Monolingual**} & \multicolumn{2}{c}{**AfriBERTa Afroxlmr-**} & ****\#arabic- BERT** & \multicolumn{2}{c}{**mBERT**} & \multicolumn{1}{c}{**Average**} \\ & **Large** & **Large** & **camelbert** & & & \\ \hline am & 50.80 & 57.30 & 50.10 & **70.00** & 54.30 & 56.50 \\ dz & 54.60 & 64.50 & **65.10** & 64.00 & 54.70 & 60.58 \\ ha & 79.50 & **81.00** & 69.20 & 66.00 & 76.80 & **74.50** \\ ig & **77.00** & 73.30 & 68.00 & 65.00 & 73.30 & 71.32 \\ kr & 50.90 & **70.60** & 51.30 & 34.00 & 65.70 & 56.50 \\ ma & 58.20 & **58.50** & **58.50** & 45.00 & **58.50** & 55.74 \\ pcm & 64.20 & **68.50** & 63.70 & 45.00 & 66.60 & 63.60 \\ pt & 66.70 & **68.50** & 51.40 & 68.00 & 64.40 & 63.80 \\ sw & **63.20** & 58.10 & 56.90 & 37.00 & 57.30 & 54.50 \\ ts & 42.90 & **50.30** & 40.40 & 37.00 & 44.00 & 44.92 \\ twi & **64.10** & 48.00 & 55.70 & 44.00 & 59.30 & 54.22 \\ yo & 62.90 & **71.90** & 60.15 & 60.00 & 67.80 & 64.55 \\ \hline
**Subtask B: Multilingual** & & & & & & \\ \hline Multilingual & 69.30 & **69.50** & 62.00 & 66.00 & 60.82 & **64.53** \\ \hline
**Subtask C: Zero Shot** & & & & & & \\ \hline Ti & **62.50** & 55.00 & 58.30 & 56.70 & 56.70 & **57.84** \\ Or & 41.20 & **46.20** & 34.50 & 39.50 & 32.80 & 38.84 \\ \hline
**Average** & 60.53 & **62.26** & 56.35 & 53.15 & 59.53 & 58.40 \\ \hline \hline \end{tabular}
\end{table}
Table 3: **Sentiment Classification Performance for the Three Subtasks.** Subtask A - Monolingual, Subtask B - Multilingual and Subtask C - Zero Shot. *Afroxlmr-large-ner-maskhaner-1.0-2.0 version of Afroxlmr is used for the Multilingual Dataset. **Bert-base-arabic-camelbert-da-sentiment
Conclusion and Future Work
In this paper, we presented our system description for the SemEval shared task on sentiment analysis for low-resource African languages using Twitter dataset. The task consists of three sub-tasks: Monolingual, Multilingual, and Zero-Shot. Several pretrained LLMs were used for fine-tuning. Afro-xlmr-large performed relatively best across the three subtasks coming top in 9 out of 15. AfriBERTa came top in 4, while Bert-base-arabic-camelbert-da-sentiment performed best in arabic datasets of Darija and Algerian Arabic. BERT and mBERT also manage to came top in 1 task each. Experimental results demonstrated that Nigerian languages: Hausa, Igbo and Yoruba achieved better performance compared to other languages due to the higher volume of data present in the languages. Our results indicate that deep learning are effective in sentiment classification in the African language given the right data, model and training. For future work, to incorporate other data sources such as moview reviews or news articles for generalizability. We also recommend fine tuning for specific individual languages to incorporate linguistic features specific to each language, such as idioms or colloquialisms.
## Acknowledgements
We would like to acknowledge the support of HausaNLP Management for providing us with the Google Colab GPU Premium Version.
|
2303.00189 | Quantum PT-Phase Diagram in a Non-Hermitian Photonic Structure | Photonic structures have an inherent advantage to realize PT-phase transition
through modulating the refractive index or gain-loss. However, quantum PT
properties of these photonic systems have not been comprehensively studied yet.
Here, in a bi-photonic structure with loss and gain simultaneously existing, we
analytically obtained the quantum PT-phase diagram under the steady state
condition. To characterize the PT-symmetry or -broken phase, we define an
Hermitian exchange operator expressing the exchange between quadrature
variables of two modes. If inputting several-photon Fock states into a
PT-broken bi-waveguide splitting system, most photons will concentrate in the
dominant waveguide with some state distributions. Quantum PT-phase diagram
paves the way to the quantum state engineering, quantum interferences, and
logic operations in non-Hermitian photonic systems. | Xinchen Zhang, Yun Ma, Qi Liu, Nuo Wang, Yali Jia, Qi Zhang, Zhanqiang Bai, Junxiang Zhang, Qihuang Gong, Ying Gu | 2023-03-01T02:36:40Z | http://arxiv.org/abs/2303.00189v3 | # Quantum Phase Diagram of PT-Symmetry or Broken in a Non-Hermitian Photonic Structure
###### Abstract
Classically, PT symmetry or broken in photonic structures is well studied, where only average effect of gain and loss on each optical mode is considered. However, in quantum, the role of gain or loss in a non-hermitian system is totally different, the specific quantum optical effect induced by which has never been studied. Here, we analytically obtained the PT-symmetry and PT-broken regime bounded by two exceptional lines in a bi-photonic structure with both gain and loss simultaneously existing. For the consideration of reality, the steady state condition under the weak gain is identified. We defined the exchange operator to represent the photon exchange between two modes and further to characterize the transition from PT symmetry to broken. Also, in the PT broken bi-waveguide system, multi-photon state can be on-demand engineered through the quantum interference. Quantum PT-Phase diagram with steady state regime is the basis to study the quantum state fabrication, quantum interferences, and logic operations in non-hermitian quantum systems.
## I I. Introduction
In the nature, the exchange of energy and particles between the physical system and the external environment is universal. Open physical system is usually described by non-Hermitian Hamiltonian [1]. Especially, the non-Hermitian system with PT-symmetry can give the eigenvalues of complete real numbers like Hermitian system [2, 3, 4]. However, the non-Hermitian system may also have the eigenvalues of conjugate complex numbers, that is, the breaking of PT-symmetry. PT-symmetry or broken depends on the relationship of system parameters and they are separated by exceptional point (EP) [5]. Among many physical branches, optical realization of PT-symmetry is relatively easy because we can obtain complex potential function by modulating the refractive index of the optical material [6]. Optical waveguides [7, 8] and whispering gallery microcavities [9, 10] with gain and loss are good candidates to realize PT-symmetry or broken directly through the two-mode coupling. Recently, PT-symmetry has also been observed on optical lattice [11] and metasurface [12]. Nearly all kinds of PT-symmetric photonic structures are non-reciprocal [13] or unidirectional visible [14], which can be used to optical isolation devices [9]. In addition, PT symmetric optics has many applications in enhanced sensing [15], laser [16, 17] and chiral optics [18].
The above statements are the properties and applications of PT-symmetry or broken under classical optics. For a single mode, we only need to consider the average effect of gain and loss. For example, in the paper [8], both waveguides have the same loss. At the same time, one of the waveguides is loaded with a gain effect that is twice the loss in value, so on average, it becomes a balanced gain-loss structure. However, if we consider the quantum light field, that is, quantum jump [19] plays an important role, the gain and loss will no longer be the temporal inverse processes [20]. Because gain will inevitably bring quantum noise, while loss will not reduce quantum noise [21]. Non-Hermitian quantum photonics is a discipline to expansively define PT-symmetry [22] and to explore the quantum properties of light in this background. It can be applied to integrated quantum optical circuits and quantum information processing [23]. At present, the main method to study non-Hermitian quantum photonics is to solve the Lindblad master equation [24] for the quantum state transmission in a non-Hermitian two-waveguide coupling structure. The Lindblad master equation controls the density matrix of quantum state with the Liouville operator. The eigenvalue degeneracy point of the Liouville operator is the EP of this system [19]. In this way, people have found the spontaneous generation of photons [25, 26], the connection between PT-symmetry and quantum interference effect [27, 28, 29], the effect of gain or loss on quantum entanglement [30, 31], and so on.
For a non-Hermitian quantum photonics, the case that only gain or loss is in a single optical mode has been widely studied. Another case, closer to reality, with both gain and loss simultaneously existing, has not been fully taken into account [32]. After considering this more general situation thoroughly, what are the conditions and some new findings for PT-symmetry or broken?
In the following, we first analytically obtain the quantum phase diagram of PT-symmetry or broken in bi-photonic structure with both gain and loss simultaneously existing. The phase diagram shows the parameter conditions for PT symmetry or broken with the steady-state solution regime [Fig. 1]. In order to verify the EP line that separating PT-symmetry or broken, we numerically calculated the eigenvalues of the Liouville operator, and found that the degeneracy point of the eigenvalues is consistent with our analytical derivation. Then, we defined a new physical quantity exchange factor, which can represent the energy exchange between two optical modes, and can also be used to characterize PT symmetry or broken.
It is worth noting that if we take the quantum PT-symmetric waveguide system as a whole, it is equivalent to a non-Hermitian beam splitter. Beam splitters are very important devices in classical and quantum optics, and they play an important role in quantum interference experiments. However, the absorption of light by materials is always unavoidable, so the non-Hermitian beam splitter came into being [33]. Non-Hermitian beam splitters also have unique properties and applications, such as quantum coherent perfect absorption [34], anti-bunching of bosons [35], preparation of squeezed states [36], and fabrication of multi-bit quantum gates [37]. So at last, we tried to transport multi-photon states in the non-Hermitian two-waveguide coupling structure. We found that some on-demand quantum states can be prepared in the PT broken regime. Our PT phase diagram and related results are the basis for the study of non-Hermitian quantum photonics and have great application potential in quantum interference, quantum state engineering, quantum beam splitting and so on.
## II II. Quantum PT phase diagram with steady state regime
Consider a bi-photonic structure with loss and gain simultaneously existing [Fig. 1(a)], whose Hamiltonian is
\[\hat{H}=\hbar\omega_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hbar\omega_{2}\hat{ a}_{2}^{\dagger}\hat{a}_{2}+\hbar\mu(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{ 2}^{\dagger}\hat{a}_{1}) \tag{1}\]
where \(\hat{a}_{i}\) and \(\hat{a}_{i}^{\dagger}\) are the boson annihilation and creation operator, respectively, and \(\mu\) is the coupling strength between two cavities. For simplicity, we let \(\omega_{1}=\omega_{2}=\omega\). With the steady state condition and weak incident light, the gain saturation effect can be neglected [38]. Then the non-hermitian system is governed by Lindblad master equation [24],
\[\begin{split}\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}, \hat{\rho}]&+\sum_{i=1,2}\gamma_{i}(2\hat{a}_{i}\hat{\rho}\hat{a} _{i}^{\dagger}-\hat{\rho}\hat{a}_{i}^{\dagger}\hat{a}_{i}-\hat{a}_{i}^{ \dagger}\hat{a}_{i}\hat{\rho})\\ &+\sum_{i=1,2}\beta_{i}(2\hat{a}_{i}^{\dagger}\hat{\rho}\hat{a}_ {i}-\hat{\rho}\hat{a}_{i}\hat{a}_{i}^{\dagger}-\hat{a}_{i}\hat{a}_{i}^{ \dagger}\hat{\rho})\end{split} \tag{2}\]
where \(\gamma_{i}\) (\(\beta_{i}\)) is the loss (gain) coefficient of the \(i\)th structure. From the Eq. (2) one can see that gain and loss in quantum regime are totally different processes, which can not be combined into one term. To construct the quantum PT phase diagram, we derive the evolution equations of \(\langle\hat{a_{1}}\rangle\) and \(\langle\hat{a_{2}}\rangle\) with \(t\) based on above Eq. (2),
\[\frac{d}{dt}\left(\begin{array}{c}\langle\hat{a}_{1}\rangle\\ \langle\hat{a}_{2}\rangle\end{array}\right)=M\left(\begin{array}{c}\langle \hat{a}_{1}\rangle\\ \langle\hat{a}_{2}\rangle\end{array}\right) \tag{3}\]
where
\[M=\left(\begin{array}{cc}-\gamma_{1}+\beta_{1}&-i\mu\\ -i\mu&-\gamma_{2}+\beta_{2}\end{array}\right). \tag{4}\]
The eigenvalues of the matrix \(M\) are
\[\begin{split}\omega_{\pm}=&\frac{1}{2}i(-\gamma_{1}+\beta_{1}- \gamma_{2}+\beta_{2})\\ &\mp\frac{1}{2}i\sqrt{[(-\gamma_{1}+\beta_{1})-(-\gamma_{2}+\beta_{2})]^{2} -4\mu^{2}}.\end{split} \tag{5}\]
The degeneracy of eigenvalues \(\omega_{\pm}\) which satisfies \(|(-\gamma_{1}+\beta_{1})-(\gamma_{2}+\beta_{2})|=2\mu\), are called exceptional lines (EPLs), shown as red lines in PT-phase diagram in Fig. 1(b). The evolution of \(\langle\hat{a_{1}}\rangle\) and \(\langle\hat{a_{2}}\rangle\) represents the properties of our system. The EPLs calculated from the matrix \(M\) are the boundaries between PT symmetry and PT broken of our system. The area between two red lines in which \(|(-\gamma_{1}+\beta_{1})-(\gamma_{2}+\beta_{2})|<2\mu\) is PT symmetric while the areas outside two lines in which \(|(-\gamma_{1}+\beta_{1})-(\gamma_{2}+\beta_{2})|>2\mu\) are PT broken. PT phase diagram in quantum system are quite different from the classical system, in that each single point on quantum PT phase diagram contains infinite cases.
Next one can see that, the EPLs that we have obtained by this new analytical method are consistent with that numerically solved by the master equation. The master equation Eq. (2) could be written as \(\frac{d\hat{\rho}}{dt}=\mathcal{L}\hat{\rho}\). With the super operator Liouvillian \(\mathcal{L}\), the quantum evolution properties of the system are described entirely. Given a set of complete quantum state basis vectors, or a superior limit of photon number, the Liouvillian \(\mathcal{L}\) can be expressed by a high dimension matrix \(L\). We can also find EPs according to the splitting of eigenvalues of the matrix \(L\)[19]. Fig. 1(c) shows the eigenvalues \(\lambda_{i}\) as functions of coupling coefficient \(\mu\). An obvious splitting point locates at \(\mu=1\), which is identical to our theoretical prediction.
Furthermore, the evolution of the mean photon number \(\langle\hat{n_{1}}\rangle=\langle\hat{a_{1}^{\dagger}}\hat{a_{1}}\rangle\), \(\langle\hat{n_{2}}\rangle=\langle\hat{a_{2}^{\dagger}}\hat{a_{2}}\rangle\) of two modes, and an exchange factor \(\langle\hat{\eta}\rangle=\langle i(\hat{a_{2}^{\dagger}}\hat{a_{1}}-\hat{a_{1} ^{\dagger}}\hat{a_{2}})\rangle\) can be written as,
\[\frac{d}{dt}\left\langle\hat{n}_{1}\right\rangle =2\left(\beta_{1}-\gamma_{1}\right)\left\langle\hat{n}_{1} \right\rangle+\mu\left\langle\hat{\eta}\right\rangle+2\beta_{1} \tag{6}\] \[\frac{d}{dt}\left\langle\hat{n}_{2}\right\rangle =2\left(\beta_{2}-\gamma_{2}\right)\left\langle\hat{n}_{2} \right\rangle-\mu\left\langle\hat{\eta}\right\rangle+2\beta_{2}\] \[\frac{d}{dt}\left\langle\hat{\eta}\right\rangle =2\mu\left\langle\hat{n}_{2}\right\rangle-2\mu\left\langle\hat{n}_{ 1}\right\rangle+\left(\beta_{1}+\beta_{2}-\gamma_{1}-\gamma_{2}\right)\left\langle \hat{\eta}\right\rangle\]
whose solutions satisfy the steady state conditions that both \(\gamma_{1}+\gamma_{2}-\beta_{1}-\beta_{2}>0\) and \((\gamma_{1}-\beta_{1})(\gamma_{2}-\beta_{2})+\mu^{2}>0\), shown as the yellow regime of phase diagram in Fig. 1(b). Under the steady state conditions, the final steady state value of mean photon number of two modes as well as \(\left\langle\hat{\eta}\right\rangle\) can be written as
\[\left\langle\hat{n}_{1}\right\rangle_{ss} =\frac{\Delta_{1}-\beta_{1}\left(\Delta_{2}+2\beta_{2}\gamma_{2} -\gamma_{2}^{2}\right)}{\Delta_{3}} \tag{7}\] \[\left\langle\hat{n}_{2}\right\rangle_{ss} =\frac{\Delta_{1}-\beta_{2}\left(\Delta_{2}+2\beta_{1}\gamma_{1}- \gamma_{1}^{2}\right)}{\Delta_{3}}\] \[\left\langle\hat{\eta}\right\rangle_{ss} =\frac{2\mu\left(\beta_{2}\gamma_{1}-\beta_{1}\gamma_{2}\right)}{ \Delta_{3}}\]
with \(\Delta_{1}=(\beta_{1}+\beta_{2})(\beta_{1}\beta_{2}+\mu^{2})\), \(\Delta_{2}=\beta_{1}\gamma_{2}+\beta_{2}\gamma_{1}-\gamma_{1}\gamma_{2}\), and \(\Delta_{3}=(\gamma_{1}+\gamma_{2}-\beta_{1}-\beta_{2})[(\gamma_{1}-\beta_{1}) (\gamma_{2}-\beta_{2})+\mu^{2}]\). Here, the parameters in the steady state region should be satisfied with the weak gain condition. From Eq. (7), one can see that, for one steady state point, there are infinite sets of parameters \(\gamma_{1}\), \(\beta_{1}\), \(\gamma_{2}\), and \(\beta_{2}\) corresponding to infinite steady state values. But owing to the decoherence effects of loss and gain, the steady state will become a thermal optical field without any quantum feature. Our following discussions are limited within the steady state regime.
## III III. Exchange factor and quadrature amplitudes
To characterize the PT-symmetry or broken, we rewrite the exchange operator \(\hat{\eta}\) as
\[\hat{\eta}=2(\hat{X}_{2}\hat{Y}_{1}-\hat{X}_{1}\hat{Y}_{2}) \tag{8}\]
with \(\hat{X}_{1,2}=(\hat{a}_{1,2}+\hat{a}_{1,2}^{\dagger})/2,\hat{Y}_{1,2}=i(\hat{ a}_{1,2}-\hat{a}_{1,2}^{\dagger})/2\). \(\hat{\eta}\) is an Hermitian operator so its expectation value \(\langle\hat{\eta}\rangle\) is a real number, so we can call it exchange factor. According to the Eq. (6), \(\langle\hat{\eta}\rangle\) can characterize the exchange between two modes, that is, \(\langle\hat{\eta}\rangle>0\) indicates the energy (photon) is flowing from mode \(2\) to mode \(1\) and vise versa. However, as an important physical quantity in PT-symmetry, exchange factor has not been defined or studied yet in the previous literature. For convenience, here we directly use letters without hats such as \(\eta,X_{1},Y_{2}\) to represent the expectation values of the corresponding operators, i.e., \(\eta=\langle\hat{\eta}\rangle,X_{1}=\langle\hat{X}_{1}\rangle,Y_{2}=\langle \hat{Y}_{2}\rangle\)
Fig. 2(a) gives the evolution of \(\eta\) with varying the loss rate \(\gamma_{1}\). Here, \(\beta_{1}=0.1,\gamma_{2}=0.1,\beta_{2}=0.1,\mu=1\). \(\eta\) experiences the phase of PT symmetry with \(\gamma_{1}=1.1\), through the EP with \(\gamma_{1}=2.1\), to PT broken with \(\gamma_{1}=3.1\), corresponding to the yellow star, gray star and red star in Fig. 1(b), respectively. It is seen that, when PT symmetry is unbroken, \(\eta\) oscillates with varying \(t\). In contrast, when PT symmetry is broken, \(\eta\) monotonically decreases after a rise and then comes to steady state. Therefore, exchange factor \(\eta\) can fully characterize the properties of
PT symmetry or broken. Moreover, according to the Eq. (7), considering a specific situation, if \(\gamma_{2}=\beta_{2}\), then the steady state value of exchange factor will be
\[\eta_{ss}=\frac{2\gamma_{2}}{\mu}. \tag{9}\]
It will only depends on \(\gamma_{2}\) and \(\mu\), but have nothing to do with \(\gamma_{1}\) and \(\beta_{1}\). For example, in Fig. 2(a), every curve finally approaches to a fixed value \(0.2\), no matter how large the loss of mode \(1\) is. Especially, the blue curve corresponding to an EP point is the fastest one to come to steady state, which is a counter-intuitive phenomenon.
Correspondingly, we explore the exchanging processing from PT-symmetry to broken in the form of the expectation values of \(\hat{X}_{1,2}\) and \(\hat{Y}_{1,2}\). From Eq. (8), the exchange occurs between \(X_{1}\) and \(Y_{2}\), or between \(X_{2}\) and \(Y_{1}\). For the PT-symmetry regime, the quadrature amplitudes \(X_{1,2},Y_{1,2}\) oscillate continuously in Fig. 2(b). While for PT-broken, they decay exponential when \(t>2\) as shown in Fig. 2(c). Inputting the Fock state \(|n\rangle\), the initial values of \(X_{1,2},Y_{1,2}\) are all \(0\). According to the Eq. (3), at any time \(X_{1,2},Y_{1,2}\) will equal to \(0\). So only using the exchange between quadrature amplitudes, one can not distinguish the PT-symmetry or broken if only inputting Fock states. While, whatever for Fock states or coherent states, one can clearly distinguish them through the exchange factor \(\eta\).
## IV IV. Quantum state engineering
The above theory is applicable to two-mode coupling in various photonic structures. In this section, let's take the coupled waveguides as an example to study the quantum state transmission or engineering. To start with, we need to change the evolution time \(t\) to the propagation distance \(z\) in the waveguides. Shown as in Fig. 3(a), the coupled waveguides with gain and loss can be looked as a non-Hermitian beam splitter. If we input quantum states from two left ports, transformed quantum states will be output from two right ports, with a part of photons absorbed or generated by material. In the following, we will mainly study the quantum state transmission in the full-parameter space of our quantum PT phase diagram.
First, we consider the case that there is no more than two photons input. For PT symmetry and PT broken, the transmission behavior will be different. This reflects the concept of "dominant waveguide". If PT is broken, photons will mainly concentrate on one of the waveguides, so that the output state basically contains only two states, \(|01\rangle\) and \(|02\rangle\). This waveguide is called "dominant waveguide". From \(\eta\) part of the Eq. (6) and Eq. (7), we can also find that which one is the "dominate waveguide" depends on the ratio of gain rate and loss rate \(\beta_{i}/\gamma_{i}\). However, PT symmetry leads to a mixture of multiple results, and the probability of \(|01\rangle\), \(|02\rangle\), \(|10\rangle\) and \(|20\rangle\) will all exceed \(10\%\). The concept of "dominant waveguide" will be absent.
Then consider the case that each port inputs one photon, that is, the HOM experiment. We find that there is an obvious depression in the probability of \(|11\rangle\) states under any parameter, and the depression probability can reach \(0\) when only with loss, which reflects the quantum interference phenomenon between waveguides. This special case can be analytically solved by Lie algebra method [27]. Moreover, in the PT broken regime, we can directly (without post-selection) obtain the mixed states of Fock state \(|1\rangle\) and \(|2\rangle\) from two Fock states \(|1\rangle\) using this passive structure. In the literature [25], vacuum state, single photon state and two photons NOON state are inputted in the gain-loss coupled waveguides. Then photons will be spontaneously generated with non-classical characters.
Next, let's consider Fock state \(|M,N\rangle\) with more pho
Figure 2: Different transmission behaviors in PT symmetry regime and PT broken regime characterized by the exchange factor \(\eta\) or quadrature amplitudes \(X_{1,2},Y_{1,2}\). (a) The evolution of \(\eta\) at different loss rate \(\gamma_{1}\). Here, \(\beta_{1}=0.1,\gamma_{2}=0.1,\beta_{2}=0.1,\mu=1\). The initial state is \(|0,1\rangle\). (b) The exchange of \(X_{1}\) and \(Y_{2}\) in PT symmetry regime, where \(\gamma_{1}=1.1\) (The yellow star in Fig. 1(b)). Inset: The exchange of \(X_{2}\) and \(Y_{1}\) in PT symmetry regime. (c) The exchange of \(X_{1}\) and \(Y_{2}\) in PT broken regime, where \(\gamma_{1}=3.1\) (The red star in Fig. 1(b)). Inset: The exchange of \(X_{2}\) and \(Y_{1}\) in PT broken regime. Other parameters in (b)(c) are the same as that in (a). The initial state in (b)(c) is coherent state \(|\alpha=1+i,0\rangle\).
tons, with the total number of photons \(M+N<10\). We study the output quantum states in the case of PT symmetry or broken. It should be emphasized that the output quantum states here are not the steady states obtained by transmitting a long enough distance, but are states directly extracted after propagating to a certain distance. This distance is usually slightly smaller than the photon exchange distance \(z_{0}=\pi/4\mu\). On this distance, the quantum state has been well transformed by the structure and has not lost too much quantum feature. As shown in Fig. 3, in the PT symmetry regime, the photon number distribution is dispersed. It is due to photons are exchanging constantly between two waveguides. While in the PT broken regime, the photon number distribution is concentrative, that is because most photons are gathered in "dominate waveguide?, which is very useful for preparing quantum states.
With the same parameters, and the same total photon number 9, there is some difference between input \(|36\rangle\) and \(|63\rangle\). The number of photons in the "dominant waveguide" plays a leading role in the output quantum state, while the number of photons in the "inferior waveguide" is only a perturbation. In Fig. 3(d), the highest probability state is \(|07\rangle\), while in Fig. 3(f), it is \(|04\rangle\). So after a photon exchange distance \(z_{0}\), no matter how many photons there are at the beginning in waveguide 1, waveguide 2 can only receive one photon. And all the photons in waveguide 1 disappear. We call it "multiple-single photon exchange". Therefore, according to the "multiple-single photon exchange", if we want a quantum state that mainly contains \(|0,N+1\rangle\), then we can choose \(|M,N\rangle\) to input. This method of quantum states engineering has potential to apply to multi-photon state generation, quantum gate and so on.
## V V. Summary
In summary, we have analytically obtained the quantum PT phase diagram with the steady state regime in non-Hermitian photonic structures. We have defined an exchange operator to characterize the PT-symmetry phase and PT-broken phase. Based on this phase diagram, we have engineered the multi-photon quantum state in the coupled waveguides structure. The present work has constructed the basic theory of quantum PT symmetry bi-photonic structure as well as its application to quantum state engineering. The established theory can be extended to study many related quantum behaviors, such as gain saturation effect, quantum entanglement, and continuous variable states, and may have potential applications in quantum interference, on-chip quantum information processing and scalable quantum networks.
_Acknowledgments._ This work is supported by the National Natural Science Foundation of China under Grants Nos. 11974032, 11525414, and 11734001, and by the Key R&D Program of Guangdong Province under Grant No. 2018B030329001.
Figure 3: Quantum state engineering based on PT symmetry or PT broken. Schematics of coupled waveguides with input Fock states (a) \(|3\rangle|6\rangle\) and (b) \(|6\rangle|3\rangle\) input. In the case of (a), photon number distribution of quantum state at \(z=0.3cm\) for (c) PT-symmetry and (d) PT broken. In the case of (b), photon number distribution of quantum state at \(z=0.5cm\) for (e) PT-symmetry and (f) PT broken. Here, the parameters for (c) and (e) are the same as Fig. 2(b), corresponding the yellow star in Fig. 1(b). The parameters for (d) and (f) are the same as Fig. 2(c), corresponding to the red star in Fig. 1(b). |
2305.04061 | We Are Not There Yet: The Implications of Insufficient Knowledge
Management for Organisational Compliance | Since GDPR went into effect in 2018, many other data protection and privacy
regulations have been released. With the new regulation, there has been an
associated increase in industry professionals focused on data protection and
privacy. Building on related work showing the potential benefits of knowledge
management in organisational compliance and privacy engineering, this paper
presents the findings of an exploratory qualitative study with data protection
officers and other privacy professionals. We found issues with knowledge
management to be the underlying challenge of our participants' feedback. Our
participants noted four categories of feedback: (1) a perceived disconnect
between regulation and practice, (2) a general lack of clear job description,
(3) the need for data protection and privacy to be involved at every level of
an organisation, (4) knowledge management tools exist but are not used
effectively. This paper questions what knowledge management or automation
solutions may prove to be effective in establishing better computer-supported
work environments. | Thomas Şerban von Davier, Konrad Kollnig, Reuben Binns, Max Van Kleek, Nigel Shadbolt | 2023-05-06T14:19:54Z | http://arxiv.org/abs/2305.04061v1 | We Are Not There Yet: The Implications of Insufficient Knowledge Management for Organisational Compliance
###### Abstract
Since GDPR went into effect in 2018, many other data protection and privacy regulations have been released. With the new regulation, there has been an associated increase in industry professionals focused on data protection and privacy. Building on related work showing the potential benefits of knowledge management in organisational compliance and privacy engineering, this paper presents the findings of an exploratory qualitative study with data protection officers and other privacy professionals. We found issues with knowledge management to be the underlying challenge of our participants' feedback. Our participants noted four categories of feedback: (1) a perceived disconnect between regulation and practice; (2) a general lack of clear job description; (3) the need for data protection and privacy to be involved at every level of an organisation; (4) knowledge management tools exist but are not used effectively. This paper questions what knowledge management or automation solutions may prove to be effective in establishing better computer-supported work environments.
## 1 Introduction
When GDPR first went into effect in 2018, Sirur et al. asked, "Are we there yet?" regarding the ability of the industry to comply with the new legislation [36]. They found limited concerns within the private sector about ensuring compliance in larger, well-equipped organisations, with additional concerns in smaller organisations [36]. However, the landscape of recent laws regarding data privacy and online safety has expanded rapidly to include GDPR, Brazil's LGPD, California's CCPA, the planned EU AI Act, and various proposed online harms legislation. As a result, the tech industry must keep track of and comply
with a significant body of laws, which motivated us to explore the experiences of relevant industry experts.
Attempts within organisations to consider privacy and data protection often have engineers and designers collaborate with lawyers and policy experts. However, studies going back as far as the early 2000s have shown that the teams building technology do not feel as though privacy and data protection are among their primary concerns and obligations [26, 6]. As a response, researchers suggested a range of value-based software engineering frameworks[37, 33, 39]. As privacy data protection laws have created new roles within organisations, notably data protection officers (DPOs) and other privacy professionals, we see the opportunity to revisit some of these past discussions, especially in light of literature exploring the benefits of knowledge management (KM) on compliance behaviour [18, 9, 42, 22].
DPOs and privacy professionals within organisations are the primary parties on the receiving end of regulations crafted by legislators. Their perspective on implementing applicable privacy laws is invaluable for crafting future regulations and ameliorating challenges and weaknesses in the current system. Furthermore, their organisations and external parties often consider these industry professionals accountable for ensuring regulatory compliance. This motivates us, in this piece, to identify what role KM plays in the DPO role and where there might be room for improvements. In our work, we focus on the three, related research questions:
**RQ:**: (Knowledge) How do data privacy experts bridge multiple knowledge bases from various teams across an organisation?
**RQ:**: (Communication) How do data protection officers maintain and communicate compliance decisions made internally and externally to the organisation?
**RQ:**: (Experiences) What are the current successes and failures data protection officers experience while working to ensure compliance?
Our preliminary discussions with experts in the industry that fill data protection (DPOs), cybersecurity, or legal roles revealed that they often assume the role of auditors and critics within their organisation. They are responsible for improving the internal practices to ensure that any data breach or other emergency scenario is handled effectively and within the standards outlined by law. They also need to occasionally oppose internal decisions where they breach relevant law.
Ultimately, our interviews revealed four categories of challenges associated with KM and the role of privacy professionals. At a high level, these categories are: (1) a perceived disconnect between how businesses operate and how regulation is implemented; (2) a lack of a clear job description; (3) a need for data protection and privacy to be involved and communicated at every level of an organisation; (4) a common availability but ineffective use of knowledge management tools. By establishing our work within the current canon of related
research in organisational compliance and privacy engineering attitudes, we will present the perspectives of our participants, revealing the need for advances in knowledge management. We will then discuss how future work in computer-aided knowledge management can improve our lives through added compliance with privacy regulations.
## 2 Related Work
There are two predominant areas of research that serve to contextualise and motivate our research. The first is existing research on organisational compliance with regulatory bodies. The second is a body of work that has explored privacy engineering attitudes with an established set of methods.
### The Role of Knowledge Management in Organisational Compliance
In response to innovation, society has started developing rules and regulations to minimise potential dangers. The history of data protection and privacy is closely tied to advances in technological innovation [20]. As our technological capabilities and data collection behaviours increase, the need for careful, privacy-preserving legislation also increases. These regulations often include detailed legal documents and associated lists of standards that need to be processed, understood, and implemented. Furthermore, ensuring compliance with these regulations is an interdisciplinary effort across an organisation.
In many ways, the need for compliance and regulation is nothing new. Specific fields like pharmaceuticals and hazardous industries have been controlled and regulated almost continuously [24, 21, 31, 16]. Naturally, there is an observed, unofficial correlation between the strength of regulation and the danger posed by the field being regulated. Similarly, we see a demonstrated history of regulation in place for the financial market [13, 2]. These fields and the associated papers demonstrate a history of organisational compliance analysis. Furthermore, these fields contain a notable body of research into the effectiveness of KM on compliance across an organisation's employees [18, 9, 42]. The findings indicate the potential for more effective KM to be correlated with improved compliance behaviours. Through rigorous study and review of the fields, we can better understand the impact of regulation and continue to steer the system away from ineffective solutions.
A notable portion of the research done on organisational compliance focuses on what regulators need to do or how they should respond to certain changes in the field [31, 13, 2, 16]. For example, some regulatory theory argues that regulators need to be "really responsive" in their process by taking into account the actions, frameworks, and attitudes of the organisations they are regulating [5]. While it is undoubtedly valuable to elevate items of concern to the attention of regulators, there is often limited analysis of what steps an organisation can take to improve their regulatory compliance [21, 24]. These analyses often highlight
the challenges organisations face by balancing compliance with expansion and improvement and the risks involved with these decisions. Some articles take a step further by implying that the decision to comply with regulations is relatively light and can be easily dismissed [10, 11]. One such article advocates for "winning over regulators" rather than working to understand the need for regulation in the first place. Another highlights the danger of compliance becoming a passive "checklist" activity. To solve the potential issue of compliance becoming a box-ticking exercise, regulatory theory argues that through discourse between organisations and legislators compliance and legitimacy can become a more interactive process and as a result, a more effective process [8, 28]. In light of this research at the organisational level, our research intends to explore the specific actions taken by professionals responsible for regulatory compliance within an organisation, especially in data protection and privacy.
Some work within the area of data protection and privacy in conjunction with industry professionals has been done within the context of energy company employees and drone operators and manufacturers [17, 12]. Here there is further support of the correlated impact between KM and compliance behaviour[22]. These research initiatives explored how KM can be used across a broad range of employees in their individual compliance tasks. Our research aims to explore the role of KM for a specific subset of data regulation and privacy specialists.
### Value-Based Privacy Engineering
While data protection and privacy have grown into regulation and legal discussions, a body of research has already been done to improve the considerations of privacy in design and engineering. An early study by Lahlou and Langheinrich opened the door to exploring how engineers view their role in ensuring data protection and privacy [26, 32]. To clarify, the general term "engineers" used in this paper and subsequent work refers to all technical product team members involved in building a system, such as designers, systems engineers, and requirement engineers. This early work led to a set of methods and approaches to design and engineering that continue to generate discourse in the field of ethical computing [25, 37, 30]. However, comparing and analysing the effectiveness and usage of these frameworks lie outside this paper's scope.
These frameworks have been explored and tested via a selection of methods aimed at understanding the attitudes and behaviours of industry professionals concerning data protection and privacy. We reviewed a collection of these methods to contextualise our approach within this canon of research. Shilton used a participant-observer method within an academic design and research team to explore what experiences and interactions were effective in introducing ethics into the design process [33]. Their combination of fly-on-the-wall observations with follow-up interviews provided fascinating insights. Nonetheless, their participants were in an open academic environment, whereas our interested population is industry professionals dealing with proprietary information. Some research on industry professionals has been done by reviewing anonymous posts on a professional discussion forum investigating privacy practices of mo
bile developers [34]. The challenge is that, unlike organised iOS or Android forums, there are limited forums for privacy professionals within the tech sector for us to use for our research.
Ultimately it was recent work that combined survey data with interview data that provided us with a potential method of investigation. To better understand the challenges experienced by engineers, there was a two-part project conducted on professionals from Ubicomp as well as select senior-level engineers [38, 6]. This mixed-method approach effectively revealed attitudes towards privacy and responsibility felt by many engineers. We decided to take a similar approach with our methodology with some adjustments to fit our particular research interests.
From this research into potential methods regarding industry professionals, we were motivated to focus on a slightly different population that has grown since GDPR came into force in 2018. This population of data protection officers are part of the technical teams that have been explored in previous work, but their roles are officially associated with regulatory compliance. Again, we can look to previous research done at the outset of GDPR [36]. They interview industry professionals before the establishment of DPOs. Their findings were hesitantly optimistic, citing that overall, organisations felt comfortable with regulatory compliance expectations, with some concerns among smaller organisations not working in the tech or security sectors [36]. Since the 2018 paper, there remains a need for further investigation into how DPOs are functioning in their roles within organisations. A 2021 survey in Croatia found 82% of their sampled DPOs reported they did not have enough education, knowledge, or training needed to feel ready for their role [29]. Thus our interview study aims to reveal specific discussions regarding attitudes and approaches to organisational data protection and privacy.
## 3 Methods
This outlines the details involved in creating and executing our qualitative exploratory study. We were inspired by the methods Spiekermann et al. and Bednar used in related work to elicit attitudes and thoughts directly from the participants [38, 6, 36]. The primary stakeholders we were interested in meeting with were data protection officers, project managers, and any developers or legal team members working with data regulation compliance.
### Recruitment
To gather our participants, we utilised professional online networks that would reach industry professionals working within organisations across the USA and UK. We recruited through direct emails, messages, and a larger post calling for participants shared on LinkedIn and Twitter. We worked to provide the participants with the option to either discuss online in real-time or fill out the interview questions as an online form.
We got insights from a total of eight participants, which builds on the numbers of previous work [26, 6]. Six participants were interviewed, with the remaining two filling out the interview questions via the form. The participants represented a range of data protection and privacy professionals from early career to C-suite (see Table 1). Their companies ranged from small startup sizes to large international organisations.
### Procedure
The primary method was to conduct a virtual interview between one of our researchers and an industry professional. The interview was designed to understand how the professionals approach regulatory compliance within their role as specialised professionals (Appendix A). Under a semi-structured interview, we could ask follow-up questions for deeper understanding. The second was an online form meant as the "lighter" of the two ways to participate in our research (Appendix A). It was essentially identical to the interview but without the possibility of a follow-up. It also provided the participants with additional anonymity.
We spent roughly 30-40 minutes with each participant in the interview. Using Microsoft Teams for the online interviews allowed us to use the automatic transcription tool. This avoided the need to capture any permanent audio or video recordings of the participant to ensure their anonymity. Furthermore, the transcripts were further anonymised to remove any reference to organisation or participant names. Our work to ensure and maintain anonymity is one of the reasons we successfully accessed individuals in privileged positions within their organisations. Section 6.2 considers why our anonymity commitment was vital for our participants. Additionally, we insisted that we were not interested in proprietary organisational information but rather in the anecdotes and learnings of the participants themselves. They were the experts dealing with internal and external policies, and we wanted to hear from them.
\begin{table}
\begin{tabular}{l l l l} \hline \hline ID (Gender) & Role & Location & Company Size \\ \hline P1 (M) & Data protection officer & USA & 250-500 \\ P2 (M) & Regulatory leader & UK & 1000+ \\ P3 (F) & Legal and security operator & UK & 50-100 \\ P4 (M) & Founder of software company & UK & 50-100 \\ P5 (F) & Associate data protection officer & EU & 1000+ \\ P6 (M) & Project manager & USA & 1000+ \\ P7 (unknown) & Legal and privacy professional & USA & 250-500 \\ P8 (unknown) & VP of privacy & USA & 1000+ \\ \hline \hline \end{tabular}
\end{table}
Table 1: This outlines the IDs and Gender of our participants, a short description of their current role, and their primary location of operation.
### Data Analysis
Following the collection of information from the participants, we conducted an affinity clustering exercise similar to that found in other HCI methods [19]. Affinity clustering involves breaking down the interview transcripts into single ideas or statements. All of the individual statements of the participants are then grouped by similarity. Each group is labelled, creating a new layer, and the process is repeated by grouping the groups until one or two coherent theses form. We reached a high degree of data saturation at around the sixth participant. In other words, we started hearing the same feedback and perspective multiple times, indicating shared experiences.
Applying affinity clustering to the transcripts, we were able to identify three layers of themes regarding the practice of ensuring compliance with regulatory bodies. This method allows us to consolidate feedback from multiple participants under a single idea which then gets grouped with other ideas to form a broader theme or message. Looking across all the themes, we could generate an overall thesis of our findings.
## 4 Findings
The feedback from the participants regarding their experiences highlighted four major categories of challenges. Within each major category, we recorded their thoughts and drew them together and highlighted how advances in knowledge management can address the experiences of our participants.
### Regulation Disconnect
Many industry professionals mentioned feeling a lack of understanding from regulators regarding the approach to real-world implementation of the policies. Our participants stated that implementing specific regulations was easier than others. For example, our participants noted how dealing with cookies is one of the most difficult aspects of their job. Our participants stated,
P3: The laws are very nicely written. But if you are operating a business, how do you put that writing into practice? There's a lot of grey area I mean. Cookies is one great example.
P4: I would say one example is the cookies... situation about cookies because it's a bit of a mess. The rules are not fully clear. They've changed over time.
On the other hand, the same participant mentioned how easy it is to establish proper privacy notices by saying,
P3: The processing is very straightforward for privacy notices.
These selections serve as examples of the differences in implementing data privacy regulation. Both privacy notices and cookie management are common aspects of digital experiences, yet our participants insist there is a difference in how easy they are to implement. While there was some frustration about having different experiences with different types of regulation, there remained a belief that understanding the regulation was important for others in the business. The professionals we met recognised the value of performing consistent regulatory practices.
P3: I think the key for DPIAs is to really get everyone on track. You know it shouldn't be just a tick. I think when it comes to product development. All the people need to really understand data impact assessments.
Data Protection Impact Assessments (DPIAs) were a common talking point during the interviews as they served as both an essential piece of regulatory compliance and a significant job responsibility for our participants. DPIAs, as stated by the participant, required involvement from multiple teams across the organisation, like cookies and privacy notices. Nonetheless, some standard regulatory practices posed greater challenges than others.
The difference in effort needed to achieve certain regulatory practices was further complicated by slight differences introduced by laws in different nations and regions. The challenge with the current regulation of the Internet is the fact that cyberspace often crosses international borders. One participant said,
P4: I do think that the transfer impact stuff is a little bit onerous because the European Union essentially said that you have to make your own judgment on the laws of a foreign country, and that is difficult. For example, if you want to send data to India, then you have to make your own judgment about Indian law, which seems a bit silly that thousands or millions of people should do that
This challenge of laws and regulations altering from one nation to another can be particularly difficult for smaller companies that might not have international regulation experts on their payroll.
P3: There needs to be a bit more help on these things especially for small organizations I would say.
A concern for small businesses reflects early concerns presented by related research at the start of GDPR [36]. Considering the feedback has not changed, there may be a significant unsolved problem for small organisations and ensuring compliance. While there exists ambiguity and assumptions as policies are translated from theory into practice, our participants noted they are open to working further with regulators to better establish this process from law to practice. Again, this mirrors previous interview research that identified the soft and hard power government entities have on the practice of privacy and data protection [6]. Our participants explained why they would want to collaborate with policymakers.
P2: So I think engaging with these kinds of bodies like standards bodies is also one of the ways to both understand the space and contribute to helping with the how these requirements are being shaped.
The privacy professionals were interested in furthering their understanding and contributing their experience to future changes. However, the path to collaboration is not always easy for privacy professionals. One participant mentioned that they could not interact directly with any regulators through their position as only the legal team from their company handles those conversations. In contrast, other participants noted that they often engage with academics and policymakers as part of their role.
P2: As part of this work with a university we started engaging with parliamentary committees. Responding to inquiries and consultations. Including for instance, when the Investigatory Powers Bill came out. It was being discussed, providing some input on that and getting invited to have interview with the Home Office for them to get a better view as to what the potential concerns might be.
These contrasting views on interacting with policymakers pointed at a larger difference between our participants, how they define their role and how they function within their organisations.
### Defining the Job
One set of findings from our interviews revealed the vast flexibility and resulting uncertainty that comes with the role of a DPO or privacy professional. While governments and legal forces define the focus of DPO work, the actual methods, tools, and job description is still defined by the individual organisations. An example provided by the participants was concerning the process of conducting proper audits.
P2: There is a lack of clarity with the methodology that will be expected. We've seen this in a number of cases, you know with the, in New York with the auditing of uh recruitment algorithms. There's no actual standard for that kind of audit.
They mentioned that the various guidelines imposed by legislation from different regions seem too broad that everyone is simply conducting audits as they see fit or as their organisation believes is correct. This finding parallels previous work in privacy engineering, where certain engineers identified the problem of "operationalising privacy" [6]. This work highlighted that regulatory forces establish the values and rules regarding privacy outside the organisation, but the actual implementation and methods are predominantly up to the employees. One participant said their strategy was to combine data protection and privacy methods with those of cybersecurity. This is one approach to solving the uncertainty of conducting a proper data audit.
P4: We review and audit internally we are also combined with our security.
Furthermore, the broader role of privacy professionals within the company can vary quite drastically depending on its position and its business model. Participants provide two contrasting examples.
P3: I remember I was in a conference and there was a big discussion on transfer impact assessments. I mean, how far do you need to go? You know if you a small business, how far would you go to look at all your subprocesses and you know it's just impossible in a sense.
This participant was sharing how the depth and scale of the work for an impact assessment may be quite different for a privacy professional within a smaller company compared to a larger one with more resources.
P2: During our work there is the need to make sure that you're not going to take a very strong position say against the business model of a major kind of client.
Here we see the impact the business model of a large, publicly traded company can have on the role of a privacy professional that is attempting to implement regulatory compliance. Between these two pieces of feedback, we see a difference in the agency offered to the professionals. In a smaller company, the DPO might have greater control in implementing a philosophy and policy of data protection and management but suffer a trade-off in terms of resources and overall security of the data being handled. On the other hand, a DPO within a large corporation may not have to worry too much about security and resources but is instead just one cog in a massive shareholder-driven machine. The explicit role of a DPO or privacy professional within a company appears to be flexible and heavily defined by the organisation with which they work and the roles taken by their colleagues in regulatory practices.
### All in This Together
All of our interview participants shared that they worked on multiple projects and initiatives across their organisation. Therefore, the work of ensuring regulatory compliance for data protection and privacy involved clear communication across multiple teams. A participant described one approach:
P4: I mean definitely you need different language and ways of presenting things with different audiences.
Most participants' primary audiences were the development, executive, legal, and marketing teams. One participant gave an example of how language needs to be used for the development team,
P3: I wouldn't go and say ohh I need this data impact assessment to my developers. You know, asking them, "What's your legal basis for
processing" I mean I really need to simplify things in their language if you wanna them to answer you back in correct ways.
Similarly, executives often need some form of simplification. One participant described these types of communications as "very consumable nuggets". Regardless of the team involved, the DPOs and privacy professionals interact with the entire organisation.
Due to this almost universal interaction, the effectiveness of the DPO's policy implementations is also dependent on the reception of others within an organisation. In some cases, data protection and privacy are established parts of the company culture.
P4: So we have a we have a very strong internal data protection policy. We have a formal policy. We train on it. Everybody has to take a test on it every year.
However, not every organisation is so clearly practised or as receptive. Another participant (P1) admitted that they do not believe their current organisation has sufficient internal data governance policies and stated that additional cyber-security and data access restrictions must be implemented first. Their interview highlighted a facet of regulatory compliance that had not been previously considered. Suppose the data structures are not adequately secured, and the access permissions are not fully established. In that case, the data is too vulnerable by default to follow the regulations' high standards. Therefore, to establish an internal policy for regulatory compliance, there needs to be some baseline of security met by an organisation in the first place. They noted having to spend a portion of their time advocating for public policy and the importance of regulatory compliance.
Whether an organisation has successfully implemented rules and regulations in their internal policies, there is a need for information dissemination and onboarding. Even with our current suite of technical collaboration and data storage tools, this poses a challenge.
### Knowledge Mismanaged
Every participant reported regardless of their years of experience or the size of their organisation, a struggle with knowledge management. In this case, it is worth expanding on the current state of KM within these organisations.
P4: I mean, we do have wikis and we do have lots of documents, but the trouble is things get out of date. So we have a huge SharePoint repository with which we've been using for, I don't know how many years, but it's probably 20 years or something like that.
KM tools, like Intranets and Wikis, work well enough as document stores and ways to share information with others. Nonetheless, there remain clear challenges regarding versioning, updating, and communicating. In particular, our participants (P1 and P6) noted the speed at which Wikis fall out of date.
The current tools used by our participants fulfil only three basic parts (package, store, and transfer) of the features proposed in the KM literature [14, 4]. In other words, the current tools act more as information repositories but do not facilitate the tasks at hand to their greatest potential.
A potential solution for this would be to turn towards automation to process all the information and perform the functions needed by the DPOs and privacy professionals. We heard from our participants that automation is primarily unexplored territory for them. While tools like OneTrust exist, our participants noted limited experience with such tools. The primary functionality of the tool that did attract their attention was the automated Data Mapping capabilities. Our participants repeatedly mentioned that automation is either an aspiration or a challenge for their work.
P3: If I have control of the future I may think about some sort of automation system. I don't know what it is yet, but mainly move a bit from manual work to automation. P7: We do not use automation in our data protection processes.
Our participants expressed the need to improve and expand the tools currently used to achieve regulatory compliance. In a world that is growing increasingly aware of the need to regulate the rapidly changing technological landscape, DPOs are working to ensure each new regulation does not require a massive overhaul for organisations.
### Findings Summarised: Knowledge Management is Key
From our findings, the four significant categories all tie back to the idea of effective internal organisation and information processing. Each of our participants also described the role of KM as essential but also the most challenging when it comes to regulatory compliance in practice. We can see how each category of feedback corresponds with a solution offered by the KM taxonomy presented by Despres and Chauvel [14]. Figure 1 shows the features, defined by Despres and Chauvel, of the KM tools for DPOs would potentially inhabit. The upper horizontal axis shows the functions a tool can play while the vertical axis identifies the level of implementation. In the first category of feedback we heard about a disconnect between regulation and practice; this may be solved by a tool able to map and capture all of the data and policies needed for an organisation to be compliant (see the upper left of Figure 1).
The second category of feedback revealed how participants see their roles nebulously defined by their employers in relation to policymakers expectations. Some participants shared that the internal process of handling regulatory compliance needs to be grounded and accessible at the heart of the organisation's workflow. In the discussion of the Bednar et al. paper, there is a division of challenges to achieving privacy engineering in practice that is labelled as the engineers' "burden", "inner conflict", and "battle with lawyers" [6]. The interviews with our participants indicate that better KM
face these challenges. For example, in the related literature, it is stated that engineers often encounter a conflict between the need to ensure privacy practices while also meeting and addressing the business model of their employer [38]. A tool able to package information for each group or team may alleviate the tension our participants shared (see middle portion of Figure 1).
In addition to the challenge of defining roles and responsibilities, the KM systems used by our participants need to process and communicate information across multidisciplinary teams. Some of the most significant challenges reported have been adjusting the language of privacy and regulatory compliance for different audiences. Once again, this is akin to similar findings from the related literature where communication between engineers and legal teams struggled [26, 6]. From previous literature and our participants' discussions, in the third feedback category, we argue that data protection and privacy are by nature collaborative. As a result, any KM tool utilised in this space must be effective for sharing and applying regulation in an effective way for each of the different teams (see the bottom-right of Figure 1).
Our participants and their peers must advocate for and develop internal policies that allow the business to comply with regulations. They described that establishing internal data protection and privacy policy in direct contact with the organisation's process will provide engineers, executives, and legal personnel with a single reference source. In our discussion, we will highlight how KM literature argues for integrating this reference source into various tasks and responsibilities inherent to an organisation.
Figure 1: Inspired by Despress and Chauvel’s chart mapping the regions of practice in knowledge management, this shows which regions are covered by a knowledge management tool that addresses the feedback presented by our participants [14].
Limitations
This paper relied on an interview-based study, and the usual limitations of this kind of study apply.
We recognize that it is challenging to draw conclusions regarding compliance behaviour across different jurisdictions and legislation (USA, UK, and EU). However, it is important to note that our participants consistently need to work across all these jurisdictions regardless of their original background or training. Their organisations operate around the world yet often have only one official DPO.
## 6 Discussion
The exploratory study presented within reveals industry professionals' lived experiences and attitudes. Their statements can help inform academics and developers in establishing future tools that will effectively address the highlighted issues. Additionally, reflecting on what has been presented may provide policymakers with questions to spark essential conversations on improving data protection and privacy legislation.
### The Future of Knowledge Management for DPOs
Concerning the challenge of KM, the issue is a matter of problem framing and terminology. The current understanding of "knowledge management" predominantly describes archiving and sharing tools that allow users to deposit, organise, and occasionally present information. However, we would argue that this does nothing to satisfy the management part of the description. Over twenty years ago, an early KM taxonomy outlined additional tasks essential for proper KM [14]. The current tools we use do not prioritise information for us; they do not highlight what has gone out of date or perhaps even unusable. This lack of actual management done by the tools results in a lack of potential usability. It avoids the original goal of computation to improve human lives.
An analogy from the physical world to further visualise our understanding of KM would say the current tools marketed as knowledge management items are more like a bank vault. We can put information inside it and organise it however we wish, but once it is in there and we leave the vault, it stays the same until we return. The KM tools we aspire to are items that act more like an asset manager. Depositing the same assets as with the vault into the care of the asset manager, providing some specifications, and then allowing the system to manage the assets in such a way as will likely benefit the user when they return.
Making the jump from the vault to the asset manager has often required the intervention of a human and the specific services they can offer in addition to the basic security of physical storage space. Early work in KM also involved the need for agents and humans before the development of modern software tools
entirely took over the field [40, 27, 15, 7]. However, with computation, we are on the cusp of facing a similar jump from large, relatively static data stores to a new future where the potential use of data is not storing it but instead letting it circulate and work for us. Let the computation handle these new tasks of processing.
Based on the information we have heard from our participants, we call for further development of KM tools to account for the lack of "innovative leverage" the tools have [14]. This topic is not radical for the field of KM, but the technical tools we have built thus far have been concerned primarily with data storage and sharing. Even recent work on new KM capabilities focuses on the storage aspect [35, 41, 1]. While cloud storage might further alleviate the challenges faced by various organisations, the need for responsive, automated KM is clear.
Covering the whole range of possible taxonomies of a KM, from knowledge mapping to innovating and transforming, will require our tools to fill previously unexplored roles. While the possibilities for new forms of KM systems are enticing, we need to consider the impact of automation on regulatory compliance. Simply reducing the role and need for compliance to an automatic box-ticking exercise would not be valuable [3]. However, combining the automation with the work of our participants as computer-supported work may alleviate the challenges we and the literature have observed while also meeting the standards of the policymakers [23]. This requires careful human-centred design approaches that consider the needs of all the stakeholders involved in the system. Therefore, in future work, we recommend developing knowledge management software that genuinely moves beyond a data store with sharing capabilities.
### Reflections on the Data Protection Officer Role
Compliance, cybersecurity, law, and data manifest themselves within organisations across the globe in a fine, often entangled, mesh. To ensure that we can meet the acceptable requirements in each field requires careful planning, effective communication, and well-organised information. Today's industry professionals in data privacy and protection are at the forefront of this growing challenge.
In this work, we interviewed a number of such industry experts to understand how they balance company and legal requirements. They expressed that while some things worked well, there were also apparent issues with how knowledge is managed by current software, especially concerning this new and growing challenge of regulatory compliance.
Our study found that current KM tools primarily function as document repositories and sharing tools. However, our participants noted that there is room for automation to drastically expand their current tools' functionality and capabilities as data protection professionals. This finding opens future research into how knowledge management software can be designed to meet better the complete taxonomy set forth by Despres and Chauvel [14]. Moving beyond the primary role of document storage and sharing to complete computer-supported work will profoundly impact this industry and other fields.
While most of this paper's work focused on applying KM to the challenge of data protection and privacy, the findings reveal profound challenges across the industry. Multidisciplinary teams need to communicate effectively about how privacy plays into every level of an organisation. Additionally, knowing what information and data are being managed is essential for developing proper security and safety measures. Finally, we noticed a clear challenge during our exploratory study's recruitment and interview setup. For the few DPOs and privacy professionals willing to meet in the first place, we had to be precise about the steps we took to anonymise and protect the information of our participants. It is unclear whether this caution comes from their knowledge as experts on data protection or the need to protect their organisation's practices. Nonetheless, accessing this population of industry professionals can be challenging and may pose problems for attempts to connect policymakers and professionals for future efforts and changes to legislation.
## 7 Conclusion
Accepting and adopting data protection and privacy methods remains challenging. Our participants mentioned that a portion of their daily efforts is working to convince and promote the importance of data protection and privacy within their organisation. Often these attempts at persuasion are formal presentations in front of executives and managers across all levels. Some of our participants would urge the importance of institutional pressure on adopting compliance software, similar to previous research [3]. To achieve data protection and privacy compliance, there needs to be future work exploring effective methods to drive corporate and institutional acceptance.
Ultimately this research is meant to provide industry professionals and legislators with examples and evidence of work being done within organisations to promote and implement data privacy practices. Collaboration and careful knowledge management are essential for discovering any solutions within the space. The need for cooperative discovery will likely increase as future regulation is drafted.
## Acknowledgement
This work was supported by the UKRI under PETRAS2 (EP/S035362/1) through the RETCON grant. We would like to acknowledge the participants that took the time to speak with us and fill out our online survey. By taking time out of their days, their insights were able to provide us with the answers to our research questions. We would also like to thank the rest of our HCAI group and all the support we have had throughout this process. |
2308.15190 | Physical and behavioral comparison of haptic touchscreens quality | Touchscreens equipped with friction modulation can provide rich tactile
feedback to their users. To date, there are no standard metrics to properly
quantify the benefit brought by haptic feedback.The definition of such metrics
is not straightforward since friction modulation technologies can be achieved
by either ultrasonic waves or with electroadhesion. In addition, the output
depends strongly on the user, both because of the mechanical behavior of the
fingertip and personal tactile somatosensory capabilities. This paper proposes
a method to evaluate and compare the performance of haptic tablets on an
objective scale. The method first defines multiple metrics using physical
measurements of friction and latency. The comparison is completed with metrics
derived from information theory and based on pointing tasks performed by users.
We evaluated the comparison method with two haptic devices, one based on
ultrasonic friction modulation and the other based on electroadhesion. This
work paves the way toward the definitions of standard specifications for haptic
tablets, to establish benchmarks and guidelines for improving surface haptic
devices. | Corentin Bernard, Nicolas Huloux, Michaël Wiertlewski, Jocelyn Monnoyer | 2023-08-29T10:17:11Z | http://arxiv.org/abs/2308.15190v1 | # Physical and behavioral comparison of haptic touchscreens quality
###### Abstract
Touchscreens equipped with friction modulation can provide rich tactile feedback to their users. To date, there are no standard metrics to properly quantify the benefit brought by haptic feedback. The definition of such metrics is not straightforward since friction modulation technologies can be achieved by either ultrasonic waves or with electrodhesion. In addition, the output depends strongly on the user, both because of the mechanical behavior of the fingertip and personal tactile somatosensory capabilities. This paper proposes a method to evaluate and compare the performance of haptic tablets on an objective scale. The method first defines multiple metrics using physical measurements of friction and latency. The comparison is completed with metrics derived from information theory and based on pointing tasks performed by users. We evaluated the comparison method with two haptic devices, one based on ultrasonic friction modulation and the other based on electrodhesion. This work paves the way toward the definitions of standard specifications for haptic tablets, to establish benchmarks and guidelines for improving surface haptic devices.
## I Introduction
Haptic feedback on touchscreens creates tactile sensations that can render human-computer interfaces tangible. With haptic feedback, the interface becomes more intuitive to use and requires less visual attention [1].
There are mainly two categories of haptic touchscreen technologies. The most common one uses low-frequency vibrations (below 800 Hz) that propagate through the plate to provide vibrotactile feedback to the user's finger [2, 3]. This paper focuses on the second category: haptic technologies that affect the frictional forces as users slide their finger across the touchscreen. Friction modulation can be achieved by different physical principles: (i) via ultrasonic friction modulation which relies on ultrasonic levitation to reduce the friction [4, 5] or (ii) via electrodhesion which relies on electrostatic attraction to increase the friction of the finger [6, 7, 8]. More recently, it has been shown that friction can also be slightly modulated through temperature changes [9].
These technologies are recent, and are confined to laboratories or startups. The generalization of haptic tablets requires that the technical specifications and performance of the haptic feedback be objectively measured and possibly standardized. However, it remains a challenge to define performance indicators since the evaluation of the haptic feedback quality is not straightforward. First, friction modulation technologies are highly dependent on the user's finger mechanical behaviour [10, 11]. Second, the tactile sensory system and its acuity to perceive friction stimulation also play an important role [12]. Finally, other technical parameters, such as the latency or fluidity of the tablet, influence the user interaction [13]. To properly evaluate these devices and compare them, user-in-the-loop measurements are needed.
Yet, measurements involving humans raise other issues in terms of variability and repeatability, which need to be addressed. Similar questions have already been considered for the comparison of force-feedback haptic devices. Although a large number of physical measures of performance can be defined [14, 15], they only partially reflect the usability [16] of the force feedback devices. Therefore, Samur proposed to assess the performances of the interfaces through the performances of users in completing a set of tasks with the device [15].
Similarly, we propose here an evaluation method for haptic touchscreen comparison composed of two parts. First, friction measurements using users' fingers exhibit physical performance metrics. Second, the performance of users in a pointing task provides behavioral comparison metrics. We experimentally assess the validity of the evaluation method by comparing two haptic tablets, a T-pad based on ultrasonic friction modulation built in our lab, and a Tanvas tablet based on electrodhesion. The results of the experiment are used to select the most relevant benchmark metrics.
## II Description of the two evaluated tablets
Fig. 1: **a.** Schematics of the testbed for the physical comparison. **b.** Pictures of the two tablets used here to assess the comparison method: (top) the T-pad build in the lab using ultrasonic friction modulation and (bottom) the Tanvas tablet using electrodhesion (_[https://tamvas.co_](https://tamvas.co_))
The comparison method is evaluated on two tablets presented in Fig. 1.b. The first one is a T-pad based on ultrasonic friction modulation built in the lab. A glass plate vibrates at 36 kHz to produce ultrasonic levitation and reduce friction. The finger position is measured with a laser light based sensor (zForce AIR Touch Sensor, Neonode) and the actuation is controlled by a microcontroller (Teensy 3.5, PJRC). The visual display is controlled by a computer connected to the microcontroller via serial communication. The other device is a TanvasTouch tablet (2018) using electrovibration. It is based on an android tablet enhanced with a haptic touchscreen developed by Tanvas (Chicago, US).
## III Physical measurements
Firstly, basic physical measurements of friction are required to evaluate the haptic surface capability. Highest and lowest constant friction levels provide the friction range metric. It reflects the maximal possible intensity of the haptic feedback. The perception of elementary stimuli such as edges is indeed directly linked to friction change amplitude [17, 18, 19]. This section presents the testbed and the signal analysis process to measure the physical metrics, and then applies the method to the evaluation of the two haptic tablets.
### _Description of the testbed_
The testbed, presented in Fig. 1, is composed of a damped table (Thorlabs) on which a 6-axis force sensor (Nano 43, ATI, Apex) is fixed to measure frictional forces. On the top of the sensor, a support with two clamps ensures a rigid connection between the force sensor and the haptic tablet we want to evaluate. Although the evaluated tablets offer their own finger tracking system, we prefer to measure the finger position externally to ensure consistent comparisons and repeatability. A small ring attached to the participant's finger is connected to a pulley-encoder system (KIS40, Kubler) that measures unidirectional finger displacements along the length of the screen. The precision of this system is approximately 0.01 mm without any significant latency. Frictional forces and finger positions are recorded with an acquisition card (USB X Series Multifunction DAQ, National Instruments) at a 10 kHz sampling rate.
### _Protocol and signal processing_
Ten participants took part in the physical measurements, 2 females and 8 males, from 22 to 41 years old (mean: 25.2). The study was approved by the Ethical Committee of Aix-Marseille Universite. The tablet screens were cleaned with an alcoholic solution and the participants washed and dried their hands 5 minutes before starting the experiment. All participants performed one measurement session with the T-pad and one with the Tanvas. On each trial, they were asked to slide their right finger from left to right and right to left on the screen during 10 s, synchronising their movement with a metronome to ensure a constant velocity of about 100 mm/s. Some trials without recording were used at the beginning to train the participants to keep their finger normal force between 0.5 and 1.5 N.
Normal force \(F_{N}\), tangential force \(F_{T}\) and finger position \(x\) measured for one typical trial are presented in Fig. 2. The friction coefficient \(\mu\) is computed as the ratio between the tangential force and the normal force \(\mu=F_{T}/F_{N}\). Finger position data are used to select only the sections between two sliding direction changes and express the friction coefficient as a function of the position \(x\). For each condition of tablet and actuation, each participant performed 3 times the 10 s finger exploration with 6 finger swipes, which led to 18 measurement repetitions per participant.
The signals from the right to left finger swipes were flipped to be treated together with left to right swipes. It appeared that all friction signals presented a constant trend with a slight increase between the start of a slide and its end. We assumed that this trend was due to mechanical crosstalk of the 6-axis force sensor. Linear regressions were performed on all trials and the mean slope (\(a=0.0036\) mm\({}^{-1}\)) was used to correct the trend by subtracting the corrective function \(\varepsilon=a(x-50)\) from the friction signals.
Force signals from 4 participants showed very high variability, even without actuation, due to stick-slip effects of the finger on the glass. Since the signals were too noisy for the analysis, their measurements were discarded from the physical comparison.
### _Constant actuation and friction range_
Friction range is an important metric to assess friction changes capability of the haptic tablet. It reflects how strong
Fig. 3: Evaluation of the constant friction levels \(\mu_{H}\) and \(\mu_{L}\), the friction range \(\Delta_{H}\), the inter-participant standard deviation \(\sigma\) and the mean intra-trial standard deviation \(\delta\). Measurements are presented for the T-pad and the Tanvas tablet for all participants. The solid lines represent the mean and the shaded zones represent the standard deviation.
Fig. 2: Forces and finger position measurements from a typical trial. The finger position is used to cut the signal in order to keep only 6 sections when the finger movement is approximately constant.
the haptic effects (like edges) can be perceived [20].
This parameter is derived from the measurement of the highest and lowest friction levels \(\mu_{H}\) and \(\mu_{L}\), i.e with constant maximal actuation and without actuation. (\(\mu_{H}\) will be obtained without actuation for the T-pad, and with constant actuation for the Tanvas). The friction range is calculated as \(\Delta\mu=\mu_{H}-\mu_{L}\) as presented in Fig. 2.a.
Constant friction measurements made on the two haptic touchscreens are presented in Fig 3. For the T-pad with actuation, we observed curves with a sinusoidal shape. This inconsancy is due to the technology of ultrasonic friction modulation and its plate vibration nodes and antinodes [21]. This highlights an important property that is the ability to provide steady stimulation to the user to render a sensation of "flatness". We therefore defined a metric assessing the variability within a finger swipe expressed by the mean friction intra-trial standard deviation \(\delta\). We also observed large differences of constant friction levels between participants. Even if these disparities are mainly due to external factors (user's finger mechanical behaviour, its angle and moisture [10]), we still propose to evaluate this aspect since some tablets could features technological solution to reduce the friction variability between users (glass processing or friction control [22] for example). Thus, we defined another metric, the inter-participant standard deviation \(\sigma\) that reflects the ability to provide repeatable stimulation across users. The measured metrics are presented in Table I for the two evaluated devices. The Tanvas tablet shows a higher friction range (Two-sample T-test: \(T_{214}=-8.74\), \(p=7.0e^{-16}\)) but the inter-participant variabilities are not statistically different (F-Test for Equality of Two Variances: \(F_{107,107}=0.853\), \(p=0.413\)).
### _End-to-end latency measurement_
The end-to-end latency corresponds to the delay between a user's action and the system final response [23]. Here we are dealing with the delay between the user's finger detection and the haptic actuation. This latency is due to several factors: the finger position measurement refresh rate, communication delays, running time of the microcontroller and the actuation duration (plate frequency response for the T-pad and plate charging time for the Tanvas, both impacted by the finger mechanics [24]). A low end-to-end latency is crucial for touch based HCI to render trustful haptic feedback [25, 13]. This problem is particularly pointed out for haptic feedback that needs to be precisely located in space, like boundaries or ridges. For example, on a tablet with a 100 ms end-to-end latency, a ridge explored with a 200 mm/s left-to-right finger swipe will be felt 2 cm too far to the right, and 2 cm too far to the left when sliding in the other direction, which causes a haptic shift that strongly reduces the realism.
The end-to-end latency is here measured using this principle. A 2 mm vertical ridge is spatially programmed on the tablet, which means that the haptic actuation should activate when the finger is tracked on the ridge (a low friction ridge for the T-pad and a high friction ridge for the Tanvas). Fig. 4 shows how the end-to-end latency \(\Delta T=t_{2}-t_{1}\) is estimated by examining the instants \(t_{1}\) when the finger crosses the ridge and \(t_{2}\) when the actuation emerges. We defined \(t_{2}\) as the point of inflection when the friction derivative reaches its extremum, which is the middle point between the point where the actuation starts and the point where it reaches its maximum, to take into account the actuation duration needed for the haptic feedback to be noticeable. The results for the two tablets are reported in Table I. The standard deviation reports the uncertainty due to the measurement method.
## IV Behavioral measurements
### _Principle_
In the previous section, we proposed to assess the potential of haptic tablets through physical measurements. However, since these interfaces are intended to be used by humans, the evaluation must be complemented by behavioral measurements. Inspired by the literature [15], we propose here to evaluate the performances of the haptic tablets through the performances of users in a one-dimensional pointing task: the user has to reach a target as quickly as possible. This classical HCI task is well described under the Fitts' law framework [26], a predictive model of human movement that describes the trade-off between precision and rapidity in
Fig. 4: Evaluation of the end-to-end latency \(\Delta T\) trough the rendering of a haptic ridge (actuation wanted on the grey area). Friction is presented for left-to-right swipes and for right-to-left swipes to exhibit the impact of the delay. Friction measurements are performed on the T-pad and the Tanvas tablet for all participants. The solid lines represent the mean and the shaded zones represent the standard deviation.
pointing at a target. Fitts' law predicts the average movement time as \(MT=a+b\times ID\), with \(a\) and \(b\) constants that depend on the interface and \(ID\) the index of difficulty. \(ID\) is expressed for interfaces by the Shannon formulation [27] as \(ID=\log_{2}(\frac{D}{W}+1)\), with \(D\) the distance to the target and \(W\) the width of the target (see Fig. 5).
Many studies have demonstrated that the addition of haptic feedback using friction modulation significantly improves the performance of pointing tasks [28, 29, 30, 31]. This is mainly shown by a reduction of the movement times, reflected by a diminution of Fitts' slope \(b\).
In this section, we propose to perform the same pointing task on the tablets (with a standardized protocol) with and without haptic feedback in order to evaluate the overall usability of the device and the gain of haptics.
### _Protocol_
For the behavioral comparison, 10 participants, 3 females and 7 males, from 22 to 46 years old (mean: 26.4) took part of the experiment. Half of the participants also participated in the physical measurements. The pointing task was performed in a 100 x 60 mm window presented in Fig. 5. At each trial, participants were asked to select the cursor with the index of their dominant hand and drag it into the green target as quickly as possible. The direction (right to left or left to right) was alternated at each trial. The distance \(D\) between the cursor and the target was fixed at 80 mm. There were 8 target width \(W\) conditions: 1, 2, 3, 4, 5, 6, 7 and 8 mm. It led to 8 difficulty indexes \(ID=\log_{2}(\frac{D}{W}+1)\): 6.3, 5.3, 4.8, 4.4, 4.1, 3.8, 3.6 and 3.4, repeated each 6 times. For the haptic condition, the tablet was actuated to deliver a high friction in the target and a constant low friction elsewhere. The experiment was performed with and without haptic feedback for the two tablets and the presentation order was balanced among participants. Overall, each participant performed 2 tablets x 2 actuation x 8 \(ID\) x 6 repetitions = 192 trials.
### _Analysis_
At each trial, the movement time (\(MT\)) is calculated as the time between the participant touching the cursor and releasing it. For each participant, the movement time is averaged over the repetitions and linear regression \(MT=a+b\times ID\) are performed to exhibit Fitts' slope \(b\) used for the comparison. The error rate is calculated by considering trials in which the cursor is released outside of the target.
### _Results_
Linear regression exhibited that the results were well in line with Fitts' law (goodness of fit \(R^{2}\subset[0.95,0.97]\)). Therefore, we propose to use Fitts' slope \(b\) as a first comparison metric. It reflects how much the movement time increases with the difficulty index. The Fitts' slope is calculated for each participant (n=10) and the standard deviation \(\sigma\) is computed to reflect the high inter-participant variability inherent in pointing tasks. Instead of the Fitts' offset \(a\) that is not relevant in our case, we rather propose as a second comparison metric the movement time for the most selective difficulty index (\(ID=6.3\)). Here, mean and standard deviation are directly calculated on the 6x10=60 trials. Those comparison metrics are reported in Table II. In the present comparison, statistical analysis reported a significant effect of the interface on the movement time (for \(ID=6.3\)) (Oneway ANOVA: \(F_{3}=3.03\), \(p=0.03\)) due to the condition with the T-pad without haptic. Performances of the Tanvas tablet with haptic (both for movement time and error rate) were not as good as expected from the physical evaluation. We hypothesized that this effect was due to oscillations of the cursor caused by a noisier finger position measure when the electrovibration actuation was on, a problem that has since been fixed in later Tanvas prototypes.
Fig. 5: Interface of the pointing task.
Fig. 6: Results of the pointing experiment. Mean movement time is plotted with respect to the difficulty index for the 4 interface conditions. Linear regressions \(MT=a+b\times ID\) are calculated to exhibit Fitts’ slope \(b\).
## V Discussion
Two experiments were presented to evaluate the comparison method. The first one assessed the haptic rendering systems by measuring physical metrics relating to the friction between the screen and different user fingers. In the second one, the haptic tablets were compared on the basis of the user's performances when performing a typical human-computer interaction task. This section discusses the results about how to improve the method and how to select a limited number of metrics to keep only the most relevant descriptors and avoid redundancy.
### _Physical measurement_
The physical experiment revealed that some participants' fingers on the screen produce stick-slip effects which makes the data difficult to exploit. A first way to address this problem is to improve the testbed to reduce its resonance. The rigidity of the measurement system could be increased by adding more force sensors, one at each side for example.
Another method would be to use an artificial finger and a robotic arm for the physical measurements. It offers the advantages to be well calibrated, to provide easily replicable measurements and to strongly reduce the variability. However, the artificial finger should present the same behaviour as human fingers both in terms of electrostatic polarizability and mechanical reaction to deformations and vibrations [32]. Moreover, this solution does not provide any information about the variability between participants, which is an important comparison metric of consistency. Indeed, we want the tablet to be able to render the same haptic feedback to any user. In addition, a larger number of participants would be needed to reflect the variability of the population. The detail of the descriptors for the highest and lowest friction is of interest to investigate the sources of differences but both are included in the friction range descriptors. The intra-trial standard deviation is still relevant to reflect the "perceived flatness" of the surface under actuation.
The friction range metric could be improved to take into account human perception. Since the perception of friction coefficient follows a Weber law [33] (with JND of friction about 20%). For example, it means that the difference between \(\mu_{L}=0.5\) and \(\mu_{H}=0.7\) is better perceived than the difference between \(\mu_{L}=0.8\) and \(\mu_{H}=1\) even if the friction range is the same. Other metrics could be used like the relative friction range \(r_{\mu}=\mu_{H}/\mu_{L}\) or the friction contrast \(FC=1-\mu_{L}/\mu_{H}\)[19].
End-to-end latency is also a crucial metric that needs to be analysed in the light of human perception. Future studies on the maximal unnoticeable latency for haptic actuation could define a threshold below which the comparison is unnecessary.
All physical metrics were measured with the same constant velocity (about 100 mm/s) and relatively steady normal force (between 0.5 and 1.5). Since frictional behaviour of the finger-glass contact is impacted by the velocity and normal force, future investigation could evaluate precisely this aspect to propose velocity and force independent metrics.
### _Behavioral measurement_
The behavioral measurements take the approach of globally evaluating the tablet through the users' performance. This part is highly dependent on the participant's motor dynamics, intention and previous experience with touch-screens. Proper comparisons should therefore include a much larger number of participants than the presented preliminary experiments to reduce bias due to participant variability. It may also be worthwhile to counterbalance the effects of age with data from the literature [34] or with a preliminary assessment of the participant's tactile sensitivity.
The linear regressions results demonstrated that pointing tasks with and without haptic feedback are well explained by Fitts' Law, in line with the literature [28, 29]. Since the objective is to highlight differences between devices, it may not be necessary to test so many difficulty indexes. User performances could be evaluated with one target size, preferably the smallest one (\(ID=6.3\)) as it is the most discriminating. It would avoid redundancy between metrics and permit a much higher number of repetition to decrease variability.
By applying the presented comparison method to a large number of haptic tablets, it would be possible to establish links between the physical and the behavioral measurement. It would be interesting to investigate how to predict the pointing performance from the physical metrics. A better understanding of the impact of each descriptor could make it possible to apply weight to the metrics to construct an overall usability score for the haptic tablet.
## VI Summary
The outcome of the study provided insight to improve the evaluation method. As argued in the discussion, some metrics are more relevant than others. It led us to propose an ideal comparison protocol with a limited selection of the essential metrics. For both experiments, the panel should be constituted of 30 participants whose age follows the distribution of the adult population. The physical comparison should be performed on a stiff testbed with force sensors on the two sides, and about twenty repetitions per measurement. We suggest that the pointing task includes only 1 target width conditions, the most discriminatory one, \(W=1\) mm (difficulty indexes \(ID=6.3\) for \(D=80\) mm) and 40 repetitions. The most relevant metrics and their descriptors are summarized in Table III. A tablet performs best when the descriptors marked with a (-) are the lowest and the descriptors marked with a (+) the highest.
We propose to report for each metric the mean, standard deviation and number of samples to enable anyone to easily perform statistical comparison of their own data, with Two-Sample T-Test for example.
## VII Conclusion
This paper is a first attempt to define objective metrics to compare different haptic tablets, even if they are based on different technologies. The proposed method was evaluated
with two different haptic devices to demonstrate its validity and to select the most relevant metrics.
Future work will investigate how the method could be extended to compare with haptic interfaces that are not based on friction modulation, such as vibrotactile tablets. This paper lays the foundation for defining generic standards for haptic touchscreens. It would allow consumers to objectively compare between devices and to request certain specifications. It would also enable manufacturers to analyze their devices and identify ways of improvement.
|
2306.12353 | The OH Megamaser Emission in Arp\,220: the rest of the story | The OH Megamaser emission in the merging galaxy Arp220 has been re-observed
with the Multi-Element Radio Linked Interferometer Network (MERLIN) and the
European VLBI Network (EVN). Imaging results of the OH line emission at the two
nuclei are found to be consistent with earlier observations and confirm
additional extended emission structures surrounding the nuclei. Detailed
information about the distributed emission components around the two nuclei has
been obtained using a concatenated MERLIN and EVN database with intermediate
(40 mas) spatial resolution. Continuum imaging shows a relatively compact West
nucleus and a more extended East nucleus in addition to an extended continuum
ridge stretching below and beyond the two nuclei. Spectral line imaging show
extended emission regions at both nuclei together with compact components and
additional weaker components north and south of the West nucleus. Spectral line
analysis indicates that the dominant OH line emission originates in foreground
molecular material that is part of a large-scale molecular structure that
engulfs the whole nuclear region. Compact OH components are representative of
star formation regions within the two nearly edge-on nuclei and define the
systemic velocities of East and West as 5425 km/s and 5360 km/s.
The foreground material at East and West has a 100 km/s lower velocity at
5314 and 5254 km/s. These emission results confirm a maser amplification
scenario where the background continuum and the line emission of the star
formation regions are amplified by foreground masering material that is excited
by the FIR radiation field originating in the two nuclear regions. | W. A. Baan, J. N. H. S. Aditya, T. An, H-R. Klöckner | 2023-06-21T15:51:22Z | http://arxiv.org/abs/2306.12353v1 | # The OH Megamaser Emission in Arp 220: the rest of the story
###### Abstract
The OH Megamaser emission in the merging galaxy Arp 220 has been re-observed with the Multi-Element Radio Linked Interferometer Network (MERLIN) and the European VLBI Network (EVN). Imaging results of the OH line emission at the two nuclei are found to be consistent with earlier observations and confirm additional extended emission structures surrounding the nuclei. Detailed information about the distributed emission components around the two nuclei has been obtained using a concatenated MERLIN and EVN database with intermediate (40 mas) spatial resolution. Continuum imaging shows a relatively compact West nucleus and a more extended East nucleus in addition to an extended continuum ridge stretching below and beyond the two nuclei. Spectral line imaging show extended emission regions at both nuclei together with compact components and additional weaker components north and south of the West nucleus. Spectral line analysis indicates that the dominant OH line emission originates in foreground molecular material that is part of a large-scale molecular structure that engulfs the whole nuclear region. Compact OH components are representative of star formation regions within the two nearly edge-on nuclei and define the systemic velocities of East and West as 5425 km s\({}^{-1}\) and 5360 km s\({}^{-1}\). The foreground material at East and West has a 100 km s\({}^{-1}\) lower velocity at 5314 and 5254 km s\({}^{-1}\). These emission results confirm a maser amplification scenario where the background continuum and the line emission of the star formation regions are amplified by foreground masering material that is excited by the FIR radiation field originating in the two nuclear regions.
keywords: masers \(-\) ISM: molecules \(-\) radio lines: ISM \(-\) galaxies:ISM \(-\) galaxies: nuclei \(-\) radio lines: galaxies
## 1 Introduction
The interacting system IC 4553/4, also known as Arp 220, is the host galaxy system of the first known hydroxyl (OH) MegaMaser (OHMM; Baan et al. 1982). The OH emission in Arp 220 was discovered at the Arecibo Observatory during a search for OH absorption in sources with strong HI absorption (Mirabel, 1982). The extraordinary properties of Arp 220 were later confirmed by the far-infrared (FIR) prominence in the Infrared Astrochemical Satellite (IRAS) data (Soifer et al., 1984; Sanders et al., 1988), and the source became the prototype of the Ultra-Luminous InfraRed Galaxy (ULIRG) population (Sanders & Mirabel, 1996). Subsequent searches for OHMM emission have resulted in about 120 systems among the ULIRG population (Baan, 1989; Darling & Giovanelli, 2002; Klockner, 2004; Zhang et al., 2014).
Arp 220 is a merger system with two radio nuclei (Baan & Haschick, 1995) embedded in a chaotic optical structure (Lockhart et al., 2015). The ongoing merger has triggered a powerful burst of star formation at each of the nuclei (Scoville et al., 1997), which results in the FIR prominence and multiple radio supernova remnants (SNRs) at each of the nuclei (Smith et al., 1998; Lonsdale et al., 2006; Varenius et al., 2019), as well as starburst (SB)-related hard X-ray emission (Clemenets et al., 2002; Iwasawa et al., 2005). Arp 220 is a (nearby) template for high redshift active galaxies with short-lived bursts of assembly and nuclear activity that ap
pear to be common at redshifts of \(\sim\)2.5 and define the characteristics of massive galaxies in the nearby Universe.
The radio positions of the two nuclei in Arp 220 are separated by 0.97\({}^{\prime\prime}\) (371 pc) on the plane of the sky (Baan & Haschick, 1995; Downes & Solomon, 1998; Sakamoto et al., 1999), while the optical images show a dust-enshrouded structure (Wilson et al., 2006; Lockhart et al., 2015). A radio continuum bridge between and below the two nuclei is the only evidence for the interactive nature of the system. However, because of the puzzling nature of the system, the systemic velocities of the two underlying nuclei remain to be determined. A reinterpretation of the early Very Large Array (VLA) A-array observations of the 1667 and 1665 MHz OH emission in Arp 220 would suggest that the features of the East and West nuclei are merged within the observed spectral signatures and that the velocities of West and East are close to 5350 km s\({}^{-1}\) and somewhat higher than 5390 km s\({}^{-1}\), respectively (Baan & Haschick, 1984). Early Multi-Element Radio-Linked Interferometer Network (MERLIN) observations confirm that the 1667 MHz emission at the velocity of the Western nucleus appears close to the Eastern nucleus and that the velocity fields of the two nuclei may be mixed (Rovilos et al., 2003). The distribution of the CO emission also indicates that the velocities at the West and East nuclei are approximately at 5370 and 5400 km s\({}^{-1}\)(Wheeler et al., 2020; Sakamoto et al., 2008; Rangwala et al., 2015). A study of the dynamics of Arp 220 based on early detection of formaldehyde and the corresponding OH emission employed similar velocities of 5346 and 5418 km s\({}^{-1}\) at the West and East nuclei for understanding the nuclear antics of the system (Baan & Haschick, 1995). The formaldehyde emission in Arp 220 is found to extend across the central molecular zone of each of the nuclei and covers the systemic velocities of both nuclei (Baan et al., 2017). Arp 220 also exhibits an OH outflow feature that extends to \(\sim\)1000 km s\({}^{-1}\) below the OH 1667 MHz feature (Baan et al., 1989).
The observed OH MM emission has been interpreted with an amplification scenario where foreground excited and masering material amplifies the background radio continuum (Baan, 1989). The OH emission would thus be superimposed on the radio structure of the source and the FIR emission regions generated by dust emission resulting from ongoing star formation, which has been suggested to serve as a pumping agent for the OH molecules in the foreground. Both compact high-brightness and extended low-brightness maser components could ensue in this manner.
The re-observations of the Arp 220 system of the complete 1667 and 1665 MHz emission spectrum presented in this paper have been taken with MERLIN and with the European VLBI Network (EVN). Previous interferometric MERLIN and EVN observations only covered the prominent 1667 MHz OH emission originating at both nuclei. The lower resolution images from the MERLIN observations provide an integrated view of the OH emission in Arp 220 without fully detailing the structural components of Arp 220. The global Very Long Baseline Interferometry observations with the EVN provide a highly resolved view of the nuclear regions and only found compact emission components. In order to identify and image the two nuclei in more detail, the two data sets will be concatenated, which will give a data base with intermediate resolution and allows mapping the spatial structure of both the 1667 and 1665 MHz OH emission regions. And this new database reveals some of the hidden secrets of the Arp 220 system.
## 2 Observations and Data Reduction
Throughout this paper a Hubble constant \(H_{0}\) = 70 km s\({}^{-1}\) Mpc\({}^{-1}\) has been assumed, which indicates that for the Arp 220 system, the angular-spatial size conversion results in 1 arcsecond corresponding to 382 pc. All results are presented using the optical helio-centric definition of velocity. For Arp 220, velocities are about 96 km s\({}^{-1}\) lower using the radio definition.
### MERLIN Observations and Data Reduction
The MERLIN observations of Arp 220 were carried out on 2003 June 24th and 25th with all seven MERLIN antennas, including the 76-metre Lovell telescope in left-hand circular polarisation mode. The data of the observing project MN/03B/22 were recorded in 128 channels covering a total bandwidth of 8 MHz, each with a channel width of 62.5 kHz, giving a velocity resolution of 11.66 km s\({}^{-1}\) at 18 cm wavelength. The sources 3C 84, 3C 286, and J 1516+1932 were used as bandpass, flux density scale, and phase referencing calibrators, respectively. The whole observation of two sessions lasted 18 hours, two third of which was spent on Arp 220.
The preliminary amplitude calibration, bandpass and antenna-based phase calibration were made in the Astronomical Image Processing System (AIPS) using the calibrator sources. The gain solutions derived from the calibrators
Figure 1: The 1.6-GHz continuum emission of Arp 220 made from the MERLIN data. The restoring beam is 0.28\({}^{\prime\prime}\)\(\times\) 0.26\({}^{\prime\prime}\), PA=\(-\)63.6\({}^{\circ}\). The peak intensity is 53.5 mJy beam\({}^{-1}\) and the rms noise in the off-source region is 0.3 mJy beam\({}^{-1}\). The contours are at 0.80 mJy beam\({}^{-1}\)\(\times\)(1, 2, 4, 8, 16, 32, 64). The two crosses representing the dust emission peak positions at 230 GHz (Sakamoto et al., 1999) are in excellent agreement with the present continuum peaks.
were applied to the Arp 220 data by interpolation. Then the Arp 220 data were exported out of the multi-source dataset and were imported into MIRIAD for self-calibration and imaging.
The data were first averaged in channels to produce a single-channel dataset, a so-called pseudo-continuum dataset. An image of Arp 220 was created using the pseudo-continuum data. After flagging some discrete bad data points induced by radio frequency interference (RFI) and other observational problems, this pseudo-continuum data was used for a few iterations of phase-only self-calibration with the time intervals starting from 5 minutes to 0.5 minutes until the dynamic range of the CLEANed image did not improve any more. A final image was produced with natural weighting (see Figure 1).
The antenna gains as a function of time determined in the self-cal procedure were applied to the line data. The Miriad task UVLIN was used to subtract the continuum emission from the visibility data by fitting a polynomial to the real and imaginary parts of the selected line-free channels across the line cube. The continuum and line data are employed separately to make the resulting maps.
### EVN Observations and Data Reduction
Arp 220 was observed with the EVN from 2003 June 24th UT18:00 to June 25th UT03:30 with observing program EB022C. Twelve telescopes participated in the observations: Jodrell Bank, Effelsberg, Cambridge, Noto, Torum, Shanghai, Westerbork, Onsala, Medicina, Urumqi, Hartebeestihoek and Robledo. The observations were made in dual circular polarisation mode and the data were recorded in 256 channels. The total bandwidth is 8 MHz; therefore, each channel has a width of 31.25 kHz (corresponding to a velocity resolution of 5.83 km s\({}^{-1}\) at 18 cm wavelength). OQ 208 was used as the bandpass calibrator and J 1613+3412 and J 1516+1932 were used as phase referencing calibrators. The whole observation lasted for 9.5 hours, during which Arp 220 was observed for 6 hours.
The calibration of the multiple-source dataset was made in AIPS. The data were first sorted in time-baseline sequence. The amplitude calibration was done using the system temperatures of the observations and the gain curves provided by each station. The phase errors induced by the ionospheric effects are corrected using the AIPS task TECOR. Fringe fitting was carried out with the compact and strong calibrators, and the derived solutions were then applied to the whole dataset to calibrate the delays, delay rates, and phases. Complex bandpass solutions were determined using the OQ 208 data. RFI was identified in the total power spectra and the affected channels were flagged. The phase, gain, and bandpass solutions derived from the calibrators were applied to Arp 220 data by interpolation. The source-rest-frame frequency was set to the line data and AIPS task CVEL was used to determine the Doppler shift correction on each baseline. The visibility amplitudes of the EVN calibrator data J 1516+1932 and OQ 208 were compared with those of the MERLIN data on common baselines, and they show consistency within 2 per cent. Considering the scattering of the EVN visibility amplitude and the variation of the telescope performance, we conservatively adopt an amplitude uncertainty of 5 per cent.
The calibrated data were exported from AIPS and imported into Miriad for further analysis. The line data were
Figure 2: Channel maps of the OH 1667 line emission from the MERLIN data. The restoring beam is 0.23* \(\times\) 0.19* at a PA=32\({}^{\rm o}\). The velocity scale is based on the rest frequency of the 1667 MHz transition. The rms noise level is 2.0 mJy beam\({}^{-1}\). The peak intensity is 157 mJy beam\({}^{-1}\) in the 5363.5 km s\({}^{-1}\) channel. For clarity, only the contour of 6 mJy beam\({}^{-1}\) is plotted in the image.
separated from the original data by using the task UVLIN, which fits the continuum emission with a linear function using line-free channel data and subtracts the fitted baseline from the line channels. Next, an iteration of self-calibration was applied to the line data on the line peak channels assuming a point source model. The line cubes were all mapped with natural weighting.
### Combining the MERLIN and EVN data
A combination of MERLIN data with 280 x 260 mas with the EVN data set with resolution 10 x 10 mas would result in a data set with an intermediate resolution such that both the extended and the compact components in Arp 220 are both identifiable in a single image.
In order to further optimise the available data, the residual phase errors on the short (and most sensitive) EVN baselines and the alignment of the EVN and MERLIN phase centres were corrected by using an EVN continuum calibration iteration employing a reference model formed by the CLEAN models derived from the MERLIN continuum data.
The images resulting from the combined MERLIN and EVN (ME) data have a restoring beam of \(41\times 38\) mas that will appropriately reveal the more extended emission regions at the nuclei as well as the very compact VLBI components.
## 3 MERLIN and EVN imaging results
Throughout this paper a Hubble constant \(H_{0}\) = 70 km s\({}^{-1}\) Mpc\({}^{-1}\) has been assumed for interpreting the imaging results. For the Arp 220 system with a systemic velocity of about 5400 km s\({}^{-1}\), the angular-spatial size conversion results in 1 arcsecond corresponding to 375 pc. All results are presented using the optical heliocentric definition of velocity. Other publications may use velocities using a radio definition, which for Arp 220 are about 96 km s\({}^{-1}\) lower.
Throughout the paper the continuum and spectral line images show two crosses representing the dust emission peak positions of the two nuclei at 230 GHz (Sakamoto et al., 1999), which are in agreement with the continuum peaks from the current data sets.
### MERLIN Imaging of the Continuum Emission
The MERLIN continuum emission displays a double-component structure embedded within a larger envelope
Figure 3: The velocity-integrated intensity (moment 0) map of the OH 1667 line emission made by integrating the line emission with the signal-to-noise ratio higher than 3 in each velocity channel. The beam for these data is 0.23\({}^{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{ \rm{ }}}}}}}}{{{{}{{{}{{}{{}{{}{{}{{}{}{{}{}{}{{}{}{}{{}{}{}{}{}{{}{}{}{}{}{}{}{{}{}{}{}{}{}{{}{}{}{}{}{{}{}{}{}{{}{}{{}{}{}{}{{}{}{}{{}{}{}{{}{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{}{{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{}{{}{{}{}{}{{}{}{{}{}{{}{{}{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{}{}{{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{{}{}{{}{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{}{{}{}{}{{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{{}{}{{}{}{}{{}{}{{}{}{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{}{{}{}{{}{{}{}{{}{}{}{{}{}{{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{}{{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{}{{}{{}{}{}{{}{}{{}{}{{}{}{{}{{}{}{{}{}{{}{{}{}{{}{}{{}{{}{}{{}{{}{}{{}{}{{}{{}{}{{}{}{{}{{}{{}{{}{}{{}{}{{}{{}{{}{{}{{}{}{{}{}{{}{{}{{}{{}{}{{}{}{}{{}{{}{{{}{{}{{}{{}{{}{{}{{}{{}{{}{{{}{{}{{{}{}{{{}{{{}{{}{{{{}{{{}{{{}{{}{{{{}{{{}{{{}{{}{{}{{}{{}{{{}{{}{{{}{{{}{{{}{{{{{}{{{{}{{{{{{}{{{{{{}{{}{{{}{{{}{{{}{{{{{{{{} 0.}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \, \, \, \, \, \, \, \, \, \, \, \, \, }}}}}}}}}}}}}}}}}}}}}}}}}}}}}\}}}\}}}\)}\)}\}\,\}}}}\}\}}\,\}}}}\{\}}\{\{\}\}\{\{\{\{\{\}}}}\{\}\{\}}\{\,\}}}}\{\{\}\{\}\{\}\{\}\}\{\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\}\{\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\}\{\}\\}\{\}\}\{\}\}\}\{\}\}\}\{\}\}\{\}\}\{\}\\}\{\}\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\{\}\}\}\{\}\}\}\{\}\{\}\}\{\}\}\}\}\{\}\{\}\}\}\{\}\}\}\{\}\}\{\}\}\{\}\}\{\}\}\}\{\}\}\{\}\}\{\}\
(Fig. 1). While the two prominent nuclear emission regions are found to be similar to those seen in the earlier less-sensitive MERLIN map (Rovilos et al., 2003), the surrounding structures are much more detailed. The two main peaks, separated by about 0\(\aas@@fstack{\prime\prime}\)97 (365 pc), are in agreement with other published high-resolution images (e.g. Sakamoto et al., 1999; Rovilos et al., 2003; Baan et al., 2017). The total flux density is estimated to be 274\(\pm\)15.2 mJy, in agreement with previous measurements at the same frequency (Rovilos et al., 2003). The West nucleus is brighter and relatively more compact than the East nucleus. Gaussian fitting with a zero baseline gives an integrated flux density of 124.2\(\pm\)5.4 mJy and 99.4\(\pm\)3.4 mJy for the West and East components, respectively (Table 1).
However, the new MERLIN continuum map of Arp 220 also shows prominent extensions to the south, the south-east, and particularly to the west. The southern bridge below the two nuclei observed in previous data (Baan & Haschick, 1995; Rovilos et al., 2003) now appears as a continuous structure extending from southeast of Arp 220E all the way to west of Arp 220W. With a peak intensity of 13.5 mJy beam\({}^{-1}\) this accounts for 45 times the off-source noise. This (arm-like) continuum bridge as well as other extensions may represent trails of star formation regions and debris resulting from the galaxy merger. This may be confirmed by the detection of a large kpc-scale structure, interpreted as a star forming disk, observed at low radio frequencies (150 MHz) with the international Low-Frequency Array (LOFAR) telescope (Varenius et al., 2016).
### The MERLIN OH 1667 Line Emission
The MERLIN 1667 MHz OH emission line data cube of Arp 220 covers a velocity range of 4900\(-\)6200 km s\({}^{-1}\) with a velocity resolution of 11.7 km s\({}^{-1}\). The 1667 MHz OH channel maps cover a range 5237 - 5447 km s\({}^{-1}\) in Figure 2 and show five distinct emission regions: the two nuclear regions, regions south and west of the West nucleus, and a region southeast of the East nucleus. These structures are clearly seen in the velocity-integrated 1667 MHz intensity map (grey scale and contours) together with the integrated 1665/1667 MHz spectra of six selected regions in the side panels (Figure 3). The current data provide more details of the emission structures at the two Arp 220 nuclei than found in earlier MERLIN data, as those data did not incorporate the 1665 MHz line emission (Rovilos et al., 2003).
Similar to the early data, the two OH components of Arp 220W straddle the continuum peak, except that now the North component appears much less prominent than the
\begin{table}
\begin{tabular}{l c c c} \hline Label & \(S_{peak}\) & \(S_{integrated}\) & maj, min, PA \\ & (mJy/b) & (mJy) & \\ \hline Arp 220W & 75.1\(\pm\)3.4 & 124.2\(\pm\)5.4 & 0.24” 0.17”108.2\({}^{o}\) \\ Arp 220E & 38.0\(\pm\)1.3 & 99.4\(\pm\)3.4 & 0.43” 0.23” 48.7\({}^{o}\) \\ \hline \end{tabular}
\end{table}
Table 1: Continuum Emission from the MERLIN data
Figure 4: The MERLIN maps or the 1667 MHz OH emission at both nuclei. (top) The velocity distribution (1Moment) and (bottom) the velocity width (2Moment) maps. The colour scale are in units of km s\({}^{-1}\).
Figure 5: Channel maps of the OH 1665 line emission from the MERLIN data. Beam = 0\(\aas@@fstack{\prime\prime}\)23 \(\times\) 0\(\aas@@fstack{\prime\prime}\)19, PA = 32\({}^{o}\). The velocity scale is based on the rest frequency of the 1665 MHz transition. Contours: 6 mJy beam\({}^{-1}\)\(\times\) (-1,1,2,3,4,5,6,7,8). The first contour represents 3 times the average off-source rms noise of 2 Jy beam\({}^{-1}\). The peak intensity is 30.8 Jy beam\({}^{-1}\)in the 5376 km s\({}^{-1}\) channel. Two crosses mark the position of two continuum peaks. The restoring beam for these maps is 0.23” \(\times\) 0.19”.
South component, which has a peak intensity of 157 mJy beam\({}^{-1}\) in the 5363.6 km s\({}^{-1}\) velocity channel. The spectrum of the Arp 220W-N region shows a narrow component at velocity 5353 km s\({}^{-1}\) and is associated with a very compact masering region that also appears in the EVN data (see Section 3.5). A third prominent component south of Arp 220W is co-located with a continuum component and appears more prominent than in the earlier data. This third component highlights the SE-NW orientation of both the continuum and line emission at Arp 220W and suggests the orientation of the nuclear disk of the source in agreement with later findings (see Section 4).
The peak in Arp 220E of 152 mJy beam\({}^{-1}\) is found in the 5410.2 km s\({}^{-1}\) velocity channel at a position close to the Eastern continuum peak with the second peak in the 5375 km s\({}^{-1}\) channel. The weaker components west and south of Arp 220W and southeast of Arp 220E seen in these high sensitivity data all lie within the confines of the continuum structure in Figure 1.
The 1667 MHz emission features in the lower resolution MERLIN data cover a broad velocity range of 5200\(-\)5600 km s\({}^{-1}\) and encompasses the suspected optical velocities of the West nucleus of about 5360 km s\({}^{-1}\) and of the East nucleus of 5425 km s\({}^{-1}\) (Fig. 3). These broad profiles also suggest that the bulk of the 1667 MHz OH emission in Arp 220 originates in more extended regions at both nuclei and shows little detail about the underlying nuclei themselves.
The velocity gradient (1Moment) map and the velocity width (2Moment) map of the 1667 MHz line emission at both nuclei are presented in Figure 4 using a 3\(\sigma\) flux cut-off. Arp 220 E shows a positive velocity gradient in a north-east direction, which suggests rotation within the nuclear regions. The Arp 220 W nuclear region and the west extension component suggest a very weak but continuous gradient towards the north. The linewidth distribution of the East component depicted in Figure 4 appears rather uniform, while those of the West component are dramatically different with linewidths up 80 km s\({}^{-1}\) in the South component below the core and as low as 20 km s\({}^{-1}\) in the North component.
### The MERLIN 1665 MHz OH line emission
The new 1665 MHz OH line cube also covers a velocity range of 5600\(-\)5900 km s\({}^{-1}\) with a velocity resolution of 11.7 km s\({}^{-1}\). The 1665 MHz channel maps and the 0Moment and 1Moment maps are presented in Figures 5 and 6. Contrary to the observed emission structure of the 1667 MHz line, the observable/prominent 1665 MHz emission appears only at the components Arp 220E and Arp 220W-South, as may also be deduced from the spectra of Figure 3. Any difference between the 1667 and 1665 MHz emission structures may result from a different dynamic range in Figure 5 and 6.
The west-east elongated 1665 MHz emission component in Arp 220E shows a continuous velocity gradient, which consistent with the 1667 MHz line findings (Fig. 3). An overall hyperfine line ratio \(R_{H}\)(67/65) = 3.6 would correspond to an amplifying optical depth \(\tau_{67}\) = 2.4 (see Sect. 5).
Arp 220W-South shows a broad 1665 MHz spectrum possibly made up of multiple velocity components (Fig. 3). The line ratio \(R_{H}\)(67/65) = 5.0 suggests a higher amplifying optical depth \(\tau_{67}\) = 3.3 but there is no clear 1665 MHz velocity gradient. Surprisingly, Arp 220W-North shows no clear 1665 MHz emission and its narrow 1667 MHz emission component appears to have a large amplifying optical depth \(\tau_{67}\) and a possible small gradient visible in the 1Moment map (Fig. 6). This narrow feature is associated with the compact source found in the EVN data (see Fig. 9 and Sect. 3.5 below).
Figure 6: The velocity-integrated 0Moment map (top) and 1Moment velocity map (bottom) of the OH 1665 line emission from the MERLIN data.
Figure 7: The 1.6-GHz continuum emission of Arp 220 made from the EVN data. The restoring beam is 0\(\aas@@fstack{\prime\prime}\)013 \(\times\) 0\(\aas@@fstack{\prime\prime}\)011, PA=74\(\aas@@fstack{\prime\prime}\)3. The peak intensity is 7.7 mJy beam\({}^{-1}\), and the rms noise in off-source regions is 0.5 mJy beam\({}^{-1}\). The contour scale is in units of mJy beam\({}^{-1}\).
### EVN imaging of the continuum emission
The continuum structure of Arp 220 at high-resolution is known to contain a number of identifiable supernova remnants (SNRs) at the two nuclei (Smith et al., 1998; Lonsdale et al., 2006; Parra et al., 2007; Varenius et al., 2019). The new EVN image shows two groups of point sources covering the East and West nuclear regions consistent with the continuum peaks (the crosses) in the MERLIN data in Figure 1. Although some side-lobe artefacts, phase errors, and RFI-related stripes may still be present in the map, the general configuration of the point sources is consistent with earlier detections even if their flux densities and positions appear not fully consistent with earlier experiments (Smith et al., 1998; Lonsdale et al., 2006; Parra et al., 2007). Our sources are found to be brighter than earlier detections but variations in flux and position are to be expected in an evolving starburst environment. The presence of the point sources in the map of Figure 7 accounts for about 90 mJy beam\({}^{-1}\), which is about 40% of the integrated flux of the East and West radio nuclei. A thorough analysis still needs to be made of the power spectra and locations of these point sources and their (re-)appearance in comparison with earlier experiments.
The configuration of the SNR point sources confirms the star formation nature of the nuclear regions. The slightly elliptical N-S source configuration appears centred in between the two OH emission regions and coincides with the H\({}_{2}\)CO emission regions (Baan et al., 2017). The West configuration appears to have a N-S absorption lane possibly related to the edge-on torus at this nucleus (see Section 4.3) It should be noted that the point sources at the Eastern nucleus appear less dense and elongated in the SW-NE direction forming a connection with the continuum bridge below and between the two nuclei.
Figure 8: Upper panel: Velocity-integrated intensity 0Moment maps of the OH 1667 and 1665 MHz line emission from the EVN data. Top: W1 component at 1667 MHz; Top Bottom: W1 component at 1665 MHz (see also Fig. 9; Bottom Top Right: W2 component at 1667 MHz and located 320 mas below W1; Bottom: E1 and E2 components at 1667 MHz separated by 126 mas. The beam size for these images is 13 x 11 mas. The colour scale is logarithmic in units of Jy beam\({}^{-1}\).
Figure 9: Spectra of the 1667 MHz OH emission from the EVN data. The spectra are for W1-N(blue), W2-S(black), double feature E1-E(red) and E2-S(green). The zero level base of these spectra covers about 300 km s\({}^{-1}\) from 5250 to 5550 km s\({}^{-1}\) with the W1 spectrum showing a plateaux at about 10 mJy reaching further down in velocity. The W1 spectrum also shows the weak 1665 MHz emission at about 5700 km s\({}^{-1}\). Velocities using an optical definition and the line widths are presented in Table 3.
### EVN Imaging of the OH Line Emission
The new high-resolution EVN observations provides more details about and confirm the existence of four compact high-brightness VLBI maser components in Arp 220, two associated with each of the nuclear regions (Lonsdale et al. 1998; Rovilos et al. 2003). Following the nomenclature based on earlier VLBI detections, they have been named W1 (north), W2 (south), E1 (east) and E2 (south). The velocity-integrated maps of these four features are presented in Figure 8 and the integrated spectra are shown in Figure 9. Together they account for about 15% of the total line emission in Arp 220.
The most prominent emission feature W1 is located at the southeastern edge of the Arp 220W-North and appears as a point source in channel maps across an optical velocity range 5416\(-\)5498 km s\({}^{-1}\) with a peak at about 5364 km s\({}^{-1}\). W1 also shows compact emission at a much lower velocity of 5360 km s\({}^{-1}\), which may result from foreground emission (see Sect. 4.3). The newly detected 1665 MHz counterpart of W1 (Fig. 8(top right) shows a point-like centre and a mysterious E-W (halo or scattering) extension and peaks at the velocity of 5586 km s\({}^{-1}\). The less compact W2 feature is only detected at 1667 MHz and appears with two weak companions at the southern edge of the South region at about 115 pc (0\(\aas@@fstack{\prime\prime}\)3) south of W1.
The compact components E1 (east) and E2 (south) at the Eastern nucleus are shown in the channel maps of Figure 10. Compared with W1, the elongated E1 component is redshifted in the range 5401\(-\)5461 km s\({}^{-1}\) with a primary peak at 5443 km s\({}^{-1}\) at the brighter head and a secondary peak at 5399 km s\({}^{-1}\) at the (slightly redshifted) eastern tail. The compact component E2 is located (0\(\aas@@fstack{\prime\prime}\)14) 49 pc southwest of E1 and is also redshifted relative to W1 with a peak at 5409 km s\({}^{-1}\). E2 shows a northwest extension resulting from a possible double structure and is more prominent than was found in earlier experiments (Lonsdale et al. 1998). The peaks of the two E features confirm a large-scale northeast velocity gradient along the structure of the East nucleus. In addition to E1 and E2, several other compact weak point sources may be identified in the field.
The relative positions of the four components has been preserved during self-calibration (see Section 2.2), their actual locations within the East and West nuclei will be evident as they re-appear in the combined EVN - MERLIN emission maps presented below (see Section 4). The spectral characteristics of the EVN-detected features are also presented in Table 3.
## 4 Combining MERLIN and EVN Data
### The MERLIN-EVN imaging of Arp 220
The continuum structure with the two nuclear components obtained from the combined ME data shows again that the
Figure 10: Channel maps of OH 1667 line emission in the Arp 220E from the EVN data. Two emission regions are identified as the extended and filamentary region E1 in the north and the compact region E2 in the south. Some additional weak regions may be found between these two regions. The restoring beam is 0.16\(\times\)0.14 mas\({}^{2}\) at PA=53\({}^{\circ}\). The rms noise level is 1.3 mJy beam\({}^{-1}\). The peak intensity of 19.8 mJy beam\({}^{-1}\) is found for E2 in the 5407.4 km s\({}^{-1}\) channel. For comparison, the peak intensity of the 1667 MHz line is 135 mJy beam\({}^{-1}\) in the 5366.6 km s\({}^{-1}\) channel. For image clarity, only the 3\(\sigma\) contour of 5.2 mJy beam\({}^{-1}\) is plotted. The velocity axis is based on the optical definition of velocity and the channel width is 5.8 km s\({}^{-1}\).
The intensity colour scale is in units of mJy beam\({}^{-1}\).
Figure 14: The combined MERLIN-EVN data of the 1665 MHz OH emission. (Top) The Moment 0 velocity-integrated intensity map of the 1665 MHz OH emission. The restoring beam size is 40 \(\times\) 40 mas. The arrows in the map indicate the positions of the four VLBI components identified within the EVN data. The unit of the logarithmic colour scale is Jy beam\({}^{-1}\) The integrated flux densities of the components are for the East component and West-South component 4.0 Jy beam\({}^{-1}\), and the West-North component 0.1 Jy beam\({}^{-1}\) with estimate error of 0.011 Jy beam\({}^{-1}\). (Middle) The Moment 1 velocity map of the OH 1665 emission. The colour bar indicates the radial velocity in km s\({}^{-1}\). (Bottom) The Moment 2 velocity width map of the OH 1665 emission. The colour bar indicates the velocity widths in km s\({}^{-1}\).
Figure 13: The combined MERLIN-EVN data of the 1667 MHz OH emission. (Top) The Moment 0 velocity-integrated intensity map of OH 1667 emission. The restoring beam size is 40 \(\times\) 40 mas. Four of the six compact components seen at the two nuclei represent components identified within the EVN data (see Fig. 8). The integrated intensity colour scale is logarithmic in units of Jy beam\({}^{-1}\). The peak values are for the East component 18.15 Jy beam\({}^{-1}\), the West-South component 11.25 Jy beam\({}^{-1}\), and the West-North component 7.44 Jy beam\({}^{-1}\) with an estimated error of 0.011 Jy beam\({}^{-1}\). (Middle) The Moment 1 velocity map of the OH 1667 emission. The colour bar indicates the radial velocity in km s\({}^{-1}\). (Bottom) The Moment 2 velocity width map of the OH 1667 emission. The colour bar indicates the velocity widths in km s\({}^{-1}\).
tion of structural components with distinct velocity systems at each nucleus in Figures 16. A position-velocity diagram in Figure 18 depicts these velocity systems using the velocity values obtained from the spectra at various locations (see Table 2). Assuming that the higher velocities of the compact and surrounding emission components at both nuclei represent the systemic velocities of the two nuclei (or what is left over of them), the lower velocity components must represent foreground gas structures in the system. This would suggest that the compact components identify a systemic velocity of 5425 km s\({}^{-1}\) for the East nucleus (see Fig. 18 and Table 3). The compact OH emissions would then be systemic emission regions that are re-amplified by excited gas within the dominant foreground structures at velocity 5314 km s\({}^{-1}\) for Arp 220E.
The curious aspect regarding the compact emission regions is that they appear as identifiable features only on the east and southeast sides of the main emission regions. As discussed further in Section 6 below, this may be related to variation in velocity-coherent amplifying column density related to a velocity gradient in the foreground material.
The position-velocity diagram in Figure 18 that the systemic components E1 - E3 show a gradient of 19 km s\({}^{-1}\) over 107 pc (0.18 km s\({}^{-1}\)pc\({}^{-1}\)). Similarly, the gradient in the foreground screen is on the order of 18 km s\({}^{-1}\) over a distance of 38 pc (0.47 km s\({}^{-1}\)pc\({}^{-1}\)). A difference exists in Arp 220E where the 1667 MHz line width is almost uniform across the region, while the 1665 MHz extension emission shows a wide section towards the East followed by a narrower eastern edge (Figs. 13 and 14)
A comparison of the observed OH emission with the large-scale CO emission shows that the optical velocity of Arp 220E is in agreement with the CO emission spectra (Figure 1 in Wheeler et al. 2020). This spectrum shows two \({}^{12}\)CO (3 - 2) emission components at optical-defined velocities of 5297 and 5603 km s\({}^{-1}\) separated by a strong absorption at 5440 km s\({}^{-1}\). The systemic (optical) velocity of 5425 km s\({}^{-1}\) for the OH emission coincides with the strong absorption feature at the central continuum source, while the velocity of the foreground material of 5314 km s\({}^{-1}\) indicates an association with a low-velocity CO component. The high-velocity CO component at 5603 km s\({}^{-1}\) has no OH counterpart and appears to be located behind the nucleus.
The most recent 0Moment maps of the \({}^{12}\)CO (3 - 2) and the optically thin \({}^{13}\)CO (4 - 3) emission data show an enhanced emission region at the location of Arp 220E within a large scale emission structure drifting in northeast direction (Wheeler et al. 2020). The \({}^{13}\)CO (4 - 3) emission at the Eastern nucleus shows a SW-NE inclined and slightly warped disk structure extending to 366 pc and covering a radio velocity range from 5160 to 5680 km s\({}^{-1}\) (Wheeler et al. 2020). The systemic OH emission features at the East nucleus are consistent with these larger scale molecular structures, although they only highlight the central region of this disc that provides the FIR pumping emission.
For comparison, the formaldehyde maser emissions at the Eastern nucleus occur at the SW side of the nuclear centre and indeed show emission at velocities close to 5400 km s\({}^{-1}\) (Baan et al. 2017). However, the western H\({}_{2}\)CO emission spectrum shows a profile that also encompasses the velocity range of the foreground component reaching down to below 5100 km s\({}^{-1}\), which is in agreement with the foreground CO data. This would suggest that at Arp 220E, the foreground component may also re-amplify the SF regions at the systemic velocity.
### Extended and compact OH emission at West
A deep 0Moment map of the 1667 MHz emission in Arp 220 also shows the dominant emission region at the West nucleus, in addition to three compact region on its eastern side (Fig. 15). Two of these regions have counterpart in the EVN data (Fig. 8) and are clearly visible in Figures 13 and 14 and are identified in Figure 15 as W1 - W3. In addition to these, a number of additional emission regions may be identified to the north and south of Arp 220W.
The spectral velocity components at all identified emission regions are displayed in Figures 16 and 17. The compact components W1-W3 at the Western nucleus have a systemic velocity of about 5360 km s\({}^{-1}\), which is 75 -140 km s\({}^{-1}\) higher than the extended West OH components with an approximate velocity of 5254 km s\({}^{-1}\). Similar to the situation at the East nucleus, the W1-W3 regions are also located on the east-southeast side of the extended emission regions (see Sect. 5). In addition, the nearby components W4 and W5 both show a low-velocity component in agreement with that of the extended main regions, and a high-velocity component in agreement with those of the W1-W3 regions. Similarly, the W-S1 and W-S2 regions show emission at the foreground velocity, except that the W-S2 region shows an emission pair (with \(R_{H}\) = 1667/1665 \(\approx\) 1) at a velocity of 5438 km s\({}^{-1}\), which corresponds to the systemic velocity of the East nucleus. The relative sizes, locations, and velocity offsets with the more extended OH emission regions suggest that any of
Figure 15: A deep Moment Zero map of the 1667 MHz OH emission in the extended region around the two nuclei based on the combined MERLIN - EVN data. The contours have been set at 0.2 mJy beam\({}^{-1}\), which is 1.5 times the rms in the map. The designations of all identified compact and extended regions have been indicated in reference to the spectral information presented in Table 3 and Figures 16 and 17. The spatial scale of the diagram of 0\(\aas@@fstack{\prime\prime}\)1 corresponds to 38 pc. All observed emission components lie within the continuum contours of the two galaxies as presented in Figures 11 and 12.
\begin{table}
\begin{tabular}{l c c c c c c c} \hline Location\({}^{1}\) & Velocity & FWHM & S1667 & S1665 & 1667/1665 & Opt.Depth & Comment \\ & (km s\({}^{-1}\)) & (km s\({}^{-1}\)) & (mJy b\({}^{-1}\)) & (mJy b\({}^{-1}\)) & ratio & \(\tau_{67}\) & \\ \hline West North1 & 5258 & 29 & 2.9 & 0.3 & 9.7 & -5.0 & foreground \\ West North2 & 5268 & 34 & -1.5 & \(<0.1\) & – & – & fg absorption \\ & 5375 & 34 & 2.5 & \(<0.1\) & \(>\)25 & \(>\)7.3 & W systemic \\ W-Main North & 5255 & 24 & 83 & \(<\)1.5 & \(>\)55 & \(>\)9.0 & foreground \\ W-Main South & 5245 & 165\({}^{2}\) & 27 & 6.0 & 4.5 & -3.0 & foreground; double peak \\ W1 – North & 5361 & 39 & 31 & \(<\)1.0 & \(>\)31 & \(>\)7.6 & W systemic \\ W1 – North EVN & 5366 & 26 & 155 & (5) & (31) & (7.6) & compact\({}^{3}\) \\ W2 – SouthEast & 5317 & 68\({}^{2}\) & 7 & 0.8 & 8.7 & -4.7 & W systemic \\ W2 – SouthEast EVN & 5325 & 65 & 12 & – & – & – & compact\({}^{3}\) \\ W3 – East & 5366 & 39\({}^{2}\) & 12 & 4.0 & 3.0 & -1.9 & W systemic \\ W3 – East & 5366 & 39\({}^{2}\) & 12 & 4.0 & 3.0 & -1.9 & W systemic \\ W4 – South & 5246 & 19 & 4.2 & 0.6 & 7.0 & -4.2 & foreground \\ & 5346 & 87 & 1.2 & 0.2 & 6.0 & -3.8 & W systemic \\ W5 – SouthWest & 5212 & 22 & 7.8 & 0.5/1.5 & 15.6 & -6.1 & foreground \\ & 5296 & 32 & 5.0 & 1.5 & 3.3 & -2.1 & W systemic \\ WS1 & 5255 & 24 & 14.0 & 0.7 & 20.0 & -6.8 & foreground \\ WS2 & 5255 & 124 & 4.0 & 1.1 & 3.6 & -2.7 & foreground \\ & 5438 & 120 & 2.0 & 2.0 & 1.0 & small & E systemic \\ East -Main West & 5304 & 78 & 93 & 11.5 & 8.1 & -4.5 & foreground \\ East -Main East & 5322 & 68 & 72 & 12.1 & 5.9 & -3.7 & foreground \\ E1 – East & 5429 & 48 & 9 & 1.1 & 8.2 & -4.5 & E systemic \\ E1 – East EVN & 5444 & 18 & 50 & – & – & – & compact double\({}^{3}\) \\ & 5411 & 23 & 20 & – & – & – & – \\ E2 – South & 5410 & 39 & 13 & 1.0 & 13.0 & -5.7 & E systemic \\ E2 – South EVN & 5410 & 35 & 42 & – & – & – & compact\({}^{3}\) \\ E3 – NorthEast & 5429 & 53 & 4.5 & 0.3 & 15.0 & -6.0 & E systemic \\ \hline \end{tabular} \({}^{1}\) Note 1: The locations of the emissions regions are identified in Figures 15, 16, and 17.
\({}^{2}\) Note 2: Narrower profile superposed on a broad base ranging from 5000 to 5400 km s\({}^{-1}\).
\({}^{3}\) Note 3: These compact components have been detected in the EVN data as shown in Figures 8 – 10.
\end{table}
Table 2: OH line properties in MERLIN - EVN emission regions
Figure 16: Spectral components in Arp 220E. The velocity range of all spectra runs from 4809 to 6046 km s\({}^{-1}\) increasing from left to right. The vertical scale of all spectra is in mJy except that this scale is not equal for all spectra in order to show the weaker components.
these compact regions correspond to star formation regions belonging to the nuclear region of the underlying galaxy.
The spectra in the composite diagram of the West nucleus again shows the superposition of two structural components with distinct velocity systems in Figures 17. The position-velocity diagram in Figure 18 depicts these velocity systems using the estimated velocity values obtained from the spectra at various locations (see Table 2). The higher velocity compact components and surrounding emission components represent the systemic velocity of West, while the lower velocity components represent foreground gas structures. The compact components at West identify a systemic velocity of 5360 km s\({}^{-1}\) (see Fig. 18), while the dominant foreground structures would be at 5254 km s\({}^{-1}\) (see Table 3.
The position-velocity diagram in Figure 18 shows a clear velocity gradient for the systemic velocity components at the West nucleus. Also taking into account the outlying components WN2 and W5, the small South to North gradient at the West nucleus has a velocity range of approximately 78 km s\({}^{-1}\), and covering a distance of 324 pc (grad = 0.24 km s\({}^{-1}\)pc\({}^{-1}\)). This gradient would confirm that the systemic components at the West nucleus are representative of rotation in a nearly edge-on disc. A weak gradient is also seen for the foreground screen when considering the data points WS3 to WN1, which covers only 28 km s\({}^{-1}\) over a distance of 800 pc (grad = 0.03 km s\({}^{-1}\)pc\({}^{-1}\)). The foreground screen is drifting slowly in front of the nuclei in a NE direction. The corresponding 1665 MHz emission in Arp 220W-South shows an apparent gradient in the opposite direction.
A comparison with the existing large-scale CO observations shows that the optical velocities are in agreement with the emission spectrum at Arp 220W (Figure 1 of Wheeler et al., 2020). This spectrum shows two \({}^{12}\)CO (3 - 2) emission components at optical-defined velocities of 5245 and 5558 km s\({}^{-1}\) separated by a strong absorption at 5424 km s\({}^{-1}\)
Figure 17: Spectral components in Arp 220W. The velocity range of all spectra runs from 4809 to 6046 km s\({}^{-1}\) increasing from left to right. The vertical scale of all spectra is in mJy except that the scale is not equal in all spectra in order to show the weaker profiles.
\begin{table}
\begin{tabular}{l c c c c} \hline Arp 220 & Systemic & Foreground & Position & Comment \\ Nucleus & Velocity & Velocity & Angle & \\ & (km s\({}^{-1}\)) & (km s\({}^{-1}\)) & (degree) & \\ \hline West & 5360 & 5354 & 168 & nearly edge-on \\ East & 5425 & 5314 & 45 & edge-on \\ \hline \end{tabular}
\end{table}
Table 3: Systemic Velocities in Arp 220
The systemic velocity of the OH emission at the West nucleus 5360 km s\({}^{-1}\) would coincide with this strong absorption feature associated with the central continuum source, while the velocity of the foreground material of 5254 km s\({}^{-1}\) is associated with the low-velocity CO component. Again the high-velocity CO component at the nucleus Arp 220W has no OH counterpart and appears to be located behind the nucleus.
The most recent \({}^{12}\)CO (3 - 2) and optically thin \({}^{13}\)CO (4 - 3) 0Moment maps show strongly enhanced emission at the central location of Arp 220W within a large scale emission structure drifting at about 2.4 km s\({}^{-1}\) pc\({}^{-1}\)in NE direction (Wheeler et al., 2020). This emission region appears as a torus structure centred on the continuum emission and consistent with the large 2Moment velocity width seen at the West nucleus (Fig. 13). The 2Moment image of this elongated and tilted S-N torus structure suggests clockwise (East blue and West red) rotation with a width of about 480 km s\({}^{-1}\) and an estimated orbital velocity of 240 km s\({}^{-1}\). The extended 1667 MHz OH emission regions have an almost South-North orientation and are separated by about 130 pc as they straddle the absorption gap centred on the peak of the radio continuum at Arp 220W. This suggests that these regions represent the tangential sections of the nuclear torus with an estimated outer diameter of 240 pc. Similarly, the velocities of the OH SF components identified for Arp 220W are consistent with the clockwise rotation of the inner CO structure and appear consistent with the N-S oriented SF regions being on the front side of a central torus.
The presence of a tilted N-S molecular torus in the nuclear region of Arp 220W with a clockwise orbital motion and a central absorption component would also be consistent with the position-velocity behaviour of the HCN emission in the nuclear region without invoking an outflow and an E-W nuclear region (see Barcos-Munoz et al., 2018).
For comparison, the formaldehyde maser emissions at the West nucleus are thought to be associated with the disk component but the emission spectra also show multiple components at the velocity of the foreground (Baan et al., 2017). This suggests that the emission of the systemic regions is also re-amplified by the foreground structure. A comparison of the spectrum of Centre and West H\({}_{2}\)CO components at Arp 220W shows that the Centre profile has a 130 km s\({}^{-1}\) lower velocity and could also be associated with the front side of the torus.
### Outflows in Arp 220
Extended blue wings have been detected in single dish OH emission spectra in Arp 220 and other OHMM that have been interpreted as outflows (Baan et al., 1989). In Arp 220 this blue wing may extend about 1000 km s\({}^{-1}\) below the systemic velocity and some evidence for such an extension may even be found in the current EVN spectrum of W1 in Figure 9. Similarly, recent CO observations of Arp 220 show the broad (1300 km s\({}^{-1}\)) emission profiles at the two nuclei, where the outer blue and red parts of these profiles have also been designated as outflows (Wheeler et al., 2020). However, considering that Arp 220 is a strongly interacting system showing multiple CO emission regions along the line of sight, one may reconsider the nature of these outflows. Are these really outflows from the nuclear regions or do they represent
Figure 18: Spectral components in Arp 220. The central velocities of the identified components of Arp 220 are presented in a position-velocity diagram. Black and red data points are found at the West and East nucleus, respectively, and green data points represent EVN detections at the two nuclei. The low-velocity data point of component W5 lies within the extended range of the broad foreground molecular CO structure.
disk material flung away from the system at a larger relative velocity during this merger interaction? Maybe indeed the more likely explanation for these high velocity components is that they result from the interactive nature of the system. The velocity in the CO data does not reveal the distance to the nuclear cores and with the right line-of-sight conditions any low-velocity foreground component can also amplify the radio background and produce a low-level blue wing in the OH profile.
## 5 Two Masering scenarios
The detailed interpretation of the Arp 220 OHMM system shows a variety of emission properties and line ratios at the different regions within this interacting system, suggesting differences of the masering environments in the emission regions. However, the basic scenario for the OH amplification is having an alignment of: 1) a radio background serving as seed radiation, 2) an FIR pumping agent with the right spectral shape to create a population inversion, and 3) and an embedded or foreground column density with a line-of-sight velocity-coherent column density. If all such conditions are fulfilled, the foreground molecular structures can amplify both an extended radio continuum background at its own radial velocity and re-amplify any maser emission originating in the underlying star-formation regions at their own radial velocity.
In a controlled environment under LTE circumstances, the optical depth of the 1667 and 1665 MHz OH transitions would vary as \(\tau_{67}=1.8\times\tau_{65}\), which suggests that the line flux ratio varies as:
\[R_{H}=S_{1667}/S_{1665}=(e^{-\tau_{67}}-1)/(e^{-\tau_{67}/1.8}-1). \tag{1}\]
While this ratio will be independent of the background (or seed) continuum radiation field behind the emission region, the variation of the amplifying optical depth across any masering region should always give a clear correlation between the two emission lines.
The lower resolution MERLIN data show an \(R_{H}\approx 4.5\) for both main East and West components suggesting relatively lower (integrated) optical depths (see Fig. 3). However, the 1667/1665 OH line ratios of components in the MERLIN-EVN data show a large range of \(R_{H}\) values ranging from 3 to 20, which indicates distinct differences in the masering conditions in the foreground gas and the systemic environments.
The OH emission from the foreground gas varies with the varying amplifying gain across the face of the foreground structure convolved with that of the background radio structure. As a result of these variable parameters, the amplifying optical depth of the foreground material is found to vary significantly with values between \(\tau_{67}\)= -2.7 and -6.8 with one apparent value \(>\)-9.0 in Arp 220W-North. Similarly, the emission of the systemic SF regions will result from the intrinsic gain in the SF region and the gain provided by a foreground column with a similar velocity. The available data of SF-related features at the systemic velocity of the two nuclei shows an optical depth range of -1.9 to -6.0 with an extreme value of \(>\)-7.6 for the W1 region in Arp 220W-North. Based on these values, the foreground regions contribute a small addition to the gain for the systemic SF regions.
RADEX simulations (van der Tak et al., 2007) show that the range of optical depths in the OH main lines required in the foreground masering regions at each of the nuclei can be achieved when the molecular gas is relatively cold at \(T_{k}\) = 20 K and is exposed to an FIR radiation field emitted by warm dust with \(T_{dust}\) = 50 K. The required OH column densities are on the order of \(10^{17}\) cm\({}^{-3}\). The masering conditions provide some diagnostics of the foreground material.
The systemic star-formation regions are curiously located at the east-southeast edges of the main emission regions at each of the two nuclear regions. The reason for this may relate to a velocity gradient in the \({}^{12}\)CO(1-0) foreground appears to run globally from southwest to northeast (Wheeler et al., 2020; Sakamoto et al., 2008). Since foreground re-amplification requires an sufficient inverted column density at the exact velocity of the systemic SF regions, the foreground velocity gradient and the density distribution in the foreground appear to provide a certain optical depth for the regions at the eastern edge of the main emissions and not for the distinct SF regions at the western side of the nuclei.
## 6 Discussion
The OH MegaMaser activity in Arp 220 appears to be more complex than was anticipated on the basis of earlier observations. While the lower resolution MERLIN data showed the larger scale foreground emission regions and did not distinguish emission components at the galactic cores, the high-resolution EVN data resolved the extended emission and detected a few of the high-brightness star-formation components at each of the galactic cores. However, the combined MERLIN - EVN data with intermediate resolution provides a much clearer view of the structural components in Arp 220. In particular, the combined data provides a consistent masering scenario, where the FIR-pumped foreground material amplifies the background continuum from within the galaxy core regions and independently re-amplifies the galactic SF-related components.
For the case of the OH emission in Arp 220 a clear velocity distinction can be made between the compact star-formation regions at the systemic velocity of the East and West nuclei and a foreground screen covering the nuclear regions at velocities about 100 km s\({}^{-1}\) below that of the nuclei. Other higher-velocity molecular CO components do not have a counterpart in the OH data and appear to be located behind the nuclei (see Wheeler et al., 2020). Assuming that the velocity of the various compact emission regions in the EVN and combined MERLIN-EVN data are part of the underlying galactic nuclei, these components accurately determine the systemic velocities of the nuclei of 5425 km s\({}^{-1}\) for the East nucleus and 5360 km s\({}^{-1}\) for the West. As expected, these systematic velocities of the two nuclei correspond closely with those of the apparent absorption regions in the large-scale molecular structures (Wheeler et al., 2020). Subsequently, the velocities of the amplifying foreground regions that produce the bulk of the OH emission are at about 5312 km s\({}^{-1}\) for the East region and at 5260 km s\({}^{-1}\) for the West region, which is 100 km s\({}^{-1}\) below the systemic velocities of the two nuclei.
Because the OH emission from the foreground dominates, very little of the systemic structures of Arp 220 can be detected except for the evidence that large scale star-formation related FIR emission serves as a pumping agent for the foreground material. The East nucleus of Arp 220 appears to have a SW-NE orientation at a position angle of about 45\({}^{\circ}\) and shows a small velocity gradient in that direction. The systemic components at the West nucleus suggest an edge-on S-N orientation with a position angle of -12\({}^{\circ}\), which is consistent with the apparent presence of a nuclear torus structure seen at the nucleus within the large scale \({}^{12}\)CO (3 - 2) emission data. No evidence can be found in the OH data for an E-W orientation of the West nuclear region. This nearly edge-on torus appears to have a clockwise rotation with an estimated orbital velocity of about 100 km s\({}^{-1}\), which would be consistent with the presence of compact SF-related emission regions on the eastern edge of the OH emission region.
The maser amplification scenario proposed early for the OH Megdaser emission is found to be clearly applicable for Arp 220 (Baan, 1985, 1989). This scenario represents a line-of-sight convolution of the variable amplifying optical depth in the foreground gas with the distributed source of molecular pumping and the radio emission in the background. Prominent MegaMaser emission lines of other molecules in extragalactic sources are likely to be generated in a similar manner. The H\({}_{2}\)COMM emission components in Arp 220 are similarly superposed on the core regions of the galaxies and show the velocity range of both the systemic and the foreground regions (Baan et al., 2017). Similarly, for prominent H\({}_{2}\)OMM sources, such as NGC 4258 and NGC 1068, the collisionally excited gas is also superposed on the continuum structures (Herrnstein et al., 1999; Gallimore et al., 2004; Baan et al., 2022). However, this type of amplified emission always depends on a geometry where the amplifying column density is aligned with a background continuum. For extragalactic sources, the probability of this happening may be low and may vary significantly for different molecules. For Galactic sources, this geometry requirement may account for the (non-)occurrence of maser action in certain environments but it may also explain the variability and dynamic behaviour observed in sources.
The dominance of the OH emission from the foreground material highlights the complex nature of the nuclear regions of Arp 220 with large amounts of molecular material at velocities below and above the systemic velocity of the system. The large line-of-sight velocity width of the CO emission in Arp 220 of some 800 km s\({}^{-1}\)(Wheeler et al., 2020) appears to be a characteristic for violent galaxy mergers as has been found in higher redshift OHMMs with OH line widths as high as 2400 km s\({}^{-1}\)(Baan et al., 1992; Darling and Giovanelli, 2002; Pihlstrom et al., 2005). Incidentally, the presence of blueshifted foreground gas may also explain the blue tails in the OH emission profiles (interpreted as outflows) reaching some 800 km s\({}^{-1}\) as in the case of Arp 220 (Baan et al., 1989).
## 7 Acknowledgement
WAB acknowledges the support from the National Natural Science Foundation of China under grant No.11433008 and the Chinese Academy of Sciences Presidents International Fellowship Initiative under grants No. 2021VMA0008 and 2022VMA0019. TA acknowledges the support of the National SKA Program of China (grant 2022SKA0120102).
The European VLBI Network is a joint facility of independent European, African, Asian, and North American radio astronomy institutes. MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of STFC.
## 8 Data Availability
The data for the experiments of Arp 220 may be obtained from the MERLIN Data Archive under project code MNRO3B-22 and the EVN Data Archive under project code EB022C. Calibration of the data has been done using the NRAO Astronomical Image Processing System (AIPS) and with the ATNF MIRIAD Data Reduction Software for self-calibration and imaging.
|
2307.05506 | Data-Driven Design for Metamaterials and Multiscale Systems: A Review | Metamaterials are artificial materials designed to exhibit effective material
parameters that go beyond those found in nature. Composed of unit cells with
rich designability that are assembled into multiscale systems, they hold great
promise for realizing next-generation devices with exceptional, often exotic,
functionalities. However, the vast design space and intricate
structure-property relationships pose significant challenges in their design. A
compelling paradigm that could bring the full potential of metamaterials to
fruition is emerging: data-driven design. In this review, we provide a holistic
overview of this rapidly evolving field, emphasizing the general methodology
instead of specific domains and deployment contexts. We organize existing
research into data-driven modules, encompassing data acquisition, machine
learning-based unit cell design, and data-driven multiscale optimization. We
further categorize the approaches within each module based on shared
principles, analyze and compare strengths and applicability, explore
connections between different modules, and identify open research questions and
opportunities. | Doksoo Lee, Wei Wayne Chen, Liwei Wang, Yu-Chin Chan, Wei Chen | 2023-07-01T22:36:40Z | http://arxiv.org/abs/2307.05506v1 | # Data-Driven Design for Metamaterials and Multiscale Systems: A Review
###### Abstract
Metamaterials are artificial materials designed to exhibit effective material parameters that go beyond those found in nature. Composed of unit cells with rich designability that are assembled into multiscale systems, they hold great promise for realizing next-generation devices with exceptional, often exotic, functionalities. However, the vast design space and intricate structure-property relationships pose significant challenges in their design. A compelling paradigm that could bring the full potential of metamaterials to fruition is emerging: data-driven design. In this review, we provide a holistic overview of this rapidly evolving field, emphasizing the general methodology instead of specific domains and deployment contexts. We organize existing research into data-driven modules, encompassing data acquisition, machine learning-based unit cell design, and data-driven multiscale optimization. We further categorize the approaches within each module based on shared principles, analyze and compare strengths and applicability, explore connections between different modules, and identify open research questions and opportunities.
metamaterials, multiscale design, machine learning, data-driven design, topology optimization
###### Contents
* 1 Introduction
* 2 Preliminaries
* 2.1 Key Concepts
* 2.2 Machine Learning
* 2.2.1 Supervised Learning
* 2.2.2 Unsupervised Learning
* 2.2.3 Semi-Supervised Learning
* 2.2.4 Reinforcement Learning
* 3
Data Acquisition * 3.1 Overview * 3.2 Shape-Centric Data Generation Method * 3.2.1 Representation of Unit Cells * 3.2.2 Reproduction from Sparse Data to Large Data * 3.2.3 Perspectives on Shape Generation * 3.3 Property-Aware Data Acquisition Strategy * 3.3.1 Sequential Acquisition with Active Learning * 3.3.2 Downsampling Representative Subsets * 3.3.3 Perspectives on Acquisition Strategy * 3.4 Data Assessment * 3.4.1 Quantitative Assessment with Metrics * 3.4.2 Qualitative Assessment Through Visualization * 3.4.3 Perspectives on Data Assessment * 3.5 Discussion * 3.5.1 Reusability of Datasets * 3.5.2 Limitations of Unit Cell Datasets * 3.5.3 Learning Global Responses vs Local Responses * 3.5.4 Determining Data Size * 3.5.5 Data Sharing Practice * 3.5.6 Other Tasks Involving Data * 3.5.7 Public Resources
* 4 Data-Driven Unit Cell Design of Metamaterials
* 4.1 Overview
* 4.2 Iterative Design Optimization
* 4.2.1 Accelerated Optimization via Data-Driven Property Prediction
* 4.2.2 Accelerated Optimization via More Efficient Design Representation
* 4.2.3 Design as Sequential Decision Making via Reinforcement Learning
* 4.2.4 Design via Physics-Based Learning
* 4.3 Iteration-Free Inverse Design
* 4.3.1 One-to-One Direct Mapping from Target to Design
* 4.3.2 Avoiding Nonuniqueness Issue via Cascaded Neural Networks
* 4.3.3 One-to-Many Mapping via Conditional Generative Models
* 4.3.4 Other Approaches
* 4.4 Discussion and Future Opportunities
* 4.4.1 Cost-Benefit
* 4.4.2 Trustworthiness
* 4.4.3 Novelty
Data-driven Multiscale Metamaterial System Design * 5.1 Overview * 5.2 Bottom-Up Framework * 5.2.1 Data-Driven Design with Single-Class Graded Unit Cells * 5.2.2 Variant 1: Considering Unit-Cell Orientation * 5.2.3 Variant 2: Increasing the Diversity of Unit Cells * 5.3 Top-Down Framework * 5.4 Discussion * 5.4.1 Comparison of Bottom-Up and Top-Down Frameworks * 5.4.2 Assumptions of Homogenization * 5.4.3 Task Specificity of Data Acquisition for Multiscale Design * 5.4.4 Other Challenges
## 1 Introduction
Engineered material structures generally benefit from some extreme or spatially varying material properties to achieve higher performance or complex functionalities [1, 2, 3, 4]. Typically, tuning material properties involves precise control of material composition and processing conditions, which can be both technically challenging and cost-prohibitive, particularly when aiming for spatially varying properties. In contrast, metamaterials are engineered architectural materials that can reach a broad range of properties by carefully designing their architectures or microstructures rather than altering the material composition itself. [5, 6, 7, 8]. Along with the recent enhancements in manufacturing capabilities, metamaterials are emerging as a new paradigmatic material system to enable unprecedented design flexibility in properties. They can advance applications in a wide range of fields, including optics [9, 10, 11], electromagnetics [12, 13], thermology [14, 15, 16, 17], acoustics [18, 19, 20, 21, 22], and mechanics [5, 23, 24, 25, 26, 27].
Nevertheless, the design of metamaterials and their multiscale structures proves to be a complex process that involves navigating an infinite-dimensional topological design space, mapping microstructures to their effective properties across multiple scales, dealing with numerous local optima in design optimization, the absence of analytical gradient information, and expensive property or performance evaluation. Most existing metamaterial designs adopt trial-and-error and heuristic methods [8, 24, 27, 28, 29, 30, 31], which rely heavily on the experience of a designer, or simple parameter optimization methods [32, 33], confining the metamaterials to a restricted selection of properties. In some specific cases with relatively simple and differentiable physical models, gradient-based topology optimization (TO) has been utilized to facilitate the automatic design of metamaterials. Nonetheless, these methods are generally not scalable to the design of multiscale systems of metamaterials that feature nested optimization loops and require numerous microscale designs at different spatial locations.
The emergence of data-driven methods has provided solutions to these challenges by enabling high-throughput property or response prediction, reducing the dimensionality of complex problems, accelerating design space exploration and design optimization, and allowing fast solutions to ill-posed inverse design problems. Data-driven metamaterials design typically includes three modules, 1) data acquisition: acquiring a precomputed dataset of unit cells; 2) machine learning-based unit cell design: using machine learning to extract information from data and help unit cell designs; 3) multiscale design: utilizing unit-cell database and machine learning models for design synthesis at the system level. In practical applications, it is possible to integrate all of these modules into a unified framework or selectively focus on specific modules based on the design requirements. However, the core idea underlying data-driven design remains consistent across these modules, which is to extract meaningful patterns from data that are unavailable or difficult to obtain with physical models, and incorporate them into the design process to simplify the complexity in traditional design approaches.. Although these capabilities come with the cost of data collection and model training, deploying the trained model has the benefits of negligible inference time, which can significantly speed up the design process. Data-driven design methods are especially useful in scenarios where the design problems are high-dimensional or the governing physics is unknown. They can also achieve unprecedented design performance owing to their capability
of encapsulating higher design freedom (e.g., heterogeneous metamaterials system design) compared to conventional design methods.
As evidence of growing attention to data-driven multiscale design, with or without the use of data, a suite of literature reviews from different communities have been published in recent years. Each review is centered on particular topics, such as design [34, 35, 36, 37, 38], manufacturing [39, 40], mechanics [41, 42, 43, 38, 44, 45, 46], and specific machine learning methods [35]. Despite the meaningful contributions of each, and of the aggregate, we recognize lack of discussion on some points that may impede researchers from unlocking the full potential of data-driven design for multiscale architectures. First, in the corpus, we observe a disconnect between two primary lines of approaches, one being the data-driven camp that harnesses pre-existing machine learning tools with minimal customization, and the other being the physics-based camp that specializes in physics with limited awareness to recent advancements of data-driven methods. Second, when covering data-driven design, the prior reviews are typically dedicated to specific aspects, e.g., machine learning methods [35], physical mechanisms [43, 41, 42], individual components of data-driven design [34, 36]. In the current state of the field, it is difficult to find a singular review that provides a holistic overview of data-driven design, a summary of the archetypal design framework, and the critical inter-dependencies between design components. Third, while some existing reviews [35, 40] discussed data preparation as a core module, i.e., a step in data-driven design, none have systematically compared how existing datasets were created and how data quality could be ensured to better support the design of multiscale architectures.
The key contributions of our review include the following:
1. We adopt a _design_-centered perspective to examine a wide range of papers, categorizing existing methods from various domains into three modules within the data-driven design framework. This synthesis of different studies allows us to present a clear and cohesive picture of _how_ data-driven design has been practiced from data acquisition to single-scale and multiscale optimization. Our key focus is on the methodological aspects of design without specific deployment goals, i.e., without filtering based on the underlying physical mechanisms, geometric families of unit cells, and fabrication methods. This holistic approach allows us to uncover the common threads and overarching principles that drive the field of data-driven design.
2. We review the common practices of current data generation strategies for metamaterial design through a standardized taxonomy, discuss key concerns in depth, and attempt to raise awareness on certain issues that are crucial yet underestimated, or even overlooked, in data acquisition.
3. We review prior data-driven design methods that can be broadly applied to metamaterials design under different physics (i.e., optical, acoustic, mechanical, thermal, etc.) and raise critical concerns that are generalizable to all types of metamaterial design problems addressed by data-driven methods.
4. We investigate the role of data-driven models and methods in the context of multiscale system design. Unlike previous studies that treat data-driven models as isolated solutions, we highlight their integration into a complete design workflow and their scalability in handling large databases, multiple scales, and combinatorial design spaces. By examining these tools within the broader design process, we aim to offer insights into how data-driven approaches can be effectively utilized to enhance the efficiency and effectiveness of metamaterial design.
We specify our scope as follows:
1. We limit our scope to only the machine learning-based design methods for metamaterials and their multiscale systems.
2. We consider a wide range of physics including optical, acoustic, mechanical, thermal, and magneto-mechanical, because the machine learning methods are usually applicable regardless of the physics that governs the problem.
3. The reviewed machine learning methods do not necessarily require prior training data (i.e., we included past works using methods such as reinforcement learning or physics-informed neural networks).
4. We encompass design for multiscale systems that are built on either a unit-cell database of metamaterials or machine learning models. It includes the use of descriptors of unit cells, surrogate models for constitutive laws, efficient simulation via machine learning, optimization, and assembly of unit cells based on the database.
This paper is structured as follows: Section 2 gives a brief definition of key concepts underpinning our review and major machine learning methods utilized in the works we will cover. In Section 3, isolating data acquisition from the pipeline of data-driven multiscale design, we anatomize the common practices of data acquisition strategies, with particular attention to the methodological procedures of shape generation and property-aware sampling. Following this, we provide a brief review on the current practices of data assessment that ensure data quality and often shed light on how it can propagate into downstream tasks. Section 4 reviews prior works using machine learning methods for
single-scale metamaterials design, discusses some critical considerations when evaluating these methods, and proposes promising future directions. Section 5 explores data-driven design methods for multiscale systems that are built on either a unit-cell database or machine learning models. It includes the use of descriptors of unit cells, surrogate models for constitutive laws, efficient simulation via machine learning, optimization, and assembly of unit cells based on the database.
## 2 Preliminaries
This section defines key terminologies and concepts used throughout this paper as well as in other data-driven metamaterials design literature. We also briefly introduce common machine learning techniques and how these techniques were applied under the context of metamaterials design.
### Key Concepts
Below, we list working definitions of key concepts to be frequently used throughout the paper.
Unit CellA unit cell is the smallest representative unit of a material used to control its properties, as shown in Figure 1. It is often referred to by interchangeable terms such as meta-atom, meta-molecule, building block, and cell [7], depending on the physical mechanism, scale and geometry.
MicrostructureA microstructure is an assembly of multiple unit cells arranged in a specific pattern to achieve more complex properties arising from their arrangement.
MetamaterialsMetamaterials is the assembly of multiple microstructures, often in a periodic pattern, to achieve unique and tailored properties that cannot be found in natural materials.
Multiscale designMultiscale design in the context of metamaterials refers to the process of designing metamaterials with desired properties at multiple length scales, from the macroscopic level of unit cell to the macroscopic level of the metamaterial structure.
ModuleForst to an independent and reusable unit or component within data-driven multiscale design. It provides a specific functionality that is required for design. An ordered sequence of modules forms a data-driven multiscale design framework. Modules of primary interest in this review include data acquisition (Section 3), machine learning-based unit cell design (Section 4), and data-driven mulitscale optimization (Section 5).
Effective PropertiesEffective property is the macroscopic property of a metamaterial that arises from the collective behavior of its microstructures [47]. These properties are often different from the intrinsic properties of the constituent materials, and can be tailored through design and optimization of the microstructure.
ClassA group of unit cells that can be generated from the same geometric motif or design parameterization [48].
RepresentationA set of parameters, or models, used to directly characterize unit cells [48]. Representations often involve the projection of high-dimensional instances into a lower-dimensional space.
Design SpaceThe space of all possible design solutions. It contains all the combinations of design variables. In the context of metamaterials design, design variables usually refer to geometric or material design variables.
Shape SpaceThe geometric design space of unit cells.
Property SpaceThe response space of unit cells.
EvaluationTo obtain system responses of concern given an architecture and loading conditions, typically through numerical analyses such as finite element methods. In machine learning literature, the evaluation process is similar to "labeling", which refers to the process of adding attributes of interest (i.e., labels) to raw data. Throughout this review, we will assume that the term evaluation is interchangeable with labeling.
Shape-Property MappingA directional mapping from instances in shape space to those in property space. It is typically learned through machine learning using labeled data.
CompatibilityCapability of neighboring unit cells to possess seamless geometric/mechanical connections, or lack of geometric frustration. It is often measured through geometric/mechanical similarities at the interface of neighboring unit cells [49; 50].
### Machine Learning
Data-driven metamaterial design processes usually include the use of machine learning models to extract useful information from data. The extracted information can then help the design process in different ways. Machine learning approaches used in this context were mainly from four categories -- supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning. We briefly introduce these categories in this section.
#### 2.2.1 Supervised Learning
In a supervised learning task, we train a machine learning model to predict attributes of interest (i.e., labels). The machine learning model is trained on a collection of data-label pairs. The data can take various forms, e.g., vectors, images, and graphs. Depending on the type of labels, supervised learning can be divided into a broad dichotomy: regression with continuous outputs and classification with discrete ones. Commonly-used machine learning models include neural networks, kernel machines, and decision trees. Selecting the type of model is generally done before training, and could be contingent upon the end task (e.g., regression/classification), data volume, need to predict uncertainty, and design applications.
Within the context of data-driven metamaterial design (DMD), supervised learning is most widely for creating shape-property relations. A data pair typically describes a shape, where the input is its parameterization (e.g., parameterized lattice) and the output is quantities of interest (e.g., elasticity components; frequency dispersion). A key motivation for using supervised learning has been to replace the resource-intensive evaluation of unit cells with a surrogate model. The types of models have been dominantly based on neural networks [51; 49; 52] and Gaussian processes [49]. Once trained on a large volume of data, a data-driven model offers on-the-fly predictions of the (effective) properties of unseen unit cells. Sometimes such a surrogate is incorporated into a larger network architecture and trained end-to-end together with other components [53; 50].
#### 2.2.2 Unsupervised Learning
In contrast to supervised learning, unsupervised learning extracts information from unlabeled data. More specifically, it addresses tasks such as clustering, anomaly detection, and dimensionality reduction. In metamaterials design, it is mainly applied to learning the representation or the distribution of complex metamaterial geometries. Unsupervised learning models commonly used in metamaterials design are autoencoders, variational autoencoders, and generative adversarial networks.
Autoencoder (AE) [54] is a type of neural network that uses an encoder-decoder architecture to extract lower-dimensional latent variables of input data. In metamaterials design, AEs were used to reduce the dimensionality of either metamaterial
Figure 1: Hierarchical terminology system in this review. Metamaterials are created by assembling multiple microstructures to achieve properties on the macroscale. A microstructure is an assembly of multiple unit cells. A unit cell is the smallest representative unit of a material in the microscale to control its properties.
geometries [55] or high-dimensional responses such as scattering parameters [56]. A variational autoencoder (VAE) [57] has a similar architecture with AEs, while being a type of deep generative model that learns the distribution of data. New data can be generated by sampling latent variables that are low-dimensional and follow a well-defined distribution. Therefore, the latent representation learned by a VAE is usually more efficient and interpretable than the original metamaterial design representation, especially when considering high design complexity and freedom [58; 59]. Same as VAEs, generative adversarial networks (GANs) [60] can also generate new metamaterial designs and learn efficient representations. A GAN models the generation as a game between its generator and discriminator. Compared to VAEs, GANs can generate higher-quality samples [61].
While VAEs and GANs have been generally used for unsupervised learning, prior works have proposed variants of these models, such as conditional GANs (e.g., [51; 62; 50], conditional VAEs (e.g., [63; 64]), and VAE-regressor models (e.g., [58]), that require supervised learning. These supervised learning models either enable the inverse design of metamaterials [51; 62; 50; 63; 64] or help construct a property-related metamaterial latent representation [58].
#### 2.2.3 Semi-Supervised Learning
Semi-supervised learning trains a machine learning model on partially-labeled data so that the model can predict the labels of any given data. Typically it is assumed that the portion of labeled data is much smaller than the counterpart. This is a commonplace scenario in machine learning due to the high labeling cost. Marrying supervised and unsupervised learning, semi-supervised learning aims to improve the performances of either. It also inherits most of the machine learning tasks above, such as semi-supervised classification and regression, and semi-supervised generative modeling.
Within DMD, the efficacy of semi-supervised learning has been demonstrated in some works. In designing photonic metasurfaces, Ma et al. used both labeled and unlabeled data when training the prediction network to improve the regression with a less computational overhead of data preparation [63]. Although the idea of semi-supervised learning could sound compelling, the efficacy comes under some conditions; in-depth discussion on this point can be found in some reviews dedicated to semi-supervised learning [65; 66].
#### 2.2.4 Reinforcement Learning
Reinforcement learning (RL) is another category of machine learning methods used in metamaterials design. It is modeled as a Markov decision process [67], where an agent takes actions in an environment to maximize a reward. The goal is to learn an optimal policy that guides the action-taking strategy. Different from supervised, unsupervised, and semi-supervised learning, RL does not require an initial dataset to learn from. Instead, the agent in RL learns from its experience of exploring the action space and receiving rewards. There have been successful applications of RL in areas such as gaming and robotics. RL is usually applied to solving sequential decision-making processes. But with a proper definition of the action space and sequential decision-making setting, the design of metamaterials can also be formulated as RL problems, as shown in prior works [68; 69; 70].
## 3 Data Acquisition
### Overview
Creating and leveraging datasets of unit cells has been a core enabler of the recent success of DMD. Data acquisition in multiscale design is a decision-making activity that determines a finite collection of structure-property pairs, which delegates the space to be explored in downstream tasks. When one strives to exploit the power of DMD, data acquisition presents multifaceted open challenges, such as exploration of high-dimensional design space of unit cells, resource-intensive design evaluation in particular for large datasets, many-to-one mapping from shape to property, distributional bias in data, and opaque compounding effects of data quality to downstream tasks. Generally taken as the first module of DMD, data acquisition was tackled through diverse strategies in past works. Despite the diversity of strategies seen across different communities, it is difficult to draw connections among them due to the absence of attempts to build a common context that bridges different strategies.
To this end, we present a review on data acquisition for DMD with a particular focus on the methodological perspectives of prior works. We propose a standardized taxonomy which which to organize the literature in a relatable and easy-to-compare manner. Based on our observations of the current research trend, our review of data acquisition consists of two parts: Shape-Centric Data Generation Method (Section 3.2) and Property-Aware Data Acquisition Strategy (Section 3.3). In the corpus, we recognize that many demonstrations of data acquisition adopted shape generation heuristics, which do not necessarily consider property, in order to incorporate domain knowledge into datasets and avoid handcrafting a large number of shape instances one by one. Section 3.2 offers a detailed review of these shape-centric
data acquisition methods. In contrast, Section 3.3 reviews acquisition strategies that take property into account. These methods could boost sampling efficiency and facilitate data customization for specific design tasks.
Following the current research trend, the scope of this section is centered on data acquisition, but we also briefly introduce exemplar demonstrations of data assessment in Section 3.4 that are key to underpinning data quality assurance and data sharing practice. Each subsection includes a discussion specific to its topic, while Section 3.5 offers a more general and comprehensive discussion that covers multiple themes in data acquisition.
### Shape-Centric Data Generation Method
Data acquisition for DMD often entails a large collection of unit cell shapes. Handcrafting individual shape instances one by one is intractable for large data, as is running inverse optimization to obtain all unit cells corresponding to a massive set of pre-specified target properties. To create large data in an effective, systematic manner, past works presented a diverse array of methodological procedures. We recognize that individual approaches commonly involved two research questions in shape generation: 1) How to specify a group of unit cells? and 2) How to grow sparse data to large data? Answering the first essentially entails a representation of unit cells (Section 3.2.1), which was usually pre-selected at the early stages of data acquisition and often justified based on criteria such as domain knowledge and fabrication methods. On the other hand, answering the second question involves the reproduction of unit cells (Section 3.2.2), which facilitates the collection of a large enough dataset from sparse data without both handcrafting and optimizing a large volume of individual samples. We remark that in literature the two questions were addressed primarily based on generation strategies driven by shape rather than property; we call these types of approaches shape-centric generation methods. Given this taxonomy, we can bridge diverse data generation approaches scattered over different communities in order to offer a comparative review.
#### 3.2.1 Representation of Unit Cells
A representation refers to a set of parameters, or models, used to directly characterize unit cells, typically by projecting high-dimensional instances into a lower-dimensional space (Section 2.1). Determining what representation to use, specifically for the unit cells in this subsection, is a key decision that should be made at the early stages of data acquisition, as it dictates the nature of resulting data distributions.
Figure 2 depicts widely-used representations reported to date, organized based on our literature survey. The list might not be exhaustive, but we believe that it provides a sufficient overview of common practices in data generation. For each category, we discuss the definition, hallmarks, and relevant works. A comparative, multifaceted discussion across all the representations covered herein can be found in Section 3.2.3.
A. Parametric MulticlassAn intuitive way to encode domain knowledge into a dataset is to directly include a set of geometric _classes_ (Section 2.1) intensely studied in the literature. Herein the seed classes are expected to serve as "pivots" of the shape generation. Each class is typically endowed with some low-dimensional explicit parameterization, e.g., the length/thickness of a bar entity [80; 81; 71; 82], volume fraction [71], angle between geometric entities [83], or rotational angle [72; 82], which supports direct design exploration either within a class [71; 83; 72] or across classes [82].
In literature, some advocated for a mixed-variable representation where qualitative (e.g., building block type) and quantitative (e.g., scaling factor) variables were used together as an explicit multiclass representation for TO (Figure 2(b)) [84]. Conceptual generalization was proposed by Liu et al., showcased by lattice-like building blocks, spinodal pattern-like ones, and multimaterial composites (Figure 2(a)), each of which involved an explicit, mixed-variable representation with ensured compatibility [72]. Meanwhile, others conceived a versatile parametric representation able to generate multiple classes without explicitly defining classes [82]. To avoid geometric frustration when assembling building blocks in the multiscale design of mechanical metamaterials, the compatibility among neighboring unit cells based on geometric and/or mechanical factors often served as a primary criterion for choosing classes [71; 72; 82; 85].
B. Implicit FunctionIt has been widely adopted to represent unit cells using implicit functions that can generate geometric families. Therein, a shape instance is implicitly represented through a surface function.
Demonstrations in literature have been centered on functions that enjoy smooth topological variations, as opposed to lattice representations, and which are tunable by a handful of parameters. The most widely used families in DMD are Triply Periodic Minimal Surfaces (TPMS) that feature zero mean curvature and large surface areas (Figure 2(c)) [73; 50]. Another isosurface representation based on linear combinations of analytical crystallographic symmetry functions was implemented by Chan et al [86]. Bodapati et al. proposed another representation that can synthesize diverse classes of quasi-free unit cells of mechanical metamaterials by a linear superposition of periodic cosine functions [87].
Inspired by the phase separation process described by the Cahn-Hilliard equation, Kumar et al. [74] reported a spinodiod representation (Figure 2(d)), which features smooth, aperiodic variations of complex topologies and tunable anisotropy.
A variant under this branch harnesses spectral decomposition. A manifestation of this for photonic metasurfaces was shown by Liu et al. [88], where Fourier transform and level-set function of shapes served as key pillars for the new representation. The spectral representation enjoys topologically rich unit cells, reconstruction capability supported by inverse Fourier Transform, and efficient symmetry handling. Another example in this line was used by Wang et al. [49], where the Laplace-Beltrami operation was employed for dimension reduction of the freeform unit cells.
Figure 2: Representations of building blocks. a) and b): Parametric Multiclass. a) The mixed-variable multiclass lattice representation of mechanical metamaterials. Reproduced from Ref. [71] with permission from ASME. b) The parametric representation of 3D multiclass building blocks of mechanical metamaterials. Reproduced from Ref. [72] with permission from American Association for the Advancement of Science. c) and d): Implicit Function. c) The representation based on Triply Minimal Periodic Surfaces of mechanical metamaterials. Reproduced from Ref. [73] with permission from ASME. d) The spinodoid representation of mechanical metamaterials. Reproduced from Ref. [74] with permission from Creative Commons CC BY. e) and f): Pixel/Voxel. e) 3D voxelated representation of mechanical metamaterials. Reproduced from Ref. [75] with permission from Association for Computing Machinery. f) 2D pixelated representation. Reproduced from Ref. [53] with permission from Elsevier B.V. g) and h): Parametric Curve/Surface. g) The parametric surface representation of photonic metasurfaces. Reproduced from Ref. [76] with permission from Optica Publishing Group. h) The parametric curve representation of photonic metasurfaces. Reproduced from Ref. [77] with permission from AIP Publishing. i) and j): Constructive Solid Geometry. i) The representation based on primitive superposition and four-fold symmetry of dielectric metasurfaces. Reproduced from Ref. [78] with permission from Wiley-VCH GmbH. j) The union-based representation of plasmonic metasurfaces. Reproduced from Ref. [79] with permission from Creative Commons Attribution 4.0.
C. Pixel/VoxelPixel/voxel representation builds on the assumption that any shape instance can be viewed as a spatial aggregate of solid/void elements. They are typically freeform. Distinct from other representations, this approach offers a direct connection with inverse topology optimization.
As an early demonstration in DMD, Zhu et al. employed the voxelated representation with TO (Figure 2(e)) [75]. Wang et al. implemented inverse TO to find hundreds of freeform unit cells to start with, each of which closely matches the target effective property specified _a priori_ (Figure 2(f)) [49, 53]. Li et al. used multimaterial TO to systemically construct a library of freeform unit cells, each of whose response was programmed to exhibit a prescribed target force-displacement behavior [89]. TO was also used for a thermal emitter design that aims for frequency selective reflectivity when generating seed instances [90]. Harnessing interpretable machine learning for band gap engineering, Chen et al. adopted the 2D pixel representation to generate unit cell templates of phononic metasurfaces [91].
In the literature, we also observe another subcategory that advocated the pixel-/voxelated representation while considering user-defined classes. For design of photonic metasurfaces, many built datasets spanning from a group of canonical classes, or meta-atoms, such as cross, bow-tie, V-shape, I-beam, split ring resonator, and others [93, 51, 63, 81, 93]. This allows one to encode the data generation procedure with domain knowledge in contrast to the optimization-based pixel representation introduced above.
D. Parametric Curve/SurfaceBoundary-based representations, also referred to as contour-based shape descriptors [94], have been commonly used to describe shapes as well. In these approaches, a shape is represented by an ordered sequence of control points on curves or surfaces.
Within DMD, this approach has been primarily favored in the design of wave-based metamaterials that pursues design exploration beyond canonical families. For metagrating design, Inampudi et al. employed a boundary representation that specified shape instances with 16 boundary Parametric Curve/Surfaces (Figure 2(h)) [77]. Li et al. used the 4-order formulation of trigonometric functions with tunable parameters to explicitly represent a boundary curve of scattering inclusions of phononic crystals [55]. Tantiover et al. also harnessed such a representation to construct a shape dataset not restricted to the canonical meta-atoms in the literature under curvature constraints [95]. As an extension, it was also shown that a higher dimensional embedding of parametric curves/surfaces can be used as an implicit representation of 2D boundaries. For example, in photonic metasurfaces, Whiting et al. conceived a representation that offers topologically diverse instances in order to generate quasi-free building blocks (Figure 2(g)) [76]. A key distinction between this and the Implicit Function method above is that the 3D embedding here is fully governed by control points, which are explicitly defined. As an example in mechanical metamaterials, Wang et al. adopted the Cassini oval curve to represent the proposed auxetic planar metasurfaces with oval holes [96].
E. Constructive Solid GeometryConstructive Solid Geometry (CGS) is a geometric modeling method to create a solid object in a syntactic manner [97]. Its underlying concept is to compose an instance by following a sequence of set-theoretic operations (e.g., union; intersection) acting on primitives (e.g., rectangle, cylinder, sphere). The semantic nature makes instances of CGS highly interpretable, and offers a seamless connection with Computer-Aided Design (CAD). Capitalizing on explicit parameterization, a similar approach, the so-called moving morphable components [98, 99], has been developed in the TO community.
Within the context of DMD, CGS has been utilized in some works, in particular for design of photonic metasurfaces. Malkie et al. [79] applied the primitive rectangle, whose presence, length, and angle were parameterized, along with the union operation to synthesize plasmonic nanostructures (Figure 2(j)). A recent work that builds on further design freedom was reported by An et al. [81, 78], where a heuristic shape composition strategy, named the needle-drop approach by the authors, was employed to produce a large volume of quasi-freeform unit cells as unions of rectangle primitives (Figure 2(i)).
#### 3.2.2 Reproduction from Sparse Data to Large Data
Once a representation is determined, a typical next step that follows is to use it to grow a large-scale shape collection, with an optional target dataset size. We will call this task reproduction throughout this review. A reproduction strategy dictates the way of producing generic instances in a shape set and the distributional nature of resulting data, thus significantly affecting the quality of downstream tasks of DMD. Through effective reproduction, DMD can enjoy a quality dataset that represents the property distribution with space-filling samples and wide coverage. Based on our survey, we observe three primary lines of reproduction methods: (i) Parametric Sweep, (ii) Multiclass Blending, and (iii) Perturbation. Figure 3 illustrates each with an example in literature.
A. Parametric SweepGiven a representation of unit cells that spans a descriptor space with low/moderate dimensionality, Parametric Sweep is often applied to explore the descriptor space as uniformly as possible with a finite number of
samples. Space-filling sampling that is effective in low-dimensional spaces has been intensely studied for a long time. Readers interested in the topic are referred to reviews [101, 102].
We observe that, perhaps not surprisingly, Parametric Sweep is the most widely-used reproduction strategy [81, 71, 50, 103, 72, 74, 77, 83, 82]. It has been combined with diverse types of unit cell representations, especially low-dimensional ones such as the mixed-variable lattice representation (Figure 3(a)) [84], the six-bar lattice representation [82], the H-shape meta-atom parameterized with six variables [81], the CGS-based representation [78], and the pixel/voxel dataset with canonical classes [92], to mention a few.
Despite its wide use, three key drawbacks are that (i) the approaches offer little design freedom, as the sweep takes place only within a selected pivot and hence cannot bridge multiple unit cell classes; (ii) as the dimensionality gets larger, the density of space-filing sampling drastically drops due to _the curse of dimensionality_[104]; (iii) sampling only in shape space, even if done well, typically leads to huge bias in property space (discussed in detail in Section 3.4.3).
B. Multiclass BlendingBlending, or interpolating, across classes offers an avenue to grow a large shape library from a few initial seed classes. This approach differs from Parametric Sweep in that multiple classes jointly contribute to a new instance. Hence, this line of reproduction approaches could be powerful for merging different classes into a unified landscape that includes unseen inter-class instances.
In DMD, some works addressing photonic metasurfaces also utilized image transformations based on Boolean operations (i.e., union, intersection, complement) among canonical classes (e.g., cross, split ring resonator, I-beam) to synthesize freeform inter-class instances [51, 63]. In the corpus, such class blending for DMD is often followed by deep generative modeling [60, 57], which distills a continuous shape manifold of multiclass unit cells [50, 51, 63]. For a recent method developed in this branch, Chan et al. proposed a versatile Multiclass Blending scheme for functionally graded structures [52, 48]. The entire scheme combines a weighted sum of seed classes and an activated soft union with lower feasible bounds. This enables the method to be directly integrated into multiscale TO with assurances on both structural integrity (i.e., connectivity within a unit cell) and feasibility (i.e., connectivity among neighbors) simultaneously, while avoiding the restriction of the unit cell to predefined shapes (Figure 3(b)).
While feasible blending operations are not tied only to set-theoretic Boolean operations (e.g., union, complement), relevant works seem to have reported either simple unions [50] or its variants [52]. For implementation one needs to specify (i) the type of operations (e.g., union; intersection); (ii) the number of classes (iii) rules for weighting factors. We point out that either justifications or analyses on the impact of the choices have been rarely reported in the literature, with a possible exception of Chan et al. [52].
C. PerturbationPerturbation-based reproduction utilizes heuristics that also allow data expansion. The core idea is to (1) either look up near-boundary or on-boundary instances in the property space of a given dataset at the current iteration, (2) apply geometric perturbations on them in the ambient shape space, often beyond the given representation space, and (3) iterate the perturbation to drive the data acquisition for on-demand purposes (e.g., coverage expansion).
Figure 3: Reproduction strategies applied to sparse data to generate large data. White arrows indicate the direction from existing shapes to new ones. a) Parametric Sweep applied to Wang et al., where six lattice classes were explored by varying their volume fraction. Reproduced from Ref. [71] with permission from ASME. b) Multiclass Blending employed in Chan et al. where the unit cells at the leftmost and rightmost columns serve as seed classes. The proposed class remixing generates inter-class instances with ensured connectivity. Reproduced from Ref. [52] with permission from Springer-Verlag GmbH Germany. c) Perturbation implemented in Wang et al [49], where radial distortion was harnessed as the perturbation method to recursively expand the coverage in property space while preserving the topology as well as axial symmetry. Reproduced from Ref. [100] with permission from ASME.
An example of this approach is the iterative database expansion proposed by Wang et al. [49, 53], where the radial distortion was recursively applied to sampled freeform building blocks for progressive growth of the property coverage. This reproduction strategy enables extensible data acquisition, which can possibly go beyond user-defined seed instances and representation, contrary to the two aforementioned strategies. At the heart of its implementation are sampling strategies that enable efficient, property-aware exploration of the shape space [75, 49].
Perturbation has been mostly applied to the Pixel/Voxel representations; yet it could be even more effective for other lower-dimensional representations, e.g., a lattice representation using the Parametric Multiclass method (Section 3.2.1), to explore new instances beyond them, as depicted in Figure 3(c).
#### 3.2.3 Perspectives on Shape Generation
Hinged on the taxonomy presented, we relate individual representations and reproduction strategies, and share our perspectives in multiple aspects.
Shape Dataset as a Design ElementIt is generally affordable to produce a bounty of unit cell shapes for DMD without obtaining their properties. Nevertheless, the shape collection needs to be judiciously prepared since (i) it primarily determines the landscape to be explored by the downstream tasks; and (ii) its utility (which we assess later in Section 3.4) is related to the resulting property distribution, which is, in general, initially unknown and resource intensive to obtain. Thus, we argue shape data for DMD is a critical design element.
A Trade-Off between Dimensional Compactness and ExpressivityAny representation is subject to the trade-off between dimensional compactness and expressivity. For example, both the Parametric Multiclass and Implicit Function methods enjoy dimensional compactness. Combined with Parametric Sweep, performing data generation with these is relatively straightforward. However, design exploration could be restricted unless supported by effective reproduction strategies. In contrast, the Pixel/Voxel representation supports freeform topologies without restrictions; yet the advantage comes with the challenges of efficient exploration in the huge design space and enforcing desirable design attributes, such as manufacturability [90, 63]. Parametric Curve/Surface allows for free boundary variations but suffers from topological restriction. As another moderate-dimensional representation, Constructive Solid Geometry offers topologically quasi-free instances, yet its coverage of possible shapes highly depends on the shape generation heuristics that are agnostic to property; hence it is prone to distributional bias in property space. This property bias could be a hurdle for a data-driven model to accurately learn and perform inference, and trigger the compounding effects of data quality issues in downstream tasks [86, 100], or, as it is known in the machine learning community, _Data Cascade_[105]. Relevant works that attempted to tackle the property bias are reviewed in Section 3.3.1.
Class-Centric vs. Class-FreeDepending on the presence of user-defined classes, the representations introduced above can be divided into two groups: class-centric approaches that typically include predefined classes (such as Parametric Multiclass and Implicit Function), and class-free (such as Pixel/Voxel, which is generally the case). By pre-specifying seed classes, class-centric approaches enjoy a database that can take advantage of desirable features inherited from the user-defined unit cell templates, i.e., it can include domain expertise. However, resorting to a particular set of user-defined classes tends to restrict the design freedom early in the procedure of DMD and bias the resulting data distribution in undesirable ways. A workaround applicable during reproduction is to consider Multiclass Blending (Section 3.2.2) introduced earlier, which offers seamless connection across seed classes [52].
Choosing Seed ClassesA crucial step for approaches under the class-centric umbrella is to choose, and justify the choice of, the classes with which to start. They can be chosen based on attributes related to shapes (e.g., topological features, mass/volume, smoothness, manufacturability) or their properties (e.g., elastic anisotropy, performance-to-mass ratio, broadband response). It is also important to ensure shape diversity among the classes, since it secures broad coverage of shape space. Last not but least, it has been pointed out that diversity in the shapes of unit cell data barely contributes to property diversity or task-awareness [86, 100]. When wider coverage and better uniformity are sought, property diversity, in addition to shape diversity, could serve as a selection criterion of seed classes [71].
Inspiration-Based Data AcquisitionNatural materials that exhibit outstanding properties supported by complicated structures evolved over a long time have been a great inspiration for innovation in design-by-analogy [106, 107]. Some works dedicated to engineering metamaterials have espoused motifs from biosystems [108, 109, 110, 111]. Although biologically inspired design provides a compelling avenue to concept generation, relevant works seem to mainly focus on proof-of-concept demonstrations with little design exploration due to grand design challenges such as scalability and repeatability [112, 113]. We believe that DMD can tackle the challenges by marrying the inspiration from bio-systems and data-driven exploration, especially in combination with effective reproduction strategies (Section 3.2.2).
Deterministic vs. StochasticTo date, most datasets prepared for multiscale architectural systems were created based on deterministic representations. On the other hand, a huge line of work addresses multiscale systems whose microstructures are either intrinsically random or subject to irreducible uncertainties associated with system deployment, e.g., material, operating conditions, and fabrications. Such scenarios can be better addressed via stochastic representations. Examples in the literature include spectral density function [114, 115] proposed for quasi-random nanophotonic structures and photovoltaic cells, and the spinododol representation [74], which was claimed to be more robust to fabrication imperfection compared to deterministic counterparts. We believe that a comparative study between deterministic and stochastic representations is yet to be explored. Relevant discussion with more focus on the trustworthiness of DMD can be found in Section 4.4.2.
Handling of On-Demand AttributesIn choosing a representation, the capability of handling desirable attributes could be decisive. Attributes of potential concern in DMD include symmetry, periodicity, invariance, volume constraint, manufacturability, connectivity among unit cells, and others. In general, the handling of those attributes is easier for explicit, low-dimensional representations, namely the Parametric Multiclass, Implicit Function, and Constructive Solid Geometry types under the proposed taxonomy. On the other hand, the Pixel/Voxel representations tend to need special techniques to enforce those attributes, typically with the aid of constraints [75, 49, 95] or data augmentation [90, 95].
### Property-Aware Data Acquisition Strategy
Once a shape collection is prepared, design evaluation of all or some of the individual shapes usually follows in order to build training data for semi-/supervised learning (Section 2.2). In literature, the exhaustive evaluation of a large amount of data has been widely employed by means of space-filling sampling, such as Latin hypercube sampling [116]. Examples in DMD include the 6-D lattice representation [82], the mixed-variable multiclass representation [71], and the Fourier transform based representation [88]. It has been widely used in a parametric space associated with reproduction (Section 3.2.2) as well, such as for exploration of the weight space of some isosurface representations [86, 50]. The popular use of space-filing design is perhaps attributable to its simplicity and generality of implementation.
However, exhaustive sampling could become intractable when (i) the relevant simulation is time-consuming (e.g., high resolution; 3D simulation), (ii) the on-demand data size is too large (e.g., more than 100k [53]), or (iii) the sampling space is too high dimensional (e.g., more than 50-D). It could also be the case that one wishes to acquire a data distribution with particular characteristics related to downstream tasks (e.g., negative Poisson's ratio or strong elastic anisotropy). Under such scenarios, it is warranted to exploit the acquisition strategies that take (estimated) property into account as a complement to the aforementioned shape generation heuristics (Section 3.2). Compared to a plethora of works on sampling in the small data regime, not many methods dedicated to DMD with large data have been reported. We introduce some within the context of data acquisition for DMD.
#### 3.3.1 Sequential Acquisition with Active Learning
Active learning [119, 120, 121, 122] refers to machine learning approaches that iteratively guide the locations of the next samples. In DMD, it is common to obtain a large pool of shape instances, but have labels for none or only a small portion of it. Under such a case, active learning offers a systematic, efficient, and general route to acquire evaluated samples, and thus could help prepare on-demand data.
In DMD literature, a few works specifically harnessed sequential, heuristic sampling as part of their data acquisition. In an early demonstration, Zhu et al. used a sequential sampling score that aims at property boundary expansion through randomly flipping voxels of near-boundary instances [75]. The estimated data density and distance to the boundary were two key criteria that constitute the sampling score. Inspired by that work, Wang et al. developed an iterative stochastic data expansion scheme that builds on a sampling rule accommodating both infilling and gamut growth in the property space [49].
With the aim to develop a method that is widely applicable to data acquisition in DMD, Lee et al. proposed a diversity-based active learning framework specialized in customizing metamaterial datasets with respect to design tasks (Figure 4(d) and (e)) [100]. As opposed to one-shot sampling where the whole samples are collected in a single iteration, sequential data acquisition powered by active learning uses metrics to monitor the growth of a dataset, thus offering a potential answer to a pressing research question in DMD: _"How much data?"_[100]. In addition to progressive dataset generation, active learning can serve as a key component for other tasks of data management, such as domain adaptation [123] and bias mitigation. An example was shown by Zhang et al. [117], where the proposed entropy-based active learning was demonstrated to substantially reduce the structure-stability bias of two public crystal datasets (Figure 4(a)-(c)).
#### 3.3.2 Downsampling Representative Subsets
All our discussion above can be summarized as how to grow a sparse, existing dataset into a large one. Some downstream tasks of data acquisition, however, do not always benefit from using an entire, massive dataset. A prevalent issue of large datasets in DMD is distributional biases. They typically present as containing more of certain shapes or properties, typically in undesirable ways, hosting the issue known as _learning under data imbalance_ during the downstream tasks [124]. To this end, Chan et al. proposed a diversity-based subset selection framework built on Determinantal Point Processes [125], a probabilistic way of modeling diversity in relation to the determinant of a similarity matrix. The key idea is to find small yet representative subsets, whose diversity in terms of shape, property, or a joint of them is tunable [86].
Such downsampling could also be useful for training the models that do not scale gracefully to large datasets, such as vanilla Gaussian processes [126] that scale cubically with respect to data size for training and inference. When large databases (e.g., more than 20k) are to be used as a ground set of downsampling based on pairwise metrics, e.g., diversity based on Euclidean distance, scalability of the downsampling algorithms becomes critical. It is often resolved through special schemes related to large-scale kernel learning [127, 128, 129]. Lastly, downsampling can be harnessed
Figure 4: Data acquisition for DMD through active learning. a)-c) Entropy-based active learning (ET-AL) [117] demonstrated by the J-CFID dataset [118]. The dataset includes 10,898 instances with seven types of crystal symmetries. a) Kernel density estimation plot of t-distributed stochastic embeddings (t-SNE), where regions with light colors are covered by sparse data. b) and c) t-SNE plots of graph embeddings of the materials selected by ET-AL and random sampling, respectively. The proposed ET-AL better covers the sparse regions, hence mitigating bias in the multiclass crystal dataset. Reproduced from Ref. [117] with permission from AIP Publishing. d)-e) Task-aware diversity-based active learning demonstrated by purposefully preferring data with strong anisotropic elasticity [100]. The test dataset includes 88,180 instances of freeform pixelated unit cells of orthotropic mechanical metamaterials [49]. The resulting property distributions of 3k datapoints in the \(C_{11}\)-\(C_{22}\) space obtained by random sampling and the proposed task-aware active learning, respectively. Reproduced from Ref. [100] with permission from ASME.
for determining a set of initial shapes to serve as seeds during shape generation, as shown by Chan et al., who leveraged Multiclass Blending (Section 3.2.2) as their data reproduction strategy [52].
#### 3.3.3 Perspectives on Acquisition Strategy
Sequential Acquisition for Generic Use: Uncertainty vs. DiversitySequential acquisition can be thought of as designing the rules with which to query an existing pool of unlabeled data, which, in our review, is typically a large number of unit cells with unknown properties. Here, we discuss and compare two key approaches to acquisition for DMD: uncertainty-based sampling [130, 131], and diversity-based sampling [86, 52, 100, 117]. Uncertainty-based sampling is centered on improving the prediction confidence of a model, typically resulting in a distributional imbalance that poorly represents the distribution of unlabeled data. Meanwhile, diversity-based sampling focuses on identifying a finite number of landmark data points to combat the distributional bias, and could include samples with little information for query. Practical implementation of either approach entails deciding the input dimensionality of data and the computational cost of the sampling algorithms.
Uncertainty is typically formulated as a model-specific, point-wise function that takes a query point as the input. Including uncertainty within sampling is effective for directly improving the predictive performance of models, whereas the computational complexity escalates as the input dimensionality increases. Within DMD, acquisition methods utilizing uncertainty could be useful when fitting machine learning models that offer uncertainty quantification, e.g., Gaussian processes or Bayesian linear regression, with a low-dimensional representation relative to a large number of data (say \(>\mathcal{O}(10^{4})\)). In general, uncertainty-based acquisition can be conducted based on either frequentist approaches, e.g., random forests and deep neural networks, or Bayesian approaches, e.g., Gaussian processes and generalized linear models. For practical guidance on which to employ, readers are referred to Zhang et al. [131].
Meanwhile, diversity is frequently modeled as a pair-wise, model-agnostic metric that involves a mapping from a pair of instances to a scalar similarity [125]. By harnessing the pair-wise kernel trick, the diversity-based acquisition is capable of handling high-dimensional input instances. However, the acquisition does not gracefully scale with respect to data size due to large storage requirement (\(\mathcal{O}(N^{2})\) where \(N\) is the data size) and matrix inversions that involve time complexity of \(\mathcal{O}(N^{3})\) unless any large-scale kernel approximations [127, 129] are employed.
Tailoring Property DistributionsData acquisition methods that are agnostic to property can lead to datasets that are prone to data bias. For example, Figure 5(a) and (b) show the highly biased distributions of formation energy in public crystal datasets. A plethora of sampling methods in low-dimensional input domains have secured datasets with decent coverage as well as uniformity [101, 133, 134, 135, 136]. The quality in the input domain, however, has been found to barely transfer to that in the output domain. Figure 5(c)-(e) shows such an example in a mechanical metamaterial dataset. Within DMD literature, this point was observed through the near-zero correlation between shape diversity and property diversity depicted in Figure 5(f) during downsampling and in Figure 5(g) under active learning [86, 100]. The implication is that property distributions are likely to be highly imbalanced even though the design in shape space is space-filling, a problem that is currently overlooked in DMD.
Furthermore, for design purposes, addressing property imbalance is only part of data quality assurance. Not all data are equally useful because users frequently have certain preferences in terms of shape, property, or both. For example, a user might wish to collect unit cells of mechanical metamaterials with negative Poisson's ratios, with packaging or shock absorption applications in mind. During data acquisition of photonic metasurfaces, broadband reflectivity might be preferred over narrowband for some defense applications. Therefore, data acquisition for DMD should, ideally, be task-aware so that more resources can be invested in the region of central interest. Figure 4(d) and (e) illustrates a methodology where batch sequential data acquisition is encouraged to favor samples with high elastic anisotropy.
It is difficult to tailor the property distribution during data acquisition without supervising the properties of interest. This supervision is non-trivial in that (1) properties of unseen unit cells are unknown before evaluation; (2) the evaluation is typically resource-intensive, particularly for large datasets; and (3) distributional control in regression tasks is more challenging and has been under-explored compared to that in classification [124]. We believe this topic calls for more research attention.
### Data Assessment
Data assessment in DMD entails quality quantification across candidate sets with respect to either general use or specific design tasks. Large sizes of data render the close inspection of individual samples intractable. Thus, their assessment is often conducted through proxy measures such as a quantitative summary of distributional characteristics, or through data visualization for qualitative interpretation. Subjectivity perhaps cannot be totally excluded in data assessment, particularly for DMD applications that involve multiple assessment criteria, e.g., data size, distributional uniformity of
data, property coverage, target design tasks, and manufacturability of unit cells. Nevertheless, protocols could help to compare datasets and make decisions on which set is best suited for the intended goal.
Below, we share feasible ideas and scenarios for data assessment that have been drawn from our survey of a large volume of existing metamaterial datasets, and a sparse number of exemplar quantitative/qualitative assessment methods. As reviewers, we hope our discussion will contribute to the establishment of assessment protocols on metamaterial datasets.
Figure 5: Illustration on distributional bias in the property space of existing DMD datasets. a) and b) Stability distributions of two different crystal datasets, 2,953-size OQMD-8 [132] and 10,898-size CFID [118], whose instances are categorized based on symmetry. Reproduced from Ref. [117] with permission from AIP Publishing. c)-e) Visualization of the six-bar parametric lattice dataset [82, 100]. c) Conceptual illustration of the unit cell shape generation based on Parametric Multiclass. d) The near-uniform space-filling design in the projected \(w_{1}\)-\(w_{2}\) shape descriptor space. e) The resulting property distribution in the \(C_{11}\)-\(C_{13}\) space. The near uniformity in the weight space leads to a strong bias in the \(C_{11}\)-\(C13\) space. f) and g) The near-zero correlation between shape diversity and property diversity observed during downsampling of the isosurface dataset and active learning applied to the orthotropic freeform dataset [49, 100], respectively. f) was reproduced from Ref. [86] with permission from ASME. g) was reproduced from [100] with permission from ASME.
#### 3.4.1 Quantitative Assessment with Metrics
Space-Filling MetricsIn general, data acquisition aims to make the data distribution as uniform as possible to ensure any local region of potential interest is equally covered by the dataset. In doing so, some works in DMD employed sequential sampling that included the density, or concentration, of data in certain regions of the design space as part of the sampling utility function [75, 49, 53]. The density in these works was not used to assess dataset quality, but can possibly be used to do so [100]. As an alternative to point-wise density, the alternative of set-wise diversity, or pairwise dissimilarity [125], can also serve as a sampling criterion to suppress distributional biases in both shape and property space, as shown in some DMD works [86, 100, 52]. Point-wise diversity can be measured through information entropy [137]. A recent demonstration in literature was performed by Zhang et al. [117], where the coverage imbalance of formation energy across seven crystal systems within the dataset was quantified through point-wise entropy and mitigated by the proposed entropy-based active learning (see Figure 5(a) and (b)).
Task-Related MetricsIn DMD, a dataset often ends up being used for a particular design scenario. In such cases, distributional metrics alone may not ensure the on-demand deployment of DMD. Even with a dataset that exhibits perfect uniformity, it could be the case that the region associated with a given design task (e.g., high performance-to-mass ratio, high stiffness anisotropy, broadband reflectivity) happens to be covered by only a few, or even none, of the datapoints. This implies, if design tasks of interest are given, the assessment on a given dataset must vary. The task-specificity of data assessment indicates that if a new task is given, data assessment must realign according to it. In a relevant work, Lee et al. specified a couple of design scenarios involving different design tasks given a shape-only dataset and showed that the resulting property distribution for each case can be tailored through diversity-driven active learning [100].
Within-Dataset AssessmentIn case multiple metamaterial datasets share the same input space, i.e., the representation space of building blocks, the quality of each can be comparatively measured based on metrics. Such comparisons could be useful for assessing across multiple data acquisition methods.
For example, Chan et al. [86] validated the proposed diversity-driven subset selection method for the isosurface representation by showing larger shape diversity of subsets against that of random, independent sampling. Assuming a similar setting but with more focus on sequential acquisition, Lee et al. [100] conceived a measure of diversity gain, which quantifies a relative ratio between the diversity of selected subsets and that of independent and identically distributed (_i.i.d._) samples with the same data size, to quantify the increase of shape diversity enabled by the proposed sequential sampling strategy. The authors also demonstrated better task-awareness in the two representation spaces, both of which were latent spaces distilled by training generative models.
#### 3.4.2 Qualitative Assessment Through Visualization
Property CoverageProperty coverage offers an intuitive, relative criterion to comparatively gauge utility across datasets, similar to how the Ashby chart visualizes a modulus-density space for disparate materials [138]. Upon valid normalization across datasets, data assessment in property space is usually less subjective compared to that in shape space due to lower dimensionality. Intuitive examples in literature include elasticity components [139, 82, 49], transmission-phase delay at a single frequency [78], and formation energy of crystal structures [117].
Figure 6 shows an example of a visual comparison between two datasets in a low-dimensional property space. Both carry 3D mechanical metamaterial instances under linear elasticity. Considering the geometrical symmetry, we only consider three components, Young's modulus (\(E\)), Poisson's ratio (\(\nu\)), and volume fraction (\(v_{f}\)). The red one denotes the 924-size TPMS dataset generated by Wang et al. [50] using Implicit Function and Parametric Sweep. The yellow one denotes the 21,684-size multiclass lattice dataset presented by Chan et al. [48], created with Parametric Multiclass and Multiclass Blending. The pairwise plots of projected properties in Figure 6(b) shed light on some insightful information in a comparative sense including: (i) Overall, the multiclass dataset has better data uniformity in terms of Young's modulus (\(E\)) and volume fraction (\(v_{f}\)); (ii) In the \(E-\nu\) space, the TPMS dataset has some regions that are covered only by sparse data, arguably attributed to the limitation of Parametric Sweep with a few classes; (iii) Although the data size of the multiclass dataset is more than 20 times larger, some property values in the \(E-v_{f}\) space are only available in the TPMS dataset.
In the case where the dimensionality of property is high-dimensional, e.g., optical spectra of transmission, dimensional reductions are necessary to visualize the data in two-dimensional space. Zandehshahvar et al. [141] shows such an example built on an autoencoder, where the latent space of optical spectra (1) visualizes the coverage dependent on design complexity of unit cells (Figure 7(d) and (e)) and (2) automatically encodes the shift of resonance frequency along the circumferential directions (Figure 7(f)).
Shape ManifoldsThe distribution of data in shape space could give another insight into data assessment in DMD. Despite high dimensionality, ranging from several (e.g., Parametric Multiclass) to millions (e.g., Pixel/Voxel), there is an array of dimension reduction schemes developed for exploratory data analysis, such as principal component analysis (PCA) [142], t-distributed stochastic neighbor embedding (t-SNE) [143], and uniform manifold approximation and projection (UMAP) [145]. The projection is conducted preferably into 2D spaces for straightforward visualization. This can uncover the underlying characteristics of the data distribution, e.g., clusters formed by data acquisition, often dictated by reproduction strategies of unit cells (Section 3.2.2).
For example, in DMD, Ma et al. [64] employed a VAE [57] and then visualized the latent space using t-SNE. The visualization projected in a 2D space reveals the clustering across the seed eight classes learned in an unsupervised manner (Figure 7). Meanwhile, Wang et al. [53] employed PCA to claim the versatility of the proposed latent representation learned by a conditional VAE. The visualization was used to delineate that the continuous, interpretable latent space offers simple interpolation across building blocks, a shape similarity measure, and intrinsic clustering of associated properties.
Even more interesting use of such high-dimensional data visualization is for comparisons across datasets. Employing the one-class support vector machine [144], Zandehshahvar et al. visually demonstrated the impact of geometric freedom in building blocks by visualizing different coverage in both shape and property space (see Figure 7(d)-(f) for illustration).
#### 3.4.3 Perspectives on Data Assessment
Task Specificity of Data Acquisition and AssessmentIn Section 3.4.1 we covered the task-specificity of data assessment with a particular focus on metrics. We now summarize our general view on data acquisition and data assessment of DMD for commonplace scenarios as follows:
Figure 6: An example visual comparison between two 3D mechanical metamaterial datasets. a) Examples of unit cells in the 924-size TPMS dataset. [50] (red, top) and the 21,684-size multiclass lattice dataset. [48] (yellow, bottom). b) Pairwise plots of the properties of interest: Young’s modulus (\(E\)), Poisson’s ratio (\(\nu\)), and volume fraction (\(v_{f}\)). The effective properties are computed by energy-based homogenization [140]. The plots located at the diagonal depict a histogram of each component. All the off-diagonal components show scatter plots of two different properties. The upper triangle plots are shown considering the symmetry. The volume fraction ranges from 0.25 to 0.8.
* Provided that no target tasks have been specified, data acquisition can aim to create a dataset for generic use, i.e., focus on uniformity and wide coverage, in both the shape and property spaces of building blocks. The data assessment can also follow the same criteria without preferring a certain region in those spaces. Ref. [86] shows an example in literature.
* Even without being given any specific on-demand properties _a priori_, instance-wise preferences related to shape (e.g., fabrication feasibility), property (e.g., high physical anisotropy), or both (e.g., performance-to-mass ratio) can be enforced during data acquisition to tailor the data distribution as desired with minimal trial-and-error. The data assessment approach must address both distributional metrics and task-related metrics. Ref. [100] is an example.
* If a target task is required at downstream, or a set of target tasks is given, the data acquisition and assessment can be aligned to the specified task(s), in addition to data uniformity. In these cases, data uniformity is useful only within the domains associated with the tasks. Moreover, the assessment is subject to the definition of target tasks. A concrete example where the task of matching target displacements at the system level is addressed in Ref. [49] and discussed in Section 5.4.3.
Assessment ProtocolsData assessment is essential for either diagnosing a dataset or choosing the best among competing ones. How to fairly measure the quality of metamaterial datasets is key to decision-making over competing datasets, as well as to minimization of iterations among the modules of data-driven metamaterials design. A general
Figure 7: Visualization of high-dimensional data. a) Visualization of the latent space in the orthotropic mechanical metamaterials dataset with the property distributions included [49]. The 16-D latent representation distilled by variational autoencoder is visualized in 2D space using principal component analysis [142]. Reproduced from Ref. [49] with permission from Elsevier B.V. b)-c) Visualization of the latent space in plasmonic metasurfaces [64]. b) The representative images of the seed classes included in the dataset. c) The resulting data distributions in the 2D latent space using t-distributed stochastic neighbor embedding [143]. Each class forms separate clusters in the latent space, projected from 20D into 2D. Reproduced from Ref. [64] with permission from Science China Press and Springer-Verlag GmbH Germany. d)-f) The latent space representation of the resonant reflection spectra in dielectric metasurfaces [141]. d) Metasurface unit cells with five different levels of geometric complexities. e) The corresponding convex hulls of shape manifolds estimated through the one-class support vector machine [144]. f) The corresponding property distribution of high-dimensional optical spectra in the property manifold. It encodes the shift of resonance frequency with respect to the traversal along counter-/clockwise directions. Reproduced from Ref. [141] with permission from American Chemical Society.
guideline for the assessment of synthetic datasets for engineering purposes was recently proposed [146]. However, not many attempts that are dedicated to data assessment for DMD have been reported, compared to the rapidly growing volume of the corpus. Without agreement upon standard protocols of data assessment, it is difficult to judge the quality of individual works that include data acquisition and to draw meaningful conclusions among them. Thus, we assert the need for more research efforts centered on data assessment.
Benchmark DatasetsThe easy access to public datasets has been an enabler of the recent surge of machine learning. Ideally, newly proposed methods on any module of DMD should be validated through diverse benchmark datasets that are suggested by the communities. In the corpus of data-driven design, however, such solid validation seems difficult to find. A profound reason that applies to general data-driven design is, arguably, a dearth of public datasets and benchmarks [35, 146]. Echoing this, we argue that securing more public datasets and setting some of them as benchmarks will be the first step towards the research practice that prompts quantitative, rigorous comparisons of relevant works and reproducible research, hence helping readers better appreciate individual works with relation to the field. In doing so, it is highly encouraged to make new datasets publicly available, preferably in online repositories that support consolidation across datasets.
### Discussion
#### 3.5.1 Reusability of Datasets
We observe that the current practice in data-driven multiscale architectural design typically starts with creating one's own dataset, rather than with existing ones; arguably, the practice is what has led to "per-task" datasets involving a similar/equivalent end-use. To avoid the trial-and-error and computational resources that data creation from scratch demands, new works can be conducted by either (i) creating a versatile, high-quality dataset that can address multiple, disparate design tasks, or (ii) customizing a public dataset with respect to new design tasks. We believe that the reusability of datasets is by no means trivial to achieve, and thus deserves further research attention.
#### 3.5.2 Limitations of Unit Cell Datasets
To date, most demonstrations of data-driven multiscale design have been built on building block datasets. Each data point, i.e., a structure-property pair, is generated based on periodic boundary conditions (PBC), which assume a given building block is surrounded by infinitely many identical neighbors. Although the assumption is valid only for periodic multiscale architectures (e.g., crystal), some recent works involving fully aperiodic design still utilized the PBC-based effective properties. They justify this choice, which can accelerate the design process, through, e.g., geometric/mechanical compatibility under linear elasticity [53], functional grading [52, 48], and uncoupled operations in photonic metasurfaces [78, 95].
We point out that while the periodicity assumption has been a backbone of the recent progress in DMD, it is also a hurdle that impedes researchers from exploring other problems where the assumption does not hold true. There are a variety of cases where the deviation of effective properties under PBC at the system level is more than acceptable [38]. Examples include the systems under: (i) large deformation [147, 148], (ii) strong local coupling among neighbors [149, 150], (iii) long-range interactions [151, 152], and (iv) heterogeneous loading conditions [153, 154, 130]. Preparing the datasets that take a supercell (i.e., a collection of neighboring unit cells) instead of a single unit cell as a datapoint could offer a simple yet powerful extension to the unit-cell-based approaches, as shown by [150, 155, 130]. The extension would involve increased computational cost and validating the boundary conditions applied to the supercell simulations. A relevant discussion at the system level can be found in Section 5.4.2.
#### 3.5.3 Learning Global Responses vs Local Responses
Many works reported in DMD include a surrogate model that directly maps the parameterized unit cells to effective, homogenized properties, such as elasticity components [49, 50, 52, 48] and scattering parameters [156, 63, 78]. The associated mapping can be relatively easily learned provided that there is enough training data and the output dimensionality is low. However, this comes at a significant loss of full field information.
A workaround that can boost the generality of the structure-property mapping is to directly learn output physical fields, e.g., displacement, electric fields, and temperature, as a function of parameterized unit cells. The goal can be achieved through either physics-informed machine learning or operator learning [157, 158, 159], where an underlying physics is either imposed as a constraint or discovered by field data. Such a mapping can capture underlying spatial correlations of fields, which could be subject to strong long-range interaction across unit cells in DMD. This approach features decent transparency, generality, and sample efficiency. These approaches need to address challenges that include (1) how to effectively regularize the learning with regard to high-dimensional output fields (e.g., adding a sparsity penalty to avoid
overfitting [157]) and (2) how to impose priors associated with physics (e.g., smoothness of fields [160]). Some works in DMD proposed to learn global behaviors [149]; however, the literature to date is sparse. There is room to place more attention on this branch of approaches. Details of this topic are reviewed in Section. 4.2.4.
#### 3.5.4 Determining Data Size
In one form or another, "_How much data?_" has been a key research question in data-driven approaches. Within the scope of DMD, this question affects the following aspects: (1) model complexity (e.g., neural networks vs Gaussian processes [126]), (2) unit cell representation (e.g., high-dimensional vs low-dimensional), (3) simulation cost/fidelity, among others. Due to the multifaceted nature of DMD problems, it is difficult to predict an "optimal" data size _a priori_, especially via one-shot sampling. The data sizes reported in the literature could be a good starting point, provided that a new design task of interest shares some attributes, e.g., model complexity and unit cell representation, with those of the reported works. Integrating active learning with data acquisition (Section 3.3.1) can be a more rigorous, general approach to determining data size since it offers metrics related to either the data themselves or model performance. Lee et al. [100] claimed that diversity metrics can be monitored to gauge the relative utility of incoming data, hence serving as a proxy to determine data size based on the gamut growth in property space. For generic scenarios, a guideline on data size of engineering datasets was proposed by Picard et al. [146].
In the corpus, we observe most prior efforts were hinged upon large data. It is equally worth investigating in the small data regime [161], as doing so will help tackle design problems that involve expensive simulations and limited computational resources. This call resonates with data-centric AI [162, 163, 164], an initiative that propels a paradigm shift of data acquisition from _more data_ to _better data_[165].
#### 3.5.5 Data Sharing Practice
The surge of DMD has inspired the emergence of open-source data sharing platforms, such as _NanoMine_[166, 167, 168], which pays special attention to polymer nanocomposites. _MetaMine_, its sister platform, currently stores 300k structure-property data of metamaterials with a diverse array of unit cell representations. Building a common platform and knowledge representation presents immense challenges. The endeavor will facilitate consolidating datasets that were acquired independently, and therefore enhance the potential of DMD beyond that what is achievable by an individual dataset.
A prerequisite to building a user-interactive data platform that supports reusability and reproducibility is sharing protocols. For example, the FAIR (Findability, Accessibility, Interoperability, and Reusability) principles [169] is a concise, domain-independent, and thus generic, guide for data sharing. Exercising the FAIR principles is built upon core elements such as standardized vocabularies, ontologies, and data formats. Despite the foundational role that such general guidelines have played in existing data-sharing platforms, they tend to only specify broad guidelines of data quality assessment at a high level, leaving the needs of discipline-specific standards unaddressed [170]. For DMD, the proper format of (meta)data could differ wildly across different domains (e.g, nanocomposites vs. mechanical/photonic metamaterials). Even within a given domain, it could be difficult to define a set of commonly structured vocabularies or knowledge representations that accommodates all datasets submitted by users. In this regard, we advocate for an extensible, dynamic platform, which starts with initial vocabularies and schema defined by humans, as showcased by _Nanomine_ for polymer nanocomposites [168], and which is then allowed to evolve without supervision as more data is ingested.
#### 3.5.6 Other Tasks Involving Data
So far we have primarily covered data acquisition. Other possible tasks include data augmentation, data consolidation, bias reduction, problem/domain adaptation, and exploratory data analysis. Each task plays a unique role that cannot be fully addressed by data acquisition itself. For example, it is highly recommended to perform data augmentation because (1) it increases the amount of training data without further evaluations; (2) it helps a machine learning model to be encoded with operational invariances, such as rotation, scaling, translation; and (3) it tends to mitigate overfitting by serving as a regularizer of model training. Within DMD the efficacy of augmentation has been demonstrated in some works [90, 95]. We believe that further research on data-related tasks other than data acquisition will underpin the future success of DMD by enhancing the generality, customizability, and reusability of data.
#### 3.5.7 Public Resources
We share a link to an online webpage of public resources that are associated with data-driven design as follows: [https://github.com/ideal-nu/Data-Driven-Design-for-Metamaterials-and-Multiscale-Systems-Status-and-Opportunities/](https://github.com/ideal-nu/Data-Driven-Design-for-Metamaterials-and-Multiscale-Systems-Status-and-Opportunities/).
Data-Driven Unit Cell Design of Metamaterials
In Section 3, we reviewed past works related to metamaterials data acquisition and discussed some challenges at the data acquisition stage. In this section, we introduce how past works used data-driven methods to solve unit-cell-level metamaterials design problems.
### Overview
The advance of machine learning has motivated researchers to seek data-driven solutions to many real-world design challenges. These challenges mainly originate from the following factors:
1. Analysis using physical experiments or high-fidelity simulations usually requires high cost. For example, numerical nanophotonic simulations can take hours or even days for complex systems [171].
2. Advanced fabrication technology (e.g., additive manufacturing, micro/nanofabrication) enables high degrees of design freedom, but exploring a high-dimensional design space to find optimal solutions is challenging.
3. While methods, like adjoint-based shape optimization and TO, can address high-dimensional design problems by using sensitivities to guide optimization, they are usually not applicable when the physics governing the problem is non-differentiable with respect to design variables.
In this section, we will introduce how past work used data-driven methods to solve these challenges, particularly in the domain of unit-cell-level metamaterials design. Note that data-driven methods have been gaining increasing attention in engineering design, mainly for shape and topological designs, to address the challenge brought by their high degrees of design freedom. For shape optimization, a large body of work looked at data-driven aerodynamic shape optimization, which mainly focused on dimensionality reduction or representation learning (e.g., [172, 173, 174, 175, 176]) and inverse design (e.g., [177, 178, 179]). Compared to only considering shape variation in shape design, metamaterials design can usually accommodate topological changes depending on their functional requirements. These extra degrees of freedom make it more difficult for traditional design methods to solve metamaterials design problems. Many works on machine learning-assisted TO serve various purposes such as reparameterization, objective function prediction, sensitivity prediction, direct prediction of TO solutions, and enhancing diversity for generative design. We refer interested readers to Ref. [36] for a summary of past contributions of neural network-based TO methods. These TO methods were usually applied to structural design problems, where well-established physics and sensitivity analysis are available. In this section, we will cover metamaterials design under different physics (e.g., mechanical, optical, acoustic/elastic, thermal) scenarios, for some of which sensitivity analysis is either unavailable or too difficult. Due to that reason, traditional gradient-based TO methods might not be applicable to certain types of metamaterials design.
The benefits of data-driven methods highly depend on (1) the cost of data collection and learning and (2) the acceleration contributed by the data-driven model. To make such methods cost-effective, the trained model needs to be reusable for different tasks. Problems like structural optimization are usually subject to task-dependent objectives and constraints. In this case, it is difficult to train a single ML model that can be reused in different tasks. In contrast, unit-cell-level metamaterial design problems under the same physics usually share common properties of interest (e.g., elasticity properties in mechanical metamaterials and transmission and phase delay in optical metamaterials), regardless of different functional goals (e.g., design for compliant mechanism, energy absorption, or noise reduction). Thus, a data-driven model trained on the same properties of interest can be reused in different metamaterials design tasks.
This section covers five types of metamaterials -- optical, acoustic/elastic, mechanical, thermal, and magneto-mechanical. Despite differences in physics, they share similar design scenarios: (1) design allows high-degree of geometric freedom; (2) physical properties are usually the target quantities of interest; (3) physical properties depend on design geometry; (4) generating random design geometries is cheap whereas computing their physics properties is expensive. For this reason, data-driven methods developed for metamaterials are usually applicable under different types of physics. Thus, we focus on the goals of data-driven methods instead of metamaterial types when characterizing past work. Specifically, we categorize the goals into two major types -- iterative design optimization (Section 4.2) and iteration-free inverse design (Section 4.3). Figure 8 illustrates the relationship between the goals of proposed data-driven methods, the physics being considered, and the machine learning models, extracted from 56 representative prior studies. These prior studies were selected from publications from 2018 to 2023, with the aim to use machine learning to assist metamaterials design. We focus on only single-scale metamaterials design in this section, where unit cell designs are arranged periodically in space. In Section 5, we will introduce multi-scale design, where aperiodic unit cell designs are considered to achieve spatially-varying material properties.
### Iterative Design Optimization
In this section, we review prior works that employed ML methods for metamaterials design optimization. In particular, ML commonly plays roles in accelerating property evaluation (Section 4.2.1), learning more efficient design representation (Section 4.2.2), sequential decision making (Section 4.2.3), and physics-informed solution generation (Section 4.2.4).
#### 4.2.1 Accelerated Optimization via Data-Driven Property Prediction
In unit-cell-level metamaterials design, material properties are usually the design target. For example, absorption spectra and dispersion relations can be properties of interest for optical and acoustic metamaterials design, respectively [180, 181]; for mechanical metamaterials design, properties of interest can be Young's modulus, Poisson's ratio, and volume fraction [50]. We can perform numerical simulations or experiments to evaluate material properties. However, depending on different physics properties, the computational cost can be prohibitive, especially when iterative design optimization is required. Data-driven models can learn complex structure-property relations and hence surrogate time-consuming simulations or experiments, allowing high-throughput property evaluation. This can accelerate the design process when combined with downstream design space exploration methods (e.g., sampling, screening, and optimization).
Based on the reviewed literature (Figure 8), most ML-accelerated design optimization works were based on this approach, among which the most commonly used ML models were convolutional neural networks (CNN) [182, 183, 184, 185, 186, 187, 188] and multilayer perceptron (MLP) [77, 188, 149, 190, 191]. Typically, CNNs were used for pixelated design representations with high geometric freedom or design dimensionality (allowing more complex designs), while MLPs were used for parametric or shape designs with lower design dimensionality. These two models were mostly used when the training data size is larger than 700. One special case is Ref. [149], where the local design variables of only 10 metasurfaces along with the local patches of their electromagnetic (EM) fields response were used as training data of a property-prediction MLP, resulting in 123,210 actual training samples.
Figure 8: Categorization of physics, goals, machine learning models, and their relations based on 56 representative prior works. (LR: logistic regression; RL: reinforcement learning; PCR: principal component regression; DT: decision tree; DGM: deep generative model; GP: Gaussian process; MLP: multi-layer perception; CNN: convolutional neural network; RNN: recurrent neural network; GNN: graph neural network; PINN: physics-informed neural network.)
While the training of most neural network-based models normally requires large datasets, Gaussian process (GP) was employed when there was a relatively small amount of training data [71, 192]. GP's ability to estimate uncertainty makes it well-suited for adaptive sampling and Bayesian optimization. These are useful design techniques especially when the computation of responses is time-consuming. However, one limitation of standard GP models is the difficulty in handling large datasets due to its \(\mathcal{O}(N^{3})\) time complexity and \(\mathcal{O}(N^{2})\) memory complexity, where \(N\) is the training data size. This also limits the dimensionality of design problems to be considered, because the required amount of data scales exponentially with the dimension due to the curse of dimensionality. Past work proposed ways to make GP more scalable with larger datasets [193, 194, 195, 196, 197, 84], which can potentially expand the use cases of GP to larger datasets and higher problem dimensions in metamaterials design.
Dimensionality reduction (DR) was also applied to reduce the dimensionality of the original design or property space before applying regression models for property prediction. Wang et al. [192] used a Gaussian mixture beta variational autoencoder (GM-\(\beta\)VAE) to reduce the dimension of the pixelated metamaterials design representation. Chen et al. [198] employed principal component regression (PCR) to reduce the 3D metamaterial design parameters, where the principal component analysis (PCA) was followed by linear regression to predict the elastic material properties of mechanical metamaterials. Zhelyeznyakov et al. [149] reduced the dimension of the near-field response of the metasurface using singular value decomposition (SVD) before applying property prediction models.
In some works, decision trees (DTs) were used as the property prediction model due to their interpretability and flexibility in learning non-linear structure-property relations. Elzouka et al. [199] predicted the spectral emissivity of dielectric and metallic particles. Chen et al. [91] used Generalized and Scalable Optimal Sparse Decision Trees (GOSDT) [200] to predict band gaps in different frequency ranges based on shape-frequency features extracted from acoustic metamaterial geometries. Particularly, the interpretability of DTs allows us to extract explicit design rules that can guide inverse design, which we will elaborate on in Section 4.3.4.
In most prior work, metamaterials design variables were either represented as a vector of design parameters or a tensor representing pixelated designs, while the properties of interest normally consisted of single or multiple scalars. Beyond these common representations, Yang et al. [201] studied 3D graphene metamaterials with a graph representation and used a graph neural network (GNN) to predict the local atomic stress distributions. Sajedian et al. [183] aimed at predicting the absorption curve of plasmonic structures and solved this problem with a recurrent neural network (RNN).
To obtain the metamaterial design solution, a common approach is to use the property prediction model as a surrogate design evaluation model and incorporate it into any iterative optimization loop [77, 187, 149, 188, 148, 192, 191]. Besides iterative optimization, past works also employed virtual screening [201] and sampling [182, 91, 189] to select design solutions, which also took advantage of the fast design evaluation capability of property prediction models.
#### 4.2.2 Accelerated Optimization via More Efficient Design Representation
Another way to accelerate design optimization is through learning an efficient design representation (i.e., latent representation) that is more compact than the original representation but still as expressive, thereby covering the same design space with fewer design variables. This benefits in three ways: (1) it mitigates the issue caused by the curse of dimensionality when training property prediction models, thus lowering the requirement for training data size and model complexity; (2) it enables more efficient optimization, since searching for global optimal solutions in a lower-dimensional latent space is easier and faster; (3) it allows easier downstream analysis such as data visualization, clustering, and arithmetic operations in the latent space. Since we introduced past work on using dimensionality reduction for property prediction in Section 4.2.1, this section will focus on the other two benefits.
As shown in Figure 8, most prior works under the category of "representation learning" used deep generative models (DGMs). Liu et al. [59] employed a variational autoencoder (VAE) to learn a lower-dimensional latent space of pixelated optical metasurface designs and leveraged an evolutionary algorithm to optimize designs over the latent space. Wang et al. [58] constructed an end-to-end model combining a VAE with a mechanical property regressor, and learned a lower-dimensional, structured latent space organized by mechanical properties of pixelated metamaterials designs. Semantic operations (e.g., moving from "low stiffness" to "high stiffness") can be achieved by simply moving in certain directions of the resulting latent space. Chen et al. [202] proposed a generative adversarial network (GAN) with hierarchical latent spaces to simultaneously represent the "as-designed" and "as-fabricated" optical metasurfaces. The model not only learned a compact latent representation for "as-designed" metasurfaces, but also captured the geometric uncertainty of "as-fabricated" designs, which enables efficient and robust design optimization under fabrication uncertainty. Shen and Buehler [203] used StyleGAN [204] to learn disentangled latent spaces that capture attributes and variations at different levels. Design optimization and geometric manipulations (i.e., projection, encoding, and mixing) can be achieved by optimizing the latent vectors.
Autoencoders (AEs) were also used for compressing the dimension of the metamaterials design space [205, 70]. Compared to deep generative models like VAEs and GANs, AEs only minimize the reconstruction errors of training samples without considering the continuity of the latent space, which may reduce the efficiency of design optimization and latent space analysis.
#### 4.2.3 Design as Sequential Decision Making via Reinforcement Learning
The design problem can also be modeled as a sequential decision-making process, which can be solved by reinforcement learning (RL). As introduced in Section 2.2.4, the main components to be defined in any RL tasks are state, action, and reward. Defining these three concepts -- especially the action -- in a design problem needs extra consideration, unlike canonical reinforcement learning problems such as gameplay and adaptive control, since it is more intuitive to treat metamaterials design as a standard optimization problem rather than a dynamic problem that requires sequential decision-making. Nonetheless, Sajedian [68] employed the Double Deep Q-Network (DDQN) [206] to design both the material type and the geometry of the optical metasurface that maximizes hologram efficiency. The actions were defined as increasing or decreasing each design parameter by a fixed amount; the state is design parameters; the reward considers both the phase-generating capability and the efficiency. Similarly, Liu et al. [69] used the DDQN to design a periodic lattice system that achieves thermal transparency. The action was defined similarly to the one in Ref. [68]; the state is the combination of design parameters and the response of heat fluxes; the reward is represented by how far the simulated system is away from achieving perfect thermal transparency. Sui et al. [70] proposed a collaborative deep Q network (DQN) that designs mechanical metamaterials (voxelated designs representing the arrangement of soft and stiff materials). The action is the "flipping process" by two agents: one agent selects a soft voxel and turns it into a stiff voxel, while the other agent does the opposite (Figure 9); the state is represented by the voxelated design; the reward is the change of the averaged equivalent modulus.
Sui et al. [70] compared deep RL with the genetic algorithm (GA) under different design dimensions. The results show that RL does not have an advantage over GA under small action spaces, but will outperform GA when the action space becomes larger, due to the generalization ability of deep neural networks on large action spaces. On the other hand, with a larger action space, more design evaluations will be required to sufficiently explore the action space, which can quickly lead to a prohibitive computational burden for RL. This was reflected by the design problem dimensions addressed by the reviewed past works, none of which exceeded 50 dimensions.
Based on these observations, we conclude that it is important to find the "sweet spot" of action space dimensions where RL can outperform classic optimization methods while not requiring prohibitive computational costs. When proposing RL methods for metamaterials design, one needs to compare it to optimization (either classic or machine learning-accelerated optimization as introduced in Secs. 4.2.1 and 4.2.2) and justify the necessity of using RL instead of optimization methods which are usually more intuitive and simpler to formulate. Despite the caveats, RL can still be a promising technique in metamaterials design because (1) compared to data-driven design optimization, RL does not require prior data and hence is not limited by the boundary of existing designs, and (2) it is easier to formulate the design problem with RL when the design has an unstructured representation (e.g., irregular truss structure represented as graphs [207]).
Figure 9: “Flipping process” of the collaborative DQN method in Ref. [70]: one agent selects a soft voxel and turns it into a stiff voxel, then the other agent does the opposite. Reproduced from Ref. [70] with permission from American Chemical Society.
#### 4.2.4 Design via Physics-Based Learning
While physics-informed machine learning has drawn huge attention in recent years, its application in metamaterials design is relatively limited. Physics-based learning in design tasks usually uses governing equations to guide the training of machine learning models which produce optimized design solutions. Jiang and Fan [208, 209] proposed a generative neural network to generate high-performance dielectric metasurfaces, where the generator training was guided by the gradients from the adjoint electromagnetic simulations of generated designs. Lu [210] proposed physics-informed neural networks with hard constraints (hPINNs) to solve the topology optimization problem in metamaterials design. The method builds on physics-informed neural networks (PINNs) [158] but further allows optimizing a design objective function as well as the governing partial differential equations (PDEs) being modeled as hard constraints. The hPINNs method was demonstrated on design problems in optics and fluids. Compared to PDE-constrained adjoint-based optimization methods, the hPINNs method achieved the same objective value, but obtained a simpler and smoother solution with faster convergence (Figure 10).
Unlike classic machine learning or data-driven methods, these physics-based learning methods require no training data and are less susceptible to the curse of dimensionality. The optimization can be guided by gradients and hence there is no need to explore the entire solution space. Therefore, design via physics-based learning can be a promising future research direction.
### Iteration-Free Inverse Design
While the reviewed works in Section 4.2 used ML methods to accelerate design optimization, they still need an iterative optimization process to obtain the final solution. In this section, we review past works that used ML to achieve iteration-free inverse design. There are mainly two application scenarios: (1) obtaining designs that meet target properties or responses; (2) obtaining near-optimal solutions under certain constraints or operating conditions. Most prior works addressed the first scenario. Existing ML methods for iteration-free inverse design primarily belong to three categories: one-to-one mapping from target to design (Section 4.3.1), cascaded neural networks (Section 4.3.2), and conditional generative models (Section 4.3.3). We will also introduce a few other works that do not fall into these three categories (Section 4.3.4).
#### 4.3.1 One-to-One Direct Mapping from Target to Design
Owing to neural networks' capability of approximating any continuous functions, it is possible and straightforward to learn a direct mapping from target quantities of interest (e.g., properties or responses) to design solutions using neural networks. Past work has used MLPs and CNNs to learn such mappings -- for example, mapping topology optimization settings (i.e., filter radius, volume fraction, and the type of design objective) to corresponding 2D density maps of mechanical metamaterials [211], target local sound fields to 1D acoustic metasurface designs [212], sets of target scattering parameters to optical metasurface patterns [56], transmission spectrum to photonic nanostructure geometry [79], and target band gaps to phononic crystal designs [55]. To train the neural network models, most of these studies used the difference between the predicted and the "ground-truth" design solutions as the training loss, quantified by metrics such as the mean squared error (MSE), mean absolute error (MAE), or binary cross entropy. This poses a problem for the faithfulness of the predicted solutions in terms of meeting the target, because even structurally similar solutions can result in very different quantities of interest (Figure 11(a)), especially when the structures are in pixelated or voxelated representations, as also illustrated in [36].
There are works that may avoid this issue by comparing the target quantities rather than the design solutions (i.e., measuring \(e_{2}\) instead of \(e_{1}\) in Figure 11(b)). Malkiel et al. [79] first trained an inverse design network to predict the photonic nanostructure geometry based on the transmission spectrum, and then fine-tuned the inverse network by combining it with a forward response prediction network. Liu et al. [213] proposed a conditional GAN-based model to
Figure 10: Designs of permittivity obtained by a) the hPINNs method, b) PDE-constrained adjoint-based optimization with the finite-difference frequency-domain (FDFD) method as the numerical PDE solvers, and c) PDE-constrained adjoint-based optimization with the finite element method (FEM) as the numerical PDE solvers. The hPINNs method achieved a simpler and smoother solution [210]. Reproduced from Ref. [210] with permission from Society for Industrial and Applied Mathematics.
map a target holographic image to a corresponding optical metasurface design. The physical operation mechanism between the electric-field distribution and the metasurface was used to reconstruct the target image. Both the MSE and an adversarial loss between the original and the reconstructed target images were employed during training. Jiang et al. [181] used a conditional GAN to map the target dispersion curves to the structural design of elastic metamaterials. A CNN-based dispersion relation prediction model was used for the fast screening of generated designs based on the predicted dispersion curves. Instead of measuring "how well the predicted design meets the ground-truth solution", these works focused on "how well the predicted design meets the target quantities", thus can lead to the prediction of designs that better match the target quantities.
#### 4.3.2 Avoiding Nonuniqueness Issue via Cascaded Neural Networks
Despite the simplicity of learning direct target-to-design mapping for inverse design, the underlying assumption of one-to-one mapping from target to design can be problematic and may lead to convergence issues during ML model
Figure 11: The issue of measuring the error in design solutions: a) Structurally similar design solutions with very different Poisson’s ratios. There is 0.16% difference in the pixelated design but 100% relative difference between their homogenized Poisson’s ratios. b) The small error \(e_{1}\) between the predicted design and the true solution compared to the large error \(e_{2}\) between the resulting Poisson’s ratio and the target Poisson’s ratio.
Figure 12: Multiple mechanical metamaterials designs correspond to the same properties. This nonuniqueness is either owing to the fact that multiple equivalent structures exist under the periodic boundary condition (top), or because there are parts of the structure that do not contribute to the properties (bottom).
training. Because the nonunique solutions will produce conflicting training instances where the same input is associated with different outputs [214]. Taking mechanical metamaterials as an example, this nonuniqueness is either owing to the fact that multiple equivalent structures exist under the periodic boundary condition (top of Figure 12), or because there are parts of the structure that do not contribute to the properties (bottom of Figure 12). Liu et al. [214] and An et al. [78] also showed a similar phenomenon for 1D nanophotonic structures and 2D optical metasurfaces, respectively. To overcome this nonuniqueness issue, past work proposed the tandem neural network (T-NN) that cascades an inverse-design network with a forward-modeling network (Figure 13) [214, 81, 74, 215, 216, 217]. The model training is separated into two phases: (1) training of the forward-modeling network, where each design corresponds to a unique property or response, and (2) training of the cascaded network by fixing the pretrained forward-modeling network, where the design produced at the intermediate layer does not necessarily belong to training data so that the model is not trained with conflicting designs.
Besides the work using T-NN, there were other model variants with a similar idea of using cascaded neural networks to solve the nonunique mapping problem. For instance, Ma et al. [80] combined two bidirectional neural networks (each of which resembles a T-NN) to learn the relation between optical metamaterial design parameters, reflection spectra, and circular dichroism (CD) spectra, aiming for on-demand inverse design of chiral metamaterials given either the full reflection spectra or the CD spectra.
All the surveyed studies using the aforementioned cascade neural networks were applied to optical metamaterial design with dimensions of design variables not higher than 25. With higher design dimensions, it is more likely that the designs produced in the intermediate layer easily fall out of the training data distribution. When this happens, the forward modeling network is not reliable anymore since it has not seen such out-of-distribution designs. Thus, the cascaded network can still have low error while producing designs that are irrelevant to the target. Besides, given one target, cascade neural networks can only predict one design solution, although there are multiple potential solutions. These limitations motivate the use of generative models (Section 4.3.3).
#### 4.3.3 One-to-Many Mapping via Conditional Generative Models
Conditional generative models' ability to learn a distribution of designs conditioned on any target quantities of interest makes them the perfect candidates for learning one-to-many mappings in inverse design applications. The conditional generative models also explicitly model the relationship between the target and the designs and thus will not produce designs irrelevant to the target. Most prior works in this direction used conditional generative adversarial networks (cGANs) [218] and conditional variational autoencoders (cVAEs) [219].
Conditional GANs are the primary model for achieving one-to-many mapping in past inverse metamaterial design studies. The original cGAN relies on the adversarial loss to ensure the generated designs possess properties or produce responses that match the given target, or show optimality under the given condition [220, 93]. However, with the purpose of reducing the distance between two distributions, the adversarial loss alone cannot promote high-accuracy matching between an individually generated design and the corresponding target or condition. To overcome this issue,
Figure 13: Tandem neural network [214, 81, 74, 215, 216, 217]. The model training is separated into two phases: (1) training of the forward-modeling network, where each design corresponds to a unique property or response, and (2) training of the cascaded network by fixing the pretrained forward-modeling network, where the design produced at the intermediate layer does not necessarily belong to training data so that the model is not trained with conflicting designs.
prior works mainly take three measures: (1) using a separate prediction network for fast screening of unqualified generated metamaterials design [78], (2) adding a prediction loss to implicitly maximize the property/response accuracy of generated designs [156, 221, 50], and (3) progressively updating the training data by adding high-performance generated metamaterial designs and removing low-performance designs [222].
Conditional VAEs were also employed to achieve the same purpose [63, 64]. While GANs have shown superior performance on approximating high-dimensional and complicated data distributions, VAEs have the advantage of stable training and are able to extract an interpretable latent space from data. Ma et al. [63, 64] showed that the latent space from cVAE automatically learns to distinguish metamaterial geometries from different classes.
Note that although some works mentioned in Section 4.3.1 used conditional GANs to generate designs based on target properties, the mapping between target properties and designs is still one-to-one [181, 213]. Because these works used a generator without random noise as its input, the generator can only produce a unique design given a fixed target.
#### 4.3.4 Other Approaches
In addition to the three main approaches mentioned above, there are other prior works aiming to achieve iteration-free inverse design. Luo et al. [223] proposed a probability-density-based neural network that predicts the acoustic metastructure design in the form of Gaussian mixture model parameters given the target transmission spectrum. By sampling from the Gaussian mixture, this approach can generate one-to-many mappings between target responses and design solutions. However, it might not work on high-dimensional design problems (e.g., topological design problems) due to the need to model more complex distributions. Elzouka et al. [199] proposed to use the decision tree to solve both the forward prediction and inverse design problem. After training a decision tree for forward prediction, one can trace up the tree branches from the target response (at leaf nodes) through all branch-splitting criteria. These criteria can be used as design rules to select designs satisfying the given target. This approach naturally captures the one-to-many mapping behavior of inverse design problems, and the extracted rules are interpretable. However, this approach does not suit high-dimensional design problems either, due to the high computational cost of training decision trees with a large depth. Gu et al. [224] trained a linear model to classify metamaterial geometries into "good" and "bad" designs based on their toughness and strength. The weights of the linear model indicate how each element in the design contributes to the performances (Figure 14), based on which new high-performing designs can be sampled. Since the continuous property prediction task was simplified to binary classification, a linear model is sufficient to achieve high predictive accuracy while having the explainability to guide the generation of new designs.
Figure 14: Weights outputted from the linear model show how much each element contributes to a) toughness and b) strength [224]. Colors represent the weight of each element: blue represents negative weights and red represents positive weights. Numbers on the elements represent the ranks in terms of weight. Reproduced from Ref. [224] with permission from Elsevier Ltd.
### Discussion and Future Opportunities
Based on Figure 8, the most frequently used machine learning models for metamaterials design are DGMs, MLPs, and CNNs, all of which are based on neural networks. Also, almost all ML models for iteration-free inverse design are based on neural networks. The flexibility and scalability of neural networks give them the versatility to address various types of problems with high complexity and ill-posedness (e.g., inverse design). On the other hand, the requirement of large datasets and low interpretability have limited the performance and applications of neural networks. To illustrate and summarize the advantages and disadvantages of different methods, in this section, we compare existing works in terms of their cost-benefit and trustworthiness.
#### 4.4.1 Cost-Benefit
The cost of using ML models mainly exists in three stages: data collection, training, and inference. The inference stage is normally much cheaper than the other two stages and hence its cost can be negligible. In most data-driven metamaterials design works, data collection includes expensive physics-based simulations. Thus, the data collection cost usually contributes to most of the total cost and highly depends on the size of the training dataset (with the exception of semi-supervised or unsupervised learning). The training cost also depends on training size, among other factors such as model complexity and the number of training epochs. Therefore, we focus on training size when analyzing the cost.
The usefulness of ML models also depends on the benefits they can provide. Since ML usually aims at accelerating the design process, we can use the reduction of computational cost as a criterion to evaluate benefits (i.e., the computational
Figure 15: Methods proposed in prior work, in relation to the training data size and the design dimension used for demonstration. Note that the training size of RL represents the number of design evaluations performed during training. (LR: logistic regression; DR: dimensionality reduction; RL: reinforcement learning; PCR: principal component regression; DT: decision tree; DGM: deep generative model; GP: Gaussian process; CNN: convolutional neural network; MLP: multi-layer perceptron.)
cost difference between conventional methods and data-driven methods at inference). However, many prior studies did not include such information. Another important factor that reflects the benefits of ML models is the complexity of the design problem they can address. The complexity highly depends on the dimensionality of the design space. Therefore, we consider the design dimension as an important factor when evaluating benefits.
Figure 15 shows the ML models employed in each prior work, in relation to the training data size and the design dimension used for demonstration. Note that the training size and design dimension are extracted from the experimental settings in prior works and do not necessarily indicate any strict model requirements. This figure shows that when the design problem has a dimension of less than 200, MLPs are the most commonly used machine learning model, usually with the existence of relatively large training datasets. An exception is Zhelyeznyakov et al. [149], where the accelerated design of high-dimensional dielectric metasurfaces was achieved by using an MLP and the data of only 10 designs. In that work, instead of treating each entire design as a training sample, local patches of the design geometry and the electromagnetic (EM) field were used. The trained MLP can then be applied to predict the EM field of the entire metasurface. Other models including GP, RL, and DT were also applied in low-dimensional cases. By combining with dimensionality reduction methods such as AEs [56, 55] and VAEs [192], both MLP and GP show the capability of addressing problems in much higher dimensions. Prior works using CNNs were applied to solving problems with a wide spectrum of dimensions, with training sizes ranging from 1,000 to 200,000.
For problems with over 1,000 dimensions, DGMs were the most commonly used models. With the capability of representation learning and modeling one-to-many mapping, DGMs are applicable to both iterative design optimization and iteration-free inverse design (Figure 8). When used for representation learning, DGMs can be trained with unlabeled data, such that a more compact design representation is learned from only geometric data [59, 202, 203], which avoids time-consuming simulations when preparing training data and produces reusable design representations for problems with different properties or responses of interest. When using DGMs for iteration-free inverse design, labeled data are normally required for the DGMs to learn the mapping from properties or responses to metamaterial designs. But there are exceptions where partially labeled or even unlabeled data were considered. Ma et al. [63, 64] proposed a framework that can use both labeled and unlabeled data for data-driven inverse design of optical metasurfaces, where adding unlabeled data was shown to improve model performance. For the same purpose of optical metasurface inverse design, Liu et al. [213] incorporated the physics-based operation between the electric-field distribution and the metasurface design into the decoder of the conditional generative model, which eliminates the need for providing labeled training data.
By infusing physics into ML models, we can even eliminate the requirement of training data. As discussed in Section 4.2.4, Jiang and Fan [208, 209] proposed a generative model whose training was guided by the gradient from the adjoint simulation, so that high-performance dielectric metasurface designs can be generated without using training data. Lu et al.[210] proposed hPINNs that can optimize a design objective under constraints of governing PDEs2.
Footnote 2: This work does not appear on Figure 15 since the design is not limited to fixed dimensionality.
Overall, compared to iterative design optimization, iteration-free inverse design trades off accuracy (i.e., how well the solution matches the true target) for time. Nonetheless, to improve accuracy while still keeping a low computation time, we can use inverse design methods to generate near-optimal solutions as warm starts and further refine the solutions by using optimization with a relatively small number of iterations [220].
Besides design dimensionality, we need to consider another important factor in evaluating the benefits of machine-learning methods -- whether the trained model is applicable to sufficiently many scenarios. To quantify this generality, Woldseth et al. [36] proposed a generality score for neural network-based methods used in topology optimization, which accounted for the required similarity of test and training problems (i.e., higher similarity indicates lower generality), in addition to other topology optimization-related criteria. This test-training similarity is also transferable to measuring generality in ML-based metamaterial design. Among the reviewed past works, the majority require the training and test problems to be similar (i.e., having the same design representation and the same properties or responses of interest). One exception is the works on representation learning, where the learned representation can be applied to design problems with different properties or responses of interest. Another exception is Zhelyeznyakov et al. [149], where the design dimension of test problems can vary since the ML model only cares about the local patches of the design geometry.
Note that although the aforementioned physics-based models have the benefit of requiring no training data, the fact that they need fully specified problem settings (e.g., design constraints, boundary conditions, and operating conditions) for training makes the trained models difficult to generalize beyond the problem specification considered during training, i.e., we need to retrain the model for different problem settings.
#### 4.4.2 Trustworthiness
The trustworthiness of ML-based design methods is important in many engineering problems, especially safety-critical and risk-sensitive ones such as metamaterials for blood vessel stents [225] and medical imaging [55]. One important aspect of trustworthiness is quantifying the uncertainty of metamaterial designs to obtain robust or reliable solutions. This uncertainty quantification, however, is understudied in past literature. Uncertainty comes from sources including operating conditions and the fabrication process. Machine learning models can also have uncertainty due to insufficient data. Data-driven design methods considering these uncertainties can make more informed decisions and generate more trustworthy solutions.
Particularly, due to fabrication uncertainties, the properties or responses of as-fabricated metamaterials can largely deviate from the as-designed ones. Figure 16(a) shows an example of geometric deviation of fabricated metasurface patterns. Figure 16(b) shows how geometric deviation can lead to response changes. The nominal design is represented as \(64\times 64\) binary pixelated images where 1 (yellow) represents material and 0 (dark blue) represents void. The perturbed design is obtained by slightly distorting the nominal design, which mimics the fabrication error. The figure shows that the absorbance spectrum changes significantly due to this small perturbation, indicating the necessity of quantifying fabrication uncertainty. Due to the high dimensionality of variables to be considered in geometric uncertainty quantification, previous metamaterials design work assumes uniform boundary variation where the boundary of the geometry is uniformly "eroded" (e.g., over-etched) or "dilated" (e.g., under-etched) [226]. To avoid making simplifying assumptions on the form of uncertainty and preserve the high degrees of freedom of geometric uncertainty, Chen et al. [202] proposed a deep generative model with hierarchical latent spaces to simultaneously model the geometric variation of nominal designs and the freeform uncertainties of fabricated designs. Chen et al. incorporated this generative model in robust design optimization and demonstrated notable improvement in as-fabricated design performances compared to only considering uniform uncertainty.
Another key aspect of trustworthiness is interpretability, where humans can understand the reasoning behind an ML model's prediction or decision. Past metamaterial design works have either used inherently interpretable models like linear models [224], decision trees [199, 91], or physics-assisted models that infuse governing PDEs [210] and physics-based solvers [208, 209] into neural networks. These methods improve interpretability in different ways: inherently interpretable models such as decision trees and generalized linear models allow us to extract design rules or investigate the importance of design variables; while physics-assisted models use physics rules to constrain the search of model parameters and solutions during model training. Existing studies using inherently interpretable models (i.e., decision trees) only considered very simple (low-dimensional) designs due to the prohibitive computational cost when using these models for high-dimensional inputs. The application and development of inherently interpretable
Figure 16: Fabrication uncertainty and its effects on design performance. a) Examples of metasurface patterns fabricated through the electron-beam lithography [227], where the nominal design is a nanocylinder with the circular cross-section (source: Balogun Research Group at Northwestern University). b) Effects of geometric perturbation on the absorbance profile of a metasurface, where the nominal design is the optimal solution of the deterministic optimization from Chen et al. [202].
models (e.g., neural network-based models [228; 229]) for complex, high-dimensional metamaterial designs were under-explored.
#### 4.4.3 Novelty
Another under-studied challenge in data-driven metamaterials design is how to create novel designs beyond just interpolating existing ones to achieve unprecedented properties or responses. Reinforcement learning can discover novel solutions, but past works only used RL to address low-dimensional design problems, due to the computational cost issue (Section 4.2.3). Data-driven methods using classic ML models are built on the _i.i.d._ assumption that training and testing data are independent and identically distributed. To extrapolate outside existing designs, the training data needs to be updated with desirable generated designs to shift the training data distribution, and the model needs to be retrained on the updated training data. One usually needs to repeat this process many times to obtain a significant improvement over originally existing designs [182]. Without retraining, data-driven models cannot learn useful information that generalizes to scenarios outside existing training data distribution, and therefore cannot lead to novel solutions beyond data distribution. More sophisticated methods are needed to distill generalizable information and create novel designs. Chen et al. [198] extracted parameterized templates of five 3D auxetic metamaterial families from data. The templates can then be used to generate new designs beyond training data, although having the same topologies as the five families. Future research may explore new methods to generate designs that break more limitations prescribed by existing data.
## 5 Data-driven Multiscale Metamaterial System Design
### Overview
Traditional structural design methods and their data-driven counterparts often focus on homogeneous material distributions. In contrast, some functional engineering structures require heterogeneous property distributions to meet spatially varying requirements, which are critical in achieving better performance and more complex functions. For instance, an invisibility cloak requires heterogeneous properties around an object to prevent it from detection with external physical fields [9; 13; 230]. Requirements on heterogeneous properties can also be found in soft robots, where the goal is to achieve local or global target postures [1; 2; 3; 4]. Recently, structural design methods have evolved to optimize both the structure and the distribution of multiple materials for heterogeneous property requirements. Topology optimization (TO) is the most flexible among these methods, enabling freeform changes to the structure and providing greater design freedom than traditional parameter or shape design methods [231; 89]. Despite its promise, it is still challenging to fabricate these multi-material structures in achieving as-designed functions. This issue is caused by the narrow selection of available materials and the constraints of manufacturing processes. In contrast, complex geometries can be more easily manufactured at fine resolutions with additive manufacturing [232]. This technical revolution has opened up new avenues to realize unprecedented and tailorable material properties by changing the geometry of microstructures rather than constituent materials [5; 6; 7; 8], as shown in Figure 17(a)-(c). Therefore, heterogeneous properties can be obtained by spatially varying the microstructures, instead of constituent materials, to assemble a multiscale metamaterials system for intricate structural behaviors (Figure 17(d)-(e)) [233; 234]. In this section, we will discuss data-driven methods for designing multiscale metamaterial systems that determine architectures at both micro and macro scales to achieve the desired metamaterial behaviors. Our focus is on the heterogeneous distribution of (effective) mechanical or thermal properties that originate from the lower material scale, as these physics are involved in most existing multiscale metamaterial designs. For designs with other underlying physics, we refer readers to Refs. [38; 41; 44; 235; 45; 46].
Multiscale structures with carefully tuned microstructures have been shown to have the edge over single-scale design (macroscale only with homogeneous materials or periodic unit cells discussed in Section 4) for engineering applications involving multi-physics or spatially varying requirements. Typical applications include strain cloaking [236; 237], target deformation design [28; 238; 239; 240], thermal-elastic property optimization [241; 242; 243], dynamic behaviors design [82; 22; 244; 245; 246], buckling resistance [247; 248] and energy absorption [249; 250; 251]. However, the design of multiscale structures is a complex two-scale problem, as shown in Figure 17(f). At the macroscale, the topology of the structure and its mechanical properties distribution are optimized to meet performance targets, while at the microscale, unit cells need to be designed at different locations to achieve the corresponding properties. Ideally, the design process for a multiscale metamaterial system should be carried out in a way that the two scales are coupled and designed concurrently. This means that the design of the microscale and macroscale architectures should be done simultaneously and interactively, rather than separately and independently. This hierarchical nature of the system poses unique challenges in the design process compared to the singlescale topology optimization methods illustrated in Section 4.
Firstly, at the macroscale, property parameters required to fully describe the physical response of materials are generally high-dimensional without strict bounds for the achievable properties. This leads to an ill-defined property design space and a complicated optimization process. As a result, most existing methods are subject to over-simplistic constraints on the design space of properties. Secondly, at the microscale, the design of microstructures is an inverse problem without much _a priori_ design knowledge (see Section 4). It is characterized by its infinite-dimensional geometrical design space and one-to-many mapping from properties to structures. This creates an irregular landscape for the design objective (macroscale properties or performance) with many local optima, making the design sensitive to the initial guess and constraints. Finally, the synthesis of micro-and macro-designs suffers from the "curse of dimensionality" induced by the hierarchical multiscale design space, complex combinatorial search associated with the unit-cell selection, and adjacent microstructures whose shapes are incompatible at their interfaces (geometric frustration). Due to these issues, traditional multiscale structural methods are either overwhelmingly time-consuming or rather restrictive in design flexibility.
Capitalizing on the growth of data resources and computational capability, data-driven design based on machine learning models is recognized as a promising tool to address the aforementioned challenges for multiscale systems. As depicted in Figure 18, we propose to divide existing data-driven multi-scale design frameworks into two main categories, i.e., bottom-up and top-down frameworks, based on the relations between design variables at the two scales. The bottom-up framework directly uses the parameters at the microscale level, e.g., volume fraction and unit cell type, as design variables. Costly nested calculation of effective properties of the unit cells is replaced by a surrogate model of the structure-property mapping. In contrast, with the top-down framework, the macroscale topology and spatial distribution of homogenized material properties are concurrently optimized first. Then, to assemble a full multiscale structure, the optimized properties will serve as targets to retrieve the corresponding building blocks. To accelerate this assembly process, ML models trained at the microscale level (Section 4) can be used to compactly represent and/or efficiently generate unit cells. In the remainder of this section, we will illustrate and review the state-of-the-art of these two frameworks.
Figure 17: Metamaterials and multiscale systems. a)-c) Materials exhibit different Poisson’s ratios \(v\) derived from different microstructures, i.e., different transverse displacement given the same pressing loading on the top, with the transparent boxes showing the original shapes before distortion. d) Multiscale orthopedic implant design. Reproduced from Ref. [233] with permission from Elsevier B.V. e) Multiscale design for shape morphing under thermal excitations. Reproduced from Ref. [234] with permission from Macmillan Publishers. f) An illustration of the multiscale metamaterial system design process.
### Bottom-Up Framework
#### 5.2.1 Data-Driven Design with Single-Class Graded Unit Cells
Most bottom-up data-driven methods assumed the same topological concept (single-class) for all the microstructures and only spatially vary, i.e., grade, the geometrical parameters as shown in Figure 19(a) and Figure 19(d). Lattice-based [252] and surface-based microstructures were commonly used due to their simplicity and good manufacturability. One could change the thickness of rods or surfaces [253] to obtain different geometries, with each corresponding to a specific volume fraction. The thickness or the volume fraction can be used as design variables in the optimization process. An advantage of this graded single-class assumption is that one could leave out the microscale details during the optimization and directly optimize the spatial distribution of geometrical parameters at the macroscale instead. This is also called homogenization-based design. The data-driven aspect of this framework is that the time-consuming homogenized properties evaluation in each iteration was replaced by a surrogate model, capturing the relation between geometrical parameters and the precomputed properties. Some examples of such models are exponential function [254], polynomial [255, 252], Kriging [253], diffuse approximation [256] and neural network [257]. After optimization, the corresponding multiscale structure can be obtained by filling the macroscale design with the microstructures specified by the optimal parameters, a process known as de-homogenization [258].
Despite its simplicity and high efficiency, single-class data-driven graded design usually leads to sub-optimal solutions since the microstructrues belong to the same topological class with fixed unit cell orientations. For example, to optimize compliance for both single-loading and multi-loading cases, microstructures must consist of oriented two and three alternating layers of orthotropic materials, known as rank-2 and rank-3 materials, respectively [259]. These microstructures are referred to as rank-2 and rank-3 materials, respectively. Achieving these designs requires spatial changes to the unit cell topology and orientation. Although most existing designs used compliance minimization as the demonstrative case, the single-class graded microstructures did not meet the optimum requirement [260]. As a result, the performance of the multi-scale design was even worse than the single-scale design. Similar observations were also reported in various applications, such as frequency response control [82] and natural frequencies maximization [139]. To increase design flexibility, many studies were dedicated to 1) increasing the diversity of microstructures and/or 2) considering unit cell orientation designs. As shown in Figure 19(c)-(d),(f)-(g), these two variants of bottom-up methods can both expand the property space of the database. The rest of Section 5.2 reviews these approaches.
Figure 18: Illustration of bottom-up and top-down frameworks. The target displacement profile (red dashed lines) is used as an example of the system-level design objective.
#### 5.2.2 Variant 1: Considering Unit-Cell Orientation
The major obstacle in designing graded structures with oriented microstructures lies in the de-homogenization process. When the unit-cell orientations vary across the macroscale structure, the corresponding microstructures need to be rotated accordingly. However, neighboring microstructures might not connect with each other after rotation (Figure 20(a) and 20(c)). This causes the multiscale structure to fail to attain the designed performance and, furthermore, impossible to manufacture. The key to mitigating this issue is to construct a smooth mapping from the stand-alone regular unit cells (e.g., unit cells shown in the upper corners of Figure 20(b)) to an assembled tiling (e.g., the multiscale structure shown in the Figure 20(b)), which can distort the micro-structure to ensure compatibility but at the same time retain their effective properties.
Conformal mapping is considered a powerful tool to realize this mapping since it can preserve the angle of the geometrical features and thus minimize the variation in effective properties [261]. Jiang et al. [262] constructed conforming mapping after homogenization-based design to morph a rectangular tiling of periodic microstructures into corresponding irregular regions in the multi-scale structure (Figure 20(b)). However, the design of unit-cell orientations was not considered in this study. Ma et al.[263] used a linear combination of a set of basis functions to parameterize the mapping from a predefined unit cell to a multi-scale oriented tiling (Figure 20(d)). The coefficients of the basis functions were used as design variables, and used to train an ANN model to predict effective properties from the local Jacobian matrix of the corresponding mapping. While this method allowed the implicit design of unit-cell orientation, the design space was restricted by the form and orders of basis functions, as well as the predefined shapes of unit cells.
In a major advance in 2008, Pantz and Trabelsi [264] focused on square microstructures with rectangular holes and proposed a method to project a homogenized design to a multiscale structure with oriented microstructures on a high-resolution mesh. They achieved this de-homogenization process by constructing implicit mapping functions from a pair of cosine fields to approximate the oriented unit cells in different spatial locations. Later, Groen and Sigmund [265] simplified this de-homogenization process by introducing the connected component labeling method to obtain a consistent orientation field, and by relaxing the optimization problem for the mapping function (Figure 20(e)) [265]. This simplified method was further extended to enable the efficient design of 3D multiscale structures [266, 267]. The de-homogenization process was accelerated by training a neural network to obtain the mapping function from the optimized unit-cell orientations without any extra optimization process [268]. Although the de-homogenization method is appealing in terms of efficiency and performance, it is still confined to simple static compliance minimization problems [269, 270]. The reason is that its original version can only handle square cells with rectangular holes, simply making the unit-cell orientation align with the principal strain direction. While these designs are optimal for static
Figure 19: Three bottom-up frameworks. a) Property space of single-class unit cell. b) Property space of oriented unit cell. The shaded regions are for unit cells with non-zero oriented angles. c) Property space of a diversified set of unit cells. The property curves are colored in accordance with the colors of unit cell classes. Corresponding multiscale designs with d) single-class unit cells, e) oriented unit cells, f) diversified unit cells.
compliance minimization under a single loading, they would become sub-optimal for general design cases, such as multi-loading, dynamic response optimization, and multi-physics problems.
To extend the applicability of de-homogenization, various enhanced methods have been proposed to accommodate more complicated unit cells. Ladegaard et al.[272] combined three cosine fields to construct the implicit mapping functions for oriented rank-3 microstructures. It enables the de-homogenization to find the optimal structures for static compliance problems in multi-loading cases. To handle freeform unit cells, Tamijian et al. [271] represented the complex unit-cell geometries as a Fourier series and optimized the spatial distribution of each Fourier basis to orient unit cells in a compatible way (Figure 20(f)). Groen [258] suggested using cosine functions to construct a similar implicit mapping from regular unit cell regions into oriented patches. Inspired by the texture mapping in computer graphics, Kumar et al. [273] adopted a finite-element mesh to parameterize this implicit mapping. The mapping was obtained by solving a set of linear equations associated with the discrete mesh. While these extensions allowed the use of more complicated unit-cell geometries, their constructed implicit mapping was not conformal and may deteriorate the performance of the multi-scale designs. In a more recent work, Wang et al. [82] proposed to construct a conformal mapping in the de-homogenization process by using Sawtooth function fields (Figure4g). Unit cells with mixed-class topologies were then used to broaden the property space, with neural networks as surrogate models.
Overall, this branch of variants enabled the de-homogeniztaion process of oriented unit cells without much loss in efficiency and has been extended to handle complex unit-cell geometries. However, the works under this category all focused on static compliance minimization problems with a few exceptions. This is due to the smoothness requirement of the orientation field and the restriction of the unit cell topologies.
Figure 20: Examples of oriented two-scale designs. a) Compatible tiling without unit cell orientation design. b) Incompatible tilling with unit cell orientation design. c) Oriented design using conformal mapping. Reproduced from Ref. [262] with permission from ASME. d) Oriented design by optimizing parameterized mapping functions. Reproduced from Ref. [263] with permission from Elsevier B.V. e) De-homogenized design via cosine-based mapping. Reproduced from Ref. [265] with permission from John Wiley & Sons, Ltd. f) De-homogenized design via Fourier-series-based mapping. Reproduced from Ref. [271] with permission from Elsevier Ltd. g) De-homogenized design via saw tooth-function-based mapping. Reproduced from Ref. [82] with permission from Elsevier B.V.
#### 5.2.3 Variant 2: Increasing the Diversity of Unit Cells
The second branch of the variants aims to increase design flexibility by considering diverse micro-structure topologies in the optimization. As illustrated earlier, the success of data-driven graded design relies on the low-dimensional descriptor of unit cell geometries. Therefore, the key challenge addressed in this branch of variants is to represent broader sets of unit-cell topologies without significantly increasing the dimension of descriptors.
As shown in Figure 19(f) and Figure 21, a relatively straightforward idea is to include multiple unit cell classes in the optimization, each with its own low-dimensional parameterization. By considering a unit cell class as a special type of discrete material, this idea naturally fits into the discrete material optimization framework. In this framework, the unit cell class was represented by one-hot encoding and relaxed to be a continuous variable by adding penalization or constraints [274]. This enabled the automatic selection of the optimal unit-cell classes for different spatial regions in the optimization. Due to its simplicity, it has become the most commonly used framework to accommodate multiple unit-classes in multiscale system design.
However, the dimension of design variables (one-hot encoding) for each microstructure grows linearly with the number of classes being considered. This will greatly increase the execution time and complexity. Meanwhile, the one-hot encoding only represents the unit-cell classes in a qualitative way without any physical meaning. As a result, the surrogate models and the optimization process cannot explicitly exploit the correlation or similarity between different classes for better performance. To address these issues, Wang et al. [71, 139, 84] proposed multi-response latent variable Gaussian process (MR-LVGP) and its enhanced variants to transform the discrete classes into continuous 2D latent variables through statistical inference. The constructed latent space captured the effects of classes on the mechanical properties, which induced an interpretable distance metric that reflects the similarity with respect to properties (Figure 21(a)). Moreover, with the nonlinear embedding, the dimension of the latent space remained when the number of unit cell classes increased. By integrating the MR-LVGP models with TO, an efficient data-driven optimization process was developed that can concurrently explore multiple classes and/or constituent materials and their associated geometric parameters for better structural performance.
To further increase the diversity in unit-cell geometries, it is desirable to enable the transition or blending between discrete unit-cell classes (Figure 21(b)-(d)). In this way, new micro-structure prototypes can be created that go beyond the geometries and properties of given classes. To achieve this, Wang et al. [275, 279] established a sophisticated parameterization method for selected classes of microstructures formed by multiple groups of rods (Figure 21(b)). By changing the relative ratio among the rod thicknesses of different groups, a smooth transition between multiple unit-cell topologies can be achieved. However, this parameterization technique is difficult to be generalized to other microstructures with freeform topologies. Chan et al. [52] proposed a more general shape blending scheme that can accommodate freeform unit cell classes with distinct and even incompatible topologies, generating smoothly graded microstructures (Figure 21(c)). The interpolation scheme only had a few extra parameters, which can be directly integrated into the data-driven graded design framework with a neural network as the surrogate model for properties. While blending multiple unit classes expands the design space, it requires a set of prespecified unit-cell prototypes. How to obtain an optimal set of prototypes is problem-dependent and generally unknown beforehand. In Chan et al., the sets were selected either using domain knowledge or an autonomous set selection method that maximizes diversity metrics [52]. Instead of using prespecified prototypes, Zhang et al. [280, 281, 245] propose to simultaneously evolve the prototypes during the optimization. The surrogate model to predict the properties of interpolated unit cells, i.e., a GP model, was also updated in each iteration after evolving the prototypes (Figure 21(d)). By doing so, the initial prototypes can change their shapes to better handle customized design scenarios.
Besides considering multiple classes and their interpolation, special types of materials, i.e., spinodal materials, are emerging as a promising choice in data-driven multiscale design. These stochastic self-assembled can easily achieve diverse microscale geometries with inherent connectivity when the volume fraction is above a given theoretical threshold [282]. Zheng et al. [277] used Gaussian random fields to describe micro-structures of spinodal materials and train a fully connected neural network to associate the field parameters with the effective mechanical properties (Figure 21(e)-(f)). By using the field parameters as design variables, a multiscale structure with spatially varying but smoothly graded microstructures can be achieved [74]. Senhora et al. [278] simplified the design of spinodal materials by focusing on selected symmetric types of candidate spinodal architected materials, extending to accommodate various complex 3D designs (Figure 21(g)).
### Top-Down Framework
The bottom-up framework illustrated in the last section depends on a properly parameterized unit cell model with low-dimensional variables. While this constrained design space greatly expedites the design process, it also shrinks the property space, which will fall short for advanced applications such as soft robots [2] and mechanical cloaking
[283]. The top-down design framework, the second branch of multiscale system design, aims to remove the constraints imposed on the microstructure geometries to allow the use of either pre-specified or freeform unit cells in assembling the full structure (Figure 22) [284, 285, 286, 287, 288, 287]. This framework, in its ideal form, can unleash the highest potential of metamaterials. Specifically, a large database of microstructures is first constructed, containing different geometries and precomputed properties (Figure 22(a)). Since the complex unit cell geometries do not have inherent low-dimensional representations as in the bottom-up framework, the property space of the database will serve as the design space for the property distribution optimization at the macroscale. After that, the property distribution at the macroscale cascades to the microscale (Figure 18) and, based on this, the corresponding microstructures are generated
Figure 21: Multiscale designs with diversified unit cells. a) Design with multiple unit cell classes. Reproduced from Refs. [84, 139] with permission from (top) ASME and (bottom) Elsevier Ltd. b) Multiscale design with unit cells that allow a smooth transition between two classes. Reproduced from Ref. [275] with permission from Springer-Verlag GmbH Germany. c) Multiscale design with blended unit cells. Reproduced from Ref. [276] with permission from Springer-Verlag GmbH Germany. d) Multiscale design with concurrently optimized unit cell prototypes. Reproduced from Ref. [245] with permission from Elsevier Ltd. e) Smooth transition of spinodal metamaterials. Reproduced from Ref. [74] with permission from Creative Commons CC BY. f) Multiscale design with spatially varying spinodal microstructures. Reproduced from Ref. [277] with permission from Elsevier B.V. under the CC BY license. g) Multiscale design with selected symmetric types of candidate spinodal unit cells. Reproduced from Ref. [278] with permission from Wiley-VCH GmbH.
or fetched from the database to fill each element in the full structure. Therefore, there is no need to do the nested optimization and property evaluation at the microscale during the design process, which significantly improves the efficiency during the design of structures. However, to ensure compatibility between adjacent unit cells, these methods still need to force unit cells to be similar in geometries or compatible on the shared boundaries, which limits the range of achievable properties.
To address this issue, Schumacher et al. [240] proposed to construct a database with different metamaterial families. Unit cell geometries are similar within each family but distinctly different across families. They recognized that these families cover different regions but would overlap in the property space. The overlapped regions contain candidates with various geometries for the same properties. The best match can then be selected from those candidates for compatible boundaries. This compatible tiling keeps the design performance at the macroscale matches with the evaluation with homogenization. Later, Zhu et al. [75, 289] discarded the concept of families and generated a larger and richer database by stochastically adding and removing materials from microstructures in the database (Figure 22(b)). In the assembly stage, a random-search based method was used to select microstructures with compatible boundaries to realize the target mechanical properties at each point in the macroscale structure.
However, due to the large amount and diverse shapes of microstructures, an immense combinatorial space needs to be explored through stochastic methods to form compatible boundaries between adjacent unit cells. Moreover, while
Figure 22: Top-down methods. a) Design framework using discrete unit cells with cubic symmetry and predefined compatibility. The top row shows two pairs of compatibly connected isotropic unit cells in the database. Reproduced with permission from the authors [239]. b) Design framework using randomly-generated isotropic unit cells with two constituents. [75] Reproduced from Ref. [75] with permission from Association for Computing Machinery. c) Design framework using randomly-generated anisotropic unit cells. Reproduced from Ref. [53] with permission from Elsevier B.V. d) Multiscale design with stochastic growth rules. Reproduced from Ref. [72] with permission from American Association for the Advancement of Science.
these studies proposed elaborate methods for database construction, they lacked an effective representation and retrieval method for microstructures. As a result, it is challenging to incorporate these large databases into the multiscale design in a scalable way. This might also be the reason that these studies only focused on unit cells with isotropic or cubic symmetry.
In contrast, Wang et al.[53] focused on anisotropic microstructures with high-dimensional stiffness tensor (Figure 22(c)). To tackle the aforementioned challenges, they simultaneously trained a VAE and an NN-based property predictor to map complex microstructures into a low-dimensional, continuous, and organized latent space. They found that the latent space of VAE provided a distance metric to measure shape similarity, enabled interpolation between microstructures, and encoded meaningful patterns of variation in geometries and properties. These characteristics enabled an effective selection of diverse unit cell candidate sets from the database to increase the chance of compatible assembly. The shape similarity metric was also utilized as a metric for geometrical compatibility between unit cells. By combining both geometrical and mechanical compatibility measures, the assembly process was formulated as an energy-minimization problem on a grid-like graph and solved efficiently by dual decomposition. This method has been successfully applied to achieve target deformation profiles [53], mechanical cloaking [236], and fracture resistance [290]. Following a similar direction, Wang et al. [50] used a GAN model to learn the distribution of implicit-surface-based geometries conditioned on given properties. By utilizing the continuity of the GAN-generated structures and a transition layer blending technique, compatible microstructures are inversely generated to achieve the designed properties in the assembled structure. Liu et al. [72] adopt a different strategy by devising a stochastic growth rule, similar to cellular automaton [291], to blend different graph-based features into irregular microstructures (Figure 22(d)). Good compatibility can be ensured by devising special local growth rules. With the relation between the homogenized properties and the growth rule, the full structure can achieve the target property distribution through an automatic growth process.
### Discussion
Both data-driven multiscale frameworks have shown promise in achieving efficient multiscale design in various applications, but they still have limitations and are not yet ideal. In this section, we will compare these frameworks to illustrate their respective strengths and weaknesses, providing insights into their applicability. We will also highlight critical knowledge gaps and challenges in existing research.
#### 5.4.1 Comparison of Bottom-Up and Top-Down Frameworks
* **Efficiency** The two types of frameworks both ignore the microscale details in multiscale optimization by using the unit cell parameters (bottom-up) or effective properties (top-down) as the design variables. This allows them to bypass nested optimization across different scales and numerous homogenization evaluations, enabling much higher computational efficiency than traditional multiscale designs. Among them, the bottom-up framework is generally more efficient than its top-down counterpart because the compatibility between neighboring unit cells is guaranteed by the parameterized unit cell or easily handled by adding local constraints. In the top-down framework, when diverse microstructures are considered, an extra tiling optimization is usually needed after the homogenization-based optimization to guarantee compatibility, which is relatively time-consuming.
* **Design Freedom** The bottom-up framework uses microscale parameters as the design variables and thus requires a parameterized model for unit cells. This restricts the change of unit cell topologies. In contrast, the top-down framework directly optimizes the effective properties, which can be adapted to any unit-cell geometries. Therefore, the top-down framework has higher design freedom than its bottom-up counterpart. The flexibility of the bottom-up framework can be improved by considering multiple classes of microstructures or the unit cell orientation, as previously illustrated. However, this will sacrifice some efficiency and may lead to a complex optimization problem with more local optima.
* **Manufacturability** The parameterization of unit cells in the bottom-up framework makes it easier to impose functionally graded constraints or filters on the geometries, which can benefit the manufacturability. The manufacturing restriction can also be considered in selecting the unit cell classes. In the top-down framework, manufacturing constraints of a single unit cell can be added to the construction of the database. However, the manufacturability of the assembled structure cannot be explicitly considered when designing the property distribution. Additional steps of compatibility optimization and post-processing are required in the assembly stage to obtain manufacturable full structures[75, 289, 53].
* **Generalizability** The top-down method assumes weak mechanical coupling between unit cells and has been confined to material designs in the realm of linear elasticity. For dynamic applications or nonlinear cases, this coupling might not be negligible [23, 21, 292, 293, 294]. Since the unit cells in the bottom-up framework are parameterized, it is easier to consider the coupling between unit cells by modeling the interaction as a function
of their parameters. The bottom-up framework can also be extended to consider strain-dependent properties in accommodating nonlinear cases [257].
#### 5.4.2 Assumptions of Homogenization
The first-order homogenization method is the basis of most data-driven multiscale designs to obtain the effective properties of unit cells in the database [47]. It assumes that the stress of each point in the macroscale only depends on its local strain value and is not affected by neighboring unit cells [295]. This is only valid when the microscopic length scale is much smaller than that of the macrostructure, and when the microstructures are periodically distributed[296]. For example, in a periodic design for compliance minimization, the ratio between the sizes of macro- and micro-structures should be of 5-6 to ensure a relatively accurate evaluation of the performance with homogenization. Most existing data-driven multiscale designs do not meet the first-order homogenization assumption due to the aperiodic tiling and the large unit cell size restricted by manufacturability. However, if neighboring unit cells have a smooth change or compatible connections within a large entire structure, homogenization can still provide relatively satisfying results as reported in some studies [52, 240]. Nevertheless, it is still advisable to present the performance metrics obtained via full-scale simulation as a validation, which was rarely reported in existing papers.
Meanwhile, reduced-order models combined with ML are becoming promising alternatives to replace homogenization in multiscale deigns which does not rely on first-order homogenization assumption and can thus simulate the response more accurately. For example, Wu and Fu et al. [297, 256] condensed the fine FE model of unit cells into a super element with only boundary nodes. Since no periodicity or scale separation is assumed, the macro-response remains accurate for heterogeneous design even with the size of the unit cell close to the macrostructure. By combing proper orthogonal decomposition and diffusion approximation methods, the relation between volume fraction and the condensed stiffness matrix can be directly obtained for the previous graded structural design framework.
Physics-based machine learning (PBML) model is emerging as another intriguing direction to accelerate the full-scale simulation without resorting to homogenization (see 4.2.4). This can be a potentially useful tool to bypass the first-order homogenization assumption. Currently, PBML mainly focuses on forward analysis instead of inverse optimization. Yao et al. [298] considered the finite element model as a special type of convolution layer to construct FEA-Net, predicting the mechanical response of metamaterials under external loading. Following similar ideas, Saha et al. [299] used neural networks to replace the interpolation functions in FEA models, which can provide fine-resolution results with lower computational expenses. While these physics-informed methods require fewer data, they are either less efficient or restricted to a single type of microstructure, compared to the aforementioned fully data-driven methods. Although many existing studies claim that PBML is more efficient and easy to use than traditional FEA, most of these models can still be considered as special FE models or PDE solvers. They used a neural network to replace those classical approximation functions, which transform the original linear weighted-form equation into a highly non-linear optimization problem. Stochastic gradient descents and their variances were then used to solve the problem, which does not guarantee convergence to the real solution. The universal applications and higher efficiency were obtained at the cost of rigorous theoretical foundations. Also, most existing studies compare PBML with naive FE models, instead of some more advanced finite element methods, which might not be a fair comparison [300, 301, 302]. It is advisable for the researchers to explore both PBML and advanced finite element methods in accelerating the data-driven designs for multiscale systems.
#### 5.4.3 Task Specificity of Data Acquisition for Multiscale Design
In Section 3.4.1 we have briefly discussed the task-specificity of data assessment within metric-level assessment. Enlarging the scope to the system-level design optimization, we reiterate the task-specificity based on some concrete results reproduced from the literature. Figure 23 shows a set of target tasks at the system level defined by Wang et al. [53]. Assuming linear elasticity deformation in 2D mechanical metamaterials, three design tasks are prepared to achieve different target displacements in Figure 23(a). Required distributions of homogenized elasticity components (\(C_{11}\), \(C_{12}\), \(C_{22}\), \(C_{33}\)) are pre-computed for each target pattern (Figure 23(b)). The 240k-size orthotropic mechanical metamaterial dataset, generated with Pixel/Voxel (Section 3.2.1) and Perturbation (Section 3.2.2), is used.
Figure 23(c) manifests that the required data distributions vary significantly contingent upon tasks. Overall, the 240-k dataset can cover all three tasks, even if no information on them was given before the data acquisition. The on-demand property distribution for each task is quite disparate across the tasks. For example, the \(50\times 50\) properties to achieve smiley face (red) are relatively clustered. This implies that the wide coverage does not benefit this particular task as much as the other two cases. But the smiley face task demands a large portion of anisotropic samples, i.e., those having large either \(C_{11}/C_{22}\) or \(C_{22}/C_{11}\), as shown in the \(C_{11}-C_{22}\) space. Thus, if known before the data acquisition, such samples can be prioritized with an associated metric during the data acquisition, as shown in Lee et al. [100] and Wang et al. [49]. Meanwhile, the required \(4\times 20\) properties to produce the bridge-like deformation (blue) tend to widely
spread in the associated property space. Among the tasks, the bridge-like deformation design benefits most from wide coverage, which is congruent with a generic goal of data acquisition. Even in this task, however, samples having negative Poisson's ratio are not used; this indicates such uniform coverage of property is not unconditionally favored, but depends on the intended tasks. We remark that herein we intentionally omit considering geometric/mechanical compatibility among building blocks to convey our point with particular focus on data.
In summary, for multiscale design purposes, the data acquisition and assessment are intrinsically open to subjectivity, in part due to task-specificity. Not all data holds equal utility for general design tasks. Given a target task(s) at the system level, the data acquisition and assessment should involve the specified task(s) and even intentionally introduce bias, in addition to data uniformity, to meet the task requirement and ensure an efficient data collection. Herein, the uniformity is meaningful only within the domains associated with the target tasks, as opposed to generic scenarios of data acquisition.
#### 5.4.4 Other Challenges
Overall, the data-driven paradigm has shown its promise in multiscale metamaterial design, with a superior capability to excavate the underlying relations between properties and geometries. However, there is a tendency in existing studies to apply off-the-shelf models as black-box tools for multiscale design, claiming that the universal fitting capability and high efficiency can benefit the design process. The underlying assumptions and constraints of the models have been constantly ignored, leaving the applicability questionable.
Before applying a specific model to multiscale design, it would be helpful to consider how it will influence the optimization solver, and whether its outputs and assumptions agree with the physics. For example, various training
Figure 23: Task specificity of data acquisition and assessment based on the case study in Wang et al. [53]. a) Illustration of three system-level design tasks associated with different target displacements: smiley face (left), mouth (center), and bridge (right). b) On-demand distributions of homogenized elasticity components \(\{C_{11}\), \(C_{12}\), \(C_{22}\), \(C_{33}\}\). c) The distributions of required property for all the individual tasks plotted with respect to that of the orthotropic dataset (gray), created through Pixel/Voxel and Perturbation. All the results are Reproduced from Ref. Wang et al. [53] with permission. Copyright 2020, Elsevier B.V.
losses have been used to evaluate the performance of ML models, which mainly focus on the mean errors. It is not guaranteed that models trained with these loss functions will always produce physically feasible outputs in the whole input domain. In some cases, this could also lead to issues that are detrimental for optimization solvers, e.g., singular and negative-defined stiffness matrix, unit cells with disconnected components, the discontinuity between different cells, non-differentiability and local fluctuations of the predicted responses. Some pre-trained deep NNs devised in computer graphics were transferred to the applications in multi-scale design. However, hierarchical features extracted by the latent layers of these models might not be suitable for the structure designs. An important characteristic in multi-scale meta-material designs is that a given macro-scale property can be achieved by multiple micro-structures [53, 240]. Nevertheless, in most existing methods, a one-to-one mapping is assumed between properties and geometries, which fails to accommodate the one-to-many nature. Most databases and ML models only consider constant property parameters of a single unit cell or periodically assembled unit cells. They do not take into account state-dependent and history-dependent properties, e.g. strain-dependent stiffness tensor in nonlinear cases, and the interactions between different unit cells. Moreover, existing research mainly focuses on simple regression models to predict the homogenized properties and generative models to reduce the dimension of shape descriptors for the unit-cell design. How to use data-driven methods to improve the optimization procedure itself, e.g., extraction of design rules and underlying physical knowledge, initial designs recommendation, iterative optimization strategy, and combinatorial assembly, generally remains unexplored.
## 6 Conclusion
We presented a comprehensive, critical review of the data-driven design of metamaterials and multiscale systems (DMD). Through our analysis, we categorized previous research endeavors into the distinct modules of a cohesive data-driven design framework, including data acquisition (Section 3), unit-cell level learning and optimization (Section 4), and multi-scale system designs (Section 5). In Section 3, we examined the common practices of data acquisition strategies, with special attention to the methodological components of shape generation and property-aware sampling, and to data assessment. In Section 4, we covered prior works that utilized datasets and ML to enable acceleration of, or higher design freedom in, unit cell design. In Section 5, we reviewed data-driven multiscale design that employed databases and ML at the unit-cell level to optimize structures across multiple scales, meeting heterogeneous properties requirements in a top-down or bottom-up manner. For these multiscale design efforts, the main focus was to replace homogenization with surrogate modeling, and increase design flexibility by accommodating diverse and oriented unit cells.
Based on our literature survey, we shared our perspectives on the current practices and suggested new avenues for future research efforts. In Section 3, we disclosed that the current research trend in DMD is arguably biased towards the final products at the downstream modules and lacks principled methods dedicated to data acquisition, which is critical to the successful and robust deployment of DMD frameworks. To address this, more benchmark datasets need to be made publicly available and standard dataset assessment protocols should be established so that the contributions of future works can be rigorously appreciated. In Section 4, we discussed the cost-benefit trade-off of the reviewed methods, which is usually neglected in prior works. In addition, we noted that the trustworthiness and creativity of ML models for unit cell designs are also under-studied. There are ample opportunities to develop interpretable or physics-informed ML methods, as well as to enhance their uncertainty quantification and generalization capabilities. In Section 5, we remarked that there are still many unexplored opportunities for data-driven multiscale system design, particularly in areas such as extracting design rules, discovering physical knowledge, providing initial design recommendations, facilitating iterative optimization strategies, and enabling combinatorial assembly.
Overall, ML has shown promise in metamaterials design with its superior capability to excavate the complex relations between properties and geometries. Despite the potential of the data-driven design approaches we reviewed, it is important to acknowledge that the field is still in its early stages and faces many grand challenges. Addressing these challenges means that the disconnect between the two primary approaches - data-driven and physics-based - needs to be resolved, which will require careful consideration of the cost-benefit and trustworthiness of ML methods, as well as collaboration between various disciplines. We believe that the tighter integration of physics and data-driven methods can unlock exciting possibilities for metamaterials design. Towards this future, we hope our review contributes to promoting interdisciplinary collaborations and bridging the gap between the two camps.
## Acknowledgements
This work was supported by the NSF BRITE Fellow program (Grant No. CMMI 2227641), the NSF CSSI program (Grant No. OAC 1835782), and the Northwestern McCormick Catalyst Award.
## Conflict of Interest
The authors declare no conflict of interest.
|
2301.06381 | Statistics of weakly nonlinear waves on currents with strong vertical
shear | We investigate how the presence of a vertically sheared current affects wave
statistics, including the probability of rogue waves, and apply it to a
real-world case using measured spectral and shear current data from the Mouth
of the Columbia River. A theory for weakly nonlinear waves valid to second
order in wave steepness is derived, and used to analyze statistical properties
of surface waves; the theory extends the classic theory by Longuet-Higgins [J.
Fluid Mech. 12, 3 (1962)] to allow for an arbitrary depth-dependent background
flow, $U(z)$, with $U$ the horizontal velocity along the main direction of wave
propagation and $z$ the vertical axis. Numerical statistics are collected from
a large number of realisations of random, irregular sea-states following a
JONSWAP spectrum, on linear and exponential model currents of varying
strengths. A number of statistical quantities are presented and compared to a
range of theoretical expressions from the literature; in particular the
distribution of wave surface elevation, surface maxima, and crest height; the
exceedance probability including the probability of rogue waves; the maximum
crest height among $N_s$ waves, and the skewness of the surface elevation
distribution. We find that compared to no-shear conditions, opposing vertical
shear ($U'(z)>0$) leads to increased wave height and increased skewness of the
nonlinear-wave elevation distribution, while a following shear ($U'(z)<0$) has
opposite effects. With the wave spectrum and velocity profile measured in the
Columbia River estuary by Zippel & Thomson [J. Geophys. Res: Oceans 122, 3311
(2017)] our second--order theory predicts that the probability of rogue waves
is significantly reduced and enhanced during ebb and flood, respectively,
adding support to the notion that shear currents need to be accounted for in
wave modelling and prediction. | Zibo Zheng, Yan Li, Simen Å Ellingsen | 2023-01-16T11:59:31Z | http://arxiv.org/abs/2301.06381v1 | # Statistics of weakly nonlinear waves on currents with strong vertical shear
###### Abstract
We investigate how the presence of a vertically sheared current affects wave statistics, including the probability of rogue waves, and apply it to a real-world case using measured spectral and shear current data from the Mouth of the Columbia River. A theory for weakly nonlinear waves valid to second order in wave steepness is derived, and used to analyze statistical properties of surface waves; the theory extends the classic theory by Longuet-Higgins [_J. Fluid Mech._**12**, 3 (1962)] to allow for an arbitrary depth-dependent background flow, \(U(z)\), with \(U\) the horizontal velocity along the main direction of wave propagation and \(z\) the vertical axis. Numerical statistics are collected from a large number of realisations of random, irregular sea-states following a JONSWAP spectrum, on linear and exponential model currents of varying strengths. A number of statistical quantities are presented and compared to a range of theoretical expressions from the literature; in particular the distribution of wave surface elevation, surface maxima, and crest height; the exceedance probability including the probability of rogue waves; the maximum crest height among \(N_{s}\) waves, and the skewness of the surface elevation distribution. We find that compared to no-shear conditions, opposing vertical shear (\(U^{\prime}(z)>0\)) leads to increased wave height and increased skewness of the nonlinear-wave elevation distribution, while a following shear (\(U^{\prime}(z)<0\)) has opposite effects. With the wave spectrum and velocity profile measured in the Columbia River estuary by Zippel & Thomson [_J. Geophys. Res: Oceans_**122**, 3311 (2017)] our second-order theory predicts that the probability of rogue waves is significantly reduced and enhanced during ebb and flood, respectively, adding support to the notion that shear currents need to be accounted for in wave modelling and prediction.
+
Footnote †: Corresponding author: [email protected]
+
Footnote †: Corresponding author: [email protected]
## I Introduction
Waves in the ocean are almost invariably affected by interaction with their surroundings, ambient currents in particular. While large-scale ocean currents may be approximately depth-independent, this is often not the case for smaller scale currents such as those driven by wind
shear, or currents in the near-shore environment including river deltas and tidal currents. Of particular interests is the role of these environmental factors on the occurrence probability of extremely large waves [1; 2; 3], known also as rogue, giant, or freak waves, defined as waves whose amplitude far exceeds that of their surrounding wave field. To this end, many formation mechanisms of rogue waves have been proposed, including (but not limited to) dispersive focusing of linear waves [1], nonlinear effects such as the modulational instability [4] and quartet resonances [5] as well as refraction by currents [6] and bathymetry [7; 8], nonlinear interaction between surface waves and depth transitions [9; 10]. In this paper, our main attention is paid to the effect of a background depth-varying current on the statistics of weakly non-linear waves, rogue wave events in particular.
In order to obtain a proper statistical description of rogue wave events, a theory for second-order interaction of waves in a random sea has been widely used in both analytical [11; 12; 13; 14; 15; 16; 17; 18] and numerical studies [19; 20]. In contrast to linear waves in a random sea for which the wave elevation can be represented as a Gaussian random process [21], second-order nonlinear waves can lead to considerable deviations from Gaussian wave statistics due to the steepened crests and flattened troughs caused by second-order (bound) waves. To describe the altered statistics, analytical models for wave crest and elevation distributions have been proposed for deep-water random waves, see, e.g., [13; 15; 17]. These generally agree well with both laboratory and field measurements for narrowband and broadband wave fields (see, e.g., [22; 23; 24; 17; 20]) with moderate steepness. In more nonlinear sea states discrepancies arise from third and higher order nonlinear effects, e.g., the well-known Benjamin-Feir instability [4] and the resonant wave quartets [5]. Hence, a second-order theory such as the one we present herein, is limited to the cases where higher-order corrections are comparatively small.
Many studies have suggested several different ways by which the probability of rogue waves is increased in the presence of currents with horizontal, but not vertical, spatial variation (c.f. [25; 26]). A current whose magnitude and direction varies slowly in space relative to the rapidly varying wave phase has mostly been considered as a (local) Doppler shift on the wave dispersion relation and as a medium of refraction in the conservation of wave action [6; 27]. Due to this, White & Fornberg [6] attribute the enhanced probability of larger wave events in currents to the local refraction by currents. Many varieties of the third-order nonlinear Schrodinger equations have been developed for slowly (horizontally) varying currents, see, e.g., [28; 29; 30]. An opposing current has been found to lead to strengthened modulational instability [7; 31] and Shrira & Slunyaev [26] found that trapped waves by a jet current can also lead to an enhanced formation probability of rogue waves while Hjelmervik and Trulsen [30] found that a wave impinging on an opposing jet
has increased significant wave height, but decreased kurtosis, and _vice versa_.
The aforementioned works have focused on a current whose velocity profile does not have significant gradients in the vertical direction. Among the studies of waves in a horizontally uniform and depth varying current, a majority have examined waves propagating along or against currents which vary linearly with depth, which in two dimensions permits the use of a velocity potential [32], considerably simplifying the analytical treatment [33; 34; 35; 36; 29; 37]. The assumption of a linearly varying current also results in significant simplification of the continuity and Euler momentum equations in three dimensions, based on which a second-order theory for three-dimensional waves was developed by Akselsen and Ellingsen [38]. A uniform vorticity plays a significant role in both the sideband instability and modulational growth rate for weakly nonlinear unidirectional Stokes waves [34; 39]. A positive vorticity, which corresponds to a following current -- i.e. \(U(z)>0\) and \(U^{\prime}(z)<0\) with \(U(z)\) the current oriented along the wave propagation direction, \(z\) the vertical coordinate, and a prime denotes the derivative -- can remove the modulational instability altogether, demonstrated experimentally by Steer _et al._[40] and Pizzo _et al._[41] (the definition of positive/negative shear in ref. [40] is different from ours due to a different choice of the coordinate system). Francis and Kharif [42] have extended [34] to two-dimensional Stokes waves where new quartet and quintet instabilities have been discovered arising from the presence of a uniform vorticity, while Abrashkin and Pelinovsky [43] derived a nonlinear Schrodinger equation for arbitrary, weak vertical shear in a Lagrangian framework, generalized in ref. [41].
Realistic natural currents have non-zero curvature in the depth direction which leads to additional effects on wave properties. A number of works, e.g., [44; 45; 46; 47; 48], have demonstrated the importance of the depth-varying curvature of a current profile in the wave action equation. Effects of the curvature are wavenumber- and depth-dependent, leading to considerable deviations of the direction and speed of the propagation of wave energy from the cases where the curvature has been neglected [48]. Experimental studies, e.g. [49; 50; 51], have confirmed the importance of curvature in wave modelling. Cummins & Swan [49] carried out an experimental study of irregular waves propagating in an arbitrarily depth varying current and the wave spectra measured showed significant differences from those in a uniform and magnitude-equivalent current. It was concluded by Waseda _et al._[50] from experiments that the variability of the ambient current affected the third-order resonant interaction of wave quartets more than its mean profile did. In field observations, ocean currents are found to have considerable effect on the significant wave height [52], estimation of Stokes drift and particle trajectories [53], and the dissipation of waves through breaking [54].
The objective of the paper is twofold. Firstly, we present a new framework to allow for the
interaction of weakly nonlinear surface gravity waves and a vertically sheared current, generalising the work of Longuet-Higgins [11]. Secondly, we implement the new theory numerically to study how a current profile's shear and curvature affect wave statistics, e.g., wave crest distribution and skewness of the surface elevation of random waves.
We highlight that the new framework presented in this paper does not rely on assumptions of weak vertical shear (such as Stewart and Joy [55], Skop [56], Kirby and Chen [57], Zakharov and Shrira [58]) or weak curvature (or 'near-potentiality', e.g., Shrira [59] and Ellingsen and Li [60]). Although these simplifying assumption may be applicable to most realistic situations in the open ocean, their validity should not be taken for granted, and must be properly ascertained [60]. Indeed the shear of a current can be strong in oceanic and coastal waters. For example, a wind-driven shear current in the top few centimetres can have very strong shear (e.g. [61; 62]) and the surface current typically takes values \(\sim 3\%\) of the wind speed [63]. Estuarine tidal flow has been found to be very strongly sheared, for instance the Mouth of the Columbia River which we use as example herein [64; 54]. We therefore choose to use the numerical Direct Integration Method (DIM) proposed by Li and Ellingsen [47] to calculate the linear wave surface and velocity fields, being equally applicable to any horizontally-uniform depth-dependent current profile regardless of its magnitude, shear, and curvature. As detailed in Li and Ellingsen [47], the computational cost of the DIM is comparable to that using analytical approximations which involve integration over the water column [55; 56; 57; 60], and unlike the aforementioned approximations, it provides an error estimate at little extra cost. The computer code used to generate the results presented in this paper is included as supplementary material online.
This paper is laid out as follows. A second-order theory based on a perturbation expansion, the Direct Integration Method for linear waves [47], and double Fourier integrals for the second-order bound waves is presented in SSII. Using the assumption of narrow-banded waves the shear current-modified wave statistics (e.g., skewness and the exceedance probability of wave crest) are derived in SSIII. With the numerical implementation of the theory detailed in SSIV, weakly nonlinear waves in a random sea are examined in SSV, for which the linear wave amplitude and phase used for random wave realisations are assumed to follow a Rayleigh distribution and a uniform distribution, respectively, following Tucker _et al._[65].
Theoretical description and methodology
### Problem statement
We consider three-dimensional surface gravity waves atop a background flow in deep water. Incompressible and inviscid fluids are assumed and the surface tension has been neglected for simplicity. The background flow propagates in the horizontal plane and varies with depth (i.e. vertically sheared). Its 3-dimensional velocity vector is described by \(\mathbf{U}_{3}^{*}(z^{*})=(\mathbf{U}^{*}(z^{*}),0)\), with \(\mathbf{U}^{*}\) the velocity vector in the horizontal plane, \(z^{*}\) the upward axis, and a vanishing vertical component. Dimensional variables are marked with an asterisk. A Cartesian coordinate system is chosen and the still water surface in the absence of waves and flow is located at \(z^{*}=0\). The surface elevation due to the background flow in the absence of surface waves is described by \(z^{*}=\eta^{*}\), which is assumed known and whose spatial and temporal variations are comparably negligible to the wave perturbed fields. Neglecting the influence of surface waves on the background flow field, the system of surface waves in a background flow can be described by the continuity and Euler momentum equations as follows (see, e.g., [27])
\[\nabla_{3}^{*}\cdot\mathbf{V}_{3}^{*}= 0, \tag{1}\] \[\partial_{t^{*}}\mathbf{V}_{3}^{*}+(\mathbf{V}_{3}^{*}\cdot \nabla_{3}^{*})\mathbf{U}_{3}^{*}+(\mathbf{U}_{3}^{*}\cdot\nabla_{3}^{*}) \mathbf{V}_{3}^{*}+\nabla_{3}^{*}\left(P^{*}/\rho+gz^{*}\right)= -(\mathbf{V}_{3}^{*}\cdot\nabla_{3}^{*})\mathbf{V}_{3}^{*}, \tag{2}\]
for \(-\infty<z^{*}<\zeta^{*}+\eta^{*}\). Here \(\nabla_{3}^{*}=(\nabla^{*},\partial_{z^{*}})\) denotes the gradient operator in three dimensions and \(\nabla^{*}=(\partial_{x^{*}},\partial_{y^{*}})\) the gradient in the horizontal plane; \(\mathbf{V}_{3}^{*}=(\mathbf{u}^{*},w^{*})\) denotes the velocity field due to surface waves in the presence of the background flow, with \(\mathbf{u}^{*}\) and \(w^{*}\) the velocity vector in the horizontal plane and vertical component, respectively, \(\mathbf{x}^{*}\) the position vector in the horizontal plane, and \(t^{*}\) is time; \(P^{*}\) denotes the total pressure; \(\rho\) and \(g\) denote the fluid density and gravitational acceleration, respectively; \(\zeta^{*}(\mathbf{x}^{*},t^{*})\) denotes the surface elevation due to additional surface waves in the presence of the background flow, \(\mathbf{U}_{3}^{*}\).
We choose the characteristic length \(L_{c}^{*}\) and velocity \(u_{c}^{*}\) to nondimensionalize the variables. In all cases we consider in SSIV, a wave frequency spectrum \(S^{*}(\omega^{*})\) is assumed which has a clear peak at a frequency \(\omega_{p}^{*}\). Therefore, we form the characteristic length, \(L_{c}^{*}=g/{\omega_{p}^{*}}^{2}\), and, characteristic velocity, \(u_{c}^{*}=g/{\omega_{p}^{*}}\) using \(g\) and \(\omega_{p}^{*}\) for convenience while our specific choice does not affect the generality of the theory derived in SSII and III. Explicitly,
\[(x^{*},y^{*},z^{*})=(x,y,z)L_{c}^{*};\ \ t^{*}=\frac{L_{c}^{*}}{u_{c}^{*}}t; \ \ \mathcal{V}^{*}=u_{c}^{*}\mathcal{V}; \tag{3}\]
Here, \({\cal V}\) represents any velocity component, and we define the wave-induced nondimensional pressure as
\[P=(P^{*}+\rho gz^{*})/(\rho u_{c}^{*2}). \tag{4}\]
The dimensionless continuity and Euler momentum equations become
\[\nabla_{3}\cdot{\bf V}_{3}= 0; \tag{5}\] \[\partial_{t}{\bf V}_{3}+({\bf V}_{3}\cdot\nabla_{3}){\bf U}_{3}+( {\bf U}_{3}\cdot\nabla_{3}){\bf V}_{3}+\nabla_{3}P= -({\bf V}_{3}\cdot\nabla_{3}){\bf V}_{3}, \tag{6}\]
for \(-\infty<z<\zeta+\eta\).
The governing equations (5) and (6) should be solved subject to the dynamic and kinematic boundary conditions at the surface, respectively,
\[P-(\zeta+\eta)=0\ \ \mbox{and}\ w=\ \partial_{t}\zeta+({\bf u}+{\bf U})\cdot \nabla\zeta\ \ \mbox{for}\ \ z=\zeta+\eta, \tag{7}\]
and the deepwater seabed condition
\[({\bf u},w)=\ 0\ \mbox{for}\ \ z\rightarrow-\infty. \tag{8}\]
### Perturbation expansion and linear wave fields
We seek the solution for unknown velocity (\({\bf V}\)) and elevation (\(\zeta\)) of the boundary value problem described by (5) - (8) in a form of power series in wave steepness denoted by \(\epsilon\); i.e. a so-called Stokes expansion. To leading order, they are given by
\[[\zeta,{\bf u},w,P]=\epsilon[\zeta^{(1)},{\bf u}^{(1)},w^{(1)},P^{(1)}]+ \epsilon^{2}[\zeta^{(2)},{\bf u}^{(2)},w^{(2)},P^{(2)}], \tag{9}\]
where the terms are kept up to second order in wave steepness and the superscript '\((j)\)' denotes the \(j\)-th order in wave steepness. Inserting the perturbed solutions (9) into the boundary value problem described by (5) - (8) and collecting the terms at the same order lead to the various boundary value problems at different orders in wave steepness. In the special case of linearly varying current, an explicit solution is available. We provide the expression, adapted from the solution by Akselsen and Ellingsen [38], in appendix C.
Linear surface elevation due to irregular surface waves can be described by
\[\zeta^{(1)}({\bf x},t)={\cal R}\left[\frac{1}{4\pi^{2}}\int|\hat{\zeta}({\bf k })|{\rm e}^{{\rm i}\psi({\bf k},{\bf x},t)}{\rm d}{\bf k}\right], \tag{10}\]
where \(\mathcal{R}\) denotes the real part, \(\mathbf{k}\) denotes a wavenumber vector in the horizontal plane, \(\hat{\zeta}(\mathbf{k})\) denotes the linear wave elevation transformed in the Fourier \(\mathbf{k}\) plane, \(\psi(\mathbf{k},\mathbf{x},t)=\mathbf{k}\cdot\mathbf{x}-\omega(\mathbf{k})t+ \theta(\mathbf{k})\) denotes the rapidly varying phase with \(\theta(\mathbf{k})\) the initial phase (angle) of the complex elevation \(\hat{\zeta}(\mathbf{k})\) at the origin, \(\omega(\mathbf{k})\) denotes the angular frequency of wave \(\mathbf{k}\). Integration is over the whole \(\mathbf{k}\) plane. Without the detailed derivations, this paper employs the Direct Integration Method (DIM) developed by Li and Ellingsen [47], which provides a shear-modified dispersion relation \(\omega=\omega(\mathbf{k})\). The dispersion relation is solved numerically together with the linear wave fields \(\mathbf{u}^{(1)}\), \(w^{(1)}\), and \(P^{(1)}\).
The linear velocity and pressure in the physical plane can be obtained through an inverse Fourier transform as follows
\[\left[\begin{array}{c}\mathbf{u}^{(1)}(\mathbf{x},z,t)\\ w^{(1)}(\mathbf{x},z,t)\\ P^{(1)}(\mathbf{x},z,t)\end{array}\right]=\mathcal{R}\left\{\begin{array}{c} \frac{1}{4\pi^{2}}\int\left[\begin{array}{c}\hat{\mathbf{u}}^{(1)}( \mathbf{k},z)\\ \hat{w}^{(1)}(\mathbf{k},z)\\ \hat{P}^{(1)}(\mathbf{k},z)\end{array}\right]\mathrm{e}^{\mathrm{i}\psi( \mathbf{k},\mathbf{x},t)}\mathrm{d}\mathbf{k}\right\}. \tag{11}\]
Arbitrary linear wave fields can then be constructed by adding monochromatic components together, in the manner of Fourier transformation. We will not consider changes in mean water level herein and set \(\eta=0\) henceforth.
### Second-order equations of motions
Inserting the solution for unknown velocity (\(\mathbf{V}\)) and surface elevation (\(\zeta\)) in a form of power series given by (9) into the boundary value problem described by (5)-(8), collecting the terms at second order in wave steepness, and eliminating the horizontal velocity (\(\mathbf{u}^{(2)}\)) and pressure (\(P^{(2)}\)) at second order leads to the following equations
\[(\partial_{t}+\mathbf{U}\cdot\nabla)\nabla_{3}^{2}w^{(2)}-\mathbf{U}^{\prime \prime}\cdot\nabla w^{(2)}=\ \mathcal{N}^{(2)}(\mathbf{x},z,t),\] (12a) for \[-\infty<z<\zeta\], \[(\partial_{t}+\mathbf{U}\cdot\nabla)^{2}\partial_{z}w^{(2)}- \mathbf{U}^{\prime}\cdot(\partial_{t}+\mathbf{U}\cdot\nabla)\nabla w^{(2)}- \nabla^{2}w^{(2)}=\ \mathcal{F}^{(2)}(\mathbf{x},z,t)\text{ for }z=0, \tag{12b}\] \[w^{(2)}=\ 0\text{ for }z\rightarrow-\infty, \tag{12c}\]
where \({\bf U}^{\prime\prime}=\partial_{zz}{\bf U}\), the forcing terms, \({\cal N}^{(2)}\) and \({\cal F}^{(2)}\), on the right hand side of (12a) and (12b) are functions of linear wave fields and are given by
\[{\cal N}^{(2)}= \nabla\cdot\left[({\bf V}^{(1)}\cdot\nabla_{3}){\bf u}^{(1)} \right]^{\prime}-\nabla^{2}\left[({\bf V}^{(1)}\cdot\nabla_{3})w^{(1)}\right], \tag{13a}\] \[{\cal F}^{(2)}= -\nabla^{2}({\bf u}^{(1)}\cdot\nabla\zeta^{(1)})-[\nabla^{2}( \partial_{t}+{\bf U}\cdot\nabla){P^{(1)}}^{\prime}-\nabla^{2}{w^{(1)}}^{ \prime}]\zeta-\zeta^{(1)}\nabla^{2}({\bf U}^{\prime}\cdot\nabla)P^{(1)}\] \[+(\partial_{t}+{\bf U}\cdot\nabla)\nabla\cdot[({\bf V}^{(1)} \cdot\nabla_{3}){\bf u}^{(1)}], \tag{13b}\]
with notation \((\cdots)^{\prime}\equiv\partial_{z}(\cdots)\). Inserting the linear solution from (11), the forcing term is then
\[{\cal N}^{(2)}= {\cal R}\left[\frac{1}{16\pi^{4}}\iint\hat{\cal N}^{(2)}({\bf k }_{1},{\bf k}_{2},{\bf x},z,t)\mathrm{d}{\bf k}_{1}\mathrm{d}{\bf k}_{2}\right], \tag{14a}\] \[{\cal F}^{(2)}= {\cal R}\left[\frac{1}{16\pi^{4}}\iint\hat{\cal F}^{(2)}({\bf k }_{1},{\bf k}_{2},{\bf x},z,t)\mathrm{d}{\bf k}_{1}\mathrm{d}{\bf k}_{2}\right], \tag{14b}\]
where \({\bf k}_{1}\) and \({\bf k}_{2}\) denote the wave vector of two different linear wave trains; the forcing terms in the Fourier space are decomposed into the two types of second-order wave interactions as (see, e.g., [11; 66])
\[\hat{\cal N}^{(2)}= \hat{\cal N}^{(2)}_{+}({\bf k}_{1},{\bf k}_{2},z)\mathrm{e}^{ \mathrm{i}(\psi_{1}+\psi_{2})}+\hat{\cal N}^{(2)}_{-}({\bf k}_{1},{\bf k}_{2},z)\mathrm{e}^{\mathrm{i}(\psi_{1}-\psi_{2})}, \tag{14c}\] \[\hat{\cal F}^{(2)}= \hat{\cal F}^{(2)}_{+}({\bf k}_{1},{\bf k}_{2},z)\mathrm{e}^{ \mathrm{i}(\psi_{1}+\psi_{2})}+\hat{\cal F}^{(2)}_{-}({\bf k}_{1},{\bf k}_{2},z)\mathrm{e}^{\mathrm{i}(\psi_{1}-\psi_{2})}, \tag{14d}\]
where the subscripts '\(+\)' or '\(\cdot\)' denote the components for the superharmonics and subharmonics, respectively; the wave phases are denoted with shorthand: \(\psi_{j}=\psi({\bf k}_{j},{\bf x},t)\); and the lengthy expressions of \(\hat{\cal N}_{\pm}\) and \(\hat{\cal F}_{\pm}\) are given in Appendix B.
With the linear velocity fields solved for by using the DIM [47], the second-order equations (12a)- (12c) for the vertical velocity \(w^{(2)}\) can be solved numerically in Fourier space. Due to the interaction of different wave components and the main harmonic components of the forcing terms (i.e. \({\cal N}^{(2)}\) and \({\cal F}^{(2)}\)) in the Fourier plane, the second-order vertical velocity
\[w^{(2)}({\bf x},z,t)={\cal R}\left[\frac{1}{16\pi^{4}}\iint\hat{w}^{(2)}({\bf k }_{1},{\bf k}_{2},{\bf x},z,t)\mathrm{d}{\bf k}_{1}\mathrm{d}{\bf k}_{2} \right]. \tag{15}\]
We can also decompose \(\hat{w}^{(2)}\) in terms corresponding to the two types of second-order wave interactions as
\[\hat{w}^{(2)}({\bf k}_{1},{\bf k}_{2},z,{\bf x},t)=\hat{w}^{(2)}_{+}({\bf k}_{ 1},{\bf k}_{2},z)\mathrm{e}^{\mathrm{i}(\psi_{1}+\psi_{2})}+\hat{w}^{(2)}_{-}( {\bf k}_{1},{\bf k}_{2},z)\mathrm{e}^{\mathrm{i}(\psi_{1}-\psi_{2})}, \tag{16}\]
Each component on the right hand side of (16) for \(\hat{w}^{(2)}\) can be solved for numerically from the boundary value problem as follows
\[\hat{w}^{(2)\prime\prime}_{\pm}-\left(|{\bf k}_{\pm}|^{2}+\frac{{\bf k}_{\pm} \cdot{\bf U}^{\prime\prime}}{{\bf k}_{\pm}\cdot{\bf U}-\omega_{\pm}}\right) \hat{w}^{(2)}_{\pm}= \frac{\hat{\cal N}^{(2)}_{\pm}}{{\bf k}_{\pm}\cdot{\bf U}-\omega_{\pm}}, \tag{17a}\]
for \(-\infty<z<0\), where \({\bf k}_{\pm}={\bf k}_{1}\pm{\bf k}_{2}\), \(\omega_{\pm}=\omega({\bf k}_{1})\pm\omega({\bf k}_{2})\), and boundary conditions
\[-({\bf k}_{\pm}\cdot{\bf U}-\omega_{\pm})^{2}\partial_{z}\hat{w}_{ \pm}^{(2)}+\big{[}{\bf k}_{\pm}\cdot{\bf U}^{\prime}({\bf k}_{\pm}\cdot{\bf U} -\omega_{\pm})+|{\bf k}_{\pm}|^{2}\big{]}\hat{w}_{\pm}^{(2)} = \hat{\mathcal{F}}_{\pm}^{(2)}({\bf k}_{\pm},z)\ \mbox{for}\ z=\eta, \tag{17b}\] \[\hat{w}_{\pm}^{(2)} = 0\ \mbox{for}\ z\to-\infty. \tag{17c}\]
In our problem setting the waves obtained from the second-order boundary value problem (17a,b,c) are bound since they do not satisfy the linear dispersion relation and can only propagate together with their linear free contents. Moreover, with the linear free waves obtained, the second-order ordinary equation (17a) with two boundary conditions (17b,c) can be solved for numerically with a finite difference method where a central Euler approximation to the second-order derivative, \(\hat{w}_{\pm}^{(2)\prime\prime}\), was used in this paper. Especially for directionally spread irregular waves in a random sea, we remark that the numerical estimation of double Fourier integrals in a form as (14a,b) is computationally expensive for statistical analysis. Nevertheless, the framework developed here can be easily reformulated such that a pseudo-spectral method for the second-order interaction of waves in a vertically sheared current can be used, following papers, e.g., [67] and [68] for a high-order spectral method and [69] for a semianalytical approach. In doing so, it allows for reducing the computational operations of \(\mathcal{O}(N_{g}^{2})\) to \(\mathcal{O}(N_{g}{\rm ln}N_{g})\), with \(N_{g}\) the total number of discrete points chosen for the grid of a computational domain.
The second-order wave surface elevation \(\zeta^{(2)}\) can be obtained from the following kinematic boundary condition
\[(\partial_{t}+{\bf U}\cdot\nabla)\zeta^{(2)}=w^{(2)}+\zeta^{(1)}w^{(1)\prime} -\frac{1}{2}{\bf U}^{\prime}\cdot\nabla\big{(}\zeta^{(1)}\big{)}^{2}-{\bf u} ^{(1)}\cdot\nabla\zeta^{(1)}, \tag{18}\]
which leads to the surface elevation \(\zeta^{(2)}\) given by
\[\zeta^{(2)}({\bf x},t) = \mathcal{R}\left[\frac{1}{16\pi^{2}}\iint\hat{\zeta}^{(2)}({\bf k }_{1},{\bf k}_{2};{\bf x},t){\rm d}{\bf k}_{1}{\rm d}{\bf k}_{2}\right]\ \mbox{with} \tag{19a}\] \[\hat{\zeta}^{(2)} = \hat{\zeta}_{+}^{(2)}({\bf k}_{1},{\bf k}_{2}){\rm e}^{{\rm i}( \psi_{1}+\psi_{2})}+\hat{\zeta}_{-}^{(2)}({\bf k}_{1},{\bf k}_{2}){\rm e}^{{ \rm i}(\psi_{1}-\psi_{2})}, \tag{19b}\]
where the elevation \(\hat{\zeta}_{\pm}^{(2)}\) is obtained from (18) in the Fourier plane through substituting the vertical velocity \(w^{(2)}\) and the linear wave fields \({\bf u}^{(1)}\) and \(\zeta^{(1)}\). It's noteworthy that for \({\bf k}_{1}={\bf k}_{2}\) the super-harmonics (\(\hat{\zeta}_{+}^{(2)}\)) reduce to the well-known second-order Stokes waves. The sub-harmonics (\(\hat{\zeta}_{-}^{(2)}\)) become a constant, which refers to a mean water level and is ignored in our experiment.
### Notation in the frequency domain
The theory in SSII so far was formulated in reciprocal horizontal (\({\bf k}\)) space. Often it is more convenient in practice to use a frequency domain formulation, for instance when working with
power spectra, from time series from wave buoys, say. In the presence of a vertically sheared current the dispersion relation \(\omega=\omega(\mathbf{k})\) is anisotropic in any reference system, i.e., \(\omega\) is always a function of the direction of \(\mathbf{k}\), not only its modulus. This introduces subtleties in interpreting nondirectional wave frequency data in the presence of a sheared current as wavelength cannot be inferred from frequency alone. We herein work in two dimensions, i.e., waves propagating with known direction either along or against the current, thus eschewing this potential complication.
The linear and quadratic-order elevations are denoted
\[\zeta^{(1)}(\mathbf{x},t)= \ \mathcal{R}\left(\int a(\omega)\mathrm{e}^{\mathrm{i}\psi}\mathrm{ d}\omega\right), \tag{20a}\] \[\zeta^{(2)}(\mathbf{x},t)= \ \mathcal{R}\left\{\iint a_{1}a_{2}\left[\hat{A}_{12}^{+}\mathrm{e }^{\mathrm{i}(\psi_{1}+\psi_{2})}+\hat{A}_{12}^{-}\mathrm{e}^{\mathrm{i}(\psi _{1}-\psi_{2})}\right]\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}\right\}. \tag{20b}\]
where \(a(\omega)\) denotes the linear (real) amplitude of a wave with frequency \(\omega\) and complex phase \(\psi(\omega)=\mathbf{k}\cdot\mathbf{x}-\omega t+\theta(\omega)\), where we solve the dispersion relation \(\omega=\omega(\mathbf{k})\) for the wave vector with a given frequency using the DIM method as noted. The following notations are used: \(a_{n}=a(\omega_{n})\), \(\psi_{n}=\psi(\omega_{n})\), \(\hat{A}_{12}^{\pm}=\hat{A}^{\pm}(\omega_{1},\omega_{2})\) with
\[\hat{A}^{\pm}(\omega_{1},\omega_{2})=\frac{|\hat{\zeta}_{\pm}^{(2)}(\omega_{1 },\omega_{2})|}{a_{1}a_{2}}, \tag{20c}\]
where \(\hat{\zeta}_{\pm}^{(2)}\) was given by (19b) with the difference that it is expressed here in the frequency domain instead.
## III Waves of a narrow bandwidth
In this section we present the skewness and probability density function of the surface displacement and wave crests in the special case where the bandwidth of the wave spectrum is narrow. We now use the frequency-domain formulation of SSII.4. Consider an ensemble of waves described in the form (20) where the amplitude \(a(\omega)\) becomes an independent random variable denoted by \(\tilde{a}(\omega)\) which follows a Rayleigh distribution based on a spectrum \(S(\omega)\) and where the phase \(\theta\) becomes another independent random variable, \(\tilde{\theta}\), which is uniformly distributed in the range \([0,2\pi)\). Therefore, \(\zeta(\mathbf{x},t)\rightarrow\tilde{\zeta}(\tilde{a}(\omega),\tilde{\theta}( \omega))\). The \(j\)-th spectral moment \(m_{j}\) is defined as
\[m_{j}=\int\omega^{j}S(\omega)\mathrm{d}\omega;\ \ j\in\{0,1,2,...\}. \tag{21}\]
Assuming zero mean water level as before, the standard deviation, \(\sigma\), and skewness, \(\lambda_{3}\), of the surface elevation are
\[\sigma=\sqrt{\langle\tilde{\zeta}^{2}\rangle}\ \ \text{and}\ \ \lambda_{3}=\langle\tilde{\zeta}^{3}\rangle/\sigma^{3},\] (22a,b)
where \(\langle...\rangle\) denotes the expectation value of random variables. Assuming the energy spectrum \(S(\omega)\) to have a narrow bandwidth (\(\nu=\sqrt{1-m_{2}^{2}/(m_{0}m_{4})}\ll 1\)), we follow the detailed derivations of Fedele and Tayfun [23] using the elevations (20a,b), and obtain to \(\mathcal{O}(\epsilon)\)
\[\sigma^{2}=m_{0}\text{ and }\lambda_{3}=6\sigma\hat{A}_{mm}^{+},\] (23a,b)
where \(\hat{A}_{mm}^{+}=\hat{A}(\omega_{m},\omega_{m})\) denotes the second-order superharmonic amplitude of the spectral mean wave, with \(\omega_{m}\) the spectral mean frequency given by
\[\omega_{m}=m_{1}/m_{0}. \tag{24}\]
The skewness given by (23b) agrees with Fedele and Tayfun [23], Srokosz and Longuet-Higgins [70] and Li _et al._[10] for waves in the absence of a shear current, which is clear when noting that the superharmonic amplitude \(\hat{A}_{mm}^{+}\) can be written as \(k_{m}/2\equiv\omega_{m}^{2}/(2g)\) in the case for second-order deepwater Stokes waves (see, e.g., [11]). It is different from Fedele and Tayfun [23] to the extent that it does not account for the effect of bandwidth as it is not so straightforward due to a shear current. Nevertheless, it allows us to take into account the effect of a shear current to some extent. Especially, if all linear waves follow the same power energy spectrum with a narrow bandwidth, i.e., \(m_{j}\) are identical for all cases, the spectral mean given by (24) is identical regardless of a shear current. A shear current affects the skewness given by (23b) through the second-order superharmonic amplitude of the spectral mean wave, compared with the cases in the absence.
Following Longuet-Higgins [12], we obtain that the normalized surface displacements follow the distribution
\[p_{\zeta}(\tilde{\zeta})=\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\tilde{\zeta}^{2}/ 2}\left[1+\frac{\lambda_{3}}{6}\tilde{\zeta}(\tilde{\zeta}^{2}-3)\right]. \tag{25}\]
For linear waves, where \(\lambda_{3}=0\), expression (25) becomes a Gaussian distribution. Different from Longuet-Higgins [12], the probability density function given by (25) can account for the effect of a shear current due to that the skewness \(\lambda_{3}\) is modified according to (23b) which considers the effect of a shear current.
Similarly, following Forristall [17], the 'exceedance probability', i.e., the probability that a randomly chosen wave crest \(X_{c}\) exceeds the value \(\tilde{\zeta}_{c}\), is found as
\[P(X_{c}>\tilde{\zeta}_{c})=\exp\left[-\frac{1}{8(\hat{A}_{mm}^{+}\sigma)^{2}} \left(\sqrt{1+\frac{16\tilde{\zeta}_{c}}{H_{s}}\hat{A}_{mm}^{+}\sigma}-1 \right)^{2}\right], \tag{26}\]
where \(H_{s}\) is the significant wave height. The exceedance probability given by (26) agrees with (2.12) by Li _et al._[10] with the same chosen notations whereas the main difference lies in that the
effect of a shear current enters here via the superharmonic amplitude of the spectral mean wave, \(\hat{A}^{+}_{mm}\). In the limit of infinitesimal wave, i.e., \(m_{0}\to 0^{+}\), the exceedance probability of wave crest becomes
\[P(X_{c}>\tilde{\zeta}_{c})=\exp\left(-8\frac{\tilde{\zeta}_{c}^{2}}{H_{s}^{2}} \right), \tag{27}\]
which is the Rayleigh distribution as expected. For second-order deepwater Stokes waves in the absence of a shear current which admits \(\hat{A}^{+}_{mm}=k_{m}/2\equiv\omega_{m}^{2}/(2g)\), the exceedance probability given by (26) is identical to eq.(4) in Forristall [17]. We will refer repeatedly to (25) and (26) in section V.2.
## IV Numerical setup
In our simulations, we generate two-dimensional (long-crested or uni-directional) waves from realistic spectra. Doing so implies that the possible triad resonant interactions in three dimensions considered in previous papers, e.g., [38; 58; 71] are assumed negligible in the simulations. We choose the characteristic velocity, \(u_{c}^{*}=g/\omega_{p}^{*}\), as defined in SSII.1. Here, \(\omega_{p}^{*}\) is the peak frequency of the spectrum; although \(\omega_{p}=1\) by definition, we find it instructive to retain it in some equations below.
We begin by defining the terms following and opposing shear for two-dimensional flow, i.e., where all waves propagate parallel or antiparallel to the mean current. We will assume that waves travel along the positive \(x\) axis. We then define
\[\bullet\quad\mbox{Following shear: }U^{\prime}(z)<0;\qquad\qquad\bullet\quad \mbox{Opposing shear: }U^{\prime}(z)>0.\]
Following (opposing) shear corresponds to the situation where the flow increases (decreases) in the direction of propagation with increasing depth.
Note carefully the distinction between following (opposing) shear and following (opposing) current. When seen in an Earth-fixed reference system, currents in nature are often strongest near the surface and decrease to zero at larger depths, such as in the Columbia River Mouth current we regard in section V.5. In such a case a "following surface current" \(U(z)>0\) would correspond to opposing shear and _vice versa_. For clarity of comparison between cases we shall work in a surface-following frame and, therefore, assume \(U(0)=0\), in which case following shear implies positive \(U(z)\) for a monotonically varying \(U\). Doing so allows us to focus only on the effects due to the profile shear and curvature of a current.
### Realisation of random seas states for linear waves
We follow Tayfun [13] and Tucker _et al._[65] for the realisation of random sea states, which assumes Rayleigh distributed amplitude of linear waves and uniformly distributed wave phases in the range of \([0,2\pi)\). The energy spectrum we choose for computation is JONSWAP spectrum [72] with a peak enhancement (or peakedness) parameter of \(\gamma=3.3\) and moderately narrow bandwidth[73; 74], which is shown in figure 1(a).
The JONSWAP spectrum is given by (recall that \(\omega_{p}=1\))
\[S_{J}(\omega)=\frac{\tilde{\alpha}_{J}}{\omega^{5}}\exp{\left[-1.25\omega^{-4} \right]}\gamma^{b(\omega)}, \tag{28}\]
where the peak enhancement factor \(\gamma\) appears with an exponent
\[b(\omega)=\exp{\left[-\frac{(\omega-1)^{2}}{2\sigma_{J}^{2}}\right]}, \tag{29}\]
and
\[\sigma_{J}=\begin{cases}0.07,&\omega\leq 1\\ 0.09,&\omega>1.\end{cases} \tag{30}\]
The parameter \(\tilde{\alpha}_{J}\) is chosen such that the JONSWAP spectrum is fixed for all numerical cases, i.e., independent of a current profile. The frequency is truncated at \(0.01\omega_{p}\) and \(2.6\omega_{p}\). The bandwidth parameter is defined as
\[\nu=\sqrt{1-\frac{m_{2}^{2}}{m_{0}m_{4}}} \tag{31}\]
and here \(\nu=0.5284\). For another widely used bandwidth parameter \(\nu_{L}=\sqrt{m_{0}m_{2}/m_{1}^{2}-1}\) proposed by Longuet-Higgins [75], the value becomes \(0.2689\). We choose bulk steepness \(\epsilon=\frac{1}{2}H_{s}=0.14\) in all cases. As noted, the peak frequency (\(\omega_{p}=1\)), significant wave height (\(H_{s}\)), and the moments (\(m_{j}\)) of the JONSWAP spectrum are fixed for all cases, regardless of the profile of a shear current. However, the spectrum peak wavenumber \(k_{p}\equiv k(\omega_{p})=k(1)\neq 1\) in the presence of a current, since the linear dispersion relation \(k(\omega)\) depends on \(U(z)\), as explained in SSII and SSIII.
Once the input spectrum is determined, the amplitudes \(a_{i}\) of a total of \(N_{s}\) linear elementary waves are generated with a prescribed significant wave height, with
\[\sum_{i=1}^{N_{s}}\frac{\tilde{a}_{i}^{2}}{2}=\int_{\omega}S(\omega)\mathrm{d }\omega\text{ and }\zeta^{(1)}(x,t)=\sum_{i=1}^{N_{s}}\tilde{a}_{i}\cos(k_{i}x-\omega_{i}t+ \tilde{\theta}_{i}), \tag{32}\]
where the energy spectrum is discretised with unequal frequency intervals and an identical area of \(N_{s}\) energy bins (i.e., constant \(S(\omega_{i})\mathrm{d}\omega_{i}\)). For a train of random waves, we assume the amplitude \(\tilde{a}_{i}\) follows a Rayleigh distribution and the phase \(\tilde{\theta}\) a uniform distribution in the range \([0,2\pi)\) similar to SSIII and Tayfun [15]. The wave numbers \(k_{i}\) are found numerically from \(\omega_{i}\) using the DIM algorithm as described. We especially computed the temporal evolution of the linear surface elevation at \(x=0\) and then, the second-order correction of the wave surface are calculated from (19a) and (19b).
We also make a flow diagram of numerical implementations, which is shown in Appendix A. In our simulations, 128 elementary waves are generated from the relevant input wave spectra and ran from \(0\leq\)t\(\leq 5638\). 2000 realizations were simulated to assure that the skewness of the wave surface elevation was converged.
### Current profiles and cases considered
We consider three different current profiles with different parameters, which are typical of the open ocean, including an exponential profile, a linearly sheared current, and one that was measured at the mouth of Columbia River from Zippel & Thomson [54], as shown in figure 1(b) and (c).
#### iii.2.1 Model profiles
The exponential and linear profile of shear current are parameterized as
\[\mathbf{U}_{\mathrm{exp}}(z)=\beta[\mathrm{exp}(\alpha z)-1]\mathbf{e}_{x} \text{ and }\mathbf{U}_{\mathrm{lin}}=Sz\mathbf{e}_{x},\] (33a,b)
respectively, where \(\mathbf{e}_{x}\) is a unit vector along the positive \(x\) axis, the subscripts 'exp' and 'L' denote the exponential and linear profile, respectively, \(\alpha\) (\(\alpha>0\)), \(\beta\), and \(S\) are dimensionless parameters that define the magnitude and shear strength of a current profile relative to the peak wave parameters. Note that we choose a reference system following the free surface so that \(\mathbf{U}(0)=0\). This eschews arbitrary Doppler shift terms which would clutter the formalism, reduces the number of free parameters, and makes results from different profiles immediately comparable. The choice also emphasizes that it is the shear \(U^{\prime}(z)\) and curvature \(U^{\prime\prime}(z)\) which cause statistics to be altered, not the strength of the current itself. The surface shear is obtained from (33)
\[\mathbf{U}^{\prime}_{\mathrm{exp}}(0)=\alpha\beta\mathbf{e}_{x}\text{ and } \mathbf{U}^{\prime}_{\mathrm{lin}}(0)=S\mathbf{e}_{x},\] (34a,b)
which denote the profile shear of an exponential and linearly sheared current at still water surface, respectively.
Recall that following (opposing) shear correspond to \(U^{\prime}(z)<0\) (\(>0\)). We wish our model current to have strong, but not unreasonable vertical shear. To determine how strongly the current
Figure 1: (a): JONSWAP power energy spectrum of linear waves with nondimensional peak frequency \(\omega_{p}=1\) and bulk steepness \(\epsilon=0.14\); (b) examples of linear and exponential (‘Exp.’) shear profiles where both opposing (‘Opp.’) and following (‘F.’) shear are shown; (c) two tidal current profiles from ref. Zippel and Thomson [54] mearured at the mouth of Columbia river (‘CR’), during ebb tide (following shear, ‘F.’), and flood (mostly opposing shear, ‘Opp.’), respectively. Note that in an Earth-fixed coordinate system (see Fig. 3 of [54]) these correspond to opposing and following surface currents, respectively. Dashed lines are extrapolations from \(z=1.35\,\mathrm{m}\) to the surface; (d) wave–averaged shear \(|\delta(k)|\) for the two profiles in panel c; (e) extract of the time series of wave surface elevation for illustration, here without current.
shear affects the dispersion of a wave of wave number \(k^{*}\) or frequency \(\omega^{*}\) (whichever is known), the proper parameter to consider is the wave-weighted depth-averaged shear [60], respectively
\[\delta=\frac{1}{c_{0}^{*}}\int_{-\infty}^{0}{U^{*}}^{\prime}(z^{*})\mathrm{e}^{2 k^{*}z^{*}}\mathrm{d}z^{*}=\sqrt{k}\int_{-\infty}^{0}{U^{\prime}}(z)\mathrm{e}^{ 2kz}\mathrm{d}z \tag{35}\]
nondimensionlized as explained in Section II.1, and \(c_{0}^{*}=\sqrt{g/k^{*}}\). Inserting \(U^{\prime}(z)=\alpha\beta\exp(2\alpha z)\) gives
\[|\delta|=\frac{|\alpha\beta|\sqrt{k}}{\alpha+2k}, \tag{36}\]
whose maximum value is found at \(k=\alpha/2\) and in either case, \(|\delta|_{\mathrm{max}}=|\alpha\beta|/\sqrt{8}\). In the following sections we use \(\alpha=2.5\) and \(|\beta|\leq 0.3\) giving \(|\delta|_{\mathrm{max}}\lesssim 0.17\).
#### ii.2.2 Profile from the Mouth of Columbia River
The profiles of tidal currents in the Mouth of the Columbia River have been used as a test-case in a wide array of studies of wave-shear current interactions (e.g. [76; 77; 78; 79; 46; 80; 47; 48; 49; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80]) due to the availability of high quality current profile measurements [54; 64] and strong vertical shear. Herein we use the profiles measured by Zippel and Thomson [54] using an acoustic Doppler current profiler (ADCP) mounted on a drifter. The currents were measured between \(1.35\,\mathrm{m}\) and \(25\,\mathrm{m}\) depth, but we require profiles ranging all the way to the undisturbed surface level. What the profile might look like in the top \(1.35\,\mathrm{m}\) is not obvious; the shear strength can drop sharply closer to the surface [81], but could also increase all the way to the top centimetres [62]. We use a polynomial extrapolation as shown in figure 1c; we show in appendix D that two other common approaches produce no discernable difference in the resulting skewness. The current profiles reported in Zippel and Thomson [54] and shown in figure Fig. 8a are fitted with a 7th order polynomial to the surface. The wave-averaged dimensionless shear \(\delta\) of Eq. (35) for the two profiles in Fig. 1c are seen in Fig. 1d, peaking near \(0.095\) for the following current.
Note that the currents taken from Zippel and Thomson [54] are not extreme for the location -- the shear current used in e.g. Li _et al._[82] taken from the measurements during the RISE project [64] peaks at a value \(\delta\approx 0.19\), more than our strongest exponential model current. For comparison with the results of Zippel and Thomson [54] for ebb and flow respectively, we choose the more conservative profiles in the latter.
We remark that Zakharov and Shrira [58] proposed a set of analytical theory for second-order wave-shear current problem with the assumptions \(U^{\prime}<0\) and \(U_{\mathrm{max}}/c\ll 1\). Here, \(U_{\mathrm{max}}\) and \(c\) refer
to the maximum velocity of shear current and phase velocity of surface wave, respectively. From Fig.1c the parameter \(U_{\rm max}/c\) of Columbia River current for peak wave could reach 0.2. Hence, the theory by Zakharov and Shrira [58] is not expected to be quantitatively accurate for the Columbia River current cases considered herein.
## V Results
We present second order statistical quantities for waves on model shear currents, generalising a number of classical results. The example for time series of wave surface elevation is shown in Fig. 1(e). All the statistical quantities are based on very long time series.
### The distribution of wave surface elevation
In this section we examine the effects of sub-surface shear on the distribution of surface elevation to second order in steepness. We compare the case of no current to cases with following and opposing shear. We also show comparisons of the same case with shear between the broadband and narrow-band theory presented in SSII and SSIII, respectively. A moderately narrowband spectrum is considered, with the linear wave field amplitudes chosen from a Gaussian distribution with zero mean and variance \(\sigma^{2}\).
Fig.2 plots the numerically calculated PDFs of wave surface elevation in the presence of a model current (equation (33)a) varying exponentially with depth, comparing our numerical results based on the broad-band theory presented in SSII, together with different theoretical predictions: a Gaussian distribution, and theoretical predictions based on a narrow-band assumption presented in SSIII. We firstly discuss the results shown by Fig.(2a). When both second-order corrections and shear are omitted, the numerically calculated PDF (diamond symbols) should coincide with the Gaussian input distribution (zero mean, variance \(\sigma^{2}\)) which indeed it does, as expected. The probability of amplitudes greater than about two standard deviations from the mean are decreased for negative values (deep troughs) and increased for positive (high crests), conforming with the known properties of second-order Stokes waves: the wave crests get higher and wave troughs get flatter.
The presence of opposing shear \(U^{\prime}(z)>0\) enhances the wave crests and flatten the wave troughs compared to no current, while following shear current has the opposite effects. The effect on second-order statistics from the shear is considerable in the range of larger wave crests (\(>2\sigma\)
but modest for wave troughs (negative elevation) in this case.
A comparison of the probability density function of surface elevation for the cases in the presence of shear is shown in Fig.(2b) comparing the numerical results based on the full theory of SSII and the narrow-band approximation in SSIII. It is seen that the narrow-band assumption agrees with the broad-band theory up to three and two standard deviations for the cases with following ('F. shear') and opposing shear ('Opp. shear'), respectively; for following shear the approximation would be good enough for most practical purposes, except extreme statistics. The narrow-band approximation underestimates the probability of the most extreme events in both cases, but to very varying degrees as the figure shows.
Figure 2: Probability density function (PDF) of wave surface elevation for a moderately narrowband Gaussian input spectrum assuming the exponential current profile (33a) with \(\beta\) the magnitude of the shear at a still water surface. Numerical results for \(\beta=-0.3\) (following shear, ‘F. shear’) and \(\beta=0.3\) (opposing shear: ‘Opp. shear’) are compared to (a) the linear prediction and the case without current, and (b) the narrow-band (N.B.) theory based on (25).
### The distribution of wave maxima and crest height
The crest height is conventionally defined as the highest surface elevation reached inside discrete time intervals. Within each time interval, the surface elevation is above the mean-surface level, \(\zeta>0\), i.e., delimited by consecutive zero crossings \(\zeta(t)=0\) so that \(\zeta^{\prime}(t)>0\) (\(<0\)) at the beginning (end). This contrasts, in general, with a _surface elevation maxima_\(\zeta_{m}\), which is any point where \(\zeta^{\prime}(t)=0\) and \(\zeta^{\prime\prime}(t)<0\). Surface elevation maxima can be negative for a broad-band spectrum, whereas for a sufficiently narrow spectrum, the two are positive and coincide: every maximum is also a wave crest.
As discussed by Goda [83, Chapter 2], when the spectrum is not narrow there is no universal and unique definition of wave height in a time series. The most common definition based on zero-crossings described above is theoretically somewhat unsatisfactory in a broadband setting; a more theoretically coherent method proposed by Janssen [5, 84] based on the envelope of \(\zeta\) is also in use [85]. For theoretical derivations the envelope procedure becomes more cumbersome for weakly non-linear waves, requiring expressions for third and fourth statistical moments, needed to adequately describe a generic wave distribution. In the following we use the customary definition using zero-crossing, as described above, bearing in mind that the identification of individual waves, and hence its distribution of maxima, will carry some dependence on the spectral shape which vanishes in the narrow-band limit.
For a narrow frequency spectrum according to linear theory, the dimensionless wave crest heights \(\tilde{\zeta}_{c}\), normalised by significant wave height \(H_{s}\), is distributed according to the Rayleigh probability function as given by (27). It is difficult, however, to determine theoretically the probability distribution of crest heights if the waves have a broad frequency spectrum. Hence, Cartwright _et al._[86] made a compromise by calculating the distribution of surface elevation maxima denoted by \(\zeta_{m}\), adapting the theory of Rice [87] from in electrical signal processing to an ocean waves setting. Their result based on linear theory for a broadband spectrum is
\[p(\xi)=\frac{1}{\sqrt{2\pi}}\nu\exp\left(-\frac{\xi^{2}}{2\nu^{2}}\right)+ \frac{\xi\sqrt{1-\nu^{2}}}{2}\exp\left(-\frac{1}{2}\xi^{2}\right)\left[1+ \mathrm{erf}\left(\frac{\xi\sqrt{1-\nu^{2}}}{\sqrt{2}\nu}\right)\right], \tag{37}\]
where \(\xi=\zeta_{m}/\sigma\) denotes the normalised maxima, the bandwidth parameter \(\nu\) is defined in (31), \(m_{j}\) is the \(j\)-th moment of the energy spectrum given by (21), and \(\mathrm{erf}\) is the error function.
Fig. 3 shows the PDF of the surface elevation maxima for linear and nonlinear results. We also plot the theoretical estimates with (37), which is given by solid line in the figure. When nonlinear effects and shear are both omitted, the numerically calculated PDF (diamond symbols)
should coincide with equation (37), which indeed it does as the figure shows. The second-order results show increased probability of large wave maxima in all cases. Notice that negative-valued surface maxima occurs for a broadband spectrum, corresponding to nonzero \(p(\xi)\) for \(\xi<0\). The probability of a negative maxima increases monotonically with bandwidth parameter \(\nu\).
The most prominent nonlinear effect in Fig. 3 is for opposing shear, where probability for large maxima above approximately two standard deviations is enhanced in our simulation, whereas maxima below this threshold are made less probable. The current with following shear has the opposite influence. This phenomenon is consistent with the PDF of wave surface elevation studied in SSV.1.
There exists a few commonly used expressions for crest height distribution obtained by empirical fitting, theoretical considerations or parameterization [17; 23; 88; 89; 90; 91; 92]. One example we use in this section is the distribution derived by Tayfun [14] for a narrow-band spectrum, which corresponds to our narrow-band equation (26) in the limiting case of no current, i.e., \(k_{m}^{*}\to k_{m0}^{*}=\omega_{m}^{*2}/g\) (shear-free dispersion relation in nondimensional units). To the best of our knowledge, theoretical expressions for wave crest distribution with a broad-band frequency spectrum have not been reported.
Fig. 4 shows the numerical PDF and exceedance probability of the scaled crest height compared to the Rayleigh and Tayfun distributions. Notice in Fig. (4a) that for very low crests \(\tilde{\zeta}_{c}\lesssim 0.1H_{s}\) the probability density of wave crest height deviates noticeably from the Rayleigh curve, consistent with Fig. 3. The reason is that finite bandwidth allows negative maxima (hence a finite probability density at zero crest height), whereas the narrow-band Rayleigh distribution only allows positive maxima. The physical significance of this difference is perhaps not so high being primarily a result
Figure 3: Probability density function of the dimensionless maxima (\(\xi=\zeta_{m}/\sigma\)) of the wave elevation. The theoretical estimates (‘Theory’) are based on (37) and the other cases shown are the same as Fig.2a.
of the definition of a crest, referring somewhat arbitrarily to the mean water level. The tail of our numerical results without shear still agrees well with those produced by the Rayleigh distribution [23], perhaps surprising in light of the linear theory for broadband waves due to Cartwright _et al._[86]. This can be explained by noting that in the context of their theory our spectrum is still
Figure 4: Numerically calculated probability density function (panel (a)) and exceedance probability (panels (b,c,d)) for wave crests. An exponential shear profile, Eq. (33a), was assumed. (a) Linear waves based on numerical simulations and the Rayleigh probability density function; (b,c) nonlinear wave fields for varying shear strength; (d) the broad-band and narrow-band results for cases with shear based on the theory in §II and §III, respectively. We used (26) with \(\beta=0\) for the Tayfun distribution. (e) Occurrence probability of rogue wave for all the exponential shear cases in panel c.
relatively narrow, since the bandwidth parameter \(\nu\approx 0.53\) as defined in Eq. (31) is considerably smaller than unity.
It can be observed in Fig. 4b and 4c that, when nonlinear second-order corrections are accounted for, the tail of the simulated curve for the case with no shear clearly exceeds the Rayleigh distribution values, yet remain lower than the Tayfun distribution curve. This observation was also made by Fedele & Tayfun [23] who considered broadband waves without current; They showed that in that case the Tayfun distribution is an upper bound for the wave crest distribution to second order in steepness.
With the additional presence of a shear current and broader spectrum, crest distributions can clearly exceed that of Tayfun. The numerical results show substantial differences between the three currents considered, consistent with the general trend observed before: opposing shear makes high crests more probable and _vice versa_. The gray dashed vertical line in Fig. 4 refers to the conventional criterion for rogue waves, which is \(\tilde{\zeta}_{c}/H_{s}=1.25\)[93]. Compared with the no-shear current case, the opposing shear current leads to significant enhancement in the occurrence probability of rogue wave, as shown in Fig. 4e. The presence of following shear current has the opposite influence. The exceedance probability increases monotonously as a function of the shear strength \(\beta\), which is shown in Figure (4b,c).
We note in passing, however, that whereas the probability of _unusually high_ (rogue) waves is decreased on following shear, the significant wave height itself will often be increased. A typical situation where this occurs is when the shear current, measured in a land-fixed reference system, has its greatest velocity at the surface. In this case the current itself is opposing in an earth-fixed frame of reference, so waves generated elsewhere will steepen as they encounter the current. Thus the expectation in many real scenarios would be that following shear makes for rougher seas overall, whereas with opposing shear, while calmer on the whole, have an increased probability of _surprisingly_ high crests. This point was discussed in depth by Hjelmervik & Trulsen [30].
Fig. 4d compares the exceedance probability of wave crest between the narrow-band predictions and numerical results for the cases with a shear current, the former of which are obtained by using (26). We observe that the narrow-band assumption leads to a small and large overestimate of the occurrence probability of wave crest for the case with a following and opposing shear current, respectively. The differences for the following current are nearly negligible, as being consistent with Fig.2b, but are much more pronounced for the opposing shear case. Fig. 4d suggests aligned conclusion with Fedele and Tayfun [23] in which it is stated that the narrow-band assumption would produce an upper bound of the exceedance probability of wave crest as aforementioned.
Since the effect of current shear on waves depend on both the shift in wavelength as reflected from the linear dispersion relation as well as the amplitude of the second-order superharmonic bound waves, the overall effect of current on waves of a broad-band spectrum will in general differ in a non-trivial way from that only on the amplitude of the spectral mean wave, \(\hat{A}^{+}_{mm}\). As a result, the assumption of narrow bandwidth seems to lead to larger overestimate for opposing shear compared to the case of a following shear.
### The distribution of maximum wave crest
Consider next the distribution of the height of the highest wave crest among a randomly chosen sequence of \(N\) consecutive waves, where a 'wave' in this context is a time interval wherein the surface elevation contains one maximum and one minimum. A long time ago Longuet-Higgins Longuet (1963) derived an expression for maximum wave crest distribution based on linear waves with a narrow band frequency spectrum. Cartwright _et al._ Cartwright et al. (2009) extended the theory to allow for a broadband spectrum, still in the linear wave regime. More recently, the Gumbel distribution was used to solve this problem up to second order Gumbel (1963); Gumbel (1963); Gumbel (1964); for a linear narrow-band process, the expressions in these references are the same. In this section we use the expression from Cartwright _et al._ Cartwright et al. (2009) for comparison:
\[\frac{\zeta_{\text{max}}}{\sigma}=\sqrt{2\ln\left[(1-\nu^{2})^{\frac{1}{2}}N \right]}+\gamma_{E}/\sqrt{2\ln\left[(1-\nu^{2})^{\frac{1}{2}}N\right]}, \tag{38}\]
where \(\zeta_{\text{max}}\) is the maximum crest height from a continuous wave train, \(\gamma_{E}\approx 0.5772\) is Euler's constant.
Fig. 5 gives the comparison of largest crest height between our numerical results and equation (38). Each point is obtained as follows: a time series containing \(2\times 10^{6}\) waves is divided into 160 segments. From each segment a sequence of \(N\) consecutive waves is chosen randomly from which the highest crest is found, then the average is taken over all the highest crests and plotted in the figure. Fig. 5a shows that, once again, our simulated results of linear wave fields fit well with the theoretical solution.
Compared with linear results, second order correction makes a considerable contribution to largest crest heights. The largest crest heights rise by around 10% to 20%. A similar phenomenon was observed by Socquet-Juglard _et al._ Socquet-Juglard et al. (2009), who used a narrow-band frequency spectrum and found the largest crest heights of nonlinear wave field increased by about 20% compared with linear wave fields. Moreover, it is clear that the additional presence of sub-surface shear also has
notable influence on largest crest heights. The opposing and following shear current increase or decrease the largest crest heights by about \(18\%\) or \(8\%\), respectively for the case with \(\beta=0.3\) and \(\beta=-0.3\), compared with the case with no shear current. Note that the comment at the end of the previous section still applies: the current will often change a free wave surface in such a way that in absolute terms, the crest heights are actually increased by opposing shear, which is a following current in the earth-fixed frame of reference, and _vice versa_.
### Skewness
In this section, we discuss the influence of a shear current on skewness, which is a measure of the lack of symmetry. Unlike skewness, kurtosis is not expected to be well approximated by second-order theory, and therefore not included in this paper.
Skewness of second-order waves can be expressed as a function of wave steepness, which is given by equation (23) in the limiting case of a narrow-band wave spectrum. The skewness should generally depend on both the bandwidth parameter (\(\nu\)) and spectrum shape, as has been shown by Srokosz and Longuet-Higgins [70].
We consider two types of shear currents, as given in equations (33a,b). From the point of view of the waves, which can "feel" the current only down to about half a wavelength's depth, the significant difference is that a linear current has the same shear at all depths, affecting the wave dispersion for all wavelengths, whereas the exponential profile is felt strongly by the short waves with \(k\gtrsim\alpha k_{p,0}\) and hardly at all for long waves \(k\ll\alpha k_{p,0}\).
Figure 5: The average of crest height of scenes containing the largest \(N\) waves. In the figure, the theoretical predictions (the black solid line) are based on (38) for linear waves.
Fig. 6a and 6b show the skewness of linear and exponential shear current cases, respectively, calculated according to its definition given by (22b). The theoretical narrow-band predictions in solid blue lines are based on (23b) with the assumption of narrow-band waves in both the absence (i.e. \(S=0\) and \(\beta=0\) in Fig. 6a and 6b, respectively) and presence of a shear current. For both linear and exponential current cases the skewness increases monotonically with \(S\) and \(\beta\), respectively. In the range of shear strengths examined in Fig. 6, the skewness always remains positive. The strongest shear current enhances the skewness by about \(86\%\) compared with the cases in the absence of a shear current. The narrow-band assumption for the cases with an exponential shear current always leads to an overestimate of the skewness, compared with the numerical simulations due to the theory in SSII applicable to arbitrary bandwidth. In contrast, it may lead to underestimated values for the linear, following current cases in the regime where \(S\leq-0.2\). The inaccuracy induced by the narrow-band assumption is obvious, which may arise from that the JONSWAP spectrum chosen is not very narrow and that the strong profile shear can lead to a considerable change in the wavelength of all waves prescribed on the JONSWAP spectrum.
### The Mouth of the Columbia River
As a real-life example we consider the real measured data described in Section IV.2.2 to demonstrate and quantify the significant misprediction of wave statistics that would result from neglecting the current's vertical shear. The currents considered, adapted from figure 3 of Zippel & Thomson [54] are shown in Fig. 8a, using the same color coding as in said figure. The surface current was subtracted and the profiles extended to the surface as explained in section IV.2.2. As input wave
Figure 6: Skewness of the wave surface elevation for the cases with a linear shear current (a) and exponential shear current (b). The narrow-band theoretical predictions in solid black lines are based on (23b). The dashed line is the no-shear case, for reference.
spectrum we fit a JONSWAP spectrum with bandwidth parameter \(\nu=0.6618\) to a representative example among the manywave spectra measured by Zippel and Thomson [54], shown in figure 7. The fit is not excellent, but sufficient to provide a representative example.
Figure 1d shows the weak-shear parameter \(\delta(\omega)\) when \(\omega\) is the given parameter; we argue in appendix E that the appropriate value in this case is \(\delta_{\omega}(\omega)=2\delta(\omega^{2}/g)\) where \(\delta(k)\) is defined in (35).
Figure 8: Skewness of wave surface elevation with Columbia River current and wave spectrum data (a) Considered current profiles, reproduced with kind permission from figure 3 of [54] with the same colour coding, shifted to the surface level and with surface current subtracted. (b) Numerically obtained skewness for the measured wave spectrum of ref. [54] on the currents in panel (a), with corresponding color coding; the ascissa is the shear-shifted peak wave number with \(k_{p}=1\) corresponding to zero shear (open circle).
Figure 7: Power energy spectrum for the Columbia River wave data.
#### iv.2.1 Skewness
The skewness of simulated results with Columbia River current data are given in Fig. 8b, where \(k_{p}\) is the dimensionless peak wavenumber which depends on the shear current as aforementioned. We chose to use \(k_{p}\) as a representation of the shear strength as it expresses the amount by which the shear changes the wavelength of the wave with peak frequency.
Failure to take into account the presence of shear causes overprediction of skewness by \(\approx 24\%\) or underprediction by \(\approx 13\%\) during ebb and flood, respectively, as is shown in Fig. 8. Absolute numbers provided by a second-order theory like ours carry significant uncertainty, particularly when the spectrum is not narrow, but show a clear and consistent trend. Held together with Zippel & Thomson's conclusion that wave steepness can be mispredicted by \(\pm 20\%\) in these waters in the same conditions if shear is not accounted for [54], there is compelling evidence that shear can be highly significant to the estimation of wave statistics from measured spectra.
#### iv.2.2 Rogue wave probability
We also carried out simulations with data from Columbia River (CR) using both the wave spectrum and shear profiles measured in this location by Zippel and Thomson [54]. As usual, rogue wave probability is defined as the probability of crests exceeding \(1.25H_{s}\).
As observed for the model currents in Figure 4, opposing shear enhances the crest heights of large waves while following shear weakens them, leading to increased and decreased exceedance
Figure 9: Exceedance probability of simulated results with the current measured by Zippel & Thomson [54] in the Columbia River (CR) shown in Fig. 1c, equal to the strongest currents in either direction in Fig. 8a. The profiles of the following and opposing CR-current are shown in Figure 1c.
probability, respectively. The rogue wave probability on opposing shear (i.e., a following surface current during ebb) is increased by \(36\%\) while on following shear (opposing surface current, during flow) it is decreased by \(45\%\); from \(1.12\times 10^{-4}\) to \(6.20\times 10^{-5}\) and \(1.52\times 10^{-4}\), respectively. Given that our theory is second order only, these numbers are not quantitatively accurate, but show clearly that shear currents must be accounted for in prediction and modelling of extreme waves.
Note carefully that the rogue wave probability is the probability of _surprisingly_ high waves, as discussed by Hjelmervik and Trulsen [30]. Although rogue waves are more than twice as probable on the wave-following flow current than the wave-opposing ebb current, the significant wave height itself is typically much greater in the former case (more than twice as high in the conditions measured in [54], for instance), making for rougher conditions overall. The effect of shear is to reduce the prevalence of very large waves during ebb, a beneficial effect with respect to sealoads and maritime safety.
## VI Conclusions
In this paper, we develop the second-order (deterministic) theory using perturbation expansion, which is extended from Longuet-Higgins [11] to allow for a depth-dependent background flow whose profile shear can be strong. The new theory can be used to investigate the wave-current interaction and applicable to waves of an arbitrary bandwidth. The linear wave field is solved with the DIM method proposed by Li & Ellingsen [47]. We derived a boundary value problem for the second-order waves, which can be solved numerically. With the additional assumption of narrow-band waves, a second-order accurate statistical model is derived for the skewness, probability density function of surface elevation, and the probability distribution of wave crest, which have accounted for the presence of a depth-dependent background flow.
We carried out numerical simulations for the analysis of wave statistics and examined effects of a shear current. We used a JONSWAP spectrum and several different shear currents as input to generate linear random waves. The second-order waves are solved for numerically based our newly derived theory. The measured wave spectrum and currents from Columbia River by Zippel & Thomson [54] were also used in our simulations.
For linear wave fields the probability distribution of wave surface elevation and wave maxima and average maximum wave crest all satisfy theoretical expressions well as expected. The nonlinear wave fields show similar properties compared with well-known second-order Stokes waves. The wave crests are higher and troughs are flatter than linear wave fields. As a result, the positive tails of
the probability density function for wave surface elevation and wave maxima from nonlinear wave fields are longer than linear wave fields while the negative tails of surface elevation are shorter. Also, the largest wave crests in nonlinear wave fields are substantially greater. We found that the opposing shear currents can strengthen such 'nonlinear properties' while the following shear currents can weaken them.
We also found that the additional assumption of narrow-band waves leads to in general negligible and pronounced differences for the following- and opposing-shear case, respectively, when comparing the second-order statistical model with the more general deterministic theory which is applicable to waves with an arbitrary bandwidth.
###### Acknowledgements.
Z.B. Zheng acknowledges the support from China Scholarship Council through project 201906060137. Y. Li is supported by the Research Council of Norway (RCN) through the FRIPRO mobility project 287389. S.A. Ellingsen is supported by the European Research Council Consolidator Grant no. 101045299 (_WaTurSheD_), and the RCN grant 325114 (_iMod_). We thank Dr. Seth Zippel and Professor Jim Thomson for the use of the data collected from the Data Assimilation and Remote Sensing for Littoral Applications (DARLA) project and the Rivers and Inlets (RIVET) program (see, e.g., [54] for details). The computer code (MATLAB) used to generate our data is included as supplementary material. We thank the anonymous referees for their valuable suggestions and comments which have improved the quality of the paper.
## Appendix A Flow diagram of numerical implementations
A flow diagram of the numerical implementation used to generate statistics is shown in Figure 10.
## Appendix B The forcing terms of the Rayleigh equation
With the linear wave fields given by (11a,b,c), the nonlinear forcing terms in (14c) are expressed as
\[\hat{\mathcal{N}}_{\pm}^{(2)}= [\mathbf{k}_{\pm}\cdot\partial_{z}\mathbf{N}_{h,\pm}+k_{\pm}^{2 }N_{Rz,+}]\cos\psi_{\pm}, \tag{16a}\] \[\hat{\mathcal{F}}_{\pm}^{(2)}= [k_{\pm}^{2}N_{F1,\pm}-N_{F2,\pm}+N_{F3,\pm+}-N_{F4,\pm}-( \mathbf{U}\cdot\mathbf{k}_{\pm}-\omega_{\pm})\mathbf{k}_{\pm}\cdot N_{h,+}] \sin\psi_{\pm}, \tag{16b}\]
with \(\psi_{\pm}=\psi_{1}\pm\psi_{2}\), \(\mathbf{N}_{h,i}=[N_{Rx,i},N_{Ry,i}]\),
\[\left[\begin{array}{c}N_{Rx,\pm}\\ N_{Ry,\pm}\\ N_{Rz,\pm}\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}-(k_{1x}\hat{u}_{1 }^{(1)}\hat{u}_{2}^{(1)}\pm k_{2x}\hat{u}_{2}^{(1)}\hat{u}_{1}^{(1)}+k_{1y} \hat{u}_{1}^{(1)}\hat{v}_{2}^{(1)}\pm k_{2y}\hat{u}_{2}^{(1)}\hat{v}_{1}^{(1)} \mp\hat{u}_{1}^{(1)\prime}\hat{w}_{2}^{(1)}-\hat{u}_{2}^{(1)}\hat{w}_{1}^{(1)} )\\ -(k_{1x}\hat{v}_{1}^{(1)}\hat{u}_{2}^{(1)}\pm k_{2x}\hat{v}_{2}^{(1)}\hat{u}_{1 }^{(1)}+k_{1y}\hat{v}_{1}^{(1)}\hat{v}_{2}^{(1)}\pm k_{2y}\hat{v}_{2}^{(1)}\hat {v}_{1}^{(1)}\mp\hat{v}_{1}^{(1)\prime}\hat{w}_{2}^{(1)}-\hat{v}_{2}^{(1) \prime}\hat{w}_{1}^{(1)})\\ k_{x1}\hat{v}_{1}^{(1)}\hat{u}_{2}^{(1)}+k_{x2}\hat{w}_{2}^{(1)}\hat{u}_{1}^{(1 )}+k_{y1}\hat{w}_{1}^{(1)}\hat{v}_{2}^{(1)}+k_{y2}\hat{w}_{2}^{(1)}\hat{v}_{1} ^{(1)}\mp\hat{w}_{1}^{(1)\prime}\hat{w}_{2}^{(1)}\mp\hat{w}_{1}^{(1)}\hat{w}_{ 2}^{(1)\prime}\end{array}\right]\] (101a) and \[N_{F1\pm}= -\tfrac{1}{2}(k_{1x}\hat{u}_{2}^{(1)}\hat{\zeta}_{1}^{(1)}+k_{1y} \hat{v}_{2}^{(1)}\hat{\zeta}_{1}^{(1)}\pm k_{2x}\hat{u}_{1}^{(1)}\hat{\zeta}_{ 2}^{(1)}\pm k_{2y}\hat{v}_{1}^{(1)}\hat{\zeta}_{2}^{(1)}) \tag{101a}\] \[N_{F2\pm}= \tfrac{1}{2}(\mathbf{k}_{1}^{2}(\mathbf{k}_{1}\cdot\mathbf{U}- \omega_{1})\hat{\zeta}_{2}^{(1)}\hat{P}_{1}^{(1)\prime}\pm\mathbf{k}_{2}^{2}( \mathbf{k}_{2}\cdot\mathbf{U}-\omega_{2})\hat{\zeta}_{1}^{(1)}\hat{P}_{2}^{(1) \prime})\] (101b) \[N_{F3\pm}= -\tfrac{1}{2}(\mathbf{k}_{1}^{2}\hat{\zeta}_{2}^{(1)}\hat{w}_{1}^ {(1)\prime}\pm\mathbf{k}_{2}^{2}\hat{\zeta}_{1}^{(1)}\hat{w}_{2}^{(1)\prime})\] (101c) \[N_{F4\pm}= \tfrac{1}{2}(\mathbf{k}_{1}^{2}\mathbf{k}_{1}\cdot\mathbf{U}^{ \prime}\hat{P}_{1}^{(1)}\hat{\zeta}_{2}^{(1)}\pm\mathbf{k}_{2}^{2}\mathbf{k}_{ 2}\cdot\mathbf{U}^{\prime}\hat{P}_{2}^{(1)}\hat{\zeta}_{1}^{(1)}) \tag{101d}\]
where \(\mathbf{k}_{1}=[k_{1x},k_{1y}]\) and \(\mathbf{k}_{2}=[k_{2x},k_{2y}]\)
Figure 10: Numerical procedures of the simulation.
## Appendix C Analytical solution for linearly sheared current
We assume the shear profile is given by \(\mathbf{U}=(S_{0}z,0)\). The linear solution can be easily solved, which is expressed as [32; 38]
\[\hat{w}^{(1)}(\mathbf{k},z)= \hat{w}^{(1)}_{0}(\mathbf{k})\mathrm{e}^{kz} \tag{101a}\] \[\hat{\mathbf{u}}^{(1)}(\mathbf{k},z)= \mathrm{i}\frac{k^{2}\mathbf{U}^{\prime}+[(\mathbf{U}\cdot \mathbf{k}-\omega)k-k_{x}S_{0}]\mathbf{k}}{(\mathbf{U}\cdot\mathbf{k}-\omega) k^{2}}\hat{w}^{(1)}_{0}\mathrm{e}^{kz}\] (101b) \[\hat{P}^{(1)}(\mathbf{k},z)= -\mathrm{i}\frac{(\mathbf{U}\cdot\mathbf{k}-\omega)k-k_{x}S_{0}} {k^{2}}\hat{w}^{(1)}_{0}\mathrm{e}^{kz}\] (101c) \[\hat{w}^{(1)}_{0}(\mathbf{k})= -\mathrm{i}\hat{\zeta}^{(1)}(\mathbf{k})\omega \tag{101d}\]
where \(\mathbf{k}=(k_{x},k_{y})\), \(k=\sqrt{k_{x}^{2}+k_{y}^{2}}\) and the subscript '0' denotes the evaluation at a undisturbed surface \(z=0\). The dispersion relation for linear waves in a linearly sheared current is given by [32; 38]
\[\omega=-\frac{S_{0}k_{x}}{2k}\pm\sqrt{k+\frac{S_{0}^{2}k_{x}^{2}}{4k^{2}}}, \tag{102}\]
where '\(+\)' and '\(-\)' denotes the waves propagating 'downstream' and 'upstream' relative to the current, respectively.
Substituting the linear solution into the forcing terms of second-order equations (17), we obtain an inhomogeneous boundary value problem for the second-order vertical velocity \(w^{(2)}\). The general solution to this boundary value problem in the Fourier space should admit the form
\[\hat{w}^{(2)}_{\pm}(\mathbf{k}_{1},\mathbf{k}_{2},z)=B_{1\pm}(\mathbf{k}_{1}, \mathbf{k}_{2})\mathrm{e}^{k_{\pm}z}+\hat{w}_{cross}(\mathbf{k}_{1},\mathbf{k }_{2},z), \tag{103}\]
where the deepwater boundary condition was used, the first term on the right hand side of the equation is due to the forcing at a still water surface and the homogeneous Rayleigh equation, and \(\hat{w}_{cross}\)is a particular solution of the inhomogeneous Rayleigh equation given by [38]
\[\hat{w}_{cross}(\mathbf{k}_{1},\mathbf{k}_{2},z)= -\frac{i}{2k_{\pm}}\frac{\hat{w}^{(1)}_{0,1}\hat{w}^{(1)}_{0,2}} {k_{\pm x}S_{0}}\frac{k_{1x}k_{2y}-k_{1y}k_{2x}}{k_{1}k_{2}}\mathrm{e}^{(k_{1 }+k_{2})z}\sum_{i,j=1}^{3}\left[\frac{\pm b_{ij}}{(\xi_{i}-z)^{j-1}}\right.\] \[\left.\times\tilde{E}_{j}[k_{\pm}(\xi_{i}-z)]\right], \tag{104}\]
with \(\hat{w}^{(1)}_{0,j}=\hat{w}^{(1)}_{0}(\mathbf{k}_{j})\) for \(j=1\) and \(j=2\),
\[b_{ij}= \sum_{m=j}^{3}\frac{-a_{im}}{(\xi_{i}-\xi_{3})^{m-j+1}},\ \ \ \ i=1,2;\ b_{31}=-b_{11}-b_{21};\ b_{32}=b_{33}=0, \tag{105a}\] \[\xi_{1}= \frac{\omega_{1}}{k_{1x}S_{0}},\ \ \ \ \xi_{2}=\frac{\omega_{2}}{k_{2x}S_{0}},\ \ \ \ \xi_{3}=\frac{\omega_{\pm}}{k_{x}S_{0}},\] (105b) \[\tilde{E}_{j}(\mu)= \mathrm{e}^{\mu}\mu^{j-1}\int_{\mu}^{\infty}\frac{\mathrm{e}^{- \tau}}{\tau^{j}}\mathrm{d}\tau. \tag{105c}\]
Assuming \(\xi_{1}\neq\xi_{2}\), the coefficients in (100) are expressed as
\[a_{i1}= (-1)^{i}\left[k_{1}k_{2}-\mathbf{k}_{1}\cdot\mathbf{k}_{2}-\frac{k_ {1}+k_{2}}{\xi_{1}-\xi_{2}}\frac{k_{1x}k_{2y}-k_{1y}k_{2x}}{k_{1}k_{2}}\tan \theta_{m}\right]\tan\theta_{i} \tag{101a}\] \[a_{i2}= (-1)^{i}\frac{1}{k_{i}}\left[k_{1}k_{2}-\mathbf{k}_{1}\cdot \mathbf{k}_{2}-\frac{k_{i}}{\xi_{1}-\xi_{2}}\frac{k_{1x}k_{2y}-k_{1y}k_{2x}}{k _{1}k_{2}}\tan\theta_{m}\right]\tan\theta_{i}\] (101b) \[a_{i3}= (-1)^{i}\frac{k_{m}}{k_{i}}\tan\theta_{i}, \tag{101c}\]
where \(i,m\in\{1,2\}\) so that \(i\neq m\) and \(\tan\theta_{i}=k_{iy}/k_{ix}\). The undetermined coefficients \(B_{1\pm}\) is solved by inserting (100) into the combined boundary condition (17b). Then, the surface elevation is obtained from (19).
## Appendix D Effects of current continuation on skewness
We here compare three alternative, physically reasonable ways in which profiles measured using ADCP can be extended from the shallowest measurement point -- \(z=-1.35\,\mathrm{m}\) for the Columbia River measurements we use [64] -- up to the surface. These are: extrapolation using a polynomial fit, shifting the profile upwards so that the shallowest measurement point is set to surface level (used, _inter alia_, in refs. [95; 82]), and the highly conservative approach of continuing the current profile to the surface with zero shear. These are referred as extended profile, shifted profile and zero surface shear profile, respectively and are shown in figure 10(a).
We compare wave skewness in these three case, the results are given in Fig. 11. Again, the \(k_{p}\) in Fig.10(b) is the dimensionless peak wavenumber as in Fig. 8, where \(k_{p}=1\) corresponds to the case without shear current whereas the modifications to the dispersion relation due to shear shifts the value. Values \(k_{p}>1\) correspond to adverse shear and _vice versa_. A plot of the calculated skewness for the different cases shows that the difference in skewness is hardly discernable.
## Appendix E Dimensionless weak-shear parameter for given \(\omega\)
Let the depth-averaged shear be small, of order a small parameter \(\delta\ll 1\). Assuming the wave number \(k\) given, Stewart and Joy [55] derived the approximate dispersion relation \(\omega(k)\) which may be written [60]
\[\omega^{*}(k^{*})\approx\sqrt{gk^{*}}[1-\delta(k^{*})]+\mathcal{O}(\delta^{2}), \tag{102}\]
with the small-shear parameter \(\delta(k^{*})\) defined in (35). It was shown [60] that a sufficient criterion for the Stewart & Joy approximation to be good is that \(\delta_{\omega}\ll 1\).
Conversely (i.e., for given \(\omega^{*}\)) the presence of shear modifies \(k\) slightly, and we write
\[k^{*}=k_{0}^{*}[1+\delta_{\omega}(\omega^{*})]+\mathcal{O}(\delta_{\omega}^{2}) \tag{100}\]
with \(k_{0}^{*}=(\omega^{*})^{2}/g\), and clearly \(\delta_{\omega}\sim\delta\). We seek to find \(\delta_{\omega}\). Inserting (100) into (100) via (35) and noting that \(\sqrt{gk_{0}^{*}}=\omega^{*}\),
\[\omega^{*}= \omega^{*}\sqrt{1+\delta_{\omega}}[1-\delta(k_{0}^{*})]+\mathcal{ O}(\delta^{2})\] \[= \omega^{*}[1+\tfrac{1}{2}\delta_{\omega}-\delta(k_{0}^{*})]+ \mathcal{O}(\delta^{2}). \tag{101}\]
Internal consistency thus demands
\[\delta_{\omega}(\omega^{*})=2\delta(k_{0}^{*}). \tag{102}\]
|
2307.02269 | SpaceNLI: Evaluating the Consistency of Predicting Inferences in Space | While many natural language inference (NLI) datasets target certain semantic
phenomena, e.g., negation, tense & aspect, monotonicity, and presupposition, to
the best of our knowledge, there is no NLI dataset that involves diverse types
of spatial expressions and reasoning. We fill this gap by semi-automatically
creating an NLI dataset for spatial reasoning, called SpaceNLI. The data
samples are automatically generated from a curated set of reasoning patterns,
where the patterns are annotated with inference labels by experts. We test
several SOTA NLI systems on SpaceNLI to gauge the complexity of the dataset and
the system's capacity for spatial reasoning. Moreover, we introduce a Pattern
Accuracy and argue that it is a more reliable and stricter measure than the
accuracy for evaluating a system's performance on pattern-based generated data
samples. Based on the evaluation results we find that the systems obtain
moderate results on the spatial NLI problems but lack consistency per inference
pattern. The results also reveal that non-projective spatial inferences
(especially due to the "between" preposition) are the most challenging ones. | Lasha Abzianidze, Joost Zwarts, Yoad Winter | 2023-07-05T13:08:18Z | http://arxiv.org/abs/2307.02269v1 | # SpaceNLI: Evaluating the Consistency of Predicting Inferences In Space
###### Abstract
While many natural language inference (NLI) datasets target certain semantic phenomena, e.g., negation, tense & aspect, monotonicity, and presupposition, to the best of our knowledge, there is no NLI dataset that involves diverse types of spatial expressions and reasoning. We fill this gap by semi-automatically creating an NLI dataset for spatial reasoning, called SpaceNLI.1 The data samples are automatically generated from a curated set of reasoning patterns (see Figure 1), where the patterns are annotated with inference labels by experts. We test several SOTA NLI systems on SpaceNLI to gauge the complexity of the dataset and the system's capacity for spatial reasoning. Moreover, we introduce a _Pattern Accuracy_ and argue that it is a more reliable and stricter measure than the accuracy for evaluating a system's performance on pattern-based generated data samples. Based on the evaluation results we find that the systems obtain moderate results on the spatial NLI problems but lack consistency per inference pattern. The results also reveal that non-projective spatial inferences (especially due to the "between" preposition) are the most challenging ones.
Footnote 1: [https://github.com/kovvalsky/SpaceNLI](https://github.com/kovvalsky/SpaceNLI)
## 1 Introduction
Natural language inference (NLI) is a popular task that evaluates NLP systems on text reasoning skills. In the task, a system has to predict an inference relation from a premise text to a hypothesis sentence/phrase. Usually, the task is three- or two-way classification, depending on whether in the inference labels of _entailment_, _neutral_, and _contradiction_, the latter two are merged into _non-entailment_. The task is intended for evaluation of NLP systems on reasoning, however, the systems with competitive results on NLI benchmarks are often exploiting dataset biases (Tsuchiya, 2018; Poliak et al., 2018; Gururangan et al., 2018; McCoy et al., 2019, _inter alia_) and their performance suffers from out-of-distribution NLI sample problems (Glockner et al., 2018).
To better evaluate the reasoning skills of NLI systems, a series of works have been (semi-)automatically or manually creating NLI datasets that specialize in certain semantic phenomena. While some of these datasets come with a training part, most of them are intended solely for evaluation. For example, several datasets have been dedicated to monotonicity reasoning (Yanaka et al., 2019, 2020, 2020), negation was targeted by Hossain et al. (2020), the dataset by Kober et al. (2019) focuses on temporal and aspectual inferences, Jeretic et al. (2020) semi-automatically generated NLI problems for implicatures and presuppositions. There are also NLI datasets that cover several semantic phenomena, having a separate section for each of the phenomena (Cooper et al., 1996;
Figure 1: Sampling NLI problem from NLI patterns (with IDs 9 and 10, Entailment and Contradiction, respectively). The problems are generated by replacing NP placeholders with definite NPs that satisfy pattern-specific selection restrictions. A system’s success rate on a pattern is defined as the accuracy on its corresponding NLI problems.
Richardson et al. 2020, _inter alia_).
While spatial reasoning has been included in several multi-modal QA datasets (Antol et al., 2015; Suhr et al., 2017; Johnson et al., 2017; Hudson and Manning, 2019) and in a couple of text-based QA datasets (Weston et al., 2016; Mirzaee et al., 2021), to the best of our knowledge, no NLI dataset has specifically covered it.2 This paper fills the gap by semi-automatically creating an NLI dataset for spatial inferences. First, we collected a diverse set of NLI problems inspired by the inference examples found in the literature on spatial semantics. Second, the NLI problems were manually converted into NLI patterns (see Figure 1), and finally, we automatically generated a large number of NLI problems from the patterns.
Footnote 2: Even the FraCaS dataset (Cooper et al., 1996; MacCartney, 2009), which was curated by linguists and semanticists, doesn’t cover spatial semantics within its nine sections.
The paper makes two main contributions:
1. SpaceNLI: the spatial NLI dataset with diverse types of spatial inferences; The inference labels of the generated problems are highly faithful (97%) to the labels of the corresponding original patterns.
2. Pattern accuracy and its curve: they measure systems' performance on patterns and the consistency in predictions on samples from the same patterns.
The conducted experiments answer the following research questions:
1. How much spatial reasoning current SOTA NLI systems are capable of?
2. We found out that the SOTA NLI systems have problems with fine-grained spatial inferences. Their performance drops at least by 24% compared to their results on common NLI datasets. Moreover, their consistency in predictions is sensitive to irrelevant lexical substitutions.
3. What types of spatial inference problems are easy or challenging for the SOTA NLI systems?
4. The results showed that the non-projective spatial relations are most challenging for the models. This was mainly due to difficulty associated with "between" and its frequent occurrence in the evaluation dataset.
## 2 Spatial expressions and inferences
### Types of spatial expressions
Spatial expressions consist of spatial prepositions and other expressions with spatial information (e.g., _far_, _the left of_, and _in front of_). They usually describe a relation between two entities, the _figure_ and the _ground_. The site or path of the figure is the focus of the discussion and is characterized with respect to the ground. For example, in (\(9_{1}\)) and (\(10_{1}\)) from Figure 1, _Mary_ is a figure and _garden_ a ground. _John_ is also a figure in the premise of (\(10_{1}\)).
Spatial expressions are roughly divided into _locative_ and _directional_ expressions, where locatives can be further classified into _projective_ and _non-projective_(Herskovits, 1986). The locative expressions describe static, locative relations between the figure and the ground while directional ones describe a more _dynamic_ relation involving a movement and/or path. An example with a directional preposition is _Cindi walked into the market_. The spatial expressions in Figure 1 are all locative except for _from_, which is directional. These locative expressions are non-projective since they require only the spatial location of the figure and the ground. In contrast, projective locatives additionally require further information from the ground in terms of a deictic frame of reference (i.e., an orientation structure). For example, the site of the house is not sufficient to interpret Mary's location in _Mary is behind the house_, it requires knowledge about the frame of reference of the house, in particular, what counts as a back side of the house.
### Types of spatial inferences
We characterize spatial inferences depending on the type of spatial expressions licensing them. An inference might depend on several spatial expressions of a different type, which makes partitioning the inferences challenging, if not impossible. We define the following classes that represent a coarse-grained partition of spatial inferences. The classes will be later referred to in SS3.3
Footnote 3: Licensing contradiction and neutral problems will be assumed from the perspective of a related entailment problem. For example, we assume that the neutral problem (16) in Table 1 is licensed in the same way as its related entailment (15). Put differently, one can see (16) as an adversary to (15) and assume that solving (15) requires competence comparable to the one required for solving (16).
Argument orientationIn spatial literature, an argument orientation entailment identifies which
argument of the verb is the figure of the spatial expression. For instance, (9\({}_{1}\)) in Figure 1 show that _Mary_ is the figure of the locative PP _in the garden_. In its original interpretation, the argument orientation entailment is not restricted to spatial expressions of a particular type. Here, we restrict the class of argument orientation to the entailment problems (and their neutral and contradiction counterparts) that come close to resolving a PP attachment. For example, correctly resolving the PP attachment in (9\({}_{1}\)) boils down to the hypothesis. The problems in this class contain a hypothesis with a copula and a predicative spatial PP, where the PP is contrasted to a tightly related PP in the premise(s). For more examples of the NLI problems in the argument orientation class, see Table 1.
DirectionalThe directional class contains spatial inferences where directional spatial expressions play the key role. Examples of such inferences are given in Table 1. Some of these NLI problems pertain to a path-place relation: (47a) shows that _walking into_ infers _being outside_,4 (41) entails _being in the tunnel_ from the premise that states that the driving path was through the tunnel. (31a) combines a part-whole relation with the movement path.
Footnote 4: Since moving along the path is related to the change of the location, sometimes spatial entailments interfere with tense and aspect.
ProjectiveThis class contains inferences that hinge on a frame of reference introduced by projective spatial expressions. In principle, the frame of reference can introduce six directions that can be referred to using the expressions like _front, behind, left, right, above, below, under, on top of_, etc. (see the examples of NLI problems in Table 1). The NLI problems that contain _on top of_ as only projective spatial expression, and when its projective interpretation is not crucial for the inference, are put in a different class.
Non-projectiveWe classify a problem as having non-projective inference if the inference is driven only by non-projective spatial expressions. Therefore, an occurrence of non-projective spatial expressions in a problem is necessary but not sufficient for assigning the problem to this class, e.g., see directional problems (31a) and (41). NLI problems that depend on spatial expressions with the semantics of order and proximity are also in this class, see _between_ (80) and _far_ (100) in Table 1.
## 3 Dataset construction
### Pattern construction
Patterns are labeled NLI problems with NPs replaced by variables as illustrated in Figure 1. The NLI patterns are obtained from the seed NLI problems. To collect the latter, we extracted the initial 56 problems from Zwarts and Winter (2000) and Nam (1995), where a majority of the problems
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**ID** & **Class** & **Premise(s)** & **L** & **Hypothesis** \\ \hline
15 & Dir & John threw the ball into the box. & E & The ball went into the box. \\ \hline
16 & Dir & John threw the ball at the box. & N & The ball went into the box. \\ \hline
31a & Dir & Los Angeles is in California. John came from California. & N & John came from Los Angeles. \\ \hline
38 & NonP & John is in the garden. The garden is in the church. & E & John is in the church. \\ \hline
41 & Dir & John drove through the tunnel. & E & John was in the tunnel. \\ \hline
47a & Dir & Cindi walked into the market. & E & Cindi was outside the market. \\ \hline
56c & Proj & The trash can is to the right of the tree from John. & C & The tree is to the right of the trash can from John. \\ \hline
70 & Proj & Mary is between the tree and the house. The tree is behind the house. & E & Mary is behind the house. \\ \hline
80 & NonP & The cat is between the house and the fence. The cat is between the fence and the tree. & C & The cat is between the house and the tree. \\ \hline
99*d & Proj & The bucket is above the bowl. The pencil is above the bowl. & N & The bucket is below the pencil. \\ \hline
96b & ArgO & Mary met John at the party. & N & Cindi was not at the party. \\ \hline
100 & NonP & The house is far from the school. & E & The school is far from the house. \\ \hline
102a & ArgO & Mary has taken the cup out of the cabinet. & C & The cup is in the cabinet. \\ \hline
102f & ArgO & Mary has hidden the cup behind the cabinet. & E & The cup is not in the cabinet. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Examples of the seed NLI problems annotated with spatial inference classes: **Di**rectional, **Pro**jective, **Non-**Projective, and **Ar**gument **O**rientation. Initial letters abbreviate the corresponding inference labels.
were labeled as entailment due to obvious biases in the semantic literature towards licensing entailment. To create a representative and challenging NLI dataset for machine learning, we applied several _revision phases_ to the problems: introducing new problems that either cover new semantic aspects of spatial expression or serve as a perturbed version of an existing problem.
In the initial revision phase, four annotators divided the extracted problems and created slightly modified versions of them with an inference label different from the original.5 This was motivated by the current trends in the literature on adversarial, stress, and debiased datasets Naik et al. (2018); Ribeiro et al. (2020); Kaushik et al. (2020); Gardner et al. (2020), _inter alia_). For example, (16) is a perturbed example of (15). Where possible, NLI problems of a new type were also created using the similar spatial expressions found in the extracted problems.
Footnote 5: The annotators for the pattern construction consist of the authors of the paper, two linguist students, and one AI student. The guideline for creating inference problems can be found in the supplementary material.
To validate the resulting pool of NLI problems (in total 162), following Zhang et al. (2017), they were labeled on a 5-point Likert scale by three annotators.6 After collecting the 5-point annotations, for each annotator, we picked a mapping of 5-point to 3-point that maximizes the inter-annotator agreement (avg. Cohen's \(\kappa=.71\)). The problems without majority labels were discarded and 111 problems remained.
Footnote 6: The question was to what extent the hypothesis sentence is true, given that the premises are true, with choices: _definitely false, most likely false, unknown, most likely true_, _definitely true_. We used two additional choices, _difficult_ (unable to annotate due to the complex reasoning it requires) and _skip_ (presence of an ungrammatical or nonsensical sentence). We used the bar annotation tool Stenetorp et al. (2012) for labeling. The annotation guideline is included in the supplementary material.
To better balance the inference labels and increase the coverage of spatial expressions, a second revision phase was carried out on the remaining problems. In several cases, problems with low annotator agreement were revised, e.g., changing the tense where it caused confusion or replacing a preposition with a weaker version (_at\(\mapsto\)near_). All the new and revised problems (in total 63) were validated based on three samples: each problem was manually converted into a pattern by replacing NPs with variables, and three random NLI samples per pattern were generated (see SS3.2 for details), which were subsequently validated by three annotators.
Finally, a third revision phase was carried out on the remaining problems to additionally decrease the overall and spatial type-specific label imbalance. The collected problems (in total 160) were treated as a seed by converting them into NLI patterns to generate a large amount of sample NLI problems from them. To illustrate the coverage of spatial expressions in the collected patterns, Table 2 gives the complete list of spatial expressions for each entailment class.
### Sample generation
We manually created NLI patterns from the initially collected NLI problems (SS 3.1) by replacing NPs with placeholders and specifying selection restrictions for them imposed by the verbs, spatial expressions, and gold inference labels (see Figure 1). The selection restrictions imposed by spatial expressions are subtle and can affect gold labels or the naturalness of sentences. For example, if the figure is much larger than the ground, it can make the sentence infelicitous: _the apple on the fridge_ and _the apple near the fridge_ are preferred to _the fridge under the apple_ and _the fridge near the apple_. Inferences driven by proximity-related spatial expressions are sensitive to the size of the objects.
\begin{table}
\begin{tabular}{l l} \hline \hline
**Class (\#patterns)** & **Spatial expression counts** \\ \hline \multirow{2}{*}{Directional (95)} & in (20), from (17), into (9), to (8), on (8), away from (7), towards (7), out of (4), back (3), \\ & through (3), across (2), at (2), outside (2), opposite (1), part of (1), by (1) \\ \hline \multirow{2}{*}{Argument orientation (67)} & in (21), at (10), from (9), away from (4), out of (4), near (3), with (3), inside (3), on (2), \\ & under (2), through (1), opposite (1), towards (1), far from (1), on top of (1), behind (1) \\ \hline \multirow{2}{*}{Projective (70)} & behind (16), between (11), in front of (10), below (6), above (6), under (6), on top of (5), \\ & front of (3), opposite (2), to the right of (2), on (2), to the left of (1) \\ \hline Non-projective (48) & between (22), in (9), far from (5), close to (4), outside (3), on top of (2), on (2), opposite (1) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The spatial expressions and their counts per entailment class in the SpaceNLI patterns
For instance, based on our conducted validations, _Cindi is opposite to the cat_ is more likely to be neutral to _Cindi is far from the cat_, but _the school is opposite to the house_ is more likely to contradict _the school is far from the house_.
To meet selection restrictions and allow relative diversity of NPs in the generated samples, we defined a mini world with a domain containing 171 entities corresponding to common and proper nouns. The entities are organized in a taxonomy with 20 subclasses covering general types of entities (e.g., person, animal, vehicle), the projections of an argument in certain argument structures (e.g., enter in \(X\), be in \(X\), throw \(X\)), compatibility with projective spatial expressions, and size categories (S for entities comparable to small objects like book and cat, M to persons, and L to vehicles). Binary and ternary relations are defined based on the set unions of the products of entity sets and subclasses.
To automatize the sampling of sound NLI problems from the patterns, we formatted the mini world in YAML and NLI patterns in XML. We implemented a procedure that samples problems from the patterns by filling in NP placeholders with definite NPs from the mini world and respecting the pattern-specific selection restrictions. For sanity checking, the procedure verifies that it can generate corresponding seed NLI problems for each pattern.
To measure how faithfully the inference labels are transferred from seed and pattern NLI problems to the corresponding NLI samples, we used sampled problems in the second phase of validation when validating new NLI problems (see SS3.1). The results showed that 79% of samples were unanimously labeled with the original label. After filtering out patterns with a relatively low agreement, this ratio increased to 97% for the samples generated from the validated patterns.
The NLI problems sampled from the same pattern or related patterns are string-wise very close to each other, sometimes differing only in terms of occurrences of a single NP. Regardless of this similarity, we expect such problems to pose a challenge for NLI systems based on large language models (LLMs) as it has been shown that their predictions can be sensitive to a single-word substitution Glockner et al. (2018); Gururangan et al. (2018). In addition to NPs, one could have allowed the replacement of other phrases in the NLI patterns, but this would have significantly complicated the definition of the mini world and generation of natural and sound NLI samples.
## 4 Experiments
### Sample dataset
We uniformly generated a spatial dataset of 32,000 NLI samples from 160 NLI patterns, i.e., 200 samples per pattern. We used the mini world as described in SS3.2. The dataset statistics are given in Table 3. The inference labels are relatively balanced: each label being represented by at least 30%
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline LLM-based & Training & snli & \multicolumn{3}{c}{SpaceNLI} \\ NLI models & data &
\begin{tabular}{c} + \\ mNLI \\ \end{tabular} & Acc & PA\({}_{0.95}\) & PA\({}_{1.0}\) \\ \hline DeBERTaV3-L\#1 & SMFA & **91.8** & 59.6 & 47.5 & **37.5** \\ Joelenhang/deberta-v3... & SMFA & 90.8 & 57.8 & **48.1** & 36.2 \\ ALBERT-XXLV2 & SMFA & 90.7 & 54.1 & 42.5 & 36.2 \\ He et al. (2021) & M & 90.6 & 55.6 & 40.0 & 31.9 \\ RoBERTa-L & SMFA & 90.4 & 55.4 & 39.4 & 29.4 \\ DeBERTaV3-L\#2 & MFALW & 90.3 & **66.5** & 44.4 & 33.8 \\ XLNet-L-Cased & SMFA & 90.3 & 55.8 & 42.5 & 30.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance of SOTA NLI systems on SpaceNLI. snli+mnli shows the average score on these datasets. Training data names are denoted with the initial letters: **SNLI**, **MNLI**, **ANLI**, Fever-NLI**, **WANLI**, and **L**ingNLI. The best system per problem accuracy on SpaceNLI, DeBERTaV3-LMEMLW (with \(\Delta\geq 6.9\%\)), doesn’t turn out to be the best at the consistency threshold \(\geq 0.95\). See the extended version of the table in Appendix A.
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline Property & E \% & N \% & C \% & All \% (\#) \\ \hline Dir & 39.6 & 35.4 & 25.0 & 30.0 & (9600) \\ NonP & 25.0 & 41.7 & 33.3 & 22.5 & (7200) \\ Proj & 29.4 & 26.5 & 44.1 & 21.2 & (6800) \\ ArgO & 47.6 & 28.6 & 23.8 & 26.2 & (8400) \\ \hline \(+\) neg & 48.0 & 28.0 & 24.0 & 15.6 & (5000) \\ \hline
1prem & 41.8 & 26.5 & 31.6 & 61.3 & (19600) \\
2prem & 25.0 & 42.9 & 32.1 & 35.0 & (11200) \\
3prem & 50.0 & 50.0 & 0.0 & 3.8 & (1200) \\ \hline All & 36.2 & 33.1 & 30.6 & 100.0 & (32000) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistics of several properties of the sampled NLI dataset. The statistics also apply to the collection of NLI patterns as the samples are evenly distributed over the patterns. The properties consist of the spatial inference types, whether including negation, and the number of premises.
of the problems. Each spatial inference type counts at least 20% of the overall problems and 23% of label-specific problems. In contrast to the common biases in NLI datasets, a majority of the problems with negation are labeled as entailment, not contradiction. This is due to perturbed problems introduced in the revision phases (SS 3.1). Around 39% of problems have multiple premises, where three-premised problems occur only in the directional problems, the argument orientation problems contain only single-premised problems, and most of the multi-premised problems are in the non-projective problems. We refer to the generated dataset as SpaceNLI and use it in subsequent experiments.7
Footnote 7: We make the collection of the patterns, the generation code, and the sample dataset publicly available upon the acceptance of the paper.
### Evaluating SOTA NLI systems
#### 4.2.1 Standard accuracy
We selected NLI models that have results comparable to the state of the art in NLI and evaluate them on SpaceNLI. The models were chosen based on their availability, tractable size, and high average accuracy (\(>90\%\)) on the SNLI (Bowman et al., 2015) and MNLI (Williams et al., 2018) datasets (see Table 4). The models are based on various large language models (LLMs) like DeBERTaV3 (He et al., 2023), BART (Lewis et al., 2020), ALBERT (Lan et al., 2020), XLNet (Yang et al., 2020), etc. (see Table 4). The LLMs are fine-tuned on several NLI train datasets: SNLI, MNLI, FEVER-NLI (Nie et al., 2019), ANLI (Nie et al., 2020), LingNLI (Parrish et al., 2021), WANLI (Liu et al., 2022). We use the models from the HuggingFace model hub8 and provide them with the corresponding hub names in Table 4.
Footnote 8: [https://huggingface.co/models](https://huggingface.co/models)
Footnote 9: The second best, DeBERTaV3-L#1, is based on the same LLM fine-tuned on a different combination of NLI datasets. Note that Laurer et al. (2022) deliberately removed SNLI from the training set as it negatively affected the accuracy of the model in their experiments.
The results in Table 4 show that DeBERTaV3-L#2 trained on a large collection of training datasets (885K problems in total) generalizes best on the spatial reasoning (66.5%), achieving a substantial improvement (\(\geq 6.9\%\)) over the other models.9
Footnote 9: The second best, DeBERTaV3-L#1, is based on the same LLM fine-tuned on a different combination of NLI datasets. Note that Laurer et al. (2022) deliberately removed SNLI from the training set as it negatively affected the accuracy of the model in their experiments.
#### 4.2.2 Consistency & pattern accuracy
To evaluate the models on the consistency of their predictions for NLI problems from the same pattern, we define the pattern accuracy (PA) score and its curve. The PA curve records the PA score of a model for each consistency threshold. Informally, the PA score with a consistency threshold \(t\) is a ratio of NLI patterns for which model gets at least \(t\) portion of the samples generated from them. For example, the PA of 50% with a threshold 90% means that there are a half of the NLI patterns such that for each pattern a model is able to correctly classify at least 90% of its sample problems. The formal definition of the PA with a threshold \(t\) is:
\[PA_{t}(\hat{Y},\mathbf{y})=\frac{1}{N}\sum_{i=1}^{N}\left[\frac{\sum_{k=1}^{M_{ i}}\delta(\hat{y}_{k}^{i}=y^{i})}{M_{i}}\geq t\right]\]
where \(\hat{Y}=(\hat{y}_{k}^{i})_{1\leq i\leq N,1\leq k\leq M_{i}}\) are predictions for \(k^{\text{th}}\) sample of \(i^{\text{th}}\) pattern, \(N\) is the number of patterns, \(M_{i}\) is the number of samples for \(i^{\text{th}}\) pattern, \(\mathbf{y}=(y^{i})_{1\leq i\leq N}\) gold labels of \(i^{\text{th}}\) pattern, and \(\delta\) is the Kronecker delta.
While DeBERTaV3-L#2 gets the best score on the SpaceNLI problems, based on the PA scores in Table 4, it shows high consistency (\(PA_{0.95}\) or \(PA_{1.0}\)) in fewer NLI patterns than the other two competing models, DeBERTaV3-L#1 and ALBERT-XXLv2. PA curves of the NLI mod
Figure 2: Pattern accuracy curves of the NLI models from Table 4. The first half, which corresponds to the scores allowing solving less than half of the samples per pattern, is omitted (see Appendix A for the complete curves).
els provide a closer look at this contrast (see Figure 2). While the curve of DeBERTaV3-L#2 outperforms other models by a margin, it is noteworthy that it does this by classifying sample problems of the patterns which it can hardly solve half of the time (this is visible in the complete curves in Appendix A). It drastically decreases after 95% of consistency while ALBERT-XXLV2 and DeBERTAV2-L#1 maintain very high consistency for \(>47\)% of NLI patterns. This demonstrates that a high-performing model is not necessarily the most consistent across patterns.
RoBERTa-L and BART-L obtain similar accuracy scores, but RoBERTa-L is more consistent in more NLI patterns than BART-L while the latter gets slightly more NLI problems for inconsistently predicted patterns. The complete curves in Appendix A shows how the curves swap places after the consistency threshold of 50. This shows that the standard accuracy (i.e., based on NLI problem samples) can blur the fine distinction in consistency between the models.
The dispersion of the curves at the lowest end of the consistency threshold is twice larger than at the highest end. This shows that the model predictions more diverge in coverage of patterns than in consistency per pattern. In other words, the contrast confirms the sensitivity of the models towards the inference-preserving word substitutions.
#### 4.2.3 Few-shot learning experiments
We measured the difficulty of the SpaceNLI problems in terms of few-shot learning experiments. We used 100 samples per pattern as a test set while other 100 samples per pattern were used for drawing a few samples for each pattern. In this way, the patterns are fully shared between the training and test sets, but no sample NLI problem is in both sets. For each number of shots, we carried out the sample drawing process three times. We used two NLI models: a high performing NLI model RoBERTa-L\({}_{\text{SMFA}}\) from Nie et al. (2020) and a _vanilla_ NLI model based on the large RoBERTa pretrained language model Liu et al. (2019). The results of the few-shot experiments are in Figure 3.
Finetuning RoBERTa-L\({}_{\text{SMFA}}\) on a single sample of each pattern increases the sample-based accuracy on the test set by 14%. Each additional sample further boosts the model's accuracy. The almost perfect accuracy (>99%) is reached when 20 samples per pattern are seen during the finetuning. The results show that the lexical variability poses a challenge to the high-performing NLI model as it needs to be finetuned on at least five samples for every pattern of the test set to achieve a high score.
The challenge coming from the lexical variability and the SpaceNLI patterns is further emphasized by the relatively low results of RoBERTa Large. Even after being finetuned on the 20 samples of each NLI pattern, the model is still far from the high performance on unseen samples (but seen patterns). The relatively low results can be also partially attributed to the low ratio between the number of training samples and the large number of the model's trainable parameters.
## 5 Analysis
To find out what type of inferences the models find challenging, we analyze the models' performance per inference type. Figure 5 shows the sample- and pattern-based accuracy scores of the models per spatial inference types as defined in SS2.2. The model ranking based on the sample accuracy varies across the inference types. For instance, the best model, DeBERTaV3-L#2, remains at the top of the rankings for all inference types with quite a margin except for the projective type. On average, non-projective spatial inferences are the most challenging for the models. The easiest of the types is argument orientation, the type that is closest to the PP attachment task. For the other inference types, projective inferences are harder than directional ones. The apparent distinction in the scores between the inference types is also preserved for the \(PA_{0.95}\) score (shown with the dark bars in Figure 5).
Figure 3: Average of three runs for each few-shot finetuning experiment. RoBERTa-L (SMFA, Nie et al.2020) is already finetuned on several large NLI datasets while RoBERTa Large Liu et al. (2019) is a pretrained language model without any previous training on NLI.
The fine-grained analysis additionally shows that the best model, DeBERTaV3-L#2, suffers least in terms of consistency on the projective inferences while its performance on this inference type is not among the best.
Based on the results in Figure 5, the non-projective NLI patterns and samples are the most challenging for the SOTA models. When looking closer at the set of non-projective problems, it turns out that it contains a high number of problems (46%) with the spatial expression "between" (as shown in Table 2), and these problems are specially challenging due to complex semantics of "between". The average accuracy of the models on such NLI samples is 41.6%. This is lower than the average sample-based accuracy (46.1%) on entire SpaceNLI and much lower than the average sample-based accuracy (54.1%) on the other part of the non-projective samples.
We further zoom in on the NLI patterns and measure a model's probabilistic predictions for the patterns. Namely, following Swayamdipta et al. (2020), we measure a model's confidence and variability. Originally the dataset cartography (Swayamdipta et al., 2020) was used to analyze the training dynamics of a model across the epochs and identify training samples that are easy or difficult for learning. In contrast, we use dataset cartography for analyzing evaluation dynamics across patterns and identifying easy and hard ones.10
Footnote 10: Put differently, iterative classification of the same training sample across epochs, is replaced with the classification of the same NLI pattern based on its samples.
Figure 4 illustrates the pattern-based evaluation dynamics of RoBERTa-L (Nie et al., 2020), an average model based on the evaluations. For instance, NLI pattern (102f) happens to have one of the most variable samples according to the model predictions: the mean and the standard deviation of the probabilities the model assigns to the entailment class of the samples of (102f) are 0.45 and 0.35, respectively.
\begin{tabular}{l l} (102f) & NP\({}_{1}\) has hidden NP\({}_{2}\) behind NP\({}_{3}\). \\ & entailment & NP\({}_{2}\) is not in NP\({}_{3}\). \\ \end{tabular} The evaluation cartography shows that the predictions vary mostly for entailment patterns (in green). Most of the hard patterns are neutral ones (in blue) and vice versa. Contradiction patterns (in red) tend to be easy with some variability.
Figure 4: Prediction cartography of RoBERTa-large from (Nie et al., 2020). NLI patterns are characterized with _confidence_ and _variability_: the mean and the standard deviation of probabilities assigned by the model to the true labels of the sample NLI problems. IDs mark NLI patterns from Figure 1 and Table 1.
Figure 5: Sample-based (in light shades) and \(PA_{0.95}\) (in dark shades) accuracy scores of the models per spatial inference type.
Related work
Several works have automatically sampled NLI problems from curated patterns/templates. Jereetic et al. (2020) generated the implicature and pre-supposition diagnostic dataset IMPPRES from pre-defined templates. McCoy et al. (2019) constructed the HANS dataset by designing templates of NLI problems that support or refute certain inference heuristics, which were later used to generate NLI problems. Richardson et al. (2020) used the template language from Salvatore et al. (2019) to produce NLI problems involving negation, Boolean connectives, quantifiers, cardinals, conditionals, and comparatives. These works all use restricted vocabulary while generating samples from the patterns.
With its pattern-based construction and restricted vocabulary, SpaceNLI comes close to the IMPPRES (Jereetic et al., 2020) and HANS (McCoy et al., 2019) datasets. Unlike these datasets, SpaceNLI involves multiple-premised problems and puts more emphasis on satisfying selection restrictions to prevent nonsensical sentences.
Based on the nature of NLI problems, SpaceNLI resembles FraCaS (Cooper et al., 1996) as both contain inference problems often found in textbooks on formal semantics. Unlike FraCaS, the inference labels of patterns in SpaceNLI are quite balanced and the number of spatial NLI patterns is twice the size of the largest section in FraCaS.
There have been attempts to identify semantic phenomena in existing NLI datasets, including aspects of spatial reasoning. By looking up certain keywords, Kim et al. (2019) automatically detect NLI problems in MultiNLI (Williams et al., 2018) that might contain spatial expressions. They create a mutated sample from the original NLI problem by negating the sentence with the potential spatial expression. Joshi et al. (2020) annotate MultiNLI problems based on the semantic aspects required by the inference label. Their taxonomic categories include the spatial subcategory, grouped with the relational, temporal, causal, and co-reference subcategories.
The problems in SpaceNLI are substantially diverse from a semantic perspective than the MultiNLI problems that were identified by Kim et al. (2019) and Joshi et al. (2020). The MultiNLI dataset is crowd-elicited and doesn't have problems with sufficient depth in spatial reasoning.
## 7 Conclusion
To the best of our knowledge, we have created the first spatial inference dataset that involves diverse spatial inference types. The structure and the evaluation protocol are unique as we focus on performance on the NLI patterns and consistency across the samples in the pattern, instead of focusing on mere quantitative accuracy based on the NLI problems/samples. The evaluation protocol tests models whether they can consistently recognize inference patterns while generalizing over _irrelevant_ lexical substitutions. The more consistent a model is in its predictions, the less unexpected its behavior becomes.
The SOTA NLI models show moderate generalization capacity on spatial problems. While the top-performing model gets the highest overall accuracy, it is ranked third when it comes to the consistency of predictions inside the patterns: predicting at least 95% of the samples per pattern.
The introduced pattern accuracy (PA) curves provide a more fine-grained distinction between the models: the models with comparable standard accuracy scores might substantially differ in the consistency of their predictions. Overall the performance of models drops ca. 10% when raising the consistency threshold to 95%. This illustrates that the predictions of the SOTA models are sensitive to lexical replacements that have no effect on the semantics of the inference.
The evaluation results revealed that the most challenging inference type is associated with non-projective locatives mainly due to the complex semantics of "between" while the argument orientation type is the easiest. The latter is somewhat expected as the problems in the argument orientation type are close to the task of PP attachment which LLMs are expected to be good at.
## Acknowledgments
This work was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 742204). We would like to acknowledge the help from three student assistants with the data annotation and thank the anonymous reviewers for their helpful comments. |
2306.02140 | Unsupervised Human Activity Recognition through Two-stage Prompting with
ChatGPT | Wearable sensor devices, which offer the advantage of recording daily objects
used by a person while performing an activity, enable the feasibility of
unsupervised Human Activity Recognition (HAR). Unfortunately, previous
unsupervised approaches using the usage sequence of objects usually require a
proper description of activities manually prepared by humans. Instead, we
leverage the knowledge embedded in a Large Language Model (LLM) of ChatGPT.
Because the sequence of objects robustly characterizes the activity identity,
it is possible that ChatGPT already learned the association between activities
and objects from existing contexts. However, previous prompt engineering for
ChatGPT exhibits limited generalization ability when dealing with a list of
words (i.e., sequence of objects) due to the similar weighting assigned to each
word in the list. In this study, we propose a two-stage prompt engineering,
which first guides ChatGPT to generate activity descriptions associated with
objects while emphasizing important objects for distinguishing similar
activities; then outputs activity classes and explanations for enhancing the
contexts that are helpful for HAR. To the best of our knowledge, this is the
first study that utilizes ChatGPT to recognize activities using objects in an
unsupervised manner. We conducted our approach on three datasets and
demonstrated the state-of-the-art performance. | Qingxin Xia, Takuya Maekawa, Takahiro Hara | 2023-06-03T15:41:59Z | http://arxiv.org/abs/2306.02140v1 | # Unsupervised Human Activity Recognition through Two-stage Prompting with ChatGPT
###### Abstract.
Wearable sensor devices, which offer the advantage of recording daily objects used by a person while performing an activity, enable the feasibility of unsupervised Human Activity Recognition (HAR). Unfortunately, previous unsupervised approaches using the usage sequence of objects usually require a proper description of activities manually prepared by humans. Instead, we leverage the knowledge embedded in a Large Language Model (LLM) of ChatGPT. Because the sequence of objects robustly characterizes the activity identity, it is possible that ChatGPT already learned the association between activities and objects from existing contexts. However, previous prompt engineering for ChatGPT exhibits limited generalization ability when dealing with a list of words (i.e., sequence of objects) due to the similar weighting assigned to each word in the list. In this study, we propose a two-stage prompt engineering, which first guides ChatGPT to generate activity descriptions associated with objects while emphasizing important objects for distinguishing similar activities; then outputs activity classes and explanations for enhancing the contexts that are helpful for HAR. To the best of our knowledge, this is the first study that utilizes ChatGPT to recognize activities using objects in an unsupervised manner. We conducted our approach on three datasets and demonstrated the state-of-the-art performance.
Human activity recognition; prompting engineering; ChatGPT +
Footnote †: c) 2018 Association for Computing Machinery, ACM ISBN 978-1-4509-30XXX-X/18/06... $15.00. [https://doi.org/](https://doi.org/)
that can be utilized for activity classification. The second stage is answer generation, which utilizes the knowledge generated from the first stage in conjunction with knowledge prompt engineering (Kang et al., 2017) to output the prediction of activities. Knowledge prompt engineering requests an explanation for the HAR result based on the knowledge in the prompt. We utilize it to enhance the contexts in the prompt that are helpful for HAR. By integrating the two-stage prompt engineering, we aim to guide ChatGPT to automatically focus on important contexts for differentiating between activities, thus improving the model's HAR performance.
The contributions of this study are listed as follows:
1. We proposed an unsupervised HAR approach using the sequence of objects. To the best of our knowledge, this is the first to utilize ChatGPT for predicting activities based on the usage of objects.
2. We proposed a two-stage prompt engineering that promotes ChatGPT to differentiate activities via objects.
3. We compare our approach to three prompt engineering baselines. Our approach demonstrates the best performance on three HAR benchmark datasets.
## 2. Related Work
**HAR using Wearables**. HAR plays an important role in monitoring human behavior, which is helpful for understanding individuals' wellness and supporting personalized systems. HAR using wearable devices (Han et al., 2017) has become one of the most important tasks in the ubiquitous community due to the cost-effectiveness of wearables compared to other sensors like video cameras, as well as their ability to collect activity data without excessively compromising privacy unrelated to HAR. There are many applications that use data collected from wearables for HAR. For example, by recognizing the workers' activities performed in a logistics center, the bottleneck activities could be identified (Kang et al., 2017). Francisco et al. (2017) utilized a hybrid model with both convolutional and recurrent structure layers to recognize human activities in daily life. Chen et al. (2018) employed a residual neural network architecture to extract features from sensor data and recognize activities. These studies can be helpful for developing individual assistant systems employed in smart homes and etc. However, in real settings, activities tend to be complex and consist of multiple actions. Training HAR models for more complex activity classes typically requires a larger number of activity labels, which can be impractical.
**Unsupervised Activity Recognition**. To reduce the number of labels used for training, many unsupervised learning techniques have been attracting attention. Maekawa et al. (2018) proposed an unsupervised method for identifying each iteration of assembly work by utilizing acceleration data collected from the workers' wrists. Xia et al. (2019) built upon the aforementioned study for estimating the starting and ending times of each activity within the assembly work by leveraging the information derived from process instructions. Hiremath et al. (2018) employed a cluster-based approach that utilized multiple sensors in a smart home setting to recognize human activities. However, the approaches proposed by these studies strongly rely on the characteristics of sensor data and the environmental settings, limiting their applicability in broader scenarios.
**HAR Relying on Usage of Objects**. Owning to the above challenges, prior studies such as Philipose et al. (2018) and Tapia et al. (2018) have suggested discriminating between many activities by using the objects that the user used because the objects strongly correlate to that of natural-language instructions (e.g., recipes). Wyatt et al. (2019) tried to model the activity of daily life using the object sequence with object labels recorded by the wristband. Patterson et al. (2018) proposed an approach to infer daily life activities from the aggregated object instance. However, the aforementioned works rely on having explicit descriptions of activities, and the quality of these descriptions directly affects the performance of the activity recognition models.
In this study, we propose an unsupervised learning approach to recognize human activities using the object sequence that the user used for each activity. The description of activities will be automatically generated through ChatGPT.
## 3. Methodology
As shown in Figure 1, given a list of objects that the user interacted with, this study employs a two-stage prompt engineering to automatically generate knowledge about activities and then recognize activities using the object sequence in an unsupervised manner.
### Problem Setup
Let \(L\) represent a sequence of objects as input, \(K_{p}\) represents the knowledge regarding \(L\), \(A\) represents the set of answers including the activity classes, and \(pG(\cdot)\) is the pre-trained language model in ChatGPT. The model aims to predict an activity \(a\in A\), which is formulated as follows:
\[\hat{a}=\operatorname*{arg\,max}_{a\in A}pG(a|L,K) \tag{1}\]
### Knowledge Generation
Because we did not manage to fine-tune the LLM for our specific task, the knowledge embedded in the prompt becomes crucial for enabling ChatGPT to generate precise activity recognition results. Merely incorporating the sequence of object names in the prompt is challenging to distinguish between similar activities that involve overlapping objects. Therefore, additional knowledge needs to be incorporated into the prompt to guide ChatGPT in focusing on important objects. In this study, we automatically generate the additional knowledge \(K\) using the pre-trained language model in ChatGPT.
As shown in Figure 1, this structure consists of two prompts. The first prompt is designed to identify several pairs of activities that are difficult to distinguish from objects of usage. Then, utilizing the outputs from the first prompt, the second prompt outputs the description of each activity using objects.
In the first prompt, we aim to generate \(k\) pairs of activities that are difficult to distinguish from each other using the implicit knowledge of ChatGPT. Let \(O\) represent all the object names in the dataset, and \(q_{p}\) represent the question in this prompt. The pairs of activities generated by the first prompt are described as follows:
\[E=\{e_{p}:e_{p}\sim pG(e_{p}|q_{p},A,O),p=1,2,\dots,k\}, \tag{2}\]
where \(e_{p}\) is a pair of activities expressed in the textual form of "activity A and activity B." According to the formula, the answer will be generated based on the object names, activity classes in the prompt, and the knowledge embedded in \(pG(\cdot)\). The implicit knowledge embedded in the \(pG(\cdot)\) can be helpful to identify similar activities based on the knowledge provided in the prompt. The details of the prompt are composed of All objects, All activities, Question, and Explain and Answer. We append the following sentences for each component: _All objects in the dataset: [All object names]; All activities in the dataset: [All activity classes]; Question: List \(k\) pair of activities in [All activities] that is difficult to distinguish; Answer and Explanation:_. In the sentences, \(\{\cdot\}\) are placeholders that are uniformly replaced with the corresponding text before being fed into the LLM.
The objective of the second prompt is to generate descriptions of activities \(K\) using the knowledge \(E\) obtained from the first prompt. For each \(e_{p}\in E\), we generate the descriptions for the pair of activities \(k_{p}\in K\) simultaneously. By asking ChatGPT to differentiate the pair of activities in the prompt, the important objects in differentiating the activities can be obtained in the output sentence. the idea of generating \(k_{p}\) for each \(e_{p}\) is formulated as follows:
\[\hat{k_{p}}=\operatorname*{arg\,max}_{k_{p}\in K}pG(k_{p}|e_{p},A,O),p=1,2, \ldots,k \tag{3}\]
The details of the prompt are similar to the first prompt with a different Question. The sentence is organized as follows: _Question: Differentiate [\(e_{p}\)] activities based on objects_. We replace the placeholder for each corresponding text of \(e_{p}\) before fed into the LLM, and then combine every output to get \(K\).
Here is an example \(k_{p}\) from the prompt: _Cleanup: During the Cleanup activity, the objects used should be put back to their original place or to the dishwasher. This includes objects such as the Bread Cutter, Knifes, Plates, Glass, Cup, and Plate._ \(\backslash\)_n Early Morning: During the Early Morning activity, the objects used can include the Switch, Drawer3 (lower), Drawer2 (middle), Drawer1 (top), Fridge, and Lazy-chair. These objects are used to perform various activities such as turning on the lights, opening drawers, and getting out of bed.
### Answer Generation
According to Knowledge Generation, ChatGPT automatically generates the descriptions of activities associated with objects. In this part, we discuss predicting activities from a sequence of object names via generated knowledge and ChatGPT. In this step, the Knowledge component in the prompt engineering becomes the descriptions of activities \(K\) generated from the first stage. Since the activity descriptions in the prompt are already generated for differentiating similar activities, when a new sequence of objects is sent to the prompt, ChatGPT can easily focus on the important objects based on \(K\). Besides, we utilize knowledge prompt engineering to improve ChatGPT's ability for HAR. By requesting explanations for HAR based on the prompt's knowledge, we emphasize the important contexts (e.g., object names) in the prompt used for activity recognition. The details of the prompt are displayed as follows: _All activities in the dataset: [All activity classes]; Name and Description of activities: [\(K\)]; Question: A list of objects a person used that ordered in time: [sequence of object names]. Output the name of the activity the person most probably performs; Answer and Explanation:_.
Here is an example output by the prompt: _Answer: Early morning: \(\backslash\)_n Explanation: The objects used in this list suggest that the person is most likely performing an early morning activity, such as turning on the lights, opening drawers, and getting out of bed._
## 4. Experimental Evaluation
### Dataset
We evaluate our method on three datasets. An overview of the three datasets is displayed in Table 1.
**Opportunity dataset**[(17)]. This is a widely used benchmark dataset for HAR from wearables, which records a variety of activities of daily life, with five classes of activities. When the user interacts with an object, the object label will be recorded.
**50 Salads**[(18)]. This dataset collects accelerometer data in a kitchen room. Three classes of activities are recorded, corresponding to different states while cooking. Object labels are collected when the user is using the objects, such as the knife, glass, etc.
**CMU-MMAC**[(4)]. This is a multi-modal database that contains multi-modal measures of the human activity of subjects performing the tasks involved in cooking and food preparation. Twenty-five subjects have been recorded cooking five recipes with the object they used during experiments.
In these datasets, each pre-segmented time period corresponds to a complete activity, and the object usage at each timestamp is recorded and converted into the corresponding object name.
Figure 1. The overview of the Proposed approach. The pink background rectangles show the input of the model, the orange rectangles show the procedures done by ChatGPT, and the green rectangles show the output results of each prompt for ChatGPT processing.
### Result
We conducted a comparative analysis of our proposed prompt engineering approach against three widely used techniques.
**Zero-shot prompting**(Kumar et al., 2017): This approach does not contain any knowledge statement in the prompt. The prompt only consists of all object names and classes of activities.
**Retrieval-based knowledge prompting**(Kumar et al., 2017): In addition to the object names and activities, this approach utilizes the knowledge statement retrieved from the dataset or the sentences in appropriate sources, such as Wikipedia.
**Few-shot prompting**(Kumar et al., 2017): In addition to the zero-shot prompting, this approach employs some examples in the knowledge statement. In this study, for each class of activity, we provide an object name sequence and the corresponding activity as question and answer. Unlike the other approaches, this approach requires costs for preparing the knowledge statement.
For a fair comparison, we implemented prompting engineering using the same parameters. We use the "text-DaVinci-003" of the GPT-3.5 family as the knowledge generator, the temperature was set to 0, and the top_p was set to 0.5.
We used the sequence of objects for each segmented time period as input and output for the predicted activity class via different prompt engineering. We evaluated the model performance using the micro average F1-score.
In Table 2, our proposed approach outperformed the baselines among all datasets. In these datasets, objects overlap between activities. According to zero-shot prompting, if only a sequence of objects was provided, it is insufficient to distinguish between activities. Retrieval-based knowledge prompting performed poorly on the Opportunity dataset, while the proposed approach always has good performances on various datasets. This result suggested that the knowledge generated by the proposed approach is more robust than the knowledge prepared by retrieval-based knowledge prompting. Few-shot prompting, which involved some examples manually prepared by humans in the prompt, demonstrated the highest performance among the three baselines. However, it did not perform as effectively as the proposed approach on the Opportunity dataset. This discrepancy can be attributed to the greater flexibility of objects used in the Opportunity dataset compared to the cook-related datasets (i.e., 50Salads and CMU-MMAC). The results imply that the proposed prompt engineering has the ability to generate high-quality knowledge based on the existing contexts within ChatGPT in an unsupervised manner.
Figure 2 presents the confusion matrix of the Opportunity dataset using different prompting engineering. In the dataset, the cleanup activity contains many objects in the other activities. By generating
\begin{table}
\begin{tabular}{c|c c c} \hline \hline & **Activity Classes** & **Object Names** \\ \hline
**Opportunity** & Relaxing, coffee time, sandwich time, cleanup, early morning & Salami, milk, fridge, bottle, glass, dishwasher, salami knife, spoon, cup, bread, plate, drawer2 (middle), drawer3 (lower), ddoor1, door2, table \\ \hline
**50Salads** & Cut\_and\_mix\_ingredients, prepare\_dressing, serve\_salad & Cucumber, tomato, cheese, bowl, lettuce, ingredients, oil, vinegar, salt, pepper, dressing, plate \\ \hline
**CMU-MMAC** & Cook: Brownie, egg, pizza, salad, sandwich & Fork, fridge, bread, egg, sink, big\_bowl, cup, frying\_pan, brownie\_box, scissors, brownie\_bag, baking\_pan, plate, oven, small\_bowl, drawer, oil, grater, cheese, pepper, knife, sausage, cucumber, caesar\_dressing, carrot, broccoli, celery \\ \hline \hline \end{tabular}
\end{table}
Table 1. An overview of the three datasets.
\begin{table}
\begin{tabular}{c|c c c} \hline \hline & **Opportunity** & **50Salads** & **CMU-MMAC** \\ \hline Zero-shot & 53.61 & 64.47 & 76.32 \\ \hline Retrieval-based & & & \\ knowledge & 48.19 & 93.87 & 71.05 \\ \hline Few-shot & 73.08 & 90.61 & 100.00 \\ \hline
**Proposed** & 91.15 & 100.00 & 100.00 \\ \hline \hline \end{tabular}
\end{table}
Table 2. The F1-score (%) of three datasets.
Figure 2. The confusion matrix of comparative prompting methods and the proposed method of Opportunity dataset.
descriptions of activities from pair of similar activities, the proposed prompting approach successfully identified cleanup activity from other activities.
## 5. Conclusion
In this paper, we propose a two-stage prompting engineering used for ChatGPT to recognize activities using objects the person used in an unsupervised manner. Our proposed prompt engineering is able to generate descriptions of activities for various datasets automatically.
|
2308.05383 | Liquid Metal Molecular Scissors | Molecules are the smallest unit in matters that can exist independently,
relatively stable, and maintain physical and chemical activities. The atomic
species, alignment commands, and chemical bonds are key factors to dominate
their structures and properties. Here we disclosed a general chemistry effect
that the liquid metals can directly cut off oxygen-containing groups in various
molecular matters at room temperature, and then recombine the remaining groups
to form functional materials including nano semiconductors. Based on this
unique mechanism, we proposed a basic tool and named it as liquid metal
scissors for molecular directional clipping and functional transformation. As
proof-of-concept, we demonstrated the capabilities of eGaIn scissors made of Ga
and In particles, and revealed that the Ga on the surface of eGaIn could
directly snatch oxygen atoms from various targeted substances such as H2O, CO2
or CH3OH molecules to form gallium oxides. As illustration, after clipping, the
remaining hydrogen atoms of H2O molecules recombined to form H2, while the
remaining groups of CH3OH lead to H2, carbon quantum dots, and other related
substances. If needed, more molecules can also be manipulated via such
scissors. This finding refreshes the basic knowledge of chemistry and suggests
easygoing ways for molecular weaving, which may break up the limitations and
single features of molecular substances. It also opens up a universal route for
innovating future molecular chemical engineering, life science, energy and
environment, and biomedicine. | Liangfei Duan, Tong Zhou, Huiqin Yang, Weihua Mu, Zhongshan Deng, Jing Liu, Qingju Liu | 2023-08-10T07:00:42Z | http://arxiv.org/abs/2308.05383v1 | # Liquid Metal Molecular Scissors
###### Abstract
We present a novel and efficient method for the adsorption of the adsorbed
###### Abstract
Molecules are the smallest unit in matters that can exist independently, relatively stable, and maintain physical and chemical activities. The atomic species, alignment commands, and chemical bonds are key factors to dominate their structures and properties. Here we disclosed a general chemistry effect that the liquid metals can directly cut off oxygen-containing groups in various molecular matters at room temperature, and then recombine the remaining groups to form functional materials including nano semiconductors. Based on this unique mechanism, we proposed a basic tool and named it as liquid metal scissors for molecular directional clipping and functional transformation. As proof-of-concept, we demonstrated the capabilities of eGaIn scissors made of Ga and In particles, and revealed that the Ga on the surface of eGaIn could directly snatch oxygen atoms from various targeted substances such as H\({}_{2}\)O, CO\({}_{2}\) or CH\({}_{3}\)OH molecules to form gallium oxides. As illustration, after clipping, the remaining hydrogen atoms of H\({}_{2}\)O molecules recombined to form H\({}_{2}\), while the remaining groups of CH\({}_{3}\)OH lead to H\({}_{2}\), carbon quantum dots, and other related substances. If needed, more molecules can also be manipulated via such scissors. This finding refreshes the basic knowledge of chemistry and suggests easygoing ways for molecular weaving, which may break up the limitations and single features of molecular substances. It also opens up a universal route for innovating future molecular chemical engineering, life science, energy and environment, and biomedicine.
**Key words:** Molecular manipulation tool; Liquid metal scissors; Surface engineering; Oxygen capture; Directional clipping; Molecular chemistry.
## Introduction
The basic constituent unit of matters mainly include atoms and molecules, and the molecules are the smallest unit in matters that can exist independently, relatively stable, and maintain physical and chemical properties. [1] The molecules are composed of atoms, while the atoms combine into a variety of molecules under certain force, alignment commands, and chemical bonds. [2] Molecular structures strongly determine the performance and application of substances, including the alignment order of atoms and chemical bond types. [3] Molecular cutting, editing and recombination are an effective strategy to construct emerging functional substances, which provides infinite possibilities for cutting-edge sciences such as biopolymers, small molecular substances, life sciences, advanced materials, biomedicine and energy environment through directional chemical editing means. [4] Historically, the ability to control and manipulate matters at molecular levels has significantly promoted human life and science advancement. [5] However, as it was pointed out, [6] current molecular editing and assembly generally rely heavily on cumbersome processes, complex reagents, and harsh reaction conditions, which turn out to be a large obstacle.
Among the many exciting functional matters, liquid metals (LMs) are emerging as new generation materials with rather unique physical and chemical behaviors. [7] LMs are safe and non-toxic, have high boiling points, reflectivities, good thermal and electrical conductivities, high flexibility, fluidity, self-healing capability and remain in liquid state at room temperature. [8] Particularly, the surface features of LMs significantly differ from solid metals, and the oxide shells with a thickness of \(\approx\) 1-3 nm are formed
on their surface spontaneously and instantaneously upon exposure to atmospheric oxygen. [9] The surface of liquid metals is sensitive to foreign matters. [10] Based on these tactics, it is speculated that LMs contain plenty of outstanding surface behaviors, and their functional capabilities and increasing application scopes would therefore be endowed through surface science discovery and practical exploration. [11] Unfortunately, investigation on the surface issues of LMs is still in the infancy and there exist big gaps within the knowledge of related mechanisms. [12]
In this article, based on a generalized finding that liquid metals can quickly achieve directional cutting, editing, and recombination of molecular substances via oxygen capture mechanism, we proposed to construct a basic tool of liquid metal molecular scissors. It can be made of gallium (Ga), its alloy with indium (In), or more other suitable low melting point alloys. In this sense, a particular tool can be called, say gallium molecular scissors. As disclosed in our experiments, the room temperature liquid metals such as gallium or its alloy have excellent surface activity, which can directionally cut off specific functional groups in molecules, and break the original stability of the chemical bonds so as to achieve the purpose of molecular weaving. As a proof-of-concept and typical example, the Ga and In particles are mixed together to form eGaIn liquid metal scissors. Without loosing any generality and also for brevity, we choose to illustrate several representative molecular substances although more candidates can also be tested. As our experiments disclosed, the excellent surface activity of eGaIn make them react with the inorganic molecule of H\({}_{2}\)O and organic molecule of CH\({}_{3}\)OH spontaneously and instantaneously. And more molecular
substances such as CO\({}_{2}\) can also be manipulated via this way. The relevant mechanisms and potential applications have been elaborated systematically. This finding opens insightful and promising route for molding future molecule chemical engineering, life science, energy and environment, and biomedicine.
## 2 Results and Discussion
### Preparation and mechanism of liquid metal molecular scissors
Living bodies are composed of substances with different functions, which are composed of various molecules with specific elements, structures and chemical bonds. [13] As shown in Figure 1a, the common substances in life are mainly divided into organic or inorganic molecular substances, including H\({}_{2}\)O and CH\({}_{3}\)OH, etc. In addition, the molecular substances also can be divided into solid, liquid and gas such as correspondingly vitamin C, CH\({}_{3}\)COOH, and CO\({}_{2}\). Most molecular substances are rich in C, H, O and other elements. [14] Figure 1b displays the eGaIn exposed to molecular substances, which can spontaneously and rapidly capture the elements with strong oxidization in molecular substances, such as oxygen. When molecular materials are clipped, it would lead to the fact that the stability of the original chemical bonds was broken, and the remaining groups can then be recombined to form new substances. [15] As schematically illustrated in Figure 1c, when liquid metals are immersed in deionized water, bubbles are generated randomly at the contact interface between liquid metals and water. The verification of liquid metal surface activity editing water molecules to produce hydrogen was achieved via two processes: (i) The metal gallium was mixed
with indium to form room temperature liquid metal of eGaIn. (ii) The eGaIn reacted with H\({}_{2}\)O to generate a gas. As schematically shown in Figure 1d, eGaIn is immersed in the methanol with Ar protection, bubbles are generated randomly at the contact interface between eGaIn and methanol. The metals on the surface of LMs was transformed into oxides with blackish-gray, and the supernatant presented obvious fluorescence properties under 254 nm ultraviolet excitation. According to a schematic in Figure 1e, the molecular matters are clipped by gallium molecular scissors via oxygen capture. The surface of eGaIn directionally snatches oxygen atoms from molecular substances to break up the stability of chemical bonds. And then, the remaining groups recombine to form new functional substances. [16]
**Figure 1 The preparation process and mechanism of gallium molecular scissors via oxygen capture mechanism.** (a) The structures, element types, and chemical bonds of common molecular substances. (b) The mechanism and process of gallium molecular scissors via oxygen capture and recombining residual groups. (c) The process and mechanism of water molecular clipping with gallium molecular scissors. (d) The process and mechanism of methanol molecular clipping with gallium molecular scissors. (e) The mechanism model of gallium molecular scissors via oxygen capture.
**Inorganic molecules editing through liquid metal scissors**
To disclose the editing capability of gallium-based liquid metals on H\({}_{2}\)O molecule at room temperature, eGaIn particles were immersed in deionized water and protected by Ar. Figure 2a displays that the surface of LMs was transformed into oxides, which then were stripped from the surface of liquid metal and make the deionized water shown cloudy and blackish-gray. Furthermore, the bubbles were generated randomly on the surface of water. This suggests that the eGaIn triggers the breakdown of water molecules at room temperature. As shown in Figure 2b, the gas composition in the reaction bottle is detected by a gas chromatograph, resulting in that with the increase of stirring time, the intensity of hydrogen chromatographic peak gradually increases, while the oxygen (O\({}_{2}\)) chromatographic peak decreases until disappears, and the nitrogen (N\({}_{2}\)) chromatographic peak changes little. This indicates that eGaIn produces hydrogen after being exposed to water ambient, absorbs oxygen, but has little effect on nitrogen. As schematically illustrated in Figure 2c, the hydrogen content in the reaction
bottles is compared and calculated, indicating that the hydrogen content has a strong positive correlation with the stirring time. These proved that eGaIn liquid metal displays obvious clipping effect on water molecules at room temperature, and the remaining hydrogen atoms combine to form H\({}_{2}\).[17] As shown in Figure 2 (d-e), the solid substances of eGaIn after reacting with water molecules is determined as GaO(OH) by IR and Raman spectrum analysis,[18, 19, 17, 18, 19] and most of GaO(OH) is converted into Ga\({}_{2}\)O\({}_{3}\) after annealing at 600 \({}^{\circ}\)C for 2.0 h, as depicted in Eqs. (1) - (2). This suggests that the eGaIn has excellent cutting capability on H-O bond in H\({}_{2}\)O molecule.
\[\text{Ga}_{(1)}\text{+H}_{2}\text{O}_{(1)}\text{=GaO(OH)}_{(3)}\text{+H}_{2( \text{g})} \tag{1}\]
\[\text{2GaO(OH)}_{(3)}\text{=Ga}_{2}\text{O}_{3(3)}\text{+H}_{2}\text{O}_{( \text{g})} \tag{2}\]
The XPS spectra of the samples were further collected to study the chemical states of each element in reaction products.[20] As shown in Figure 2f, the spectrum of Ga 2p is fitted into four sub-peaks, Ga2p\({}_{3/2}\) at 1115.6 eV and 2p\({}_{1/2}\) at 1142.44 eV, Ga\({}^{3+}\) 2p\({}_{3/2}\) at 1118.08 eV and 2p\({}_{1/2}\) at 1142.44 eV. Figure 2g displays that the In 3d spectrum is fitted into four sub-peaks, In 3d\({}_{5/2}\) at 442.73 eV and 3d\({}_{3/2}\) at 450.26 eV, In\({}^{3+}\) 3d\({}_{5/2}\) at 443.77 eV and 3d\({}_{3/2}\) at 451.22 eV, respectively. According to Figure 2h, the O 1s spectrum is fitted into two sub-peaks, Ga-O at 530.78 eV and Ga-OH at 531.71 eV. As shown in Figure 2i, the products of the eGaIn liquid metal reacted with water were annealed at 600\({}^{\circ}\)C for 2.0h, the proportion of Ga-O bond increased and the proportion of Ga-OH bond decreased. It was further confirmed that annealing transformed GaO(OH) into Ga\({}_{2}\)O\({}_{3}\).[21] This indicates that gallium scissors serve well to generate hydrogen by trapping oxygen from H\({}_{2}\)O, and converted gallium into functional semiconductors of GaO(OH)
and \(\beta\)-Ga\({}_{2}\)O\({}_{3}\). It has pretty practical values in the fields of energy, electronic information, optoelectronic devices and biomedicine.
As schematically illustrated in Figure 3a, liquid metals and their immersion in water all lacked crystalline lattice. The crystalline of products were determined by X
Figure 2: **The effect of gallium molecular scissors on H\({}_{2}\)O molecules.** (a) Appearance of eGaIn reacting with H\({}_{2}\)O. (b) Gas chromatogram of eGaIn reacting with H\({}_{2}\)O. (c) The hydrogen production of eGaIn reacting with water varying with time. (d) IR spectra of GaO(OH) and \(\beta\)-Ga\({}_{2}\)O\({}_{3}\). (e) Raman spectrograph of GaO(OH) and \(\beta\)-Ga\({}_{2}\)O\({}_{3}\). (f) XPS spectra and fitting result for Ga2p. (g) XPS spectra and fitting result for In3d. (h) XPS spectra and fitting result for O1s. (i) XPS spectra and fitting result for O1s of the products by eGaIn reacting with water after annealing.
-ray diffraction (XRD) analysis, and the corresponding data is depicted in Figure 3b. The results showed that GaO(OH) was mainly generated after the eGaIn liquid metal was clipped to water molecules, but GaO(OH) was gradually transformed into \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) after annealing at 600 \({}^{\circ}\)C for 2.0 h (Figure 3c). These all confirmed that Ga on the surface of eGaIn liquid metal has excellent and directional editing effect via oxygen capture for H\({}_{2}\)O molecule, and the remaining hydrogen atoms rebuild to form hydrogen. As shown in Figure 3d, the core-shell structure is an important and ubiquitous characteristic of liquid metals. Once exposed to even trace amounts of water, a thin layer of oxidation would form on the surface of liquid metals almost immediately. [22] As depicted in Figure 3e, after centrifugation and vacuum drying, the solid reaction products from eGaIn liquid metal editing water are mainly powdered, and their surfaces are covered by linear structural materials. As shown in Figure 3f, TEM observation, the supernatant of eGaIn clipping water molecules is mainly GaO(OH) nanowires with a diameter of about 10-25 nm. [23] Therefore, the gallium molecular scissors via oxygen capture editing water molecules is also an effective strategy for fabricating semiconductor nanomaterials.
XRD patterns of liquid metals and their immersion in water. (b) XRD patterns of the products by eGaIn reacting with water. (c) XRD patterns of the products by eGaIn liquid metal reacting with water after annealing at 600 \({}^{\circ}\)C for 2.0 h. (d) SEM image of eGaIn droplet exposed to the water. (e) SEM image of GaO(OH) powders. (f) TEM image of GaO(OH) nanowires. (g-i) EDS spectra of products by eGaIn liquid metal reacting with water.
### Organic molecules editing through liquid metal scissors
Methanol is a typical oxygen-containing organic molecule, and a vital chemical raw material.[24] The editing function of gallium-based liquid metal to organic molecules was further disclosed. As performed in our experiments, eGaIn was soaked in methanol and stirred continuously. As shown in Figure 4a, eGaIn gradually becomes turbid in
Figure 3: **Editing effect of liquid metal scissors on water molecules at room temperature.** (a) XRD patterns of liquid metals and their immersion in water. (b) XRD patterns of the products by eGaIn reacting with water. (c) XRD patterns of the products by eGaIn liquid metal reacting with water after annealing at 600 \({}^{\circ}\)C for 2.0 h. (d) SEM image of eGaIn droplet exposed to the water. (e) SEM image of GaO(OH) powders. (f) TEM image of GaO(OH) nanowires. (g-i) EDS spectra of products by eGaIn liquid metal reacting with water.
methanol solution, accompanied by bubbles generation. Clearly, the methanol solution was clipped by gallium-based liquid metal scissors, and the remaining groups recombine to form new products and exhibit obvious fluorescence characteristics under 254nm ultraviolet excitation. This suggests that the eGaIn has a significant editing effect on methanol molecule, which allowed them to transform into new substances with fluorescence properties at room temperature. [25] As shown in Figure 4b, the gas composition in reaction bottles were analyzed by a gas chromatograph. The results indicated that the intensity of hydrogen chromatographic peak gradually increases with the stirring time. [26] In addition, ultrasonic treatment was more favorable for eGaIn to clip methanol and increase hydrogen content, while the intensity of oxygen chromatographic peak gradually decreases until disappears, and the intensity of nitrogen chromatographic peak changes little. This suggests that exposure of the liquid metal to oxygen-containing organic molecules breaks the stability of their chemical bonds through oxygen capture, and free hydrogen atoms inter-combined to form hydrogen gas. As shown in Figure 4c, the hydrogen content in the reaction bottles is compared and calculated, resulting in that the hydrogen content gradually increases with the stirring time. This proved that eGaIn has a significant cutting effect on methanol molecules by oxygen capture at room temperature. As shown in Figure 4d, the phase structures for the reaction products of eGaIn reacting with methanol were analyzed, resulting in that the products mainly display the diffraction bulge of the material structures of carbon and GaO(OH), which confirmed that the crystallinity of the products were poor. As schematically illustrated in Figure 4e, IR analysis, the
reaction products of eGaln and methanol mainly included GaO(OH), C-O-C, C-C, C-H, COO', C-O, and OH' groups, [27] and we infer that the new products with fluorescence characteristics are carbon quantum dots (C-CDs). [28] As shown in Figure 4f, Raman spectrum analysis, it is further confirmed that the products of eGaln liquid metal reacting with methanol mainly include GaO(OH) and carbon materials. [29] This confirmed that eGaIn liquid metal can lead to the formation of carbon quantum dots with fluorescence behaviors after editing methanol molecule. The XPS spectra of the samples were further collected to study the chemical states of each element in reaction products, and the corresponding data is shown in Figure 4(g-i). As indicated in Figure 4g, the spectrum of Ga 2p is fitted into two sub-peaks, Ga\({}^{3+}\) 2p\({}_{3/2}\) at 1117.74 eV and 2p\({}_{1/2}\) at 1144.56 eV. Figure 4h displays that the O 1s spectrum is fitted into two sub-peaks, Ga-O at 530.69 eV, -OH at 532.14 eV, and OH' at 532.64 eV. As shown in Figure 4i, C 1s spectrum is fitted into three sub-peaks, C-C at 284.8 eV, C-OH at 286.42 eV, and COOH' at 288.78 eV. According to XPS analysis, it further confirmed that after the eGaIn liquid metal reacting with methanol, the oxygen-containing groups and the gallium on the surface of eGaIn formed gallium oxides, a variety of new substances can be formed through random combinations of remaining groups, including hydrogen, carbon quantum dots, etc. The cutting process can be described as Eqs. (3):
\[\text{Ga}_{(0)}+\text{CH}_{3}\text{OH}_{(0)}=\text{GaO(OH)}_{(\text{s})}+ \text{H}_{2(\text{g})}+\text{C-CDs(s)} \tag{3}\]
The methanol solution presented obvious fluorescence characteristics under UV excitation after eGaIn editing. Figure 5a showed the photoluminescence (PL) spectra for all samples measured in the region around 500 nm. The intensities and positions of the PL peaks shifted with changes in reaction time. The PL spectra was fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. 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The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. 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The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green, respectively. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green, respectively. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, green, and green. The PL spectra were fitted using Gaussian model, revealing multiple light emission peaks corresponding to blue, green, and green.
yellow, and red wavelengths, respectively (Figure 5b). Figure 5c illustrated variations in the fluorescence decay profiles in the same strategy, the PL decay spectra can be fitted by a bi-exponential function providing in real time the (\(\tau_{1}\)) and (\(\tau_{2}\)) components, [30] and the average recombination lifetime. Figure 5c indicated that the maximum fluorescence lifetime of methanol solution after eGaIn editing is 2.51 ns. The position, area, and FWHM (Full-width half-maximum) of the peaks varied with the reaction time, which was consistent with the shifts in the relevant CIE coordinates. Figure 5d depicted the CIE chromaticity diagrams of the three samples, and the shift in luminescence color with reaction time. The CIE coordinates were correspondingly CIEx,y = 0.41, 0.53, CIEx,y = 0.30, 0.61, and CIEx,y = 0.31, 0.61, respectively. These suggested that the types and properties of the methanol molecular editing products as tested have a strong dependence on time and methods. As shown in Figure 5e, after vacuum drying, a core-shell structure is formed between the reaction products, which is tightly connected and not easy to fall off. Further, the contents of Ga, O, C, and In elements in the core-shell structure are shown in Figure 5f. The shells and their texture are mainly rich in C (72.7 wt.%), with little amounts of Ga (16.9 wt.%) and O (10.4 wt.%). But the core and their texture are mainly rich in Ga (65.9 wt.%) and O (26.9 wt.%), with little amounts of C (7.2wt.%). This suggests that the cores are mainly composed of GaO(OH), and the shells are formed by the deposition of carbon quantum dots during the drying process. As shown in Figure 5g, the core-shell structure formed by Ga(OH) and carbon materials is about 10-50 nm in size. Figure 5g also displays the EDS mapping acquired on the core-shell structure particles, resulting in that the distribution of the Ga, O, and C
elements on the surface of core-shell structure particles was relatively broad. The EDS spectrum also showed enrichment of C, O elements with significant correlation in shell region, and the Ga, O elements overlapped with the enriched region in core. This indicated that after eGaIn combing with methanol molecule, the thin layers stacked with Ga(OH) and carbon quantum dots were formed instantaneously and spontaneously at their contact interface, meanwhile hydrogen gas is released. The composite shells of Ga(OH) and carbon quantum dots covered on the surface of eGaIn, which prevented the liquid metal particles from binding to each other during agitation. In addition, combined with all the studies and analyses, the outer layer of the carbon quantum dot shell also adsorbed -H, -OH, -COOH and other groups.
PL spectra of eGaIn reacting with methanol. (b) Multi-peak fitting with Gaussian model of eGaIn
Figure 5: **Luminescence properties and micromorphology of eGaIn reacting with methanol.** (a) PL spectra of eGaIn reacting with methanol. (b) Multi-peak fitting with Gaussian model of eGaIn
reacting with methanol fitted using a biexponential model. (c) TPL decays fitted using a biexponential model. (d) CIE coordinates of eGaIn reacting with methanol. (e) SEM image for the core-shell structure of reaction products. (f) EDS spectra for the core-shell structure of reaction products. (g) TEM and EDS spectra for the core-shell structure of reaction products.
## Conclusion
In summary, the phase structures and surface chemical activity of room temperature liquid metals are essential for achieving unique molecular scissors on demand. A basic way was discovered and demonstrated to realize the gallium molecular scissors via oxygen capture. Except for the tested eGaIn liquid materials by mixing of Ga with In particles under Ar protection, more scissors can also be developed in the coming time. At this stage, the transition of the liquid phase structure significantly increased the surface chemical activity of Ga, leading to its spontaneous and rapid capture of oxygen atoms in oxygen-containing molecular matters. As a typical experimental demonstration, the eGaIn spontaneously and rapidly reacted with water to form GaO(OH), along with hydrogen production. This outputs quite useful byproduct GaO(OH), which is in fact a semiconductor material with convenient structure and performance regulation. We also found that the eGaIn could react with methanol to generate GaO(OH), H\({}_{2}\), carbon quantum dots, and other related products. It should be pointed out that, gallium molecular scissors via oxygen capture is a spontaneous process, which avoids the traditional harsh conditions such as high temperature, high vacuum and catalyst, breaking through the limitations of performance and application
singularization facing many molecular substances. Clearly, the abundance of organic and inorganic molecules, and the fluidity and activity of gallium-based liquid metals guarantee tremendous opportunities for tackling the tough challenges in diverse areas from energy, environment, to advanced materials etc. This work established a general, tunable, and scalable liquid metal scissors for cutting and editing molecules via a rather easygoing way. The scissors are also expected to be a powerful tool for innovating future chemical engineering, life science, energy and environment, and biomedicine where various molecules are playing indispensable roles everywhere.
## Materials and Methods
### Syntheses and processing of materials
The LMs (eutectic gallium and indium, eGaIn) were made by high purity gallium (Ga) and indium (In) metals (with purity of 99.9%) as raw materials. The mass ratios of Ga, In in the eGaIn were 75.5% and 24.5%, respectively. The Ga and In metals were determined by an electronic balance (METTLER TOLEDO 1.0 \(\times 10^{-5}\) g). Then the pre-weighed Ga and In metals were mixed and stirred to prepare the liquid metals at room temperature for 30 min in argon (Ar) protection.
### Processing of molecular editing
Gallium molecular scissors via oxygen capture triggers the molecular clipping at room temperature which was synthesized by the following steps: Firstly, the 50 ml glass reaction bottles were filled with 30 ml H\({}_{2}\)O or CH\({}_{3}\)OH, respectively. Secondly, the air in the reaction bottles was removed using a gas controller, and the argon (Ar) was filled
as a protective gas to isolate the external air into reaction bottles. Then the eGaIn (10g) is injected into H\({}_{2}\)O or CH\({}_{3}\)OH from the seal of reaction bottles. Finally, The eGaIn is constantly exposed to H\({}_{2}\)O or CH\({}_{3}\)OH to produce hydrogen (H\({}_{2}\)) and any solid materials.
**Characterization methods**
The X-ray diffraction (XRD) patterns were acquired on a SmartLabSE X-ray diffractometer (CuK\(\alpha\)1, k = 1.54056 A, 40 kV, 50 mA), in the scanning range of 2\(\theta\) = 5\({}^{\circ}\) to 90\({}^{\circ}\). X-ray photoelectron spectra (XPS) were acquired using ESCALAB250Xi X-ray photoelectron spectrometer (USA), wherein the binding energy of the C 1s peak at 284.8 eV was used as the internal reference. Scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDS) were conducted on a TESCAN AMBER and Nova NanoSEM 450 scanning-electron microscope at 20 keV voltage. The infrared spectroscopy patterns were acquired on a FTIR-2000 Fourier transform infrared (FTIR) spectroscopy, and the scanning range is 400 cm-1 to 4400 cm-1. The Raman spectroscopy patterns were acquired on a LabRam HR Evolution high-resolution confocal laser micro-Raman spectrometer, and the scanning range is 200 cm-1 to 3250 cm-1. The gas in the reaction bottle was detected and analyzed by a GC9790II gas chromatography. Photoluminescence (PL) spectra were obtained using a FLS 1000 fluorescence spectrometer (Edinburgh Instruments, UK), with excitation wavelength of 254-365 nm, scanning speed of 1 nm s-1, and the widths of both the excitation slit and emission slit are 2.0 nm.
## Acknowledgement
This work was funded by the National Key Research and Development Program of China (2022YFB3803600), National Natural Science Foundation of China Projects (51890893), the Key Research and Development Program of Yunnan Province (202302AF080002), Yunnan Yunling Scholars Project, Young and Middle-aged Academic and Technical Leaders Reserve Talent Project in Yunnan Province (202005AC160015). Yunnan Industrial Technology Innovation Reserve Talent Project (No. 202105AD160056). Yunnan Basic Applied Research Project (No. 202101AT070013, 2019-1-C-25318000002171). Authors thank the Electron Microscopy Center, the Advanced Computing Center, the Advanced Analysis and Measurement Center of Yunnan University for the sample testing and computations service.
## Author contributions:
Conceptualization: L. Duan, J. Liu, Q. Liu. Methodology: L. Duan, T. Zhou. Investigation: L. Duan, T. Zhou, H. Yang, W. Mu. Project administration and supervision: Q. Liu, J. Liu. Writing - original draft: L. Duan, T. Zhou. Writing - review & editing: L. Duan, J. Liu, Q. Liu. All authors have discussed and given approval to the final version of the manuscript.
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2306.14722 | FC-KBQA: A Fine-to-Coarse Composition Framework for Knowledge Base
Question Answering | The generalization problem on KBQA has drawn considerable attention. Existing
research suffers from the generalization issue brought by the entanglement in
the coarse-grained modeling of the logical expression, or inexecutability
issues due to the fine-grained modeling of disconnected classes and relations
in real KBs. We propose a Fine-to-Coarse Composition framework for KBQA
(FC-KBQA) to both ensure the generalization ability and executability of the
logical expression. The main idea of FC-KBQA is to extract relevant
fine-grained knowledge components from KB and reformulate them into
middle-grained knowledge pairs for generating the final logical expressions.
FC-KBQA derives new state-of-the-art performance on GrailQA and WebQSP, and
runs 4 times faster than the baseline. | Lingxi Zhang, Jing Zhang, Yanling Wang, Shulin Cao, Xinmei Huang, Cuiping Li, Hong Chen, Juanzi Li | 2023-06-26T14:19:46Z | http://arxiv.org/abs/2306.14722v1 | # FC-KBQA: A Fine-to-Coarse Composition Framework for Knowledge Base Question Answering
###### Abstract
The generalization problem on KBQA has drawn considerable attention. Existing research suffers from the generalization issue brought by the entanglement in the coarse-grained modeling of the logical expression, or inexecutability issues due to the fine-grained modeling of disconnected classes and relations in real KBs. We propose a Fine-to-Coarse Composition framework for KBQA (FC-KBQA) to both ensure the generalization ability and executability of the logical expression. The main idea of FC-KBQA is to extract relevant fine-grained knowledge components from KB and reformulate them into middle-grained knowledge pairs for generating the final logical expressions. FC-KBQA derives new state-of-the-art performance on GrailQA and WebQSP, and runs 4 times faster than the baseline. Our code is now available at GitHub [https://github.com/RUCKBReasoning/FC-KBQA](https://github.com/RUCKBReasoning/FC-KBQA).
## 1 Introduction
Question answering over knowledge bases (KBQA) aims to provide a user-friendly way to access large-scale knowledge bases (KBs) by natural language questions. Existing KBQA methods Zhang et al. (2023) can be roughly categorized into retrieval-based and semantic-parsing (SP) based methods. The former Feng et al. (2021); He et al. (2021); Zhang et al. (2022) directly scores the relevance between the question and answer candidates, thus it is difficult to resolve the complex questions. On the contrary, some KBQA approaches, such as Das et al. (2021); Kapanipathi et al. (2021); Qiu et al. (2020); Sun et al. (2020), are based on semantic parsing (denoted as SP-based), which can address complex questions and achieve promising results on i.i.d. datasets. SP-based methods first translate the questions into logical expressions such as SPARQL and then execute them against KB to yield answers. As illustrated in Figure 1, a logical expression consists of multiple components such as classes and relations. Most existing SP-based approaches fail with logical expressions that contain unseen compositions of components (called compositional generalization) or unseen components (called zero-shot generalization).
To address the above problem, GrailQA-Rank Gu et al. (2021) proposes a BERT-based rank model to match the given question with each logical expression candidate, which leverages the generalization abilities of the pre-trained language models. On top of that, RN
Figure 1: Illustration of generalization tasks in KBQA. Each question is paired with a logical expression that consists of different components. Components involved in the training data are colored in non-green color, while unseen components are colored in green.
Figure 2: Results of the pilot study. The coarse-grained method directly matches the question with the logical expression (i.e., the composition of components), while the fine-grained method matches the question with each component candidate and then composes them to derive the logical expression. The exact match accuracy of logical expressions on compositional generalization test data and zero-shot generalization test data is shown on the right of the figure.
2022) further uses a pre-trained generation model, which takes top-5 ranked logical expressions as the additional input beyond the question to generate the target logical expression. Behind these mainstream models, a logical expression is viewed as an inseparable unit during modeling. Actually, logical expressions are coarse-grained because they can be decomposed into relatively fine-grained components including relations, classes, entities, and logical skeletons (See examples in Figure 3). Such coarse-grained modeling entangles representations of fine-grained components, thereby overfitting the seen compositions during the training process, which weakens the model's compositional generalization ability. Meanwhile, even though pre-trained language models can deal with zero-shot components to some extent, compositional overfit reduces their ability to identify individual unseen components with zero-shot generalization.
To demonstrate the above idea, we perform a pilot study (Cf. the detailed settings in Section 4.1) with two preliminary experiments: one calculates the similarity score between a question and each coarse-grained logical expression to obtain the most relevant one, and the other searches the most relevant fine-grained components to form the final logical expression of a question. We observe that **the fine-grained modeling derives more accurate logical expressions on both the compositional task and zero-shot task (Cf. Figure 2)**. It could be explained that fine-grained modeling focuses exclusively on each component, avoiding overfitting of seen compositions in the training data. Although some studies attempt to leverage fine-grained components, they only consider partial fine-grained components such as relations, classes, and entities (Chen et al., 2021), or suffer from inexecutability due to disconnected fine-grained components in real KBs (Shu et al., 2022).
Thus, to both ensure the generalization ability and executability of logical expressions, we propose a Eine-to-Coarse composition framework for KBQA (FC-KBQA), which contains three sub-modules. The overview of our model is shown in Figure 4. The first module is fine-grained component detection, which detects all kinds of fine-grained component candidates from Freebase by their semantic similarities with the question. Such component detection guarantees the generalization ability in both compositional and zero-shot tasks. The second module is the middle-grained component constraint, which efficiently prunes and composes the fine-grained component candidates by ensuring the components' connectivity in the KB. The final module is the coarse-grained component composition, which employs a seq-to-seq generation model to generate the executable coarse-grained logical expression. In addition to encode the fine-grained components, the middle-grained components are also encoded to enhance the model's reasoning capacity, so as to improve the executability of the generated logical expression. In contrast to previous work (Cao et al., 2022; Chen et al., 2021; Shu et al., 2022) that only uses the knowledge constraints to guide the decoding process, we emphasize injecting them into the encoding process, because the encoder which learns bidirectional context could better suit natural language understanding (Du et al., 2022).
We conduct extensive experiments on widely used GrailQA, WebQSP, and CWQ datasets. GrailQA (Gu et al., 2021) is a KBQA benchmark focusing on generalization problems. FC-KBQA derives new state-of-the-art performance on GrailQA-Dev (+7.6% F1 gain and +7.0% EM gain respectively). Meanwhile, FC-KBQA also obtains good performance on WebQSP and CWQ. Moreover, FC-KBQA runs 4 times faster than the state-of-the-art baseline RNG-KBQA. The ablation studies demonstrate the effect of our middle-grained encoding strategy.
**Contributions.** (1) We conduct a pilot study to reveal an intriguing phenomenon -- a fine-grained understanding of the logical expression helps enhance the generalization ability of SP-based KBQA methods, which is rarely discussed before. (2) We propose a fine-to-coarse composition framework FC-KBQA to address the generalization problem, which takes advantage of the idea of fine-grained modeling. (3) We devise a middle-grained component constraint that is injected into both the encoder and the decoder to guide the seq-to-seq model in producing executable logical expressions. (4) FC-KBQA not only maintains efficiency but also achieves significant improvement on GrailQA.
## 2 Related Work
**Coarse-Grained SP-based Methods.** Many efforts are paid to solve generalization problems on SP-based KBQA. Some approaches, such as (Lan and Jiang, 2020; Gu et al., 2021), use a rank-based model that takes advantage of a coarse-level match
between the question and the logical expressions or query graphs. They first enumerate numerous query graph candidates based on KBs and then they rank them according to how relevant they are to the question. Another line of approaches, in addition to the rank-based ones, makes use of a generation model. KQAPro (Cao et al., 2022) leverages BART to directly convert questions into logical expressions. Additionally, RNG-KBQA (Ye et al., 2022) further injects top-k ranked logical expressions as an additional input to the question. CBR-KBQA (Das et al., 2021) injects analogous questions and their corresponding logical expressions from the training data to increase the generalization. All of the aforementioned methods are pure coarse-level frameworks that treat each coarse-grained logical expression as a separate unit.
**Fine-Grained SP-based Methods.** Many researchers have been motivated to address the generalization issue by the notion of utilizing decomposed components, such as class, relation, and logical skeleton. Some approaches (Wang et al., 2020; Zhao et al., 2022; Li et al., 2023) retrieve the relevant schema item such as relation and column as additional fine-grained input information, while another line of approaches (Dong and Lapata, 2018) extracts the skeleton of logical expression as the decoder guide. Such methods primarily concentrate on the grammar of logical expression and often ignore the knowledge constraint, which is essential in large-scale KB. They usually focus on KBs or DBs that contain a small number of relations where a logical expression can be easy to be executable. Program Transfer (Cao et al., 2022), R-Track (Chen et al., 2021), and TIARA (Shu et al., 2022) simply apply KB constraints to control the generation of the decoding process. As opposed to them, we make use of middle-grained KB constraints during both the encoding and the decoding processes to help the model better adapt to KB and ensure executability.
## 3 Problem Definition
**Knowledge Base (KB).** A KB is comprised by ontology \(\{(C\times R\times C)\}\) and relational facts \(\{(E\times R\times(E\cup C))\}\), where \(R,C,\) and \(E\) denote relation set, class set, and entity set respectively. Notably, we consider literal as a special type of entity. Specifically, an ontology triple \((c_{d},r,c_{r})\) consists of a relation \(r\in R\), a domain class \(c_{d}\) which denotes the class of the subject entities, and a range class \(c_{r}\) which denotes the class of the object entities. Each class has multiple entities, thus an ontology triplet can be instantiated as several relational facts. For example, both \((e_{1},r,e_{2})\) and \((e_{3},r,e_{4})\) correspond to \((c_{d},r,c_{r})\), where \(e_{1},e_{3}\in c_{d}\) and \(e_{2},e_{4}\in c_{r}\). Figure 3 illustrates a KB subgraph.
**SP-based KBQA.** Given a natural question \(q\), KBQA models aim to find a set of entities denoted by \(A\subseteq E\) from KB as the answers to \(q\). Instead of directly predicting \(A\), SP-based KBQA models translate \(q\) to an executable logical expression denoted by \(s\) such as SPARQL, lambda-DCS (Liang et al., 2013), query graph (Lan and Jiang, 2020), and s-expression (Gu et al., 2021).
We select s-expression as our used logical expression since it could provide a good trade-off on compactness, compositionality, and readability (Gu et al., 2021). The **logical skeleton** of an s-expression can be derived by removing all the relations, classes, and entities in the expression and only keeping function operators and parentheses. Specifically, we replace relations, classes, entities, literals with special tokens "<rel>", "<class>", "<entity>", "<literal>" respectively. Figure 3 shows an executable logical expression on the KB and its corresponding logical skeleton. We unitedly name the relations, classes, entities, and logical skeleton in an s-expression as the **fine-grained component**, while the complete s-expression is the **coarse-grained logical expression**.
## 4 Approach
### Pilot Study
As analyzed in Section 1, considering the logical expression as a unit will lead to entangled representations of fine-grained components and thus weakens generalization ability. Here we study the necessity of fine-grained modeling by testing how coarse-grained and fine-grained matching methods perform when selecting a question's logical expression from the corresponding candidate pool.
**Dataset.** To simplify the experiment, we extract a toy dataset that only involves 1-hop logical expressions from GrailQA. Then, for the relation \(r\) and the class \(c\) in such logical expressions, we study the compositional generalization where the composition \((r,c)\) is unseen or zero-shot generalization where the individual \(r\) or \(c\) is unseen in the training data. For each question with its ground-truth logical expression, we select 100 logical expressions
that share the same domain as the ground truth as the coarse-grained expression candidates. For fair comparison, we separate all of the relations, classes, and logical skeletons from the coarse-grained candidates as the fine-grained component candidates.
**Methods.** We aim to find the target logical expression of a given question by a ranking model trained with a contrastive loss (Chen et al., 2020), which is also used by RNG-KBQA (Ye et al., 2022). The coarse-grained method concatenates a question and a candidate logical expression to feed into BERT, then the output embedding of [CLS] is fed into a linear layer to compute the similarity score. The fine-grained method follows the above pipeline, but the input is the concatenation of a question and a fine-grained candidate component, then scores each logical expression candidate by summing up the normalized question-component similarity scores. For both methods, we compute accuracy by evaluating whether the ground-truth logical expression owns the highest score in the candidate pool.
**Observation -- Fine-grained modeling can better solve the generalization problems on KBQA.** The matching accuracy is reported in Figure 2. The fine-grained method outperforms the coarse-grained method in both composition generalization and zero-shot generalization tasks. A possible explanation is the fine-grained matching focuses solely on each component and is simple to learn, which better capture the semantic information of each component and also well adaptable to express the various compositions of components. The coarse-grained matching, on the other hand, attempts to describe all of the components as a whole composition, limiting the ability to express unseen compositions and components. Inspired by this, we propose FC-KBQA in the next section.
### Model Overview
We propose a fine-to-coarse composition framework FC-KBQA bridged by a middle-grained KB constraint. Figure 4 illustrates the overall framework, which contains three parts:
**Fine-grained Component Detection.** Given a question, we extract relation candidates and class candidates from the whole KB based on semantic similarity. Simultaneously, we adopt an entity linker to detect mentioned entities and use a seq-to-seq model to generate logical skeletons.
**Middle-grained Component Constraint.** Based on the detected components, we devise an efficient way to check the connectivity of component pairs on the KB, including class-relation pairs, relation-relation pairs, and relation-entity pairs. We only keep the executable component pairs to guarantee the executability of final logical expression.
**Coarse-grained Component Composition.** Finally, a seq-to-seq model takes the concatenation of the question and the reformulated components as input to generate the logical expression. In particular, the middle-grained components are injected into both the encoder and the decoder to ensure the executability of the final logical expressions.
### Fine-grained Component Detection
**Relation and Class Extraction.** Taking the relation extractor as the example, given a question \(q\), we aim to extract relations in \(q\). First, we apply BM25 (Robertson et al., 2009) to recall the relation candidates from the KB based on the surface overlaps between relations' names and \(q\). Then we apply BERT (Devlin et al., 2019) as the cross-encoder to measure the semantic similarity between \(q\) and each relation candidate \(r\). We describe \(r\) using the relation domain, the relation name, and the relation range and let the BERT input be "[CLS] q [D] domain(r) [N] name(r) [R] range(r) [SEP]", where [CLS], [SEP], [D], [N], and [R] are the special tokens. To better distinguish the spurious relations,
Figure 3: Illustration of a KB subgraph and an executable logical expression, where the ovals denote the entities, the rectangles denote the classes, the solid lines denote the relations, and the dashed lines connect the entities and their classes. The upper part of the subgraph illustrates examples of ontology triplets, while the bottom illustrates relational facts.
we sample the relations that share the same domain as the ground-truth relation as the negatives for training. The trained model is used to retrieve the set of top-\(k\) relations, denoted by \(R_{q}\).
The class extractor works in the same way as the relation extractor. We represent the class using its name and domain, and use other classes in the same domain as negatives. \(C_{q}\) represents the set of the top-\(k\) relevant classes.
**Entity Linking.** A common paradigm of finding topic entities in KBQA methods is to first leverage a NER tool (Finkel et al., 2005) to detect mentions and then apply an entity disambiguation model to link them to entities in KB. However, some nounphrase mentions such as "rich media" are hard to be detected by the NER tool, and some ambiguous entities could not be distinguished by the pure entity names. To address both issues, we equip the NER tool1 with a trie tree-based mention detection method and propose a relation-aware pruning method to filter the mentions.
Footnote 1: We follow GrailQA which utilizes an open BERT-NER tool on GitHub ([https://github.com/kamalkraj/BERT-NER](https://github.com/kamalkraj/BERT-NER)).
Specifically, we build a trie tree (Fredkin, 1960) with the surface names of all entities in the KB. Then we can search noun phrase mentions in the question efficiently and link them to the KB by BLINK (Wu et al., 2020) to obtain the corresponding entities \(E_{q}\). After that, we propose a relation awarded pruning strategy to prune \(E_{q}\) by removing the entities that could not link to any relations in \(R_{q}\). Finally, following GrailQA (Gu et al., 2021), we choose the entity with the highest popularity. We define regular expressions to extract literals such as digits and years appearing in \(q\).
**Logical Skeleton Parsing.** Logical skeleton parsing aims to transform a given question \(q\) into a logical skeleton \(l\). Because the logical skeleton is domain-independent, the parsing process could be generalized across domains. We adopt T5 (Raffel et al., 2020), a state-of-the-art generation model to parse logical skeletons. Since many entity names contain tokens such as "and" and "of" that may cause the logical skeleton to be incorrectly determined, we mask each mention \(m\in M_{q}\) with the special token "entity0>", "entity1>",..., in order of appearance. For example, we change "Thomas was the designer of what ship?" to "entity0> was the designer of what ship?". We notice that a common error is parsing out logical skeleton with wrong relation numbers, for example "<rel>" instead of "<rel><rel>". Instead of increasing beam numbers, we manually add grammar rules, such as add "<rel><rel>" as the second candidate when
Figure 4: Overview of FC-KBQA. In the step of fine-grained component detection, we perform class extraction, relation extraction, entity linking, and logical skeleton parsing to obtain the most relevant components of the question. Then we utilize the KB-based constraint to obtain middle-grained component pairs that are connected in the KB. Finally, a T5-based seq-to-seq model encodes the reformulated fine-grained and middle-grained candidates (reformulation unit), and employs a controllable decoder with dynamic vocabulary (control unit) to generate the executable target logical expression.
"<rej>" is T5's top-1 prediction. The set of the top-2 logical skeleton candidates is denoted as \(L_{q}\).
### Middle-grained Component Constrain
After deriving the candidate components according to Section 4.3, the KB-based constraint is required to guarantee the composed logical expression is executable. A straightforward idea is to fill the logical skeleton with candidate relations, classes, and entities, and execute them one by one to check executability. However, such enumeration is inefficient, since all combinations of candidate components should be considered. Therefore, we incorporate the middle-grained component pairs which are connected in KB. Such pairs can be produced efficiently to keep the model's efficiency.
The middle-grained component pairs include class-relation pairs, relation-relation pairs, and relation-entity pairs. For each class \(c\in C_{q}\) and each relation \(r\in R_{q}\), if \(r\) is connected with the domain class \(c\), we add \((c,r)\) into the class-relation pair set \(P_{c-r}\). For example in Figure 3, the class "railway.railway" is linked with the relation "rail.railway.terminus", so the pair (railway.railway, rail.railway.terminus) is executable and will be added into \(P_{c-r}\). If the range class of \(r\) is \(c\), we add the pair of \(c\) and the reverse relation of \(r\). We construct executable relation-relation pair set \(P_{r-r}\) by checking each relation pair \((r_{1}\in R_{q},r_{2}\in R_{q})\). If \(r_{2}\)'s domain class does not match \(r_{1}\)'s range class, we directly remove this pair to maintain efficiency, otherwise, we reformulate \((r_{1},r_{2})\) to a logical expression and execute on KB to check its connectivity. For each relation-entity pair \((r,e)\), we first check whether the logical skeleton candidates contain the <entity> placeholder or not. If not, we leave \(P_{r-e}\) empty; otherwise we directly take the result of the relation-pruning strategy for entities in Section 4.3.
### Coarse-grained Component Composition
We apply a generation model based on T5 to compose all the above fine-grained and middle-grained component candidates and output an executable logical expression by a controlled decoder.
**Encoding Process.** Before feeding the fine-grained and middle-grained component candidates into the generator, we sort the middle-grained candidates according to their similarity scores to the question. By doing this, the order can reveal the pattern of which pair is more likely to appear in the ground-truth logical expression. In intuition, such a pattern will help to generate more accurate logical expressions. To accomplish this, we take the logits of the fine-grained component detection in section 4.3 as the similarity score between the question and each class/relation component, and then calculate the similarity score between the question and a middle-grained component pair by summing the scores of contained single components. The encoding of such middle-grained component improves the generator's reasoning capacity in terms of capturing the knowledge constraints.
We use ";" to separate each element (a component or a component pair). To explicitly inform the model the type of each component, we place "[REL]", "[CL]", "[ENT]", and "[LF]" before each relation, class, entity, and logical skeleton respectively. For example, we organize the input of encoder as "query:[CL]\(c_{1}\)[REL]\(r_{1}\);[REL]\(r_{1}\) [REL]\(r_{2}\);[CL]\(c_{2}\)[REL]\(r_{3}\);[ENT]\(e_{1}\);[LF]\(l_{1}\);[LF]\(l_{2}\)".
**Decoding Process.** The middle-grained components are also used to produce a dynamic vocabulary to constrain the decoding process. The generated token \(y_{t}\) is confined to the tokens involved in the dynamic vocabulary at each step \(t\). We initialize the dynamic vocabulary with the union of tokens from the detected entities, tokens from the detected classes in \(P_{c-r}\), i.e., usually the answer type, and the keywords such as "JOIN" in logical skeleton. Then we update the dynamic vocabulary by the relations paired with \(r\) in \(P_{r-r}\) if the last generated component is \(r\) or by the relations paired with \(c\) in \(P_{c-r}\) if it is \(c\).
## 5 Experiment
### Experimental Settings
**Dataset.** We evaluate our method on GrailQA (Gu et al., 2021), WebQSP (Yih et al., 2016), and CWQ (Talmor and Berant, 2018), all of which are based on Freebase. GrailQA focuses on generalization problems which involved up to 4-hop logical expressions and complex operations. WebQSP is an i.i.d. benchmark that required 2-hop reasoning. Although CWQ is not designed to solve generalization problem, we can still separate out the zero-shot test set with all the unseen relations and classes, yielding 576/3519 zero-shot/all test set.
**Evaluation Metrics.** To measure the accuracy of logical expression, we use the well-adopted exact match (EM) which measures the exact equivalence
between the query graph of the predicted and the gold logical expression. We also calculate the F1 score based on the predicted and gold answers.
Baselines.On GrailQA, we mainly compare with the published works on the leaderboard, including GrailQA-Rank Gu et al. (2021), GrailQA-Trans Gu et al. (2021), Retrack Chen et al. (2021), RNG-KBQA Ye et al. (2022). They are all SP-based models that target generalization problems in KBQA. On WebQSP and CWQ, we compare our method with the retrieval-based models including GraphNet Pu et al. (2018),PullNet Sun et al. (2019) and NSM He et al. (2021), and the SP-based models including QGG Lan and Jiang (2020), RNG-KBQA Ye et al. (2022), and PI Transfer Cao et al. (2022). We evaluate F1 for the retrieval-based models, while evaluate both F1 and EM for the SP-based methods. We compare all the baselines that have the results on the two datasets or publish the codes that can be executed.
### Overall Evaluation
Performance.In Table 1 and Table 2, we evaluate the performance of FC-KBQA on different datasets. For the baselines, we directly take their results reported in the original papers. To be noted, on the extracted zero-shot test set of CWQ, the results for some models remain empty because their full codes are not released. As shown in Table 1, our model outperforms all the baselines, especially on the compositional and zero-shot test tasks. Compared with RNG-KBQA, the state-of-the-art published model, we have an absolute gain of 4.3% and 4.4% in terms of F1 score and EM respectively. We also outperform on the extracted zero-shot CWQ test set by 11.3% in terms of F1, as for an unseen complex question, parsing out correct knowledge components and logical skeletons is much easier than directly parsing the coarse-grained logical expression correctly. Since the fine-grained module solely focuses on each component and thus leads to a higher component accuracy, FC-KBQA also outperforms on the i.i.d test set of WebQSP. On the original test set of CWQ, we only under-perform PI Transfer which leverages a pre-train process on a large-scale wiki data that is out scope of CWQ.
class-relation pairs and relation-relation pairs with the corresponding fine-grained candidates, such as classes and relations. (3) **-Logical Skeleton** gets rid of the logical skeleton. (4) **-Decode Constraint** deletes the dynamic vocabulary created with the middle-grained components.
The results show that removing "knowledge" reduces model performance by 60% F1 score, and replacing "knowledge pairs" with pure fine-grained components also reduces model performance by 28% F1, indicating that encoding the middle-grained components can significantly improve the model's reasoning capacity. To further demonstrate that encoding such middle-grained components can also help improve other model's performance, we create Enhanced RNG-KBQA by taking the top-10 ranked results from its ranking model and formulating them into middle-grained component pairs to be injected into its encoder. The results in Table 3 show that middle-grained reformulation improves the performance of RNG-KBQA. Middle-grained component pairs, like coarse-grained logical expressions, can guarantee connectivity, but they are more compact and much shorter. As a result, because PLMs have a maximum input length, the middle-grained formulation can inject more components and is more likely to cover the components involved in the target logical expression.
Removing "logical skeleton" can result in a 3.0% F1 drop, indicating that skeleton is useful for guiding the question understanding even though it is less important than the knowledge.
Removing "decode constraint" in the decoder can also have an effect on model performance, but is much weaker than removing "knowledge pairs" in the encoder, indicating that injecting the knowledge constraints in the encoding process is more useful than in the decoding process, because the encoder learns the bidirectional context, which is better suited to natural language understanding. This is also a significant difference from the existing knowledge constrained decoding methods.
Both "Knowledge Pairs" and "Decode Constraint" are proposed for addressing the in-executability issue, which guarantee all generated logical expressions are executable. Removing either reduces the accuracy, which indicates that high executability can improve the model performance.
### Error Analysis
We randomly select 50 error cases on GrailQA and summarize the error into three main categories: error entity (60%), error relation and class (35%), and error logical skeleton (40%). We also analysis the error cases while our model fails but some baseline methods can answer successfully resolve them. A typical mistake is on logical expressions that involve KB-specific component composition. For example, in Freebase, "coach" is represented by the join of "sports.sports_team_coaches" and "sports.sports_team_coach_tenure.coach".
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline & \multicolumn{2}{c}{Overall} & \multicolumn{2}{c}{I.I.D.} & \multicolumn{2}{c}{Compositional} & \multicolumn{2}{c}{Zero-Shot} \\ \cline{2-9} & EM & F1 & EM & F1 & EM & F1 & EM & F1 \\ \hline T5-base & 22.7 & 23.4 & 61.8 & 64.1 & 28.3 & 29.0 & 0.3 & 0.3 \\ RNG-KBQA & 71.4 & 76.8 & 86.5 & 88.9 & 61.6 & 68.8 & 69.0 & 74.8 \\ Enhanced RNG-KBQA & 72.8 & 78.2 & 86.6 & 90.2 & 61.7 & 69.3 & 71.5 & 76.7 \\ \hline FC-KBQA & **79.0** & **83.8** & **89.0** & **91.5** & **70.4** & **77.3** & **78.1** & **83.1** \\ –Knowledge & 23.1 & 24.0 & 62.1 & 64.2 & 29.5 & 31.0 & 0.3 & 0.3 \\ –Knowledge Pairs & 53.6 & 55.6 & 70.2 & 72.3 & 44.0 & 46.0 & 50.3 & 52.2 \\ –Logical Skeleton & 78.0 & 80.8 & 85.2 & 86.8 & 68.5 & 71.9 & 79.2 & 81.8 \\ –Decode Constraint & 77.5 & 83.1 & 88.3 & 91.1 & 67.8 & 76.3 & 76.8 & 82.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ablation studies on GrailQA-Dev (%).
Figure 5: Inference time on GrailQA. “Overall” denotes the total inference time of each model. Specially, GrailQA-Rank has no composition step, we record the corresponding time as zero.
Our fine-to-coarse model only predicts the previous relation but is unable to recall "sports.sports_team_coach_tenure.coach", while some coarse-grained methods are able to memorize such composition and provide the correct answer.
## 6 Conclusion
This paper proposes FC-KBQA, a Fine-to-Coarse composition framework for KBQA. The core idea behind it is to solve the entanglement issue of mainstream coarse-grained modeling by the fine-grained modeling, and further improve the executability of logical expression by reformulating the fine-grained knowledge into middle-grained knowledge pairs. Benefiting from this, FC-KBQA achieves new state-of-the-art performance and efficiency on the compositional and zero-shot generalization KBQA tasks. This fine-to-coarse framework with middle-grained knowledge injection could be inspiring for generalization on other NLP tasks.
## 7 Limitations
Although our model achieves good performance in solving the compositional and zero-shot generalization problems, there is still room for improvement on the i.i.d datasets. The fine-grained module in our framework cannot take advantage of explicit composition information when the component compositions in the testing set and training set significantly overlap. For example, in Freebase, "Who is the coach of FC Barcelona?" is answered by the join of relation "sports.sports_team.coaches" and "sports.sports_team_coach_tenure.coach". Our fine-grained extractor may fail to recall "sports.sports_team_coach_tenure.coach" and instead select "base.american_football.football_coac -h.coach" as the candidate since 'football coach" is more relevant to the question than "coach tenure" in semantics. The only coarse-grained model, however, can directly memorize the pattern because such composition appears frequently in the training data. Therefore, compared to conventional models that completely memorize composition patterns, our model may only have minor advantages.
Another limitation is that we cannot guarantee the generalization on other KBs such as WikiData because gaps between KBs may bring negative impact. For example, relations in Freebase are often more specific (ice_hockey.hockey_player.hockey_position, soccer.football_player.position_s), while relations in Wikidata are more general (position_played_on_team). We consider it as a direction for our future work.
## 8 Ethics Statement
This work focuses on the generalization issue of knowledge base question answering, and the contribution is fully methodological. Hence, there are no direct negative social impacts of this work. For experiments, this work uses open datasets that have been widely used in previous work and are without sensitive information as we know. The authors of this work follow the ACL Code of Ethics and the application of this work have no obvious issue that may lead to the risk of ethics.
## Acknowledgments
This work is supported by National Natural Science Foundation of China (62076245, 62072460, 62172424,62276270); Beijing Natural Science Foundation (4212022).
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2307.03177 | PanoDiffusion: 360-degree Panorama Outpainting via Diffusion | Generating complete 360-degree panoramas from narrow field of view images is
ongoing research as omnidirectional RGB data is not readily available. Existing
GAN-based approaches face some barriers to achieving higher quality output, and
have poor generalization performance over different mask types. In this paper,
we present our 360-degree indoor RGB-D panorama outpainting model using latent
diffusion models (LDM), called PanoDiffusion. We introduce a new bi-modal
latent diffusion structure that utilizes both RGB and depth panoramic data
during training, which works surprisingly well to outpaint depth-free RGB
images during inference. We further propose a novel technique of introducing
progressive camera rotations during each diffusion denoising step, which leads
to substantial improvement in achieving panorama wraparound consistency.
Results show that our PanoDiffusion not only significantly outperforms
state-of-the-art methods on RGB-D panorama outpainting by producing diverse
well-structured results for different types of masks, but can also synthesize
high-quality depth panoramas to provide realistic 3D indoor models. | Tianhao Wu, Chuanxia Zheng, Tat-Jen Cham | 2023-07-06T17:57:02Z | http://arxiv.org/abs/2307.03177v6 | # IPO-LDM: Depth-aided 360-degree Indoor RGB Panorama Outpainting via Latent Diffusion Model
###### Abstract
Generating complete 360\({}^{\circ}\) panoramas from narrow field of view images is ongoing research as omnidirectional RGB data is not readily available. Existing GAN-based approaches face some barriers to achieving higher quality output, and have poor generalization performance over different mask types. In this paper, we present our 360\({}^{\circ}\) indoor RGB panorama outpainting model using latent diffusion models (LDM), called IPO-LDM. We introduce a new bi-modal latent diffusion structure that utilizes both RGB and depth panoramic data during training, but works surprisingly well to outpatient normal depth-free RGB images during inference. We further propose a novel technique of introducing progressive camera rotations during each diffusion denoising step, which leads to substantial improvement in achieving panorama wraparound consistency. Results show that our IPO-LDM not only significantly outperforms state-of-the-art methods on RGB panorama outpainting, but can also produce multiple and diverse well-structured results for different types of masks. The code will be released soon. And our project page is at [https://smOkywu.github.io/ipoldm/](https://smOkywu.github.io/ipoldm/).
## 1 Introduction
Omnidirectional 360\({}^{\circ}\) RGB panoramas are helpful for various applications, such as lighting estimation [9, 8, 32] and new scene synthesis [31] in AR and VR. An obvious limitation, however, is that capturing, collecting and restoring such extensive datasets with 360\({}^{\circ}\) images is a high-effort and high-cost undertaking [1, 2], while manually creating a 3D space from scratch can be a demanding task [18, 4, 24].
To reduce the cost of collecting large 360\({}^{\circ}\) datasets, the latest learning methods [1, 31, 2, 25] have been proposed, focusing on _generating omnidirectional RGB panoramas from narrow field of view (NFoV) images_. These methods are typically built upon Generative Adversarial Networks (GANs) [10], which have achieved remarkable success in creating new content. However, GAN architectures face some notable problems, including 1) mode collapse (seen in Fig. 1(c)), 2) unstable training [30], and 3) difficulty in generating multiple structurally reasonable objects [6], which hinder its performance on synthesizing complex scenes (Fig. 1).
In this paper, we propose an alternative method for 360\({}^{\circ}\) indoor RGB panorama outpainting via the latest latent diffusion models (LDMs) [29], called IPO-LDM. An important insight here is that a diffusion model directly adds noise
to the spatial images or features through a Markov Chain over \(T\) steps, which results in a _stable training of a generative model with consistent spatial resolution in each step_. This characteristic is critical in our 360\({}^{\circ}\) panorama scenario, as it preserves the spatial information necessary for generating structurally reasonable objects. Although recent works have already applied diffusion models in image inpainting tasks [21, 19], it remains a challenge to directly apply them in our setting.
Unlike previous inpainting works [26, 35, 21], generating a 360\({}^{\circ}\) panorama from an NFoV image faces greater challenges: 1) the outpainting mask is _significantly larger_ than traditional inpainting and 2) _semantically reasonable objects have to be generated_ within the scene, instead of filling in with generic background textures which will create empty rooms (as shown in Fig. 1 (c)). To achieve this, we creatively introduce depth information when _training_ the diffusion model to aid the RGB generation. Our _key motivation_ for doing so is that the depth information is crucial for helping the network understand the physical structure of objects and the layout of the scene [28].
Conversely, our proposed IPO-LDM _does not_ depend on depth input at all during _inference_ (Fig. 2(b)), which enables applications such as panoramic outpainting from normal photos taken by casual users. Despite this lack of depth input, when compared to the state-of-the-art BIPS [25], which uses depth for both training and testing, our method was able to achieve significant improvement on RGB outpainting (as seen in Fig. 1). While we recognize BIPS as being more focused on outpainting RGB-D panoramas, it is nonetheless rather interesting that even in the extreme case when the full ground-truth depth is provided to BIPS, and no depth is provided to our method, it is still able to achieve generally better RGB outpainting performance (Table 2b).
Another challenge of this task is the unique characteristic of panorama images: 3) the two ends of the image must be aligned to ensure the integrity and _wraparound consistency_ of the entire space, _i.e_. the indoor scene itself does not have a beginning and an end. We present two strategies to enhance this property in the generated results. During the training process, a _camera-rotation_ approach is used to randomly crop and stitch the images for data augmentation (Fig. 4). It encourages the networks to capture information from different views in a 360\({}^{\circ}\) panorama. More importantly, a _two-end alignment_ mechanism is applied at each step of the denoising process (Fig. 5), which explicitly enforces the two ends of an image to be wraparound-consistent.
We evaluate the proposed method on the Structured3D dataset [37]. Experimental results demonstrate that our IPO-LDM not only significantly outperforms previous state-of-the-art 360\({}^{\circ}\) RGB panorama outpainting or inpainting methods, but is also able to provide multiple and diverse well-structured results for different types of masks (Fig. 1). In summary, our main contributions are as follows:
* A new bi-modal latent diffusion structure that utilizes both RGB and depth panoramic data to better learn spatial layouts and patterns during training, but works surprisingly well to outpaint normal depth-free RGB images during inference;
* A novel technique of introducing progressive camera rotations during _each_ diffusion denoising step, which leads to substantial improvement in achieving panorama wraparound consistency;
* Our IPO-LDM not only significantly outperforms state-of-the-art methods on RGB panorama outpainting, but can also produce diverse well-structured results for different mask types.
## 2 Related Work
### Image Inpainting/Outpainting
Early image inpainting approaches [5, 12] tended to focus on mining the input image for low-level patterns or clues to fill in missing areas. However, these methods often assume that the missing patches can be replicated from visible parts. These models usually do not work well in generating new content, nor capable of performing large-scale completion.
Since the introduction of GANs [10], most in/outpainting methods [16, 20, 26, 17] started adopting the encoder-decoder + adversarial training approach, resulting in substantial progress. More recently, the denoising diffusion probabilistic model (DDPM) is an emerging alternative for generative modeling [14], outperforming even GAN-based methods [3] for image synthesis. When applied to image outpainting, we can consider methods to fall into the two categories below:
Mask-conditional.Here the DDPM is trained to outpaint conditioned on the input mask. The image is typically masked before the denoising stage, with the DDPM then trained to generate results that are visually consistent with the original image. A disadvantage is that such approaches can be sensitive to the training mask distribution, responding poorly to out-of-distribution masks.
Unconditional.RePaint [21] used an unconditionally trained DDPM. Specifically, instead of learning a mask-conditional generative model, the generative process is conditioned by sampling from the given pixels during the reverse diffusion iterations. Therefore, the model is not trained for the inpainting task itself and can better handle out-of-distribution masks.
### 360\({}^{\circ}\) Panorama Image Outpainting
Unlike normal images, 360\({}^{\circ}\) images are subjected to equirectangular projection. As a result, all objects and layouts in the images are distorted to varying amounts depending on placement, more so nearer the top and bottom poles. The generated image has to not only maintain the distorted structure but also be visually plausible, with the two ends also needing to be wraparound-consistent. Some works [1, 31] are focused on deterministic completion of 360\({}^{\circ}\) RGB images, with BIPS [25] further extending this to RGB-D panorama synthesis. In order to generate diverse results, SIG-SS [11] uses a symmetry-informed CVAE, while OmniDreamer [2] uses transformer-based sampling. Both require additional mechanisms to achieve this goal. For DDPM, every reverse diffusion step is inherently stochastic since it incorporates noise from a Gaussian distribution, giving diverse results.
### Latent Diffusion
LDMs [29] can be trained on larger image scales because they perceptually compress images into a smaller latent space with lower diffusion cost. Given an image in RGB space, the encoder \(\mathcal{E}\) maps \(x\) into a latent representation \(z=\mathcal{E}(x)\in\mathbb{R}^{h\times w\times c}\). In contrast to previous works [7, 27] that relied on an arbitrary 1D ordering of the learned space \(z\) to model its distribution autoregressively, the inherent spatial structure of the image does not change but is downscaled during this process, with the decoder \(\mathcal{D}\) used for returning to higher resolutions. Two different kinds of regularization, KL-reg and VQ-reg, were experimented with in [29]. In our work, we chose to use the VQ model, which uses a vector quantization layer [34] within the decoder. As stated in [29], it can be interpreted as a VQGAN [7] but with the quantization layer absorbed by the decoder.
## 3 Methods
Given a 360\({}^{\circ}\) image \(x\in\mathbb{R}^{H\times W\times C}\), degraded by a number of missing pixels to become a masked image \(x_{m}\), our main goal is to infer semantically meaningful content for the missing regions, while simultaneously generating visually realistic appearances. This task is conceptually similar to conventional learning-based image inpainting, but this setting faces greater challenges due to the following three differences: 1) our **output** is a _360\({}^{\circ}\) panorama image with wraparound consistency_, rather than a typical NFoV image; 2) the outpainting **mask** is _significantly larger_ than those used in traditional inpainting; 3) our **goal** is to _generate multiple appropriate objects_ within a scene, instead of simply replacing objects with generic background.
To address these challenges, we propose a novel framework, called IPO-LDM. As depicted in Fig. 2(a), the training stage starts with two branches for RGB and depth. In each branch, the input is embedded into the latent space prior to the discrete layer in the corresponding pre-trained VQ model, following [29]. These representations are then combined to form \(z_{rgbd}\), which undergoes diffusion to obtain \(z_{T}\). The resulting \(z_{T}\) is inversely denoised back to the original latent domain through a trained UNet+attention structure. Finally, the pre-trained decoder is employed to rebuild the full RGB-D results.
Figure 2: **The overall pipeline of our proposed IPO-LDM method.** (a) During training, no masks are used, and depth information is applied to aid in completing RGB panorama synthesis. (b) However, during inference, the depth information is no longer needed for masked RGB panorama outpainting. (c) Additionally, a super-resolution model is implemented to further enhance the high-resolution outpainting. Note that the VQ-based encoder-decoders are pre-trained in advance, and fixed in the rest of our framework (identified as “locked”).
During inference, our system takes a masked RGB image, and conducts panoramic outpainting. Note that our proposed model does _not_ require harder-to-acquire depth maps as input, needing only a noise map (Fig. 2(b)). The output is then super-resolved into the final image in a refinement stage (Fig. 2(c)).
### Latent Diffusion Outpainting
As mentioned, current diffusion-based inpainting methods [21, 15] are often restricted to small image sizes, typically up to 256\(\times\)256. Additionally, these approaches do not ensure _wraparound consistency_ during completion, which is crucial for 360\({}^{\circ}\) panoramas. Finally, they do _not_ work well for producing multiple objects within large masks.
In order to perform our task on 512\(\times\)1024 panoramas, we extend RePaint [21] to latent space outpainting. This is possible because the partially visible regions are not changed during perceptual image compression. Note that the 360\({}^{\circ}\) wraparound consistency is still preserved in both the pixel and latent domains, which is important for our setting. To further ensure such a wraparound consistency, a rotational outpainting mechanism is introduced in Sec. 3.2.
The diagram of our latent diffusion outpainting method is shown in Fig. 3. Let \(x\) denote the original visible image, while \(m\odot x\) and \((1-m)\odot x\) represent the missing and visible pixels, respectively. The latent input \(z\) is then defined as \(z=\mathcal{E}_{\theta}((1-m)\odot x)\). In the completion task, we expect the model to _generate plausibly reasonable content for the missing regions, while preserving the visible information as much as possible_. Therefore, we add a step-dependent amount of Gaussian noise to the known regions, while denoising the previous latent vector for one step. To combine them, the mask is first downscaled to the latent vector size \(m_{d}\), with the noised and denoised results then processed by the mask and inverse mask respectively. For each outpainting step, the process can be described by the following expressions:
\[z_{t-1}^{known} \sim q(z_{t}|z_{t-1}), \tag{1}\] \[z_{t-1}^{unknown} \sim p_{\theta}(z_{t-1}|z_{t}),\] (2) \[z_{t-1}=m_{d}\odot z_{t-1}^{known}+(1-m_{d})\odot z_{t-1}^{unknown}. \tag{3}\]
Here, \(q\) is the forward distribution in the diffusion process and \(p_{\theta}\) is the inverse distribution. After \(T\) iterations, \(z_{0}\) is restored to image space using the pre-trained VQ decoder.
### Camera-rotation and Two-end Alignment Mechanism for 360\({}^{\circ}\) Panorama
Since 360\({}^{\circ}\) panoramas are meant to be wraparound consistent, we apply a _circular shift_ data augmentation, called camera-rotation, to the panorama image dataset (examples shown in Fig. 4), to enhance the model's performance. In particular, we randomly select a rotation angle, and use it to crop and re-stitch the patch to produce a new panorama.
While camera-rotation may improve the model's implicit understanding of the expected wraparound consistency by providing a large number of data-augmented examples, it still does not impose strong enough constraints on wraparound alignment of the results. Therefore, in the inference processing, we propose a _novel two-end alignment mechanism_ that can be naturally combined with our latent diffusion outpainting process.
The denoising process of DDPM consists of a number of iterations, rather than a single step. During _each iteration_, we apply the camera-rotation operation to rotate both the latent vectors and masks by 90\({}^{\circ}\), before performing an outpainting step. This procedure more effectively connects the two ends of the panorama from the previous step, which encourages the model to take into account the condition that the two ends are actually connected and generate aligned ends. Instead of changing the size of the images, generating overlapping content, or introducing extra loss functions, we provide 'hints' to the model by rotating the panorama horizontally, thus enhancing the effect of alignment at both ends (examples shown in Fig. 5).
### Bi-modal Latent Diffusion Model
In order to introduce depth information to aid RGB generation, perhaps a simple idea would be to use depth infor
Figure 4: **Camera-rotation.** During training, we randomly crop \(\theta\) degree from the left and stitch to the right for data augmentation.
Figure 3: **LDM outpainting structure.** In each step, we sample the known region from the encoded latent input (above) and the unknown part from the DDPM output (below). The depth map is _not_ necessary during the inference, which is set as random noise.
mation as an explicit condition during training and inference. The depth information may be compressed into latent space and then introduced into the denoising process of the RGB images via cross-attention. However, through experiments, we have found that such an approach often leads to blurry results (Fig. 8). Meanwhile, using two parallel LDMs to reconstruct depth and RGB images separately, together with a joint loss, may also appear to be an intuitive solution. However, this idea is difficult to implement due to the computational resource requirements of multiple LDMs.
Therefore we designed a bi-modal latent diffusion structure to introduce depth information while generating high-quality RGB output, but which is _needed only during training_. Specifically, we trained two separate VQ models for RGB and depth images, and then concatenate \(z_{rgb}\in\mathbb{R}^{h\times w\times 3}\) with \(z_{depth}\in\mathbb{R}^{h\times w\times 1}\) at the latent level to get \(z_{rgbd}\in\mathbb{R}^{h\times w\times 4}\). The training of VQ models are exactly the same as in LDM with downsampling factor \(f\)=4. Then we follow the standard process to train an unconditional DDPM with \(z_{rgbd}\) via a variant of the original LDM loss:
\[\begin{split} L_{RGB-D\,LDM}:=&\mathbb{E}_{z_{rgbd },e\sim\mathcal{N}(0,1),t}[\|\epsilon-\epsilon_{\theta}(z_{t},t)\|_{2}^{2}], \\ z_{rgbd}=& Cat(\mathcal{E}_{1}(x_{rgb});\mathcal{ E}_{2}(x_{depth}))\end{split} \tag{4}\]
Reconstructed RGB-D images can be obtained by decoupling \(z_{rgbd}\) and decoding. It is important to note that during training, we use the full RGB-D image as input, _without masks_. Conversely during the inference stage, the model can perform outpainting of the masked RGB image directly _without any depth input_, with the fourth channel of \(z_{rgbd}\) replaced by random noise.
### RefineNet
Although mapping images to a smaller latent space via an autoencoder prior to diffusion can save training space and thus allow larger size inputs, the panorama size of 512\(\times\)1024 is still a heavy burden for LDM [29]. Therefore, we adopt a two-stage approach to complete the outpainting task. Initially, the original input is downscaled to 256\(\times\)512 as the input to the LDM. Correspondingly, the image size of the LDM output is also 256\(\times\)512. Therefore, an additional module is needed to upscale the output image size to 512\(\times\)1024. Traditional interpolation methods often lead to blurry results. Also, since panorama images are distorted and the objects and layouts do not follow the regular image patterns, we trained a super-resolution GAN model specifically for panoramas, in order to produce visually plausible results at a higher resolution.
## 4 Experiments
### Experimental Details
Dataset.We estimated our model on the Structured3D dataset [37], which provides 360\({}^{\circ}\) indoor RGB-D data with a 512\(\times\)1024 resolution. We split the dataset into 16930 train, 2116 validation, and 2117 test instances.
Metrics.Due to large masks, we should not require the completed image to be exactly the same as the original image, since there are many plausible solutions (_e.g_. new furniture and ornaments, and their placement). Therefore, we mainly report the following dataset-level metrics: 1) Fr\(\acute{e}\)chet Inception Distance (FID) [13], 2) Spatial FID (sFID) [23], 3) density and coverage [22]. FID compares the distance between distributions of generated and original images in a deep feature domain, while sFID is a variant of FID that uses spatial features rather than the standard pooled features. Additionally, density reflects how accurate the generated data is to the real data stream, while coverage reflects how well the generated data generalizes the real data stream.
Mask Types.Most works focused on generating omnidirectional images from NFoV images (Fig. 6(a)). However, partial observability may also occur due to sensor damage in 360\({}^{\circ}\) cameras. Such masks can be roughly simulated by randomly sampling a number of NFoV camera views within
Figure 5: **An example of our two-end alignment mechanism. During inference, we rotate the scene for 90\({}^{\circ}\) for _each_ denoising step, so that the full denoising diffusion process will effectively achieve wraparound consistency.**
Figure 6: **Examples of various mask types. See text for details.**
the panorama (Fig. 6(b)). We also experimented with other types of masks, such as randomly generated regular masks (Fig. 6(c)). Finally, the regions with floors and ceilings in panoramic images are often less interesting than the central regions. Hence we also generated layout masks whichuffle all areas except floors and ceilings, to more incisively test the model's generative power (Fig. 6(d)).
Baseline Models.We mainly compared with the following state-of-the-art methods: including inpainting models LaMa [33]WACV-2022 and TFill [36]cvPR-2022, panorama outpainting models BIPS [25]ECCCV-2022 and OmniDreamer [2]CVPR-2022. All models are retrained on the Structured3D dataset using their publicly available codes.
Implementation Details.To verify the auxiliary effect of depth training input on RGB image generation, we trained two different versions of IPO-LDM, 1) IPO-LDM (RGB) and 2) IPO-LDM (RGB-D). The RGB-D version follows Fig. 2, while the RGB version excludes depth altogether.
### Main Results
Table 1 shows the quantitative comparison of RGB panorama outpainting on the Structured3D dataset with camera masks. For a fair comparison, all models are evaluated without depth maps during testing. As can be seen, the proposed IPO-LDM model significantly outperforms all state-of-the-art models. Specifically, the FID score is substantially better (relative 67.0\(\%\) improvement).
The effectiveness is also clearly visualized in Fig. 7. For BIPS [25] and OmniDreamer [2], the generated areas hold obvious gaps to the original visible regions. As for LaMa [33] and TFill [36], they output blurry results for large invisible areas. Compared to them, our IPO-LDM produces more natural results in terms of transition, as well as more realistic floor texture and a more logical and clearer sofa in the central area. Comparing the RGB and RGB-D versions of our IPO-LDM, in Fig. 7(c), there are some step-like artifacts on the ground in the RGB version. In contrast, the same region of RGB-D result (Fig. 7(d)) appears more structurally appropriate. The transitions between the items, walls, ceiling, and floor are also more natural. Such improvement proves the advantages of jointly learning to synthesize depth data along with RGB images, _even when depth is not used during test time_.
### Ablation Experiments
We ran a number of ablations to analyse the effectiveness of each core component in our IPO-LDM. Results are show in Tables 1(a), 1(b), and 1(c), and Figs. 8 and 9.
Depth Maps.We first evaluated the importance of depth maps in panoramic outpainting, reported in Table 1(b). We also compared with the state-of-the-art BIPS [25], as it was also trained with RGB-D images. Besides, we have
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Methods** & Training Data & FID \(\downarrow\) & FID \(\downarrow\) & Density \(\uparrow\) & Coverage \(\uparrow\) \\ \hline BIPS & RGB+D & 68.79 & 42.62 & 0.306 & 0.412 \\ OmniDreamer & RGB & 65.47 & 37.04 & 0.14 & 0.175 \\ LaMa & RGB & 115.92 & 107.69 & 0.034 & 0.082 \\ TFill & RGB & 83.84 & 61.40 & 0.075 & 0.086 \\ IPO-LDM (RGB) & RGB & 24.33 & 29.00 & 0.667 & 0.635 \\ IPO-LDM (RGB-D) & RGB+D & **21.55** & **26.95** & **0.867** & **0.708** \\ \hline \hline \end{tabular}
\end{table}
Table 1: **Quantitative results for RGB outpainting.** All models were tested without depth input.
Figure 8: **Outpainting with a basic depth-conditioned LDM**. This leads to blurry results, see text for details.
Figure 7: **Qualitative comparison for RGB panorama outpainting.** Our IPO-LDM generated more objects with appropriate layout, and with better visual quality. Please zoom in to see the details. More comparisons are provided in the supplementary material.
also considered the depth-conditioned LDM, mentioned in Section 3.3. However, it always produced blurry results (Fig. 8), so we leave out its quantitative results.
As can be seen, BIPS's performance appears to deteriorate significantly when the input depth visual area is reduced. Conversely, our IPO-LDM is not sensitive to these depth maps, indicating that the generic model has successfully handled the modality. Interestingly, we noticed that having fully visible depth at test time did _not_ improve the performance of our IPO-LDM, and in fact the result deteriorated slightly. A reasonable explanation for this situation is that during the training process, the signal-to-noise ratios (SNR) of RGB and depth pixels are roughly the same within each iteration, since no masks were used. However, during outpainting, the SNR balance will be disrupted when RGB input is masked and depth input is fully visible. Therefore, the results are degraded, but only slightly because IPO-LDM has effectively learnt the distribution of spatial visual patterns across all modalities, without being overly reliant on depth. This also explains why our model is more robust to depth inputs with different degrees of visibility.
Mask Types.As described in previous sections, our model is trained unconditionally, with masks only used during the inference phase. Therefore, our model is supposed to _be able to handle a greater range of mask types_, with performance less affected by the specific mask shape, and mainly by the area of the mask. To prove this, we tested and compared the performance of our model and the baseline models under four different mask types, as described previously. As the performance of BIPS varies considerably depending on whether depth is visible or not, we listed both its performance when depth is partially visible (with depth), and when it is not visible at all (w/o depth).
The results are shown in Table 1(a), which shows that both RGB and RGB-D IPO-LDM significantly outperform the baseline models for all types of masks, while RGB-D IPO-LDM achieved the best results. This indicates that LDM is able to perform outpainting of panoramas best, and also proves that the introduction of depth information significantly improves the quality of the generated RGB panorama.
Conversely, the performance of baseline models can vary considerably between mask types. For BIPS, although the difficulty between outpainting with camera masks and NFoV masks is not significant, the performance on camera masks is significantly better. This is likely due to BIPS using camera masks in the training process. In contrast, IPO-LDM has a more robust performance, producing high-quality and diverse output images for all mask distributions.
Two-end Alignment.Currently, there is no corresponding quantitative metric to evaluate the performance of aligning the two ends of an image. To make a more reasonable comparison, we make one side of the input image be fully visible, and the other side fully masked. Then compare the output with/without our rotational outpainting using the same model. To compare as many results, we only show the
\begin{table}
\end{table}
Table 2: **IPO-LDM ablations**. All models are trained and evaluated on the Structured3D dataset.
end regions that are stitched together to highlight the contrast. The same tests were also performed on BIPS [25] and OmniDreamer [2].
The comparison results are shown in Fig. 9. They show that the consistency of the two ends of the results is improved after the use of rotational outpainting, especially the texture of the walls and the alignment of the layout. Still, differences can be found with rotated outpainting. We believe it is mainly due to the fact that rotational denoising is based on the latent level, which may introduce extra errors during decoding.
RefineNet.As described in Section 3.4, we trained a super-resolution model to increase the resolution of the input from 256\(\times\)512 to 512\(\times\)1024 as the second stage of our framework. Quantitative results on camera and NFoV masks are shown in Table (c)c. The results show an overall increase in performance.
## 5 Discussion
Experimental results show that our model significantly outperforms baseline models on the 360\({}^{\circ}\) indoor RGB panorama outpainting task. Nevertheless, we consider several aspects of the model that can be further improved.
Depth Outpainting.We believe that IPO-LDM not only performs well for RGB outpainting but can also be used for depth image outpainting. The architecture we designed should in theory also be able to naturally synthesize RGB-D panoramas. In our experiments, however, depth generation did not perform well and contained non-negligible noise. From our analysis, we believe the reason for this issue is that the VQ model for depth modeling is not robust enough. Since the depth datasets used to train the LDM and VQ models are the same, latent depth can provide accurate information to assist RGB generation during training. However, during inference, the VQ model may not be able to accurately compress test depth input into the latent space. This may also explain why our model is inferior to BIPS in terms of coverage and density under the fully visible depth condition, as the depth information is not fully exploited. We hope that this problem can be solved in our subsequent work, which will lead to the simultaneous generation of consistent RGB-D panoramas.
Manipulable Panorama Outpainting.Even though our model is capable of generating diverse plausible results, it will be more meaningful to have a manipulable generative process. We will investigate such capabilities in the future. By using some simple prompts as conditions, the user can manipulate the generation of results, which enhances the usability of the model.
## 6 Conclusion
In this paper, we show that our proposed method, the two-stage RGB-D IPO-LDM, achieves state-of-the-art performance for indoor RGB panorama outpainting. The introduction of depth information via our bi-modal LDM structure significantly improves the performance of the model. Such improvement illustrates the effectiveness of using depth during training as an aid to guide RGB panorama generation. In addition, we show that the alignment mechanism we employ at each step of the denoising process of the diffusion model enhances the wraparound consistency of the results. With the use of these novel mechanisms, our two-stage structure is capable of generating high-quality RGB panoramas at 512\(\times\)1024 resolution.
|
2308.12827 | The LISA Data Challenge Radler Analysis and Time-dependent Ultra-compact
Binary Catalogues | Context. Galactic binaries account for the loudest combined continuous
gravitational wave signal in the Laser Interferometer Space Antenna (LISA)
band, which spans a frequency range of 0.1 mHz to 1 Hz.
Aims. A superposition of low frequency Galactic and extragalactic signals and
instrument noise comprise the LISA data stream. Resolving as many Galactic
binary signals as possible and characterising the unresolved Galactic
foreground noise after their subtraction from the data are a necessary step
towards a global fit solution to the LISA data. Methods. We analyse a simulated
gravitational wave time series of tens of millions of ultra-compact Galactic
binaries hundreds of thousands of years from merger. This data set is called
the Radler Galaxy and is part of the LISA Data challenges. We use a Markov
Chain Monte Carlo search pipeline specifically designed to perform a global fit
to the Galactic binaries and detector noise. Our analysis is performed for
increasingly larger observation times of 1.5, 3, 6 and 12 months.
Results. We show that after one year of observing, as many as ten thousand
ultra-compact binary signals are individually resolvable. Ultra-compact binary
catalogues corresponding to each observation time are presented. The Radler
Galaxy is a training data set, with binary parameters for every signal in the
data stream included. We compare our derived catalogues to the LISA Data
challenge Radler catalogue to quantify the detection efficiency of the search
pipeline. Included in the appendix is a more detailed analysis of two corner
cases that provide insight into future improvements to our search pipeline. | Kristen Lackeos, Tyson B. Littenberg, Neil J. Cornish, James I. Thorpe | 2023-08-24T14:34:37Z | http://arxiv.org/abs/2308.12827v1 | # The LISA Data Challenge _Radler_ Analysis
###### Abstract
Context:Galactic binaries account for the loudest combined continuous gravitational wave signal in the Laser Interferometer Space Antenna (LISA) band, which spans a frequency range of 0.1 mHz to 1 Hz.
Aims:A superposition of low frequency Galactic and extragalactic signals and instrument noise comprise the LISA data stream. Resolving as many Galactic binary signals as possible and characterising the unresolved Galactic foreground noise after their subtraction from the data are a necessary step towards a global fit solution to the LISA data.
Methods:We analyse a simulated gravitational wave time series of tens of millions of ultra-compact Galactic binaries hundreds of thousands of years from merger. This data set is called the _Radler_ Galaxy and is part of the LISA Data challenges. We use a Markov Chain Monte Carlo search pipeline specifically designed to perform a global fit to the Galactic binaries and detector noise. Our analysis is performed for increasingly larger observation times of 1.5, 3, 6 and 12 months.
Results:We show that after one year of observing, as many as ten thousand ultra-compact binary signals are individually resolvable. Ultra-compact binary catalogues corresponding to each observation time are presented. The _Radler_ Galaxy is a training data set, with binary parameters for every signal in the data stream included. We compare our derived catalogues to the LISA Data challenge _Radler_ catalogue to quantify the detection efficiency of the search pipeline. Included in the appendix is a more detailed analysis of two corner cases that provide insight into future improvements to our search pipeline.
Conclusions:
## 1 Introduction
Ultra-compact binaries (UCBs) are compact or degenerate star systems with orbital periods of a few hours or less. They emit continuous gravitational radiation with frequencies in the mHz range. Circularised compact binaries of the Milky Way Galaxy are expected to be the most numerous type of gravitational wave (GW) signal below \(\sim 5\) mHz. Double white dwarfs (WDs) are the most common type of UCB, although UCBs can also involve neutron stars or black holes, some possibly with non-zero eccentricity.
Here we analyse a simulated data set from a future space-based GW detector LISA (Hils et al., 1990; Amaro-Seoane et al., 2017) and present time-evolving UCB catalogues for 1.5, 3, 6, and 12 months of simulated data. We resolve as many as 10,000 binaries from a 12-month observing period. More than 400 of these are constrained to a sky localisation area of 10 deg\({}^{2}\) or better. The simulated data span a 24-month period. Here, we do a full analysis up to 12 months. In the Appendix we discuss a few cases which have poor convergence of the sampler after 12 and 24 month analyses. These corner cases were identified when comparing the the 6 and 12-month catalogues.
LISA is a European Space Agency led mission, in collaboration with NASA and an international consortium of scientists 1 designed to explore the uncharted Universe of low-frequency GWs and promises 'answers to fundamental questions in physics and astronomy:2. Ground-based detectors are insensitive to low frequency GWs due to gravity gradient noise from terrestrial sources. One must use space-based detectors to observe frequencies below 1 Hz.
Footnote 1: [https://lisa.pages.in2p3.fr/consortium-userguide/](https://lisa.pages.in2p3.fr/consortium-userguide/)
LISA will monitor the observable Universe with a triangular constellation of spacecraft separated by 2.5 million km. Each spacecraft houses two free flying test masses (TM) and two lasers linking the two other spacecraft. Heterodyne laser interferometry will be used to observe picometer level changes in TM separations (Weise et al. (2017)). The observatory lies in a plane inclined 60\({}^{\circ}\) with respect to the ecliptic and will be in a heliocentric orbit with a period of one year. This arrangement is sensitive to GWs spanning four decades in frequency from 0.1 mHz to 1 Hz (Baker et al. (2019)). A sampling of LISA's most anticipated sources are a stochastic GW background (SGWB) from cosmological sources (Christensen (2019)), late-time coalescence and merger of massive black hole binaries (MBHB) (Klein et al. (2016)), and stellar (Sesana (2016)), intermediate and extreme mass ratio inspirals (Amaro-Seoane et al. (2007)). Finally, there will be the unanticptied or, yet, unknown astrophysical sig
nals (Cornish et al. (2019)). LISA will observe this multitude of overlapping signals simultaneously.
Resolving UCB signals from a noisy LISA data stream is the focus of this paper. Tens of millions of UCBs are expected to emit GWs below \(\sim\) 5 mHz, with the overwhelming majority forming a Gaussian confusion, or foreground, noise in excess of LISA's instrumental noise. At higher frequencies, where there are fewer UCBs, the foreground is reduced substantially (Nissanke et al., 2012; Nelemans, G. et al., 2001). For frequencies below \(\sim\) 5 mHz, however, characterisation of the UCB foreground component will be essential to disentangling the SGWB, transient, and continuous extragalactic signals that overlap in frequency and time with each other and with the numerous UCB signals. It is necessary to perform a global fit (Crowder and Cornish, 2007; Cornish and Crowder, 2005) to the data, where the GW signals and noise sources are fit simultaneously and the number of GW signals in the data is an unknown variable. A LISA global fit pipeline for UCB parameter estimation has been in development for several years (Littenberg et al. (2020)) and has recently been extended to include MBHBs (Littenberg and Cornish (2023)).
The generation and analysis of simulated LISA data streams is driven by a series of data challenges issued to the broader data analysis community. The first collection of pipelines were developed as part of the Mock LISA Data Challenge (MLDC; Arnaud et al. 2007, 2008; Babak et al. 2008, 2008, 2010, 2010). The MLDC was designed to facilitate and coordinate the development of data analysis pipelines, and signal waveforms. The most recent incarnation of this effort, begun in the Spring of 2018, is now known as the LDC3, with an aim to improve existing algorithms, and create new ones, to generate a common platform to evaluate and compare the performance of different algorithms, to address science requirements, and with overarching goal of developing'mission ready' end-to-end data analysis pipelines.
Footnote 3: [https://lisa-ldc.lal.in2p3.fr/ldc](https://lisa-ldc.lal.in2p3.fr/ldc)
The first LDC data set _Radler_ comprises four separate challenges, each focusing on extracting a different type of gravitational wave source from a noisy data stream: stochastic signals, single MBHB, single EMRI, and a Galactic binary (GB) or UCB population. The latter is referred to as 'the Galaxy' 4. The LDC Galaxy was constructed from the simulations of Toonen et al. (2018). In this paper we report on the analysis of the LDC Radler Galaxy using data set LDC1-4_GB_v1.
Footnote 4: [https://lisa-ldc.lal.in2p3.fr/challenge1](https://lisa-ldc.lal.in2p3.fr/challenge1)
In recent years, a number of techniques have been developed for the analysis of gravitational waves from a simulated population of UCBs. Using the _Radler_ verification binary data set as a starting point Strub et al. (2022) inject overlapping signals and use differential evolution to find a maximum likelihood estimate (MLE) and then apply the Metropolis-Hastings algorithm to sample the posterior distribution around the MLE of each candidate detection to determine parameter uncertainties. To generate posteriors, Strub et al. (2022) uses Gaussian Process Regression with the existing 'FastGB' LISA response simulation (Cornish and Littenberg (2007)) to model the log-likelihood function. This helps further decrease computation time, since the approximated LISA response is not simulated for each sample. We say more about 'FastGB' in Section 2, when we introduce our signal model. The authors of Zhang et al. (2021) also use MLE, but with particle swarm optimisation and evaluation of the \(\mathcal{F}-\)statistic to resolve UCBs. Different techniques have their own strengths and weaknesses. For multi-modal distributions, MLE is vulnerable to missing the modes which contain the true parameter values. However, for targeted searches of known EM binaries (for example, known in optical, X-ray, \(\gamma\)-ray, or radio) this limitation can be mitigated by designing priors informed by the EM observations.
Thousands of detached-binaries and a few ten to hundreds of interacting systems will be detected (Nissanke et al. (2012)). These systems provide unique laboratories for fundamental physics. For example new bounds on the graviton mass competitive with those produced by ground based GW observatories and pulsar timing arrays are expected using eclipsing (Cooray and Seto (2004)) and eccentric (Jones (2004)) compact binaries. Modifications to GR in low velocity regimes can be constrained with high mass white dwarf binaries (Littenberg and Yunes (2019)). With an observed population of relativistic binaries the Milky Way (MW) potential will be mapped using GWs (Adams et al. (2012)), and regions previously obscured by intervening material will be revealed (Korol et al. (2018)). Models of MW globular cluster formation and evolution will be further constrained (Benacquista and Downing (2013)). In Danielski et al. (2019) the authors discuss the prospect of gravitational waves affirming the existence of post-main sequence exoplanet and brown dwarf populations in the MW. Breivik et al. (2020) demonstrate that a catalogue of thousands of MW UCBs will help constrain binary star formation and evolution. Finding a WDs binary system near the threshold of merging would reveal new insights into the precursor physics of Type Ia supernova (Webbink (2010)). Multi-messenger astrophysics of compact binary stars will be enriched with a GW perspective. Follow-up electromagnetic (EM) searches of newly detected LISA binaries will confirm relativistic binaries missed by traditional MW globular cluster searches (Kremer et al., 2018, 2019). For known multi-messenger binaries, that is systems that have been observed electromagnetically, joint EM-GW observations will provide improved physical constraints on masses, orbital parameters and dynamics, beyond what independent EM or GW observations achieve on their own (Shah and Nelemans, 2014; Littenberg and Cornish, 2019). On classes of UCBs unobserved electromagnetically, Sberna et al. (2021) predict with simulation and semi-analytic evolution models that WD-black hole binaries will be detectable and could inform follow-up searches in X-ray. LISA detections of Galactic black hole/white dwarf-neutron star binaries will inform and increase the computational efficiency of radio searches for pulsars in these systems (Kyutoku et al., 2019; Thrane et al., 2020). It is also possible to use UCBs as phase/amplitude standards for self-calibration of the data (Littenberg (2018)). There is also a technique to use the WD binary annual modulation to extract an isotropic astrophysical SGWB (Adams and Cornish (2014), Lin et al. (2022)), which depends on first resolving and remove as many UCBs as possible. There is even a proposal to use UCBs as a GW timing array to indirectly detect GWs in the low frequency regime (mHz to \(\mu\)Hz) (Bustamante-Rosell et al. (2022))! Looking even further to the future, UCB analysis has been investigated using a coherent network of at least three independent space-based gravitational wave detectors (Zhang et al. (2022)).
Gravitational waves from individual and populations of UCBs are interesting in their own right, _i.e._ beyond considering them a foreground noise source. Their characterisation and extraction is an integral part of the 'global fit' solution (Littenberg et al. (2020)) for extragalactic source detection at mHz frequencies. Listed above are just a few reasons why the analysis presented here is vital to achieving the widest possible scientific impact for the LISA mission. The main motivation of this paper is to further test and develop the Galactic Binary Markov Chain
Monte Carlo (GBMCMC) search pipeline in preparation for a global fit.
The individual sections of our paper are as follows. In Section 2 we provide an introduction to the likelihood used in our analysis, the noise and signal models, and the UCB parameterisation. The computational resources used are also discussed there. We present GBMCMC catalogues for the observation times analysed in Section 3. For each catalogue we make various signal-to-noise (\(S/N\)) cuts to the data, and identify well-localised UCBs to target for multi-messenger studies. In addition to resolving as many UCBs as possible, we quantify the efficacy of our search pipeline by comparing the GBMCMC catalogue to the LDC _Radler_ UCB catalogue, classifying each catalogue UCB as either matched, confused or false alarm (Section 4). Lastly, we summarise our results in Section 5. In Appendix A, two catalogue UCBs classified as confused are examined in more detail. These serve as case studies for future development of the sampler.
## 2 GBMCMC analysis
The GBMCMC search pipeline (Littenberg et al. (2020)) uses Bayesian model selection to optimise the number of detectable UCBs. In a nutshell, GBMCMC performs a global fit to the resolvable binaries using a trans-dimensional (reversible jump) (Green (1995)) MCMC algorithm with parallel tempering (Swendsen & Wang (1986)). At the same time, it fits a model to the residual confusion noise.
Parallel tempering is used to prevent the sampler from becoming trapped in sub-dominate modes of the posterior, by sampling with parallel chains of different temperatures, with exchanges of parameters between chains subject to detailed balance. Higher temperature chains are more freely able to sample the parameter space. For example, a chain given an infinite temperature will simply sample the prior distribution (Littenberg & Cornish (2010)).
Trans-dimensional MCMC algorithms addresses the model selection aspect of the problem. The MCMC stochastically transitions between models, where each model contains a different number of UCBs, while satisfying detailed balance. Therefore, the number of iterations the chain spends in a particular model is proportional to the marginalised likelihood, or evidence, for that model. Before saying more about the waveform and noise models, we describe a likelihood for Bayesian inference adapted for LISA science analysis.
For our analysis we used 100,000 MCMC steps, after the burn-in phase. Convergence time depends critically on sampling from customised proposal distributions (Littenberg et al. (2020)). In the Appendix we discuss a few cases which have poor convergence of the sampler after 12 and 24 month analyses.
### Likelihood and noise constructions
The three LISA spacecraft communicate with each other via laser links forming the interferometric arms of the detector. The arms of a space-based detector will have different lengths, varying on the order of a few percent of their length over the course of a year. This occurs due to the solar wind, the gravitational coupling of the Earth-moon system and the influence of the other planets in the solar system on the spacecraft orbits, causing the test masses to deviate from their Keplerian orbits. In an equal arm detector, laser frequency noise experiences the same delay in each arm and will cancel at the detector. For time-varying armlengths, Time-Delay Interferometry (Prince et al. 2002; Adams & Cornish 2010) (TDI) has been developed to algorithmically remove the otherwise dominating laser frequency noise by generating virtual equal-armlength interferometers, performed on the ground in post-processing. In general, many different TDI combinations of interferometer output signals, or observables, are possible (Estabrook et al. (2000)). For the LISA mission two quasi-independent Michelson interferometer data streams and a third null-stream (the LISA "Sagnac" observable) will be constructed in post-processing5.
Footnote 5: [https://www.cosmos.esa.int/documents/678316/1700384/SciRD.pdf](https://www.cosmos.esa.int/documents/678316/1700384/SciRD.pdf)
The likelihood function (1) depends on the TDI observables, or 'channels', used in the algorithm. The LISA signal is a superposition of two parts: the frequency response of the \(l^{\text{th}}\) channel to all the gravitational wave signals incident on the detector, \(\mathbf{h}_{I}\), and the combination of all the noise sources impacting that channel, \(\mathbf{n}_{I}\): \(\mathbf{d}_{I}=\mathbf{h}_{I}+\mathbf{n}_{I}\). The "noise" term is a superposition of instrument noise and gravitational wave signals that are individually too quiet to extract from the data (forming a confusion noise below \(\sim 5\) mHz). The detectable gravitational wave signal is recovered using a signal model \(\mathbf{h}_{I}\) such that the residual \(\mathbf{r}_{I}=\mathbf{d}_{I}-\mathbf{h}_{I}\) is consistent with the noise model. For Gaussian noise the likelihood is:
\[p(\mathbf{d}|\mathbf{h})=\frac{1}{(2\pi\,\det\mathbf{C})^{1/2}}\,\exp\left(- \frac{1}{2}(d_{Ik}-h_{Ik})C^{-1}_{(lk)(lm)}(d_{Jm}-h_{Jm})\right)\,, \tag{1}\]
where \(\mathbf{C}\) is the noise correlation matrix. Indices \(k\) and \(m\) correspond to the data samples for the \(I\) and \(J\) channels respectively, where there is an implicit sum over the TDI channels \(I=\{X,Y,Z\}\) and data samples.
If the fluctuations in the data are stochastic the noise correlation matrix in terms of frequency becomes partially diagonalised \(C_{(lk)(Jm)}=S_{IJ}(f_{L})\delta_{\text{km}}\), where \(S_{IJ}(f)\) is the cross-power spectral density between channels \(I\), \(J\)Adams & Cornish (2010). Since the noise levels are equal and uncorrelated on each spacecraft, noise orthogonal TDI variables \(I=\{A,E,T\}\)(Prince et al. (2002)) are constructed such that the cross-spectral density matrix is diagonalised by performing a linear transformation in the space of TDI variables. See Adams & Cornish (2010) for complete expressions for the instrument noise contributions to the cross spectra \(S_{IJ}(f)\), where the realistic scenario of unequal noise levels in each spacecraft is treated.
The \(\{A,E,T\}\) combination also results in signal orthogonality for frequencies below the inverse round-trip light travel time along the arms of the instrument (c/2\(\pi\)L), \(f_{\ast}\simeq 19.1\) mHz, such that \(A\sim h_{\ast}\), \(E\sim h_{\ast}\) (corresponding to the two virtual Michelson interferometer channels) and \(T\sim h_{\ast}\) (the null channel). This is not restricted to equal arms; Adams & Cornish (2010) derived a combination that maintains this insensitivity for unequal arm length detectors. The gravitational wave response of the null \(T\) channel is highly suppressed for \(f<f_{\ast}\). For this reason the Sagnac data combination is particularly valuable for noise characterisation and the detection of stochastic backgrounds (Tinto et al. 2001; Hogan & Bender 2001) and unmodelled signals (Robson & Cornish (2019)).
For this analysis we have made a number of simplifying assumptions, to be relaxed in the future. We take the noise to be stationary and assume the noise correlation matrix is diagonal in the frequency domain. In reality, the confusion noise is cyclo-stationary, with periodic amplitude modulations imparted
by LISA's orbital motion (Seto (2004)). Here we neglect off-diagonal terms in the frequency domain noise correlation matrix \(\mathbf{C}\). Since an overwhelming majority of signals have frequencies well below frequency \(f_{*}\), we only use the \(A\) and \(E\) data combinations in the analysis and we assume that the noise in these channels is uncorrelated. The instrument response includes finite arm-length effects of the LISA constellation and arbitrary spacecraft orbits. The TDI prescription currently implemented treats the arm lengths as equal and unchanging with time, saving on computational cost. We split the analysis into 3317 sub-bands [\(f_{i},f_{i}+N/T_{\rm obs}\)], where \(f_{i,\rm min}=0.00813\) mHz. The noise in each band is approximate as an undetermined constant \(S_{i}\). The noise level in each band becomes a free parameter explored by the Reversible-jump Markov chain Monte Carlo algorithm, resulting in a piece-wise fit to the instrument noise over the full analysis band.
### Compact binary parameterisation and signal model
A compact binary orbit is modelled with eight parameters, \(N_{\rm p}=8\), \(\lambda\rightarrow(\mathcal{A},f_{0},f,\varphi_{0},\iota,\psi,\theta,\phi)\), where \(\mathcal{A}\) is the amplitude, \(f_{0}\) is the observed GW frequency, which is twice the orbital frequency of the binary, \(\dot{f}\) is the (constant) time derivative of the GW frequency, \(\varphi_{0}\) is the initial GW phase at the observation start time, \(\iota\) is the inclination of the orbital plane relative to the line of sight, the wave polarisation axes in the Solar System barycentre are determined by \(\psi\), and \(\theta,\phi\) are the ecliptic latitude and longitude, respectively. See Shah et al. (2012) for a description of parameter correlations and degeneracies.
When at least 90 % of \(\dot{f}\) MCMC samples are positive, we take the orbital evolution to be GW-dominated evolution and use (2) to estimate the chirp mass \(\mathcal{M}\) and luminosity distance (\(D_{L}\)) of the binary.
\[\dot{f} = \frac{96}{5}\pi^{8/3}\mathcal{M}^{5/3}f_{0}^{11/3}\] \[\mathcal{A} = \frac{2\mathcal{M}^{5/3}(\pi f_{0})^{2/3}}{D_{L}} \tag{2}\]
We also have optional settings to include the second derivative of the frequency (Littenberg and Cornish (2019)) as an additional parameter, in which case the frequency derivative is no longer constant, so the parameter \(\dot{f}\to f_{0}\) is fixed at the same fiducial time as \(f_{0}\) and \(\varphi_{0}\).
The detector response of the \(I^{\rm th}\) data channel to the signal from a galactic binary with parameters \(\lambda_{a}\) is \(\mathbf{h}_{I}(\lambda_{a})\), and the superposition of individual UCBs forms the signal model:
\[\mathbf{h}_{I}(\mathbf{\Lambda})=\sum_{a=1}^{N_{\rm GW}}\mathbf{h}_{I}(\lambda_{a}). \tag{3}\]
The number of binaries in each sub-band, \(N_{\rm GW}\), is _a priori_ unknown, and has to be determined from the analysis. A probability distribution for \(N_{\rm GW}\) is established, essentially producing a catalogue for each dimension. Here we build our catalogue with the dimension having the highest Bayesian evidence. Individual binary systems are modelled as isolated point masses on slowly evolving quasi-circular orbits. Orbital eccentricity (Seto (2001)), tides (Fuller and Lai (2012)) and third bodies (Robson et al. (2018)) are not included in our model. The signals are modelled using leading order post-Newtonian waveforms, and the instrument response \(\mathbf{h}_{I}\) is computed with the fast-slow decomposition method, in terms of frequency (Cornish and Littenberg (2007)). FastGB, or the "fast-slow" method, is the decomposition of the relative path length variation of the detector arm, \(\delta l(t)/L\). Namely, the product between the rapidly varying exp(\(i_{\rm orb}l\)), where \(\omega_{0}\) is the instantaneous angular frequency of the GW, and a slowly varying amplitude factor that depends on the LISA spacecraft orbits and the GW amplitudes, that is part of the LISA instrument response.
The LISA instrument response is modulated as the detector rotates about its centre and orbits the Sun. The sensitivity pattern is anisotropic, so as the detector moves the sensitivity pattern evolves with time with respect to a given source. This modulation is imprinted on the detected amplitude. Additionally, as LISA orbits around the Sun, the frequency of the GW is Doppler-shifted, resulting in a time-dependent phase shift of the instrument response (Petersem et al. (1997)). Both effects introduce a spread in the power of the source such that it is no longer monochromatic when viewed from a LISA-based frame and depends on the direction and orientation of the source. The power is reduced relative to the instrumental noise as it is spread over a series of side-bands, offset from the GW frequency at integer multiples of the modulation frequency \(f_{m}=(1\ {\rm yr})^{-1}\). The subdominant harmonics lead to secondary maxima in the likelihood surface which are dealt with using tailored proposals (Crowder and Cornish 2007; Littenberg 2011). The detector velocity with respect to the Solar System barycentre evolves with time. Therefore, the phase modulation depends on a coupling between frequency and sky location. This effect helps one localise the source on the sky and determine its orientation, because each source has a unique modulation pattern. GBMCMC implements multi-modal proposals to account for degeneracies and symmetries in the likelihood surface to improve chain convergence time (Littenberg et al. (2020)).
### Data segmentation and computational resources
Four time periods were searched over: \(T_{\rm obs}=1.5\), 3, 6 and 12 months, each with the same starting time. A catalogue is produced for each time period searched. The catalogue data include a point estimate of the UCB parameters, the waveforms, and posterior distributions for the parameters \(\lambda\). The search is done in terms of frequency, where the full LISA band is divided into 3317 'analysis segments' or 'frequency segments'. The LISA band is divided into frequency segments of width \(2^{5}/T_{1.5\rm mno}\), \(2^{5}/T_{3\rm mno}\), \(2^{7}/T_{6\rm mono}\) and \(2^{8}/T_{12\rm mno}\) for the 1.5, 3, 6 and 12-month analyses, respectively. The \(f_{\rm min}\) and \(f_{\rm max}\) for each of the 3317 analysis segments are the same for each \(T_{obs}\). In Figure 1, we see how the waveforms in a particular frequency segment evolve with observing time. After 12 months of analysis all of the LDC _Radler_ UCB signals in this segment have been recovered. An example of frequency spreading due to detector motion is also apparent in Figure 1, where the waveforms do not appear monochromatic.
Each analysis segment is padded with data amounting to the typical bandwidth of a source. This creates a certain amount of overlap with neighbouring analysis segments. This allows the MCMC to explore the data in the padded region, which is especially useful for sources with long tails that extend beyond the hard boundary of the analysis segment. During catalogue production, only samples fitting sources in the original analysis window are retained. This prevents the same source from appearing in the catalogue more than once. In the next section, we discuss how the raw chain samples from the MCMC analysis are sorted
into individual UCB catalogue entries and present the results of our analysis in the form of evolving catalogues as a function of \(T_{\rm obs}\).
To perform the wholesale analysis of \(\sim 3000\) frequency segments in a reasonable amount of time, we used Amazon Web Service (AWS) cluster computing resources 6. Each segment was analysed in an 'embarrassingly parallel' way, such that there is no communication between segments being analysed. There is interest in using cloud-based computing resources for the actual LISA mission, so our analysis is a first test run in using this infrastructure.
Footnote 6: [http://aws.amazon.com/what-is-aws/](http://aws.amazon.com/what-is-aws/)
GBMCMC and the software used to produce the catalogues are downloadable from GitHub (Littenberg et al. (2020)). Our catalogue data are available upon request. Additionally, a Python package dedicated to exploring these data is available (Thorpe et al. (2021)).
## 3 The recovered GBMCMC-Radler catalogue
The development of UCB catalogues for the LISA mission is the process of transforming the search data products into a form that is useful to the greater astronomy community. In our case, this means filtering the GBMCMC parameter chain outputs into individual catalogue entries using the maximum likelihood model. Before moving on to the catalogue results, we summarise the process of filtering posterior samples to construct catalogue entries. The details of this process are also found in Littenberg et al. (2020) and Littenberg and Cornish (2023).
To build a catalogue for a particular frequency segment, we start with the highest evidence model chain. Namely we select the \(N_{\rm GW}\)-source model which has the highest evidence. The correlation, or overlap, value between waveforms \(\mathbf{h}(\lambda_{i})\) and \(\mathbf{h}(\lambda_{j})\) for binaries with parameters, \(\lambda_{i}\) and \(\lambda_{j}\), Equation (4), is used as a metric to cluster parameter samples.
\[M_{ij}=\frac{\langle\mathbf{h}(\lambda_{i})|\mathbf{h}(\lambda_{j})\rangle}{\sqrt{ \langle\mathbf{h}(\lambda_{i})|\mathbf{h}(\lambda_{i})\rangle\langle\mathbf{h}(\lambda_{ j})|\mathbf{h}(\lambda_{j})\rangle}}. \tag{4}\]
We set a threshold \(M_{*}=0.5\) above which parameter sets are interpreted as describing the same UCB template.
The first \(N_{\rm GW}\) samples in the chain correspond to the first \(N_{\rm GW}\) UCB catalogue entries, in no particular order. Waveforms for these first \(N_{\rm GW}\) entries are generated and used as reference entries for cross-correlating with other samples waveforms. If the correlation value between a given sample waveform and a reference entry waveform exceeds the default threshold value of \(M_{*}\), the sample parameters are appended to the entry parameters. Correlations are only computed when a sample has an \(f_{0}\) that is within a range of \(10/T_{\rm obs}\) bins of the reference entry \(f_{0}\). If a chain sample is not within range of an existing entry or does not have a correlation \(M_{ij}>M_{*}\) with an existing entry, a new entry is created and added to the list of reference entries to be matched against. This process continues until all chain samples are grouped.
Each entry has an associated evidence that is used to further filter the number of entries. The evidence for an entry \(p(\mathbf{d})=\int p(\mathbf{d}|\lambda)\,d\lambda\) is proportional to the total number of chain samples. The evidence for an entry is computed as the number of chain samples in the entry divided by (\(N_{\rm total}/N_{\rm GW}\)), where \(N_{\rm total}\) is the total number of samples in the chain. A threshold evidence of 0.5 must be exceeded for a particular entry to be included in the final catalogue.
The filtered parameter chains for each entry are used to form additional catalogue products. An entry's point-estimate is chosen to be the sample that corresponds to the median of the marginalised posterior on \(f_{0}\). We also compute the full multi-modal \(N_{\rm P}\times N_{\rm P}\) covariance matrices for each mode of the posterior. These are then used for covariance matrix proposals as more data are acquired. From the point-estimates of each entry, waveform entries are computed. Finally, metadata about the catalogue are stored including the total number of above-threshold entries, their weights and S/N and the full set of posterior samples for each entry. History data are also included, which simply links catalogue entries with preceding \(T_{\rm obs}\) catalogue entries, if such a link exists. This is dependent on the correlation of entry waveforms with preceding \(T_{\rm obs}\) catalogue entry waveforms. A default threshold of \(M_{*}=0.5\) is again used to make an association, but the user can adjust this as needed.
The numbers of catalogue UCBs for our 1.5, 3, 6 and 12 month LDC analyses are shown in Table 1, along with UCB number as a function of different \(S/N\) cuts in Table 2. In Table 3 we make various parameter cuts on catalogue UCBs that have frequencies above 5 mHz. Catalogue UCBs with at least 90 % of their samples meeting the particular parameter cut criteria are included in the count. In Figure 2 we show parameter posteriors, \(f_{0}\), \(\mathcal{A}\), sky location, \(\dot{f}\), \(t\), for a well-localised eclipsing UCB as a function of observing time. In Figure 3, we graph the chirp mass and luminosity distance posteriors using Equation ((2)). The LDC values of chirp mass and distance, \(0.4004M_{\odot}\) and \(14.39\) kpc, are within \(1\sigma\) uncertainty of the the derived chirp mass and distance, \(0.4073^{+0.0040}_{-0.0057}M_{\odot}\) and \(14.12^{+0.86}_{-0.77}\)kpc.
In Figure 4 we show the power spectral density for the full frequency band that was analysed. The residual becomes smoother with time, and one can clearly see the excess confusion noise below \(\sim\)4 mHz. In Figure 5 are joint posteriors for sky location in ecliptic coordinates, for each observation time, 1.5, 3, 6, and 12-months, from left to right starting from the top. The sky location posteriors are graphed using every tenth sample.
One use of the catalogue data are posterior-based proposals for individual UCBs for use in future global fits to the LISA data. Updates to UCB parameters are proposed, independently of other UCBs in the catalogue. The cadence of applying new proposals to the global fit is to be determined for the LISA mission.
For this analysis, we applied covariance matrix proposal updates to the 3, 6, and 12 month analysis, respectively using the 1.5, 3 and 6-month catalogue UCB parameter distributions. Along with a given catalogue, history tree data are produced linking the UCBs of a catalogue to UCBs in the preceding \(T_{\rm obs}\) catalogue. The history data will be useful to the observer when determining which catalogue UCBs are potentially confused. Where this is especially useful is for two UCB nearby in frequency and sky location, as discussed in the second corner case of the Appendix.
The overlap integral (4) used in the catalogue production step is also used to cross correlate LDC injected waveforms and GBMCMC catalogue waveforms, to determine which catalogue UCBs have a matching LDC injection, with \(M_{ij}>0.8\). We see the results of this in the next section where we compare our catalogues to the population of LDC injections to quantify the efficacy of our search.
## 4 Comparing the GBMCMC catalogues to LDC injections
Since we are dealing with a simulated data stream which comes with the parameter values describing the waveforms for every LDC UCB in the data, it is possible to check the efficacy of our search pipeline by cross correlating our catalogue waveforms with the LDC waveforms using Equation (4). The methods and results of this process are presented next. These results inform the efficacy of our search pipeline across observing time. Each catalogue UCB is classified as a matched, confused or false alarm detection. The different classifications are explained below.
When the correlation coefficient \(M_{ij}\) of a catalogue-injection pair exceeds a threshold of 0.8 we regard these as a'matched' pair. There is typically one LDC injection meeting this criterion for the given catalogue UCB. Though it occurs less often, we shall see that it is possible for one catalogue UCB to have a match with two injections. In general, we refer to catalogue UCBs that have a match with one LDC injection, or more, as matched.
The majority of catalogue UCB that are not matched are classified as confused. Confused catalogue UCB can be further distinguished as two sub-categories of blending: (1) two UCBs each have a positive overlap with the same injection and the sum of the two UCB waveforms has a larger overlap with that injection; (2) a single UCB has a larger overlap with the sum of two injected waveforms than with each injected waveform alone.
A histogram of the number of catalogue UCBs in each of the three categories: matched, confused and false alarm, as a function of observing time is shown in Figure 6. The fraction of catalogue UCBs which have a match with an LDC injection is 0.74, 0.88, 0.82, and 0.79 for the 1.5, 3, 6 and 12-month catalogues, respectively. The fraction of catalogue UCBs which are confused is 0.26, 0.12, 0.18, and 0.21 (for 1.5, 3, 6 and 12-month catalogues). The larger the fraction of confused sources in each catalogue, the smaller the fraction of matched sources. Figure 7 shows the cumulative distribution of cross correlation values between the \(T_{\rm obs}\)-month catalogue UCB and LDC injections. The fraction of catalogue UCBs with match below \(M_{ij}=x\) is shown
\begin{table}
\begin{tabular}{|c|c|} \hline Observation time (months) & GBMCMC catalogue UCBs \\ \hline
1.5 & 1998 \\ \hline
3.0 & 2758 \\ \hline
6.0 & 6196 \\ \hline
12.0 & 10027 \\ \hline \end{tabular}
\end{table}
Table 1: Total number of GBMCMC catalogue UCBs as a function of observation time.
\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline Observation time (months) & SN/N5 7 & SN/N5-17 & SN/N5-40 & SN/N5-100 \\ \hline \multicolumn{5}{|c|}{\(M_{ij}>0.8\)} \\ \hline
1.5 & 826 & 181 & 31 & 6 \\ \hline
3.0 & 1819 & 594 & 118 & 11 \\ \hline
6.0 & 4356 & 1481 & 357 & 55 \\ \hline
12.0 & 7255 & 2989 & 780 & 128 \\ \hline \multicolumn{5}{|c|}{\(0.5<M_{ij}<0.8\)} \\ \hline
1.5 & 182 & 30 & 5 & 0 \\ \hline
3.0 & 357 & 83 & 6 & 0 \\ \hline
6.0 & 643 & 75 & 6 & 0 \\ \hline
12.0 & 1107 & 115 & 13 & 4 \\ \hline \end{tabular}
\end{table}
Table 2: Same as 1 but with S/N cuts on \(M_{ij}>0.8\) catalogue UCBs (top) and catalogue UCBs with \(0.5<M_{ij}<0.8\) (bottom). Catalogue UCB that have \(M_{ij}<0.5\) are not included in the S/N cut.
Figure 1: Waveform evolution as a function of observing time. Individual waveforms are plotted over the input A-channel PSD (solid, red curve), noise (constant, black curve stretching across the frequency window) and residual (dashed, blue curve) curves. The top graphs show a single 1.5-month (left panel) and 3-month (right panel) catalogue waveform, in green. The bottom panels show, in total, three 6 and 12-month catalogue waveforms, two of which did not appear in any earlier catalogue. All injected signals in this frequency window were recovered in a 6 and 12 month analysis.
on the vertical axis, and one notices the fraction of 1.5-month catalogue UCBs below the match threshold of 0.8 is significantly larger than the other \(T_{\rm obs}\) catalogues. For such a short observation time, a larger fraction of confused sources is expected.
Even though blending occurs more often for confused catalogue UCB, it is possible for matched UCB to exhibit blending also. For matched 12-month catalogue UCBs we find the second type of blending occurs exclusively. In Figure 8, we show the sky location of blended 12-month catalogue UCBs with \(M_{ij}>0.8\) and in the bottom graph are the sky locations for blended catalogue UCBs with correlation values in the range \(0.5<M_{ij}<0.8\). The blended UCBs are represented as '\(\mathbf{x}\)' markers with a black border. In each graph, the underlying distribution of points are the catalogue UCBs meeting a correlation threshold of \(M_{ij}>0.8\) and \(0.5<M_{ij}<0.8\), for the top and bottom graphs, respectively.
In Figure 9 we show the \(\mathcal{A}\)-\(f_{0}\) plane of all matching catalogue UCBs in blue. Catalogue UCBs that do not have a match are graphed with a colour-bar indicating the largest correlation value with an LDC injection. Each graph displays a different correlation range. The top graph highlights non-matching UCBs with \(M_{ij}<0.5\) and the bottom graph highlights non-matching UCBs with \(0.5<M_{ij}<0.8\). From both graphs, it is clear that non-matching catalogue UCBs are primarily below 5 mHz. Some of these catalogue UCBs also suffer from blending, that is they also have \(M_{ij}<0.5\) with nearby LDC injections that are matched with other catalogue UCBs, which is one reason to set a higher match threshold, in our case to 0.8.
There is one more type of catalogue UCB that is not classified as confused or matched, according to the categories described above. UCBs are classified as a 'false alarm' when no LDC injection exists within 10/\(T_{\rm obs}\) of the UCB frequency. How
Figure 2: A well-localised, eclipsing and chirping UCB for EM follow-up. The LDC parameter values are shown as black markers on the GBMCMC posteriors. The 1\(\sigma\) and 2\(\sigma\) posterior curves are graphed in this corner plot. Colours pink, purple, and green are the 3, 6, and 12 month posteriors, respectively.
ever, the false alarms identified in our analysis are due to boundary effects. As \(T_{\rm obs}\) becomes larger, the bandwidth of a UCB signal is also wider, leading to long waveform tails. When a long waveform tail also extends beyond the boundaries of an analysis segment, into a neighbouring analysis segment, a false alarm can emerge in the neighbouring segment's catalogue. In post-processing we explored the highest frequency catalogue UCB false alarm in the 12-month catalogue. This false alarm UCB is located at a \(f_{0}\sim 16.64\) mHz. This frequency is on the boundary between two analysis segments and the false alarm UCB waveform template overlaps with the waveform tails of two matched, or recovered, bright UCBs on either side of the boundary. Further examination of the other false alarm UCBs in the catalogue data reveals that each is the symptom of this boundary effect. This symptom is alleviated by allowing communication between neighbouring analysis segments as they run in parallel, such that the residual curve is consistent across the boundary (Littenberg and Cornish (2023)).
We identified all 12-month catalogue UCBs that have \(M_{ij}>0.8\), are eclipsing and well-localised and graph their chirp mass as a function of distance and sky locations in Figure 10. From the group of matching, eclipsing and well-localised 12-month catalogue UCBs, we selected the lowest frequency source and graphed its parameter posteriors in Figure 11. It has \(\iota=88.50^{\circ}\pm 0.17^{\circ}\) and an orbital period of \(\sim\)8 minutes. The frequency derivative is constrained at 12 months and is positive \(\dot{f}=\left(63.4^{+11.6}_{-8.8}\right)\times 10^{-17}s^{-2}\). Galactic sky location in degrees is \((\theta,\phi)=(-61.70^{+0.18}_{-0.13},195.55^{+0.16}_{-0.13})\). The distance to the system is relatively close at \(1.37^{+0.25}_{-0.22}\) kpc, and its chirp mass is \(0.751^{+0.074}_{-0.078}\)\(M_{\odot}\). Even though this recovered UCB has a high correlation value (\(M_{ij}=0.9995\)), the joint \(\mathcal{A}_{t}\) posterior to the injected amplitude and inclination reveals that these parameters were not accurately recovered. No correlation is visible in the \(\mathcal{A}\)-\(t\) plane, indicating insufficient sampling of the parameter space. Moreover, this 12-month UCB system is also found in the 1.5, 3 and 6-month catalogues with correlation values \(>0.8\). For reference, this low frequency catalogue UCB is overlapping with more than \(\sim 30\) UCBs. All of these are packed within a frequency range of only \(\pm 2^{8}/T_{12\rm{mo}}\). There are 12-month catalogue UCB with even lower frequencies, and with \(S/N\gtrsim 8\), but have poorly constrained sky location and distances due to confusion with tens of thousands of unresolved UCB. More observation time is required to determine if the lowest frequency systems are suitable for EM follow-up observations.
Posteriors for a distant (\(D_{L}=16.1\pm 2.4\) kpc) and reasonably well-localised (22 deg\({}^{2}\)) UCB in the 12-month catalogue, with a correlation value M=0.99 and S/N of 36, are shown in Figure 12. One sees that GW parameters \(\mathcal{A}\) and \(\iota\) are strongly correlated. This high S/N 12-month UCB system is matched in the 1.5, 3 and 6-month catalogues, with S/N = 2, 18, 22, respectively. The 3D location of this binary system places it in the part of the Milky Way that is inaccessible to optical telescopes due to intervening dust and gas obscuration, called the Zone of Avoidance. LISA will complement radio and infrared surveys in providing a new view to this part of the Milky Way Galaxy.
In the Appendix, we put the data obtained from comparing the GBMCMC catalogues to the injections to further use by examining a few 12-month frequency segments that contain confused catalogue UCBs. In particular, we explore catalogue UCBs that have a match at six months but do not appear as matched UCBs in the 12-month catalogue. Each corner case involves blended 12-month catalogue UCBs of the first type discussed in the beginning of this section.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline Observation time & sky localisation & \(f>\)0 & \(f<\)0 & eclipsing & eclipsing with \(f>\)0 & eclipsing with \(f>\)0, \\ & 10 sq. deg. & & \(70^{\circ}<i\)(deg.)\(<\)110\({}^{\circ}\) & & and well-localised \\ \hline
1.5 & 2 & 279 & 5 & 91 & 16 & 0 \\ \hline
3.0 & 3 & 553 & 1 & 372 & 48 & 0 \\ \hline
6.0 & 25 & 1950 & 10 & 1027 & 275 & 4 \\ \hline
12.0 & 441 & 4239 & 20 & 1873 & 735 & 58 \\ \hline \end{tabular}
\end{table}
Table 3: Same as Tables 1 and 2, but with different cuts on the catalogue UCBs with frequencies above 5 mHz to identify UCBs worthy follow-up analysis by EM observatories. A catalogue UCB is added to each column when at least 90% of its MCMC samples satisfy the given condition. In the first cut we state the number of UCBs that are ‘well-localised’, with sources contained within a sky location area of 10 deg\({}^{2}\). In the next two columns, we have the number of UCB with positive and negative frequency derivatives. The number of eclipsing binaries is shown in column 5, and we further subdivide this category into the last two columns. Eclipsing with GW dominated frequency evolution, and the number of UCBs which are eclipsing and well-localised. From this final category we select a high frequency 12-month catalogue UCB, as an example of a target for follow-up EM observations and archival searches.
Figure 3: Luminosity distance and chirp mass for a well-localised, eclipsing and chirping UCB. The 12-month posteriors \(\dot{f}\) and \(A\) the binary from Figure 2 were re-sampled using Equation ((2)) to form the luminosity distance \(D_{L}\) and chirp mass \(M_{\odot}\) posteriors. The black marker on the graph corresponds to the \(D_{L}\) and \(M_{c}\) derived by substituting the injected LDC parameters into (2). See Figure 2 for a description of the three colours.
## 5 Discussion
This paper is a report on the analysis of the LISA Data Challenge _Radler_ data with the GBMCMC code. This work is a necessary step in our efforts to prepare for the global fit required for the LISA mission. The LDC simulated data stream contains millions of white dwarf binary signals. We divided the time series data into different observation time increments, \(T_{\rm obs}\), of 1.5, 3, 6 and 12 months and produce catalogues for each \(T_{\rm obs}\) by performing a global fit to the resolvable binaries.
GBMCMC, is a trans-dimensional reversible jump MCMC algorithm with parallel tempering. The MCMC sampling was done in frequency, where the full frequency band has been divided into a total of 3317 frequency segments. Bayesian inference is used to select the highest evidence model to build catalogues for each of the 3317 frequency segments as a function of observing time. These are all combined for each observing time to create 1.5, 3, 6 and 12-month catalogues.
The UCB catalogue waveforms are then cross correlated with the known LDC injected waveforms to determine the efficacy of our search pipeline. For each observation time, we quantify the number of matching, confused, and false detections in the catalogues. We recover more than 10,000 binaries after 12 months of observing. We found that 7,255 of these 12-month catalogue UCB have a match with an LDC injection (with a correlation value \(M_{ij}>0.8\)) and have S/N \(>7\). Of these, there are 128 UCBs with a S/N \(>100\).
We identify two interesting corner cases for in-depth follow-up analysis. For the first Appendix corner case we investigate two confused 12-month catalogue UCBs, the sum of which are a match with a single LDC injection. Moreover, the two catalogue UCBs have a common ancestor in the 6-month catalogue. Namely, they both have a match value greater than 0.5 with a single 6-month catalogue UCB. This indicates that the 12-month MCMC analysis has not converged. Smaller time-jumps between catalogues is a target for future investigation.
In the second corner case, we examined a different type of confused UCB in the 12-month catalogue. There are two UCBs occupying the same region of the \(\mathcal{A}\)-\(f_{0}\) and \(\mathcal{B}\)-\(\phi\) planes. In this region of parameter space, there are two LCD injections. After a 12-month analysis neither of the injections are separately a match with the confused UCBs; however, the sum of the UCB waveforms is a match with the sum of the LDC injected waveforms. Similar to the previous corner case, we find the 12-month MCMC analysis for this analysis segment has not converged. Additional incremental analysis is necessary between the 6-month and 12-month catalogues to disentangle these two LDC injections.
These corner cases are part and parcel of a broader discussion of a strategy for creating and publishing catalogues. Namely, how often will the catalogues be updated and released, and the type and format of information dispensed in the catalogues. Alerts and low-latency analysis and outputs will be in
Figure 4: The power spectra graphs for the analysed LISA band, for the 1.5, 3, 6, and 12-month catalogue. The red curve is the A-channel input data, and the dashed, blue curve is the residual, after the catalogue UCBs have been subtracted. The black curve plotted on top of the data and residual is the noise level.
formed by our time-evolving catalogues. Further in-depth analysis of blended catalogue UCBs in data challenges will be essential for answering some of these questions. Checking the convergence of analysis segments that have catalogue UCBs which share the same parent or are close in sky location and frequency-amplitude will be necessary. Accurate UCB science and low frequency GW science in general depend heavily on validation of the data products output from search pipelines.
We make one last comment regarding future work. Binary UCBs that make up the _Radler_ LDC simulated data stream have zero eccentricity, which is likely the case for most resolvable binaries. However, eccentric UCBs do exist, of course, and one of the future upgrades to GBMCMC will be to incorporate eccentric UCB models into GBMCMC. The harmonics of low frequency eccentric binaries will be put to use in the search routine. Holding an arbitrary number of parameters fixed at values determined, for example, with radio, optical, X-ray or \(\gamma\)-ray observations of known binaries (Littenberg & Cornish (2019)) will be of particular interest when checking to see if a newly timed pulsar is observable in the LISA band.
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Figure 10: Chirp mass versus distance for eclipsing and well-localised catalogue UCB that are matched. The graph on the left shows chirp mass versus distance for all matched 12-month catalogue UCBs that are eclipsing and have positive \(\dot{f}\), coloured by GW frequency. Eclipsing UCB are defined here as having more than 90% of their inclination angle samples constrained within 70\({}^{\circ}<\) \(\iota\)(deg.) \(<\) 110\({}^{\circ}\). The graph on the right shows the ecliptic coordinates of the UCB from the left graph, now coloured by distance. These are plotted over all 12-month catalogue UCB that have \(M_{ij}>\)0.8 (light blue).
Figure 11: The 1 and 2\(\sigma\) parameter posteriors for the lowest frequency and 12-month matching catalogue UCB that is eclipsing and localised to within 10 deg\({}^{2}\) on the sky. The black diamonds on the 2D posteriors represent the LDC injected parameter values. One can see that the amplitude and inclination angle have not been recovered, and no correlation is visible in the \(\mathcal{A}\)-\(\iota\) plane. Increasing the number of MCMC steps and searching over a longer observation time are needed to better determine these parameters and recover the expected correlation between them. |
2310.07906 | Power Tracking Control of Heterogeneous Populations of TCLs with
Partially Measured States | This paper presents a new aggregate power tracking control scheme for
populations of thermostatically controlled loads (TCLs). The control design is
performed in the framework of partial differential equations (PDEs) based on a
late-lumping procedure without truncating the infinite-dimensional model
describing the dynamics of the TCL population. An input-output linearization
control scheme, which is independent of system parameters and uses only partial
state measurement, is derived, and a sliding model-like control is applied to
achieve finite-time input-to-state stability for tracking error dynamics. Such
a control strategy can ensure robust performance in the presence of modeling
uncertainties, while considerably reducing the communication burden in large
scale distributed systems similar to that considered in the present work. A
rigorous analysis of the closed-loop stability of the underlying PDE system was
conducted, which guaranteed the validity of the developed control scheme.
Simulation studies were performed while considering two TCL populations with a
significant difference in their size, and the results show that the developed
control scheme performs well in both cases, thereby confirming the
effectiveness of the proposed solution. | Zhenhe Zhang, Jun Zheng, Guchuan Zhu | 2023-10-11T21:24:52Z | http://arxiv.org/abs/2310.07906v1 | # Power Tracking Control of Heterogeneous Populations of TCLs with Partially Measured States
###### Abstract
This paper presents a new aggregate power tracking control scheme for populations of thermostatically controlled loads (TCLs). The control design is performed in the framework of partial differential equations (PDEs) based on a late-lumping procedure without truncating the infinite-dimensional model describing the dynamics of the TCL population. An input-output linearization control scheme, which is independent of system parameters and uses only partial state measurement, is derived, and a sliding model-like control is applied to achieve finite-time input-to-state stability for tracking error dynamics. Such a control strategy can ensure robust performance in the presence of modeling uncertainties, while considerably reducing the communication burden in large scale distributed systems similar to that considered in the present work. A rigorous analysis of the closed-loop stability of the underlying PDE system was conducted, which guaranteed the validity of the developed control scheme. Simulation studies were performed while considering two TCL populations with a significant difference in their size, and the results show that the developed control scheme performs well in both cases, thereby confirming the effectiveness of the proposed solution.
A +
Footnote †: journal:
agregate power tracking control, finite-time input-to-state stability, input-output linearization, partial differential equations, thermostatically controlled loads.
## 1 Introduction
In the context of today,'s smart grids, it is widely recognized that demand response (DR) programs have great potential in dealing with ongoing demands, while enhancing the energy efficiency and resilience of the power grid [4, 7, 28, 40]. As a promising demand-response enabled resource, thermostatically controlled loads (TCLs), such as air conditioners (ACs), space heating devices, refrigerators, and water heaters, are attracting increasing attention. Although a single TCL unit has very limited power regulation capability, ensembles of a large number of TCLs, when managed in an orderly and controllable manner, can have a significant impact on the entire power grid [12, 38, 26]. It has been shown that a large TCL population can be managed to support demand response tasks, including peak load shaving and load following [9, 25, 34], and to provide ancillary services, such as primary or secondary frequency controls [20, 36, 33, 21].
The present work focuses on load tracking control, which allows the aggregate power of a TCL population to follow a desired consumption profile. The control design is based on a model of the dynamics of the TCL population described by partial differential equations (PDEs). Specifically, we consider a set of TCLs in which the dynamics of every individual device are modeled by a lumped stochastic hybrid system (SHS) operated through thermostat-based deadband control. The aggregate dynamics of such a TCL population can be modeled by two coupled Fokker-Planck equations (see, e.g, [37, 2, 19]) describing the evolution of the probability distribution of TCLs in the ON and OFF states over the temperature. Note that the same form of PDE-based models can also be derived by assuming that the dynamics of individual TCLs are described by deterministic systems while considering population heterogeneity [1, 5, 23].
Another widely adopted method to build the aggregate dynamical model of TCL populations is to divide a fixed range of temperatures into several segments, called state-bins, each of which is associated with the number of TCLs with their temperature fitting in this bin. The dynamics of state-bin transactions can be described by a Markov chain (see, e.g., [14, 18, 27, 29, 35]) or state queue (see, e.g., [17, 32]), which leads to finite-dimensional state-space models. It is worth noting that discretizing a PDE with respect to (w.r.t.) the space variable (temperature) also leads to a finite-dimensional state-space model. However, as the considered Fokker-Planck equation is a semi-linear time-varying PDE, its discretization results in a finite-dimensional nonlinear time-varying system. Consequently, a model described by the linear time invariant (LTI) system, which is the most used state-bin model in the existing literature, may be equivalent to that derived from PDEs only locally around particular equilibrium points and operational conditions (e.g., temperature set-point, ambient temperature, deadband), even with a variety of extensions. Therefore, the PDE provides a more generic framework for modeling the aggregate dynamics of TCL populations, which allows handling nonlinearity, time-varying operational conditions, and parametric uncertainties with often very simple control algorithms. However, the PDE control system design procedure generally involves more complex mathematical analysis and is more challenging.
The main objective of TCL population control is to manipulate the total power consumption of the entire population, which can be achieved by changing the temperature set-point, moving the deadband, or interfering with the probability distributions of the TCLs via forced switches (see, e.g., [1, 2, 30, 36, 18, 39]). Because a TCL population usually contains a large number of units that may spread over a large geographical area, only decentralized or distributed schemes are applicable control strategies. In fact, a remarkable amount of work on the control of TCL populations has been reported in the literature, and the majority of the proposed solutions are based on lumped models by applying optimization theory and optimal control techniques, in particular model predictive control (see, e.g, [14, 17, 18, 20, 27, 29, 30, 32, 35, 36]). It should be noted that, owing the nature of the considered problem, control schemes requiring the state measurement of the entire population in real-time are practically infeasible (see, e.g., [31] and the references therein). This problem can be addressed using state observers [20, 22]. Nevertheless, it is still very challenging to assess the performance of model-based state estimation algorithms because it depends heavily on the accuracy of the system parameters.
The load tracking control algorithm developed in the present work is a decentralized scheme in which the rates for set-point temperature adjustment generated by a central unit are broadcast to the TCLs over the population. Emphasis is placed on solving issues arising in practical applications, particularly communication restrictions and modeling uncertainties for large scale TCL populations. The control system design is carried out in the framework of PDE-based modeling and control techniques. It should be noted that the two basic paradigms in PDE control system design and implementation, namely early-lumping and late-lumping procedures, have all been applied to the control of the coupled Fokker-Planck equations associated with TCL populations. The early-lumping method discretizes the underlying PDEs to obtain a lumped model, and then applies the techniques for finite-dimensional control system design [1, 2, 30, 5, 23]. In contrast, with the late-lumping method, the controller is designed using the PDE model and then discretized for implementation [6, 39]. A significant advantage of the late-lumping method is that it can preserve the essential properties of the PDE model and no approximation is required in the control design. However, some issues remain open. More specifically, the schemes developed in [39] and [6] are based on input-output linearization by state feedback control, which may incur a communication burden. In addition, these control schemes require an accurate knowledge of the system parameters, for example, the diffusion coefficient in Fokker-Planck equations, which are not easy to determine from both theoretical and practical viewpoints considering the nature of the problem under investigation. Finally, although taking a weighted power load as the system output proposed in [39] can avoid the controllability issue introduced by the use of the total power load of the in-band TCLs as the system output in [6], such a choice lacks physical interpretation and is unsuitable for practical operation.
In this paper, we developed a new control algorithm based on the input-output linearization technique, which results in a system composed of finite-dimensional input-output dynamics and infinite-dimensional internal dynamics. The control design amounts then to finding a robust closed-loop control law that stabilizes the finite-dimensional input-output dynamics while guaranteeing the stability of the infinite-dimensional internal dynamics. Specifically:
* A new system output for power tracking control is proposed that can guarantee the controllability of the input-output dynamics.
* A linearization control law, which is independent of system parameters, e.g., the diffusion coefficient, while requiring only knowledge of the states of TCLs near the deadband boundaries, is derived.
* To tackle modeling uncertainties while making the control scheme computationally tractable, a sliding model-like tracking control scheme that can achieve finite-time input-to-state stability (FTISS) [8, 16], is designed.
* The non-negativeness of the solution to the Fokker-Planck equations under the developed control law and other properties required to ensure closed-loop stability are rigorously validated.
The main contribution of the present work lies in the simplicity, scalability, and applicability of the control strategy developed
under a generic framework. In addition, it is worth noting that as the developed control algorithm requires only measuring the state of the TCLs on the end-points of the deadband, TCLs need to notify their state only when switching occurs. Because the cyclic rate of the TCLs is much slower than the controller sampling rate, the communication burden can be significantly reduced. Obviously, it is very difficult for state feedback control schemes based on lumped aggregate models to achieve such features, which is critical for practical implementations.
The remainder of this paper is organized as follows. Section 2 introduces the notations used in the study and preliminaries on FTISS. Section 3 presents the first-order equivalent thermal parameter (ETP) model for a single TCL unit and the coupled Fokker-Planck model for the aggregate dynamics of the TCL population. Section 4 presents the power tracking control design and closed-loop stability analysis. The experimental validation of the developed control strategy and the simulation results are reported in Section 5, followed by concluding remarks in Section 6. Finally, the proof of one of the main theoretical result is presented in the appendix.
## 2 Notations and preliminaries
### Notations
Let \(\mathbb{R}:=(-\infty,+\infty),\)\(\mathbb{R}_{\geq 0}:=[0,+\infty)\), \(\mathbb{R}_{>0}:=(0,+\infty)\), and \(\mathbb{R}_{\leq 0}:=(-\infty,0].\) Denote by \(\partial_{s}f\) the derivative of the function \(f\) w.r.t. argument \(s\). Note that, for notation simplicity, we may omit the arguments of functions if there is no ambiguity.
By convention, we denote by \(|\cdot|\) the module of a function. For positive integers \(m,n\) and a given (open or closed) domain \(\Omega\subset\mathbb{R}^{n}\), let \(L^{\infty}(\Omega;\mathbb{R}^{m}):=\{\phi:\Omega\to\mathbb{R}^{m}|\,\phi\) is measurable in \(\Omega\) and satisfies \(\text{ess sup}_{s\in\Omega}|\phi(s)|<+\infty\}.\) For \(\phi\in L^{\infty}(\Omega;\mathbb{R}^{m})\), the norm of \(\phi\) is defined by \(\|\phi\|_{L^{\infty}(\Omega)}:=\text{ess sup}_{s\in\Omega}|\phi(s)|.\) Let \(L^{\infty}_{loc}(\Omega;\mathbb{R}^{m}):=\{\phi:\Omega\to\mathbb{R}^{m}|\,\, \phi\in L^{\infty}(\Omega^{\prime};\mathbb{R}^{m})\text{ for any }\Omega^{\prime}\subsetneqq\Omega\}\)
For given (open or closed) domains \(\Omega_{1},\Omega_{2}\subset\mathbb{R}^{n}\) and \(\Omega_{3}\subset\mathbb{R}\), let \(C\left(\Omega_{1};\Omega_{3}\right):=C^{0}\left(\Omega_{1};\Omega_{3}\right): =\{\phi:\Omega_{1}\to\Omega_{3}|\,\,\phi\) is continuous w.r.t. its all augments in \(\Omega_{1}\}.\) For positive integers \(i,j\), let \(C^{i}\left(\Omega_{1};\Omega_{3}\right):=\{\phi:\Omega_{1}\to\Omega_{3}|\,\,\phi\) has continuous derivatives up to order \(i\) w.r.t. its all augments in \(\Omega_{1}\}\), and \(C^{i,j}\left(\Omega_{1}\times\Omega_{2};\Omega_{3}\right):=\{\phi:\Omega_{1} \times\Omega_{2}\to\Omega_{3}|\,\,\phi\) has continuous derivatives up to order \(i\) w.r.t. its augments in \(\Omega_{1}\) and up to order \(j\) w.r.t. its augments in \(\Omega_{2}\). In particular, if \(\Omega_{3}=\mathbb{R}\), we denote \(C\left(\Omega_{1}\right):=C^{0}\left(\Omega_{1};\mathbb{R}\right)\) and \(C^{i}\left(\Omega_{1}\right):=C^{i}\left(\Omega_{1};\mathbb{R}\right)\) for \(i>0\).
As in [16] and [11], we define the following sets of comparison functions. Let \(\mathcal{K}:=\{\vartheta:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}|\,\,\vartheta(0)=0,\vartheta\) is continuous, strictly increasing\(\}\); \(\mathcal{L}:=\{\vartheta:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}|\,\,\vartheta\) is continuous, strictly decreasing, \(\lim_{s\to+\infty}\vartheta(s)=0\}\); \(\mathcal{KL}:=\{\beta:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\to\mathbb{R}_{ \geq 0}|\,\,\beta(\cdot,t)\in\mathcal{K},\forall t\in\mathbb{R}_{\geq 0}\), and \(\beta(s,\cdot)\in\mathcal{L},\forall s\in\mathbb{R}_{+}\};\)\(\mathcal{K}_{\infty}:=\{\vartheta:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}|\,\,\vartheta\in \mathcal{K}\text{ and }\lim_{s\to+\infty}\vartheta(s)=+\infty\};\)\(\mathcal{GKL}:=\{\beta:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}|\,\, \beta(\cdot,0)\in\mathcal{K},\) and for each fixed \(s\in\mathbb{R}_{>0}\) there exists \(\widetilde{T}(s)\in\mathbb{R}_{\geq 0}\) such that \(\beta(s,t)=0\) for all \(t\geq\widetilde{T}(s)\}.\)
### Finite-time input-to-state stability of finite dimensional systems
Consider the following nonlinear system
\[\dot{z}(t)= f(z(t),d(t)),\,\,\,\forall t\in\mathbb{R}_{\geq 0}, \tag{1a}\] \[z(0)= z_{0}, \tag{1b}\]
where \(z:=[z_{1},z_{2},...,z_{n}]^{T}\in\mathbb{R}^{n}\) is the state, \(z_{0}\in\mathbb{R}^{n}\) is the initial datum, \(d\in\mathcal{D}:=L^{\infty}_{\text{loc}}(\mathbb{R}_{\geq 0};\mathbb{R}^{m})\) is the input (disturbance) to the system, \(f:\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}\) is a nonlinear function that is continuous w.r.t. \((z,d)\), ensures the forward existence of the system solutions, at least locally, and satisfies \(f(0,0)=0\), and \(m\geq 1\) and \(n\geq 1\) are integers.
**Definition 2.1**: _System (1) is said to be finite-time input-to-state stable (FTISS) if there exist functions \(\vartheta\in\mathcal{K}\) and \(\beta\in\mathcal{GKL}\) such that for any \(x_{0}\in\mathbb{R}^{n}\) and \(d\in\mathcal{D}\) its trajectory satisfies_
\[|z(t)|\leq\beta(|z_{0}|,t)+\vartheta(\|d\|_{L^{\infty}(0,t)}),\,\,\,\forall t \in\mathbb{R}_{\geq 0}. \tag{2}\]
**Remark 2.1**: _Note that FTISS is defined in a similar way to the definition of input-to-state stability (ISS) in [11, Chapter 4] via the norm of \(d\) over the interval \((0,t)\) rather than \((0,+\infty)\). Thus, the FTISS presented here is a refined notion of the one introduced in [8, 16], where the second term in the right-hand side of (2) is under the form \(\vartheta(\|d\|_{L^{\infty}(0,+\infty)})\), which describes the influence of the global bounds of \(d\) instead of the bounds of \(d\) over the finite time interval \((0,t)\)._
**Definition 2.2**: _A continuously differentiable function \(V:\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}\) is said to be an FTISS Lyapunov function for system (1) if there exist functions \(\mu_{1},\mu_{2}\in\mathcal{K}_{\infty}\), \(\chi\in\mathcal{K}\) and constants \(c>0\) and \(\theta\in(0,1)\) such that for all \(x\in\mathbb{R}^{n}\) and all \(d\in\mathcal{D}\) it holds that_
\[\mu_{1}(|x|)\leq V(x)\leq\mu_{2}(|x|),\] \[|z|\geq\chi(|d|)\Rightarrow DV(z)\cdot f(z,d)\leq-cV^{\theta}(z),\]
_where \(DV(z):=\left[\frac{\partial V}{\partial z_{1}},\ldots,\frac{\partial V}{ \partial z_{n}}\right].\)_
The following Lyapunov-like lemma gives a sufficient condition for the FTISS.
**Lemma 2.1**: _System (1) is FTISS if it admits a finite-time ISS Lyapunov function._
**Proof.** Setting \(\mathcal{V}:=\{z|V(z)\leq\mu_{2}(\chi(|d|))\}\) in the proof of [8, Theorem 1(a)], the lemma statement follows immediately. \(\blacksquare\)
## 3 Mathematical model and problem specification
### Dynamics of individual TCLs
In the present work, we focus on modeling the population of residential air conditioners (ACs). While, its extension to other cooling and heating devices is straightforward. We consider the case where all ACs are operated by thermostats hence, every AC switches between the ON and OFF states whenever it reaches the prescribed lower or upper temperature bounds. For simplicity, we ignore the solar irradiation and internal heat gains and assume that the ACs operate at a fixed frequency. Then, the dynamics of the indoor temperature, denoted by \(x\), for a representative load can be modeled by the following SHS (see, e.g., [2, 19, 30]):
\[\mathrm{d}x(t)=\frac{1}{CR}\left(x_{a}(t)-x(t)-s(t)RP\right)\mathrm{d}t+ \sigma\,\mathrm{d}w(t), \tag{3}\]
where \(x_{a}(t)\) is the ambient temperature, \(R\), \(C\), and \(P\) are the thermal resistance, capacitance, and power, respectively, and \(s(t)\) is the switching signal. In (3), \(w(t)\) is a standard Wiener process, which, along with the parameter \(\sigma\), represents modeling uncertainties, such as unaccounted heat loss or heat gain, parameter variations, and disturbances.
For a thermostat-controlled AC, the switching signal \(s(t)\) takes a binary value from \(\{0,1\}\), representing the OFF and ON states. We consider a hybrid control scheme, as shown in Fig. 1, in which the device always switches at the endpoints of the deadband. In addition, forced switches at any moment, denoted by \(r(t)\), may also occur to alert the probability distributions of the TCL population. Let \(r(t)\) take a binary value from \(\{0,1\}\), with 1 representing the occurrence of switching and 0 otherwise. Letting \(\underline{x}\) and \(\overline{x}\) be the prescribed lower and upper temperature bounds, respectively, the deadband control for an AC can then be expressed as
\[s(t)=\begin{cases}1,&\text{if }x\geq\overline{x};\\ 0,&\text{if }x\leq\underline{x};\\ (s(t^{-})\wedge r(t))+(s(t^{-})\lor r(t)),&\text{otherwise};\end{cases}\]
where "\(+\)" is the one-bit binary addition with overflow. In addition, the notations \((\cdot)^{-}\) and \((\cdot)^{+}\) denote the left and right limits of the scalar variable, respectively. Note that different actions, such as random switches to avoid power demand oscillations due to synchronization within a TCL population, mechanisms for blocking the switches to protect the ACs, etc., can be integrated in the design of forced switching schemes.
### Dynamics of aggregate TCL population
As mentioned previously, the dynamics of an aggregate TCL population can be characterized by the evolution of the distributions of the TCLs over temperature. When the number of TCLs in the population tends to be infinite, this population can
be modeled as a continuum whose temperature distribution is governed by the coupled Fokker-Planck equations [2, 5, 19, 23]. Specifically, we denote by \(f_{1}(x,t)\) and \(f_{0}(x,t)\) the probability density functions (PDFs) of the TCLs in the ON and OFF states at temperature \(x\) and time \(t\), respectively. As illustrated in Fig. 2, we assume that all the loads are confined in a fixed temperature range \((x_{L},x_{H})\) along all possible operations, where \(x_{L}\) and \(x_{H}\) are constants, which is a reasonable assumption for practical application. Moreover, owing to the nature of thermostat-based control, there must be that \(f_{1}(x,t)=0\) for all \(x\leq\underline{x}\) and \(t\in\mathbb{R}_{>0}\), and that \(f_{0}(x,t)=0\) for all \(x\geq\overline{x}\) and \(t\in\mathbb{R}_{>0}\). Therefore, we can divide the range \((x_{L},x_{H})\) into three segments:
\[I_{a}:=(x_{L},\underline{x}),I_{b}:=(\underline{x},\overline{x}),I_{c}:=( \overline{x},x_{H}),\]
which will be used in the upcoming study.
Suppose that the dynamics of each load in the TCL population are described by (3). Let further
\[\alpha_{0}(x,t):= \frac{1}{CR}\left(x_{a}(t)-x\right),\] \[\alpha_{1}(x,t):= \frac{1}{CR}\left(x_{a}(t)-x-RP\right).\]
The evolutions of \(f_{0}(x,t)\) and \(f_{1}(x,t)\) are governed by the following coupled Fokker-Planck equations [2, 19, 30]:
\[\partial_{t}f_{0}= \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{0}-( \alpha_{0}-u)f_{0}\bigg{)}\text{ in }I_{a}\times\mathbb{R}_{>0}, \tag{4a}\] \[\partial_{t}f_{0}= \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{0}-( \alpha_{0}-u)f_{0}\bigg{)}-g(f_{0},f_{1})\text{ in }I_{b}\times\mathbb{R}_{>0},\] (4b) \[\partial_{t}f_{1}= \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{1}-( \alpha_{1}-u)f_{1}\bigg{)}+g(f_{0},f_{1})\text{ in }I_{b}\times\mathbb{R}_{>0},\] (4c) \[\partial_{t}f_{1}= \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{1}-( \alpha_{1}-u)f_{1}\bigg{)}\text{ in }I_{c}\times\mathbb{R}_{>0}, \tag{4d}\]
where \(g(f_{0},f_{1})\) represents the net probability flux due to the switches occurring over segment \(I_{b}\), that is, the so-called forced switches. Hence, the signs of \(g(f_{0},f_{1})\) in (4b) and (4c) should be opposite to each other, which implies a mass conservation property as claimed in Theorem 4.3 in Section 4.3. Note that (4b) and (4c) have a general form compared to that given in [30]
Figure 1: Hybrid thermostat-based deadband control scheme.
Figure 2: Illustration of probability density functions of a TCL population at a given time.
(see (19a) and (19b) of that paper), where an explicitly linear function \(g(f_{0},f_{1})\) was used to model a switching rate control scheme.
Following [30], we introduce the notation of probability flows \(\mathcal{F}_{i}\). When there is no additional flux from the forced switches, i.e., \(g=0\), \(\mathcal{F}_{i}\) is the integral of the probability fluxes \(\partial_{t}f_{i}\) over the temperature (\(x\)-) coordinate:
\[\mathcal{F}_{i}(x,t):=\frac{\sigma^{2}}{2}\partial_{x}f_{i}(x,t)-(\alpha_{i}(x,t)-u(t))f_{i}(x,t),i=0,1.\]
The boundary conditions can then be written as
\[\mathcal{F}_{0}(x_{L}^{+},t)= 0,\ \ \forall t\in\mathbb{R}_{>0}, \tag{5a}\] \[\mathcal{F}_{0}(\underline{x}^{-},t)= \mathcal{F}_{0}(\underline{x}^{+},t)+\mathcal{F}_{1}(\underline{ x}^{+},t),\ \ \forall t\in\mathbb{R}_{>0},\] (5b) \[f_{0}(\underline{x}^{-},t)= f_{0}(\underline{x}^{+},t),\ \ \forall t\in\mathbb{R}_{>0},\] (5c) \[f_{0}(\overline{x}^{-},t)= 0,\ \ \forall t\in\mathbb{R}_{>0},\] (5d) \[f_{1}(\underline{x}^{+},t)= 0,\ \ \forall t\in\mathbb{R}_{>0},\] (5e) \[f_{1}(\overline{x}^{-},t)= f_{1}(\overline{x}^{+},t),\ \ \forall t\in\mathbb{R}_{>0},\] (5f) \[\mathcal{F}_{1}(\overline{x}^{+},t)= \mathcal{F}_{0}(\overline{x}^{-},t)+\mathcal{F}_{1}(\overline{x}^ {-},t),\ \ \forall t\in\mathbb{R}_{>0},\] (5g) \[\mathcal{F}_{1}(x_{H}^{-},t)= 0,\ \ \forall t\in\mathbb{R}_{>0},\] (5h) \[\mathcal{F}_{0}(\underline{x}^{-},t)> \mathcal{F}_{0}(\underline{x}^{+},t),\ \ \forall t\in\mathbb{R}_{>0},\] (5i) \[\mathcal{F}_{1}(\overline{x}^{+},t)< \mathcal{F}_{1}(\overline{x}^{-},t),\ \ \forall t\in\mathbb{R}_{>0}. \tag{5j}\]
The initial data of \(f_{0}\) and \(f_{1}\) defined over \(\overline{I}_{a0}:=[x_{L},\underline{x}(0)],\overline{I}_{b0}:=[\underline{x} (0),\overline{x}(0)]\), and \(\overline{I}_{c0}:=[\overline{x}(0),x_{H}]\) are given by
\[f_{0}(0,x)= f_{0}^{a0}(x),\ \ \forall x\in\overline{I}_{a0}, \tag{6a}\] \[f_{0}(0,x)= f_{0}^{b0}(x),\ \ \forall x\in\overline{I}_{b0},\] (6b) \[f_{1}(0,x)= f_{1}^{b0}(x),\ \ \forall x\in\overline{I}_{b0},\] (6c) \[f_{1}(0,x)= f_{1}^{c0}(x),\ \ \forall x\in\overline{I}_{c0}. \tag{6d}\]
The total power demand of the TCL population at time \(t\in\mathbb{R}_{\geq 0}\) is given by
\[y_{\text{total}}(t):=\frac{P}{\eta}\int_{\underline{x}(t)}^{x_{H}}f_{1}(x,t)\, \mathrm{d}x, \tag{7}\]
where \(\eta\) is the load efficiency coefficient.
**Remark 3.1**: _We provide remarks on the boundary conditions presented in (5)._
* _For continuous functions_ \(\alpha_{0},\alpha_{1}\)_, and_ \(u\)_, the boundary conditions in (_5_) are equivalent to:_ \[\frac{\sigma^{2}}{2}\partial_{x}f_{0}(x_{L}^{+},t)= (\alpha_{0}(x_{L}^{+},t)-u(t))f_{0}(x_{L}^{+},t),\] (8a) \[\partial_{x}f_{0}(\underline{x}^{-},t)= \partial_{x}f_{0}(\underline{x}^{+},t)+\partial_{x}f_{1}( \underline{x}^{+},t),\] (8b) \[f_{0}(\underline{x}^{-},t)= f_{0}(\underline{x}^{+},t),\] (8c) \[f_{0}(\overline{x},t)= 0,\] (8d) \[f_{1}(\underline{x},t)= 0,\] (8e) \[f_{1}(\overline{x}^{-},t)= f_{1}(\overline{x}^{+},t),\] (8f) \[\partial_{x}f_{1}(\overline{x}^{+},t)= \partial_{x}f_{0}(\overline{x}^{-},t)+\partial_{x}f_{1}(\overline {x}^{-},t),\] (8g) \[\frac{\sigma^{2}}{2}\partial_{x}f_{1}(x_{H}^{-},t)= (\alpha_{1}(x_{H}^{-},t)-u(t))f_{1}(x_{H}^{-},t),\] (8h)
\[\partial_{x}f_{1}(\underline{x}^{+},t)> 0, \tag{8i}\] \[\partial_{x}f_{0}(\overline{x}^{-},t)< 0. \tag{8j}\]
2. _It is worth noting that this set of boundary conditions ((_5_) or (_8_)), with possible variations, is commonly used in the literature_ _[_2, 19, 30_]__, which captures the basic properties of the considered problem, for example, impenetrable wall reflections (_(_8a_) and (_8h_)), absorbing actions due to thermostat switching (_(_8d_)) and (_8e_)), and probability conservation at the boundaries of the deadband (_(_8b_) and (_8g_)). Note that because of the absorbing property and the continuity of the PDFs on the boundaries of the deadband, the conditions (_8b_) and (_8g_) remain the same as those originally derived in_ _[_19_]__, even though the considered problem in the present work contains control actions._
### Problem statement and basic assumptions
In this work, we study the dynamics described by the PDE model (4) under the boundary and initial conditions (5) and (6). Based on (7), a new output function will be defined and specified in Section 4. With these dynamics, a continuous time controller that considers the convergence time and robustness is designed to stabilize the tracking process.
In the sequel, we assume that \(x_{a}\in C(\mathbb{R}_{\geq 0})\ \underline{x},\overline{x}\in C^{1}( \mathbb{R}_{\geq 0};\mathbb{R}_{>0})\), and denote
\[S_{ab} :=\left(C^{2,1}(I_{a}\times\mathbb{R}_{>0})\cap C(\overline{I}_{a }\times\mathbb{R}_{\geq 0})\right)\cup\left(C^{2,1}(I_{b}\times\mathbb{R}_{>0}) \cap C(\overline{I}_{b}\times\mathbb{R}_{\geq 0})\right),\] \[S_{bc} :=\left(C^{2,1}(I_{b}\times\mathbb{R}_{>0})\cap C(\overline{I}_{b }\times\mathbb{R}_{\geq 0})\right)\cup\left(C^{2,1}(I_{c}\times\mathbb{R}_{>0}) \cap C(\overline{I}_{c}\times\mathbb{R}_{\geq 0})\right).\]
Based on the physical properties of the problem, we impose the following structural conditions and basic assumptions on the solution and control for the system:
\(\bullet\) The function of net probability flux \(g\) belongs to \(C^{1}(\mathbb{R}^{2};\mathbb{R})\) and satisfies
1. \(g(0,\tau)\leq 0\) for all \(\tau\in\mathbb{R}\);
2. \(g(s,0)\geq 0\) for all \(s\in\mathbb{R}\);
3. \(|g_{s}(s,\tau)|+|g_{\tau}(s,\tau)|\leq 1\) for all \((s,\tau)\in\mathbb{R}^{2}\).
\(\bullet\) The pair of solution \((f_{0},f_{1})\) and the control \(u\) satisfy
1. \(u\in C(\mathbb{R}_{\geq 0};\mathbb{R})\) such that \(\dot{\underline{x}}=\dot{\overline{x}}=u\) in \(\mathbb{R}_{\geq 0}\);
2. \(f_{0}^{a0}\in C(\overline{I}_{a0};\mathbb{R}_{\geq 0})\), \(f_{0}^{b0}\in C(\overline{I}_{b0};\mathbb{R}_{\geq 0})\), \(f_{1}^{b0}\in C(\overline{I}_{b0};\mathbb{R}_{\geq 0})\), \(f_{1}^{c0}\in C(\overline{I}_{c0};\mathbb{R}_{\geq 0})\);
3. \(f_{0}\in S_{ab}\) and has derivatives \(\partial_{x}f_{0}(x_{+}^{+},t)\), \(\partial_{x}f_{0}(\underline{x}^{\pm},t)\) and \(\partial_{x}f_{0}(\overline{x}^{-},t)\) for any fixed \(t\in\mathbb{R}_{>0}\);
4. \(f_{1}\in S_{bc}\) and has derivatives \(\partial_{x}f_{1}(x_{H}^{-},t)\), \(\partial_{x}f_{1}(\overline{x}^{\pm},t)\) and \(\partial_{x}f_{1}(\underline{x}^{+},t)\) for any fixed \(t\in\mathbb{R}_{>0}\).
**Remark 3.2**: _It should be mentioned that for \(f_{0}=0\) (or \(f_{1}=0\)), condition (G1) (or (G2)) guarantees \(-g(f_{0},f_{1})\geq 0\) (or \(g(f_{0},f_{1})\geq 0\)) in (4b) (or (4c)). This indicates that forced switching, which generates additional fluxes, is only possible from the \(f_{1}\) system into the \(f_{0}\) system when \(f_{0}\) is zero._
_Condition (G3) indicates that the change in the probability density of the additional flux cannot be too fast for practical applications. This is in accordance with the suggestion in [30]._
_Condition (F1) indicates that the initial data are assumed to be nonnegative and continuous over the given domains. Conditions (F2) and (F3) describe the regularity of the solutions at the endpoints of the given domains at any time \(t\)._
## 4 Control design and stability analysis
In this section, we design a feedback control to ensure that the output of the system (4)-(6) tracks a reference power curve, and assess the stability of the error dynamics in the framework of FTISS theory. Moreover, we study the mass conservation and non-negativeness properties of the solutions to the considered system, which allows further clarification of the physical meanings of the mathematical model.
### Control design
The control objective is to drive the power consumption of the population to track the desired regulation signal. To this end, we choose an output of the power tracking control scheme as
\[y(t):= y_{\text{total}}(t)+\frac{P}{\eta}\int_{\overline{x}(t)}^{x_{H}}f_{1}(x,t )\,\mathrm{d}x-\frac{P}{\eta}\int_{x_{L}}^{\overline{x}(t)}f_{0}(x,t)\,\mathrm{ d}x,\ \ t\in\mathbb{R}_{\geq 0}. \tag{9}\]
It is worth noting that, as the probability flows of \(f_{0}\) and \(f_{1}\) always move towards the deadband, \(y(t)\) defined in (9) converges to the aggregated power demand \(y_{\text{total}}(t)\) in the steady state. The motivation to add two extra terms to \(y_{\text{total}}(t)\) is to ensure the controllability of the input-output dynamics.
The regulation of power consumption of the TCL population is achieved by moving the mass of the temperature distribution, and the control signal is chosen to be the set-point temperature variation rate \(\dot{x}_{sp}\), which may induce a change in the probability flux [2, 37]. As we consider a control scheme with a fixed deadband width, denoted by \(\delta_{0}\), we have \(\overline{x}=x_{sp}-\frac{\delta_{0}}{2},\ \underline{x}=x_{sp}+\frac{\delta_{0}}{2}\). Thus, the actual control signal is given by \(u(t):=\dot{x}_{sp}=\dot{\underline{x}}=\overline{\dot{x}}\).
Let \(y_{d}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}\) be the desired power profile, which is sufficiently smooth, and define the power tracking error as
\[e(t):=y(t)-y_{d}(t).\]
In what follows, we introduce a nonlinear control law and derive the corresponding tracking error dynamics.
**Theorem 4.1**: _Consider the system given in (4) and (9) under the boundary conditions in (5) (or equivalently) (8). Let the control input be defined as_
\[u(t):= \frac{k|e(t)|^{\gamma}\mathrm{sgn}(e(t))+\Phi(t)}{2\left(f_{1}( \overline{x},t)+f_{0}(\underline{x},t)\right)}, \tag{10}\]
_where \(k\in\mathbb{R}_{>0}\) and \(\gamma\in(0,1)\) are constants, \(\mathrm{sgn}(e)\) is the sign function defined by_
\[\mathrm{sgn}(e):=\begin{cases}-1,&e<0,\\ 0,&e=0,\\ 1,&e>0,\end{cases}\]
_and_
\[\Phi(t):= -\frac{\eta}{P}\dot{y}_{d}(t). \tag{11}\]
_Then, the power tracking error dynamics are given by_
\[\dot{e}(t)= -\frac{P}{\eta}k|e(t)|^{\gamma}\mathrm{sgn}(e(t))+\Gamma(t), \tag{12}\]
_where_
\[\Gamma(t):= \frac{P}{\eta}\left(\alpha_{1}(\overline{x},t)f_{1}(\overline{x}, t)+\alpha_{0}(\underline{x},t)f_{0}(\underline{x},t)\right)-\frac{\sigma^{2}P}{2 \eta}\left(\partial_{x}f_{1}(\underline{x}^{+},t)+\partial_{x}f_{1}( \overline{x}^{+},t)\right)\] \[-\frac{\sigma^{2}P}{2\eta}\left(\partial_{x}f_{0}(\underline{x}^ {-},t)+\partial_{x}f_{0}(\overline{x}^{-},t)\right)+\frac{P}{\eta}\int_{ \underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x. \tag{13}\]
**Remark 4.1**: \(\Gamma(t)\) _defined in (13) captures the terms depending on the diffusion coefficient or requiring instantaneous state measurements and will be treated as a disturbance thereafter. Moreover, the control law given in (10) involves only the measurement of the states (probability distributions \(f_{0}\) and \(f_{1}\)) on the end-points of the deadband (\(\underline{x}\) and \(\overline{x}\)), which results in a control scheme with significantly reduced communication burden compared to control schemes that require full-state measurements._
**Proof of Theorem 4.1.** Note that
\[\dot{e}(t)= \dot{y}(t)-\dot{y}_{d}(t)\] \[= \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{P}{\eta}\int_{\underline {x}(t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x+\frac{P}{\eta}\int_{\overline{x}(t)}^{x _{H}}f_{1}(x,t)\,\mathrm{d}x-\frac{P}{\eta}\int_{x_{L}}^{\underline{x}(t)}f_{0 }(x,t)\,\mathrm{d}x\right)-\dot{y}_{d}(t)\] \[= \frac{P}{\eta}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x} (t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x+\frac{P}{\eta}\frac{\mathrm{d}}{\mathrm{d }t}\int_{\overline{x}(t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x-\frac{P}{\eta}\frac{ \mathrm{d}}{\mathrm{d}t}\int_{x_{L}}^{\underline{x}(t)}f_{0}(x,t)\,\mathrm{d} x-\dot{y}_{d}(t).\]
Hence, we decompose the whole computation process into three steps.
_Step 1:_ Compute \(\frac{\mathrm{d}}{\mathrm{d}t}\int_{\overline{x}(t)}^{x_{H}}f_{1}(x,t)\, \mathrm{d}x\). It follows immediately from Leibniz's integral rule and (4d) that
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{\overline{x}(t)}^{x_{H}}f_{1} (x,t)\,\mathrm{d}x\] \[= 0-\frac{x}{x}(t)f_{1}(\overline{x},t)+\int_{\overline{x}(t)}^{x _{H}}\partial_{t}f_{1}(x,t)\,\mathrm{d}x\] \[= -u(t)f_{1}(\overline{x},t)+\int_{\overline{x}(t)}^{x_{H}} \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{1}(x,t)-(\alpha_{1}(x, t)-u(t))f_{1}(x,t)\bigg{)}\,\mathrm{d}x\] \[= -u(t)f_{1}(\overline{x},t)+\left(\frac{\sigma^{2}}{2}\partial_{x }f_{1}(x_{H}^{-},t)-(\alpha_{1}(x_{H}^{-},t)-u(t))f_{1}(x_{H}^{-},t)\right)- \left(\frac{\sigma^{2}}{2}\partial_{x}f_{1}(\overline{x}^{+},t)-(\alpha_{1}( \overline{x},t)-u(t))f_{1}(\overline{x},t)\right).\]
Using boundary condition (8h), it follows that
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{\overline{x}(t)}^{x_{H}}f_{1} (x,t)\,\mathrm{d}x= -2u(t)f_{1}(\overline{x},t)-\frac{\sigma^{2}}{2}\partial_{x}f_{1 }(\overline{x}^{+},t)+\alpha_{1}(\overline{x},t)f_{1}(\overline{x},t). \tag{14}\]
_Step 2:_ Compute \(\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{x_{H}}f_{1}(x,t)\, \mathrm{d}x\). Since
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{x_{H}}f_{1} (x,t)\,\mathrm{d}x= \frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{\overline{x }(t)}f_{1}(x,t)\,\mathrm{d}x+\frac{\mathrm{d}}{\mathrm{d}t}\int_{\overline{x }(t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x,\]
and \(\frac{\mathrm{d}}{\mathrm{d}t}\int_{\overline{x}}^{x_{H}}f_{1}(x,t)\, \mathrm{d}x\) is given by (14), we only need to compute \(\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{\overline{x}(t)}f_{1} (x,t)\,\mathrm{d}x\). It follows from (4c) and (8e) that
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{\overline{x }(t)}f_{1}(x,t)\,\mathrm{d}x= u(t)f_{1}(\overline{x},t)+\int_{\underline{x}(t)}^{\overline{x }(t)}g(f_{0},f_{1})\,\mathrm{d}x+\int_{\underline{x}(t)}^{\overline{x}(t)} \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{1}(x,t)-(\alpha_{1}(x, t)-u(t))f_{1}(x,t)\bigg{)}\,\mathrm{d}x\] \[= u(t)f_{1}(\overline{x},t)+\int_{\underline{x}(t)}^{\overline{x}(t )}g(f_{0},f_{1})\,\mathrm{d}x+\left(\frac{\sigma^{2}}{2}\partial_{x}f_{1}( \overline{x}^{-},t)-(\alpha_{1}(\overline{x}^{-},t)-u(t))f_{1}(\overline{x}^{ -},t)\right)\] \[-\left(\frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t) -(\alpha_{1}(\underline{x}^{+},t)-u(t))f_{1}(\underline{x}^{+},t)\right)\]
\[= u(t)f_{1}(\overline{x},t)+\frac{\sigma^{2}}{2}\partial_{x}f_{1}( \overline{x}^{-},t)-(\alpha_{1}(\overline{x},t)-u(t))f_{1}(\overline{x},t)- \frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t)\] \[+\int_{\underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{ d}x\] \[= 2u(t)f_{1}(\overline{x},t)+\frac{\sigma^{2}}{2}\partial_{x}f_{1} (\overline{x}^{-},t)-\alpha_{1}(\overline{x},t)f_{1}(\overline{x},t)-\frac{ \sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t)+\int_{\underline{x}(t)}^ {\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x. \tag{15}\]
Combining (14) and (15) we obtain by (8h)
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{\underline{x}(t)}^{x_{H}}f_{1}(x,t)\, \mathrm{d}x= -\frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t)- \frac{\sigma^{2}}{2}\partial_{x}f_{0}(\overline{x}^{-},t)+\int_{\underline{x} (t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x. \tag{16}\]
_Step 3:_ Compute \(\frac{\mathrm{d}}{\mathrm{d}t}\int_{x_{L}}^{\underline{x}(t)}f_{0}(x,t)\, \mathrm{d}x\). According to (4a) and (8a), we have
\[\frac{\mathrm{d}}{\mathrm{d}t}\int_{x_{L}}^{\overline{x}(t)}f_{0} (x,t)\,\mathrm{d}x= u(t)f_{0}(\underline{x},t)+\int_{x_{L}}^{\underline{x}(t)} \partial_{t}f_{0}(x,t)\,\mathrm{d}x\] \[= u(t)f_{0}(\underline{x},t)+\int_{x_{L}}^{\underline{x}(t)} \partial_{x}\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{0}-(\alpha_{0}-u)f_{0} \bigg{)}\,\mathrm{d}x\] \[= u(t)f_{0}(\underline{x},t)+\bigg{(}\frac{\sigma^{2}}{2}\partial_ {x}f_{0}(\underline{x}^{-},t)-(\alpha_{0}(\underline{x},t)-u(t))f_{0}( \underline{x},t)\bigg{)}\] \[-\bigg{(}\frac{\sigma^{2}}{2}\partial_{x}f_{0}(x_{L}^{+},t)-( \alpha_{0}(x_{L}^{+},t)-u(t))f_{0}(x_{L}^{+},t)\bigg{)}\] \[= 2u(t)f_{0}(\underline{x},t)+\frac{\sigma^{2}}{2}\partial_{x}f_{0 }(\underline{x}^{-},t)-\alpha_{0}(\underline{x},t)f_{0}(\underline{x},t). \tag{17}\]
Finally, by combining (14), (16), and (17), we obtain:
\[\dot{e}(t)= \frac{P}{\eta}\left(-\frac{\sigma^{2}}{2}\partial_{x}f_{1}( \underline{x}^{+},t)-\frac{\sigma^{2}}{2}\partial_{x}f_{0}(\overline{x}^{-},t )\right)+\frac{P}{\eta}\left(-2u(t)f_{1}(\overline{x},t)-\frac{\sigma^{2}}{2} \partial_{x}f_{1}(\overline{x}^{+},t)+\alpha_{1}(\overline{x},t)f_{1}( \overline{x},t)\right)\] \[-\frac{P}{\eta}\left(2u(t)f_{0}(\underline{x},t)+\frac{\sigma^{2 }}{2}\partial_{x}f_{0}(\underline{x}^{-},t)-\alpha_{0}(\underline{x},t)f_{0}( \underline{x},t)\right)-\dot{y_{d}}(t)+\frac{P}{\eta}\int_{\underline{x}(t)}^ {\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x\] \[= -\frac{2P}{\eta}u(t)\left(f_{1}(\overline{x},t)+f_{0}(\underline {x},t)\right)-\frac{\sigma^{2}P}{2\eta}\partial_{x}f_{1}(\underline{x}^{+},t )-\frac{\sigma^{2}P}{2\eta}\partial_{x}f_{1}(\overline{x}^{+},t)-\frac{\sigma^ {2}P}{2\eta}\partial_{x}f_{0}(\underline{x}^{-},t)\] \[-\frac{\sigma^{2}P}{2\eta}\partial_{x}f_{0}(\overline{x}^{-},t)+ \frac{P}{\eta}\int_{\underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{ d}x+\frac{P}{\eta}\alpha_{1}(\overline{x},t)f_{1}(\overline{x},t)+\frac{P}{\eta} \alpha_{0}(\underline{x},t)f_{0}(\underline{x},t)-\dot{y_{d}}(t).\]
The error dynamics can then be expressed as
\[\dot{e}(t)=-\frac{2P}{\eta}u(t)\left(f_{1}(\overline{x},t)+f_{0}(\underline{x}, t)\right)+\frac{P}{\eta}\Phi(t)+\Gamma(t).\]
Let
\[u(t):=\frac{v(t)+\Phi(t)}{2\left(f_{1}(\overline{x},t)+f_{0}( \underline{x},t)\right)},\]
where \(v(t)\) is an auxiliary control input, then
\[\dot{e}(t)=-\frac{P}{\eta}v(t)+\Gamma(t). \tag{18}\]
Considering an auxiliary control of the form:
\[v(t):=k|e(t)|^{\gamma}{\rm sgn}(e(t)), \tag{19}\]
the tracking error dynamics in the closed loop are then given by (12).
**Remark 4.2**: _Note that for the given initial data (see (F1)), it can be shown that the term \(f_{1}(\overline{x},t)+f_{0}(\underline{x},t)\) is strictly positive (see Theorem 4.4 (iii) in Section 4.2). Therefore, the control signal \(u\), given in (10) is well-defined. In addition, \(u\) is continuous due to the fact that \(\gamma\in(0,1)\) and the assumptions on the continuity of \(\dot{y}_{d}(t)\) and \(f_{1}(\overline{x},t)+f_{0}(\underline{x},t)\) (see (F2) and (F3)). It is also worth noting that, as \(f_{1}(\overline{x},t)\) and \(f_{0}(\underline{x},t)\) describe the probability density of TCLs in the ON and OFF states at the prescribed upper and lower temperature boundaries \(\overline{x}\) and \(\underline{x}\), respectively, it is impossible in practice that \(f_{1}(\overline{x},t)+f_{0}(\underline{x},t)\to 0\) as \(t\to+\infty\)._
### Finite-time input-to-state stability of the tracking error dynamics
In this section, we assess the robust stability of the tracking error dynamics in the sense of FTISS, with \(\Gamma\) as the input (disturbance). One of the main properties of the closed-loop system is stated below.
**Theorem 4.2**: _The power tracking error dynamics (12) under the control law given in (10) are FTISS w.r.t. \(\Gamma(t)\) for any \(\gamma\in(0,1)\)._
**Proof.** Consider a Lyapunov candidate of the form \(V(e)=\frac{1}{2}c^{2}\). The time derivative of \(V\) along the trajectory of the tracking error dynamics (12) is given by:
\[\dot{V}=e\dot{e}=e\left(-\frac{P}{\eta}k|e|^{\gamma}{\rm sgn}(e)+\Gamma\right) =-\frac{P}{\eta}k|e|^{1+\gamma}+e\Gamma=-\frac{P}{\eta}k\left(\sqrt{2V}\right) ^{1+\gamma}+e\Gamma=-\frac{P}{\eta}k\left(2V\right)^{\frac{1+\gamma}{2}}+e\Gamma,\]
which implies that
\[DV(e)\cdot f(e,\Gamma)\leq-\frac{P}{\eta}k(2V)^{\frac{1+\gamma}{2}}+|e||\Gamma| \tag{20}\]
with \(f(e,\Gamma):=-\frac{P}{\eta}k|e(t)|^{\gamma}{\rm sgn}(e(t))+\Gamma(t)\).
Let \(C_{0}\in(0,k)\) be a constant. Then, for any \(|e|\geq\left(\frac{\eta}{PC_{0}}|\Gamma|\right)^{\frac{1}{\gamma}}\), i.e., \(|\Gamma|\leq\frac{P}{\eta}C_{0}|e|^{\gamma}\), we deduce by (20) that
\[DV(e)\cdot f(e,\Gamma)\leq-\frac{P}{\eta}k(2V)^{\frac{1+\gamma}{2}}+\frac{P} {\eta}C_{0}|e|^{1+\gamma}=-\frac{P}{\eta}k(2V)^{\frac{1+\gamma}{2}}+\frac{P}{ \eta}C_{0}(2V)^{\frac{1+\gamma}{2}}=-\frac{P}{\eta}(k-C_{0})2^{\frac{1+\gamma }{2}}V^{\frac{1+\gamma}{2}}.\]
Note that \(\frac{P}{\eta}(k-C_{0})2^{\frac{1+\gamma}{2}}>0\), \(\frac{1+\gamma}{2}\in(\frac{1}{2},1)\), and that \(\chi(s):=(\frac{\eta}{PC_{0}}s)^{\frac{1}{\gamma}}\) is a \(\mathcal{K}\)-function w.r.t. \(s\in\mathbb{R}_{\geq 0}\). The FTISS of system (12) is then guaranteed by Lemma 2.1.
### Properties of the governing PDEs
In practice, we can assume that the number of TCLs in a population remains unchanged within a specific DR control period. Therefore, the mass conservation property of the solutions to the system (4)-(6) should be verified under the imposed boundary conditions, thereby conforming the compliance of the mathematical model with the imposed condition. Moreover, non-negativeness of the solutions is also required.
**Theorem 4.3** (Mass conservation property): _The solution to the initial-boundary value problem (IBVP) (4)-(6) is conservative in the sense that_
\[\int_{x_{L}}^{\overline{x}(t)}f_{0}(x,t)\,\mathrm{d}x+\int_{\underline{x}(t)}^{x _{H}}f_{1}(x,t)\,\mathrm{d}x=1\;\;\forall t\in\mathbb{R}_{\geq 0}, \tag{21}\]
_provided that_
\[\int_{x_{L}}^{\underline{x}(0)}f_{0}^{a0}(x)\,\mathrm{d}x+\int_{\underline{x}(0 )}^{\overline{x}(0)}f_{0}^{b0}(x)\,\mathrm{d}x+\int_{\underline{x}(0)}^{ \overline{x}(0)}f_{1}^{b0}(x)\,\mathrm{d}x+\int_{\overline{x}(0)}^{x_{H}}f_{1 }^{c0}(x)\,\mathrm{d}x=1. \tag{22}\]
**Proof.** Using (4a), (4b), (8a), (8b), and (8d), and noting (U) and (F2), we have
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{x_{L}}^{\overline{x}(t) }f_{0}(x,t)\,\mathrm{d}x\right)\] \[= \frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{x_{L}}^{\underline{x}( t)}f_{0}(x,t)\,\mathrm{d}x+\int_{\underline{x}(t)}^{\overline{x}(t)}f_{0}(x,t)\, \mathrm{d}x\right)\] \[= \int_{x_{L}}^{\underline{x}(t)}\partial_{t}f_{0}(x,t)\,\mathrm{d }x+f_{0}(\underline{x}(t),t)\underline{\dot{x}}(t)+\int_{\underline{x}(t)}^{ \overline{x}(t)}\partial_{t}f_{0}(x,t)\,\mathrm{d}x+f_{0}(\overline{x}(t),t) \dot{\overline{x}}(t)-f_{0}(\underline{x}(t),t)\dot{\underline{x}}(t)\] \[= \int_{x_{L}}^{\underline{x}(t)}\partial_{x}\Big{(}\frac{\sigma^{ 2}}{2}\partial_{x}f_{0}(x,t)-(\alpha_{0}(x,t)-u(t))f_{0}(x,t)\Big{)}\,\mathrm{ d}x+\int_{\underline{x}(t)}^{\overline{x}(t)}\partial_{x}\Big{(}\frac{ \sigma^{2}}{2}\partial_{x}f_{0}(x,t)-(\alpha_{0}(x,t)-u(t))f_{0}(x,t)\Big{)}\, \mathrm{d}x\] \[-\int_{\underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{ d}x\] \[= \frac{\sigma^{2}}{2}\partial_{x}f_{0}(\underline{x}^{-}(t),t)-( \alpha_{0}(\underline{x}(t))-u(t))f_{0}(\underline{x}(t),t)-0+\frac{\sigma^{ 2}}{2}\partial_{x}f_{0}(\overline{x}^{-}(t),t)-(\alpha_{0}(\overline{x}(t))-u (t))f_{0}(\overline{x}(t),t)\] \[-\left(\frac{\sigma^{2}}{2}\partial_{x}f_{0}(\underline{x}^{+}(t ),t)-(\alpha_{0}(\underline{x}(t))-u(t))f_{0}(\underline{x}(t),t)\right)-\int _{\underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x\] \[= \frac{\sigma^{2}}{2}(\partial_{x}f_{0}(\underline{x}^{-}(t),t)- \partial_{x}f_{0}(\underline{x}^{+}(t),t))+\frac{\sigma^{2}}{2}\partial_{x}f_ {0}(\overline{x}^{-}(t),t)-\int_{\underline{x}(t)}^{\overline{x}(t)}g(f_{0},f _{1})\,\mathrm{d}x\] \[= \frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t)+\frac {\sigma^{2}}{2}\partial_{x}f_{0}(\overline{x}^{-}(t),t)-\int_{\underline{x}(t) }^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x. \tag{23}\]
Similarly, we infer from (4c), (4d), (8e), (8g), (8h), (U) and (F3) that
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\underline{x}(t)}^{x_{H }}f_{1}(x,t)\,\mathrm{d}x\right)= \frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\underline{x}(t)}^{ \overline{x}(t)}f_{1}(x,t)\,\mathrm{d}x+\int_{\overline{x}(t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x\right)\] \[= -\frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+},t)- \frac{\sigma^{2}}{2}\partial_{x}f_{0}(\overline{x}^{-}(t),t)+\int_{ \underline{x}(t)}^{\overline{x}(t)}g(f_{0},f_{1})\,\mathrm{d}x. \tag{24}\]
By (23) and (24), we obtain
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{x_{L}}^{\overline{x}(t)}f_{0}(x,t) \,\mathrm{d}x+\int_{\underline{x}(t)}^{x_{H}}f_{1}(x,t)\,\mathrm{d}x\right)=0 \;\;\forall t\in\mathbb{R}_{\geq 0},\]
which along with (22) implies (21).
**Theorem 4.4** (Non-negativeness): _The following statements hold true for the solution to IBVP (4)-(6):_
1. \(f_{0}(x,t)\geq 0\) _for all_ \(x\in[x_{L},\overline{x}(t)]\) _and all_ \(t\in\mathbb{R}_{\geq 0}\)_:_
2. \(f_{1}(x,t)\geq 0\) _for all_ \(x\in[\underline{x}(t),x_{H}]\) _and all_ \(t\in\mathbb{R}_{\geq 0}\)_;_
3. \(f_{0}(\underline{x}(t),t)+f_{1}(\overline{x}(t),t)>0\) _for all_ \(t\in\mathbb{R}_{>0}\)_._
The proof of this theorem is provided in Appendix.
## 5 Experimental Validation
In this section, we present simulation results to demonstrate the effectiveness of the proposed control scheme. Note that the control law given in (10) is derived from the coupled Fokker-Planck equations, which assume a population of an infinite number of TCLs. As the number of TCLs in a real-world TCL population is always finite, and considering the fact that the larger the population size, the more accurate the PDE model, we present a comparative study of two heterogeneous populations with 1,000 and 100,000 TCLs.
### Simulation setup
A numerical simulation is conducted to validate the proposed control scheme and evaluate its performance. Table 1 lists the physical parameters of the AC units utilized in the simulation, which are the same as those in [2]. The thermal resistances and thermal capacitances are random variables following a log-normal distribution with average mean values of \(2\ ^{\circ}\)C/kW and \(10\) kWh/\({}^{\circ}\)C, respectively. The level of heterogeneity is parameterized by the standard deviation \(\sigma_{p}\). In our experiment, the initial temperatures of the AC units are uniformly distributed around the initial set-point \(x_{sp}^{0}=20\ ^{\circ}\)C over the deadband, and initially \(40\%\) of the AC units are set randomly in "ON"-state. This setting causes the population to begin running from an almost steady state.
The disturbances brought into the system come mainly from the following three sources. First, all AC units operate under the same varying outside temperature, as depicted in Fig. 3, which rises from \(30^{\circ}\)C at 11:30 to \(23^{\circ}\)C at 12:30 and then drops back from 14:30 to 15:30. Second, a forced random switch mechanism is added to desynchronize AC operations. The number of forced interrupts per hour can be adjusted through the hyper-parameter \(p_{f}\). Moreover, a safe border distance of \(5\%\) of the deadband width is incorporated to prevent forced switches from happening when an AC is around \(\overline{x}(t)\) and in "ON" state or around \(\underline{x}(t)\) and in "OFF" state. Finally, because frequent switching leads to reduced energy efficiency and more rapid compressor wear out, a lockout time, \(t_{\mathrm{lock}}\), is included for each AC. Thus, an AC unit remains inactive to the control signals when it is locked.
The reference power is a predefined curve, as shown in Fig. 4. From 10:30 to 11:30, the normalized desired power is maintained constant at \(0.4\). From 11:30 to 12:00, the reference power drops to \(0.2\) and keeps constant for the following two and a half
\begin{table}
\begin{tabular}{|l|l|c|} \hline
**Parameter** & **Description (Unit)** & **Value** \\ \hline \(R\) & average thermal resistance (\({}^{\circ}\)C/kW) & 2 \\ \(C\) & average thermal capacitance (kWh/\({}^{\circ}\)C) & 10 \\ \(P\) & electric power (kW) & 14 \\ \(\eta\) & load efficiency & 2.5 \\ \(x_{sp}^{0}\) & initial temperature set-point (\({}^{\circ}\)C) & 20 \\ \(\delta\) & temperature deadband width (\({}^{\circ}\)C) & 0.5 \\ \(\sigma_{p}\) & standard deviation of lognormal distributions & 0.2 \\ \hline \(p_{f}\) & forced switch probability per hour (\%) & 3 \\ \(t_{cl}\) & control interval (second) & 30 \\ \(t_{\mathrm{lock}}\) & locked time of each TCL (minute) & 6 \\ \hline \end{tabular}
\end{table}
Table 1: Simulation parameter
hours. From 14:30, the desired power rises to \(0.5\) in 30 minutes and remains constant until 16:30. During the rising and dropping phases, the desired power is specified by a smooth polynomial with the endpoint constraints given below:
\[y_{d}(t)=\left(y_{d}(t_{f})-y_{d}(t_{i})\right)\tau^{5}(t)\sum_{l=0}^{4}a_{l} \tau^{l}(t),t\in[t_{i},t_{f}], \tag{25}\]
\[\dot{y}_{d}(t_{i})=\dot{y}_{d}(t_{f})=\ddot{y}_{d}(t_{i})=\ddot{y}_{d}(t_{i})= \dddot{y}_{d}(t_{i})=\dddot{y}_{d}(t_{f})=0, \tag{26}\]
where \(\tau(t)=(t-t_{i})/(t_{f}-t_{i})\). By a direct computation, the coefficients can be determined as follows:
\(a_{0}=126\), \(a_{1}=420\), \(a_{2}=540\), \(a_{3}=315\), and \(a_{4}=70\).
In the simulation, the control signal is updated every 30 seconds (\(t_{ci}\) in Table 1). The control signal that every AC receives is the set-point variation rate. Each AC computes then its set-point temperature offset for the next control interval starting from \(t_{k}\). To compute the denominator of the controller given in (10), a mid-point rectangular method with a temperature bin width \(\delta_{x}\) is used to estimate \(f_{1}(\overline{x}(t_{k}),t_{k})\) and \(f_{0}(\underline{x}(t_{k}),t_{k})\). The percentage of ACs falling in the rectangular region is used as \(f_{1}(\overline{x}(t_{k}),t_{k})\times\delta_{x}\) or \(f_{0}(\underline{x}(t_{k}),t_{k})\times\delta_{x}\). In general, \(\delta_{x}\) should not be too large because the underlying system has complex nonlinear dynamics. On the other hand, considering the limited number of ACs involved in the simulation, the bin width \(\delta_{x}\) should not be too small, which may introduce larger biases. In our implementation, histogram bin widths of \(0.008\)\({}^{\circ}\)C, \(0.004\)\({}^{\circ}\)C, and \(0.002\)\({}^{\circ}\)C are used, which are reasonable and provide reliable estimations of \(f_{1}(\overline{x}(t_{k}),t_{k})\) and \(f_{0}(\underline{x}(t_{k}),t_{k})\).
### Simulation results
First, we present the test results for the population with 1,000 TCLs. The control cycle lasts for 6 hours, from 10:30 to 16:30. The test is performed continuously for \(10\) episodes, and the tracking performance is measured by the root mean square error
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|} \hline
**Episode** & 1 & 2 & 3 & 4 & 5 \\ \hline
**RMSE** (\(\%\)) & 0.948 & 0.923 & 0.844 & 0.834 & 0.935 \\ \hline
**Episode** & 6 & 7 & 8 & 9 & 10 \\ \hline
**RMSE** (\(\%\)) & 0.880 & 0.923 & 0.890 & 0.925 & 0.861 \\ \hline \end{tabular}
\end{table}
Table 2: Tracking performance of 10 episodes for the population with 1,000 TCLs
Figure 4: Desired power profile.
Figure 3: Ambient temperature.
(RMSE), as reported in Table 2. In the test, the controller parameters in (10) are set to be \(k=8\) and \(\gamma=0.5\), respectively. The final result shows that the mean RMSE for this setting is \(0.896\%\), and the standard deviation (STD) of the dRMSEs is \(0.040\%\).
Fig. 5 shows a sample of the control results corresponding to the episode with an RMSE of \(0.948\%\). It can be seen from Fig. 4(a) that the proposed control strategy is effective. The temperature evolution of \(10\) randomly selected ACs in the population is presented in Fig. 4(b). It can be observed that all of them, unless forced switches occur, operate smoothly inside the deadband between the turning on and turning off points. Fig. 4(c) shows the control signal generated during this episode. During the first 30 minutes (from 10:00 to 10:30), the controller is inactive, and the system operates in an open-loop mode.
When the number of ACs increases, the model of the coupled Fokker-Planck equations becomes more accurate. To evaluate the effectiveness of the proposed control strategy, tracking control performance is examined for a population of 100,000 ACs. The RMSE values for 10 continuous tests are shown in Table 1, which gives a mean RMSE of \(0.497\%\) and an STD of \(0.004\%\). In this test, \(k=15\) and \(\gamma=0.5\) are used. Fig. 6 illustrates one of the control samples corresponding to the episode with an RMSE of \(0.505\%\). The normalized power consumption is shown in Fig. 5(a), and the temperature evolutions of \(10\) ACs are shown in Fig. 5(b). The control signal is shown in Fig. 5(c).
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|} \hline
**Episode** & 1 & 2 & 3 & 4 & 5 \\ \hline
**RMSE** (\(\%\)) & 0.505 & 0.500 & 0.496 & 0.491 & 0.490 \\ \hline
**Episode** & 6 & 7 & 8 & 9 & 10 \\ \hline
**RMSE** (\(\%\)) & 0.497 & 0.499 & 0.495 & 0.498 & 0.500 \\ \hline \end{tabular}
\end{table}
Table 1: Tracking performance of 10 episodes for the population with 100,000 TCLs
Figure 5: Control performance for a population of 1,000 TCLs
The results of the comparative study show clearly that the tracking control system performs better for the population of larger size with smaller RMSE, smoother power trajectory, and less "noisy" control signals. This is consistent with the nature of the PDE model on which the proposed control scheme is based. Nevertheless, the performance is not significantly degraded for a population with a significantly smaller size. This demonstrates the robustness and potential applicability of the developed control strategy to practical systems.
## 6 Conclusion
In this work, we have developed a strategy for the power tracking control of heterogeneous TCL populations. The control scheme can ensure a robust performance in the presence of modeling uncertainties in the sense of FTISS and requires measuring the states of the system only on the end-points of the deadband. The simulation results provided encouraging evidence that the proposed control approach is highly effective. From a practical application viewpoint, we can consider in our future work other types of devices, such as battery charging systems, and other demand-response tasks, such as frequency regulation or transaction control [10, 13, 24]. Control of populations of TCLs described by the second-order equivalent thermal parameters model [3, 15] may also be a research direction worthy of exploration.
## Appendix: Proof of Theorem 5
We first prove statement (i). Given any \(T>0\), it suffices to show that \(f_{0}\geq 0\) over \([x_{L},\overline{x}(t)]\times[0,T]\) for all \(t\in[0,T]\).
Fig. 6: Control performance for a population of 100,000 TCLs
Indeed, the transformations of variable \(y:=\frac{x-x_{L}}{\overline{x}-x_{L}}:=\frac{x-x_{L}}{h}\) and \(f_{0}(x,t)=f_{0}(yh+x_{L},t):=\tilde{f}_{0}(y,t)\) yield
\[\partial_{x}f_{0} =\frac{1}{h}\partial_{y}\tilde{f}_{0},\] \[\partial_{xx}f_{0} =\frac{1}{h^{2}}\partial_{yy}\tilde{f}_{0},\partial_{t}f_{0}= \partial_{t}\tilde{f}_{0}+\partial_{y}\tilde{f}_{0}\frac{\partial y}{\partial t }=\partial_{t}\tilde{f}_{0}-(x-x_{L})\frac{\dot{\overline{x}}}{h^{2}}\partial_ {y}\tilde{f}_{0}=\partial_{t}\tilde{f}_{0}-\frac{1}{h}yu\partial_{y}\tilde{f}_ {0}.\]
Note that
\[x\in[x_{L},\underline{x}]\Leftrightarrow y\in[0,1],\] \[0<\delta_{0}\leq h(t)\leq x_{H}-x_{L},\forall t\in[0,T].\]
The PDEs (4a) and (4b) are equivalent to
\[\partial_{t}\tilde{f}_{0}-\frac{1}{h}\left(\frac{\sigma^{2}}{2h }\partial_{yy}\tilde{f}_{0}+\left((1+y)u-\tilde{\alpha}_{0}\right)\partial_{ y}\tilde{f}_{0}-\tilde{\alpha}_{0y}\tilde{f}_{0}\right)= 0,\ \ \forall y\in\left(0,z(t)\right),\forall t\in(0,T], \tag{27a}\] \[\partial_{t}\tilde{f}_{0}-g(\tilde{f}_{0},\tilde{f}_{1})-\frac{1 }{h}\left(\frac{\sigma^{2}}{2h}\partial_{yy}\tilde{f}_{0}+\left((1+y)u-\tilde {\alpha}_{0}\right)\partial_{y}\tilde{f}_{0}-\tilde{\alpha}_{0y}\tilde{f}_{0} \right)= 0,\ \ \forall y\in\left(z(t),1\right),\forall t\in(0,T], \tag{27b}\]
respectively, where \(\tilde{\alpha}_{0}(y,t):=\alpha_{0}(yh(t)+x_{L},t)\), \(f_{1}(x,t)=f_{1}(yh+x_{L},t):=\tilde{f}_{1}(y,t)\), and \(z(t):=1-\frac{\delta_{0}}{h(t)}\).
Note that (8) is equivalent to (5), and (8a), (8b), and (8d) become
\[\frac{\sigma^{2}}{2}\partial_{y}\tilde{f}_{0}(0^{+},t)-(\tilde{ \alpha}_{0}(0^{+},t)-u(t))h(t)\tilde{f}_{0}(0^{+},t)= 0,\ \forall t\in(0,T], \tag{28a}\] \[\partial_{y}\tilde{f}_{0}\left(z^{-}(t),t\right)-\partial_{y} \tilde{f}_{0}\left(z^{+}(t),t\right)= \sigma_{0}(t),\ \forall t\in(0,T],\] (28b) \[\tilde{f}_{0}(1^{-},t)= 0,\ \forall t\in(0,T], \tag{28c}\]
where, for the given solution \(f_{1}\), \(\sigma_{0}(t):=\frac{\sigma^{2}}{2}\partial_{x}f_{1}(\underline{x}^{+}(t),t)\) is a well-defined function w.r.t. \(t\), and \(\sigma_{0}(t)>0\) for all \(t\in[0,T]\) owing to (F3) and (8i).
The initial data of \(\tilde{f}_{0}\) over the domain \([0,z(t)]\) and \([z(t),1]\) are given by
\[\tilde{f}_{0}^{a0}(y):=f_{0}^{a0}(yh(0)+x_{L})\geq 0,\]
and
\[\tilde{f}_{0}^{b0}(y):=f_{0}^{b0}(yh(0)+x_{L})\geq 0,\]
respectively.
Let \(\phi(y):=\mathrm{e}^{m(y-\frac{1}{2})^{2}}\) and \(\tilde{f}_{0}:=\phi\,\mathrm{e}^{\gamma t}\,\hat{f}_{0}\) with \(m>0\) and \(\gamma>0\) being constants that will be chosen later. Then (27) and (28) lead to
\[\partial_{t}\tilde{f}_{0}-\frac{\sigma^{2}}{2h^{2}}\partial_{yy} \hat{f}_{0}+\mathcal{B}(y,t)\partial_{y}\hat{f}_{0}+\mathcal{C}(y,t)\hat{f}_{0}= 0, \forall y\in\left(0,z(t)\right),\forall t\in(0,T], \tag{29a}\] \[\partial_{t}\hat{f}_{0}-\frac{\sigma^{2}}{2h^{2}}\partial_{yy} \hat{f}_{0}+\mathcal{B}(y,t)\partial_{y}\hat{f}_{0}+\mathcal{C}(y,t)\hat{f}_{0}+ \frac{e^{-\gamma t}}{\phi(y)}g(\tilde{f}_{0},\tilde{f}_{1})= 0, \forall y\in\left(z(t),1\right),\forall t\in(0,T],\] (29b) \[\frac{\sigma^{2}}{2}\partial_{y}\hat{f}_{0}(0^{+},t)-k(t)\hat{f}_ {0}(0^{+},t)= 0, \forall t\in(0,T],\] (29c) \[\partial_{y}\hat{f}_{0}\left(z^{-}(t),t\right)-\partial_{y}\hat{f}_ {0}\left(z^{+}(t),t\right)= \hat{\sigma}_{0}(t),\forall t\in(0,T],\] (29d) \[\hat{f}_{0}(1^{-},t)= 0, \forall t\in(0,T], \tag{29e}\]
where
\[\mathcal{B}(y,t):= -\frac{1}{h}\left(\frac{\sigma^{2}}{2h}\frac{2\partial_{y}\phi}{ \phi}+(1+y)u-\tilde{\alpha}_{0}\right),\] \[\mathcal{C}(y,t):= \frac{1}{h}\left(\gamma-\frac{\sigma^{2}}{2h}\frac{\partial_{yy} \phi}{\phi}-\frac{\partial_{y}\phi}{\phi}\left((1+y)u-\tilde{\alpha}_{0} \right)+\tilde{\alpha}_{0y}\right),\] \[k(t):= \frac{m\sigma^{2}}{2}+(\tilde{\alpha}_{0}(0^{+},t)-u(t))h(t),\] \[\hat{\sigma}_{0}(t):= \frac{\mathrm{e}^{-\gamma t}}{\phi(1)}\sigma_{0}(t).\]
The initial data for the \(\hat{f}_{0}\)-system over the domain \([0,z(t)]\) and \([z(t),1]\) are given by
\[\hat{f}_{0}^{a0}(y):=\frac{\tilde{f}_{0}^{a0}(y)}{\phi(y)}\geq 0\;\;\text{and}\;\;\hat{f}_{0}^ {b0}(y):=\frac{\tilde{f}_{0}^{b0}(y)}{\phi(y)}\geq 0, \tag{30}\]
respectively.
Note that \(u,\tilde{\alpha}_{0}\), and \(\tilde{\alpha}_{0y}\) are continuous in \([0,1]\times[0,T]\). Letting first \(m\) and then \(\gamma\) be sufficiently large, there must be positive constants \(k_{0}\) and \(c_{0}\) such that
\[k(t)\geq k_{0},\forall t\in(0,T], \tag{31}\] \[\mathcal{C}(y,t)-1\geq c_{0},\forall(y,t)\in(0,1)\times(0,T]. \tag{32}\]
To prove the non-negativeness property of \(f_{0}\), it suffices to show that \(\hat{f}_{0}\geq 0\) in \([0,1]\times[0,T]\). We now proceed with the proof by contradiction. Assume that there exists a point \((y_{0},t_{0})\in[0,1]\times[0,T]\) such that
\[\hat{f}_{0}(y_{0},t_{0})=\min_{(y,t)\in[0,1]\times[0,T]}\hat{f}_{0}(y,t)<0.\]
Considering (29e) and (30), we have \(y_{0}\neq 1\) and \(t_{0}\in(0,T]\).
_Case 1_: \(y_{0}\in(0,z(t_{0}))\). At point \((y_{0},t_{0})\), it holds that
\[\partial_{t}\hat{f}_{0}(y_{0},t_{0})\leq 0,\partial_{y}\hat{f}_{0}(y_{0},t_{0})= 0,\partial_{yy}\hat{f}_{0}(y_{0},t_{0})\geq 0.\]
Then (29a) and (32) imply that
\[0>(c_{0}+1)\,\hat{f}_{0}(y_{0},t_{0})\geq\partial_{t}\hat{f}_{0}(y_{0},t_{0}) -\frac{\sigma^{2}}{2h^{2}(t_{0})}\partial_{yy}\hat{f}_{0}(y_{0},t_{0})+ \mathcal{B}(y_{0},t_{0})\partial_{y}\hat{f}_{0}(y_{0},t_{0})+\mathcal{C}(y_{0 },t_{0})\hat{f}_{0}(y_{0},t_{0})=0,\]
which leads to a contradiction.
_Case 2_: \(y_{0}\in(z(t_{0}),1)\). At the point \((y_{0},t_{0})\), it also holds that
\[\partial_{t}\hat{f}_{0}(y_{0},t_{0})\leq 0,\partial_{y}\hat{f}_{0}(y_{0},t_{0})= 0,\partial_{yy}\hat{f}_{0}(y_{0},t_{0})\geq 0.\]
In addition, using the Mean Value Theorem, (G1), and (G2), we obtain:
\[g(\tilde{f}_{0}(y_{0},t_{0}),\tilde{f}_{1}(y_{0},t_{0}))=g(0,\tilde{f}_{1}(y_{ 0},t_{0}))+\tilde{f}_{0}(y_{0},t_{0})g_{s}(s,\tilde{f}_{1}(y_{0},t_{0}))|_{s= \xi}\leq|\tilde{f}_{0}(y_{0},t_{0})|,\]
where \(\xi\) is between \(0\) and \(\tilde{f}_{0}(y_{0},t_{0})\).
It follows that
\[\frac{\mathrm{e}^{-\gamma t_{0}}}{\phi(y_{0})}g(\tilde{f}_{0}(y_{0},t_ {0}),\tilde{f}_{1}(y_{0},t_{0}))\leq |\tilde{f}_{0}(y_{0},t_{0})|\frac{\mathrm{e}^{-\gamma t_{0}}}{\phi(y_{0}) }=-\hat{f}_{0}(y_{0},t_{0}). \tag{33}\]
From (29b), (32), and (33), we obtain:
\[0> c_{0}\hat{f}_{0}(y_{0},t_{0})\] \[\geq \left(\mathcal{C}(y_{0},t_{0})-1\right)\hat{f}_{0}(y_{0},t_{0})\] \[\geq \mathcal{C}(y_{0},t_{0})\hat{f}_{0}(y_{0},t_{0})+\frac{\mathrm{e }^{-\gamma t_{0}}}{\phi(y_{0})}g(\tilde{f}_{0}(y_{0},t_{0}),\tilde{f}_{1}(y_{0 },t_{0}))\] \[\geq \partial_{t}\hat{f}_{0}(y_{0},t_{0})-\frac{\sigma^{2}}{2h^{2}(t_ {0})}\partial_{yy}\hat{f}_{0}(y_{0},t_{0})+\mathcal{B}(y_{0},t_{0})\partial_{ y}\hat{f}_{0}(y_{0},t_{0})+\mathcal{C}(y_{0},t_{0})\hat{f}_{0}(y_{0},t_{0})+ \frac{\mathrm{e}^{-\gamma t_{0}}}{\phi(y_{0})}g(\tilde{f}_{0}(y_{0},t_{0}), \tilde{f}_{1}(y_{0},t_{0}))\] \[= 0,\]
which leads to a contradiction.
_Case 3_: \(y_{0}=0\). It follows that \(\partial_{y}\hat{f}_{0}(0^{+},t_{0})\geq 0\), which, along with (29c) and (31), yields
\[0<-k_{0}\hat{f}_{0}(0^{+},t_{0})\leq-k(t_{0})\hat{f}_{0}(0^{+},t_{0})\leq \frac{\sigma^{2}}{2}\partial_{t}\hat{f}_{0}(0^{+},t)-k(t_{0})\hat{f}_{0}(0^{+},t)=0.\]
We get a contradiction.
_Case 4_: \(y_{0}=1\). It follows that \(\partial_{y}\hat{f}_{0}(1^{+},t_{0})\leq 0\), which along with (29c) and (31) yields
\[0<-k_{0}\hat{f}_{0}(0^{+},t_{0})\leq-k(t_{0})\hat{f}_{0}(0^{+},t_{0})\leq \frac{\sigma^{2}}{2}\partial_{y}\hat{f}_{0}(0^{+},t)-k(t_{0})\hat{f}_{0}(0^{+},t)=0.\]
We get a contradiction.
_Case 5_: \(y_{0}=z(t_{0})\). It follows that \(\partial_{y}\hat{f}_{0}(z^{-}(t_{0}),t_{0})\leq 0\) and \(\partial_{y}\hat{f}_{0}(z^{+}(t_{0}),t_{0})\geq 0\), which along with (29d) and \(\hat{\sigma}_{0}(t)>0\) yields
\[0\geq\partial_{y}\hat{f}_{0}(z^{-}(t_{0}),t_{0})-\partial_{y}\hat{f}_{0}(z^{+} (t_{0}),t_{0})=\hat{\sigma}_{0}(t_{0})>0,\]
leading to a contradiction.
Because we always obtain a contradiction in each case, we have shown that \(\hat{f}_{0}\geq 0\) over the domain \([0,1]\times[0,T]\), which implies the non-negativeness property of \(f_{0}\) over the domain \([x_{L},\overline{x}(t)]\times[0,T]\) for all \(t\in[0,T]\) and all \(T\in\mathbb{R}_{>0}\).
Because the proof of statement (ii) can proceed in the same way as above, we omit the details of the proof.
Finally, suppose that statement (iii) fails to be true; then, for any given \(T\in\mathbb{R}_{>0}\) there must be a \(t_{0}\in(0,T]\) such that
\[f_{0}(\underline{x}(t_{0}),t_{0})+f_{1}(\overline{x}(t_{0}),t_{0})=0,\]
which, along with the non-negativeness property of \(f_{0}\) and \(f_{1}\), implies that \(f_{0}\) and \(f_{1}\) attain their minima at \((\underline{x}(t_{0}),t_{0})\) and \((\overline{x}(t_{0}),t_{0})\), respectively. Then, using the same argument as that in _Case 5_, we obtain a contradiction. Therefore, statement (iii) holds true. |
2302.08824 | Coherence build up and laser thresholds from nanolasers to macroscopic
lasers | We detail the derivation of nanolaser models that include coherent and
incoherent variables and predict the existence of a laser threshold,
irrespective of cavity size and emitter number, for both single- and
multi-electron systems. The growth in photon number in the lasing mode is
driven by an increase in correlation between absorption and emission processes,
leading to the onset of self-sustained stimulated emission (laser threshold),
followed, in turn, by a correlation decrease and ending with the dominance of
coherent emission. The first-order coherence $g^{(1)}$ steadily increases, as
the pump grows towards the laser threshold value, and reaches unity at or
beyond threshold. The transition toward coherent emission becomes increasingly
sharp as the number of emitters and of the coupled electromagnetic cavity modes
increase, continuously connecting, in the thermodynamic limit, the physics of
nano- and macroscopic lasers at threshold. Our predictions are in remarkable
agreement with experiments whose first-order coherence measurements have so far
been explained only phenomenologically. A consistent evaluation of different
threshold indicators provides a tool for a correct interpretation of
experimental measurements at the onset of laser action. | Mark Anthony Carroll, Giampaolo D'Alessandro, Gian Luca Lippi, Gian-Luca Oppo, Francesco Papoff | 2023-02-17T11:46:04Z | http://arxiv.org/abs/2302.08824v1 | # Coherence build up and laser thresholds from nanolasers to macroscopic lasers
###### Abstract
We detail the derivation of nanolaser models that include coherent and incoherent variables and predict the existence of a laser threshold, irrespective of cavity size and emitter number, for both single- and multi-electron systems. The growth in photon number in the lasing mode is driven by an increase in correlation between absorption and emission processes, leading to the onset of self-sustained stimulated emission (laser threshold), followed, in turn, by a correlation decrease and ending with the dominance of coherent emission. The first-order coherence \(g^{(1)}\) steadily increases, as the pump grows towards the laser threshold value, and reaches unity at or beyond threshold. The transition toward coherent emission becomes increasingly sharp as the number of emitters and of the coupled electromagnetic cavity modes increase, continuously connecting, in the thermodynamic limit, the physics of nano- and macroscopic lasers at threshold. Our predictions are in remarkable agreement with experiments whose first-order coherence measurements have so far been explained only phenomenologically. A consistent evaluation of different threshold indicators provides a tool for a correct interpretation of experimental measurements at the onset of laser action.
## I Introduction
The rapid advancement in the design and manufacturing of laser resonators over the past few decades has allowed the construction of lasing devices with mode volume \(V\propto\lambda^{3}\), where \(\lambda\) is the emission wavelength [1]. Such small devices are far more compact and less energy hungry compared to standard lasers, as lower input power is required to achieve coherent emission. In addition to nanolasers, microlasers (e.g., \(V=a\lambda^{3}\), \(2\lesssim a\lesssim 40\)) hold promise for a number of uses spanning multiple research disciplines and industrial applications, such as integrated optical interconnects, sensing and biological probes, to name a few [2]. Photon number squeezing is also expected to naturally emerge before the transition to coherent emission, leading to cw photon fluxes for non-classical applications [3]. The complexity of the transition between incoherent and coherent emission in micro- and nanolasers is at the origin of interpretative problems, and gives rise to new opportunities. The difficulties in threshold identification in nanolasers [4] come from the intrinsic physical properties of the transition in small systems rather than from technical measurement limitations.
As the mode volume of a device decreases, so does the number of electromagnetic cavity modes available for a spontaneous transfer of the energy stored in the medium. In the Cavity Quantum Electrodynamics (CQED) regime this number is significantly reduced. This "number" is characterized by the spontaneous emission factor, \(\beta\), which quantifies the ratio between the spontaneous emission rate into the lasing mode and the total spontaneous emission rate. For macroscopic devices \(\beta\lesssim 10^{-6}\), while the ideal nanolaser limit corresponds to the asymptote \(\beta=1\). In other words, \(\beta\) is inversely proportional to the systems size.
The laser threshold is typically identified in macroscopic lasers by inspecting the output power as a function of the input power. The input-output (I-O) curves display a characteristic S-shape on a log-log plot with a steep growth. The laser threshold is located at the inflection point of these curves [5]. As the cavity volume decreases, the steep growth is progressively smoothed, leading, in the nanolaser limit of \(\beta=1\), to a straight line. The extrapolation of this linear dependence down to zero pump power has ushered the questionable concept of thresholdless lasers [5; 6].
In spite of the equivalence of the intracavity light-matter interaction in macroscopic and microscopic lasers, two different approaches have emerged, each with its own limitations. For macroscopic systems, the well established semi-classical Maxwell-Bloch equations [7] describe coherent emission above the laser threshold by considering the expectation values of the classical coherent field amplitude and the standard medium polarization. The application of classical factorization schemes to expectation values, which describe the light-matter interaction, neglects quantum correlations [8], thus limiting the theoretical description to above-threshold coherent emission, with no access to the incoherent regimes below it. Quantum models for nanolasers, on the other hand, neglect the classical variables associated with coherent emission [9; 10; 11; 12] and apply factorization techniques, such as the cluster expansion [13], keeping only the slowly varying quantum correlations.
In recent papers [3; 14] we combined these two approaches by including the slowly-varying quantum correlations as well as the coherent variables into a single
Coherent-Incoherent (CI) model, whilst neglecting quantum correlations between electromagnetic field and those medium operators which oscillate on a fast timescale, as done in semi-classical theories [7]. We then used the Linear Stability Analysis (LSA) of the CI model's incoherent solution to calculate analytically the laser threshold for all two-level emitter nanolasers, including the so-called _thresholdless lasers_ (\(\beta=1\)).
In this paper we examine in detail the derivation and predictions of the CI model and extend it to multi-electron systems. We discover that - for both single and multi-electron systems - the critical point identified by the bifurcation analysis is the threshold beyond which stimulated emission becomes self-sustained. This analysis is corroborated by experimental measurements [15]. It is important to stress that our approach further contributes to solving the confusion, first identified in [5], reigning around the concept of a _thresholdless laser_ by further dispelling the concept that an ideal CQED laser would be a truly thresholdless device. In a future paper we will consider models that retain fast quantum correlations between field and medium operators, including more quantum aspects than those considered here at the price of a significantly larger number of equations. These models confirm the existence of lasing solutions in all nanolasers [16; 17] and predict laser thresholds associated to the establishment of self-sustained stimulated emission that become increasingly close to those calculated in this paper as the number of emitters increases.
This paper is structured as follows. In Section II we outline the structure of the system Hamiltonian. Section III covers the cluster expansion technique needed to close the model equations and presents the derivation of the CI model for single and multi-electron systems. Section IV presents the Linear Stability Analysis, including a discussion of the conditions required for the existence of instabilities in the system. Sections V and VI detail the effects of emitter number \(N\) and cavity "size" \(\beta\). Section VII introduces the characterization of coherence and conclusively interprets existing experimental results in the framework of the models here introduced, while Section VIII offers a brief overview of the work and conclusions.
## II The structure of the system Hamiltonian
Our investigation starts with writing the fully quantized Jaynes-Cummings Hamiltonian [18] generalized to describe light-matter interaction between two interacting levels with lasing and non-lasing modes,
\[H=H_{free}+H_{int}, \tag{1}\]
where \(H_{free}\) is the non-interacting part and \(H_{int}\) the interacting part of the Hamiltonian, respectively. The non-interacting part of the Hamiltonian is itself made up of contributions from the free electromagnetic field, \(H_{E}\), and the free electrons in the quantum dot, \(H_{QD}\),
\[H_{free}=H_{E}+H_{QD}. \tag{2}\]
The photon operators of the system Hamiltonian obey the Bosonic commutation relations, and the carrier operators obey the Fermi anti-commutation relations. The Bosonic operators \(b\) and \(b^{\dagger}\) correspond to single-particle operators. It can be shown that \(2N\) Fermi operators are formally equivalent to \(N\) Bosonic operators under the requirement that the compound Fermi operators contain equal numbers of creation and annihilation operators [8]. Examples are the population of the excited state, \(c^{\dagger}c\),and the standard material polarization, \(v^{\dagger}c\). Therefore, we refer to the coherent field operator \(b\) and standard polarization \(v^{\dagger}c\) as single particle operators and to the photon number operator \(b^{\dagger}b\) and photon assisted polarization \(bc^{\dagger}v\) as two particle operators.
The first term in Eq. (2) reads
\[H_{\rm E}=\hbar\sum_{q}\nu_{q}\left(b^{\dagger}_{q}b_{q}+\frac{1}{2}\right), \tag{3}\]
where \(\nu_{q}\) is the frequency of a photon in the \(q\)-th mode and the quantum mechanical operators \(b_{q},b^{\dagger}_{q}\) annihilate and create a photon in the \(q\)-th mode, respectively. The sum over \(q\) includes both lasing and non-lasing modes.
The free electron part of the Hamiltonian describes charge carriers in the conduction and valence band states of the \(n\)-th quantum dot with respective energies \(\epsilon_{c,n}\) and \(\epsilon_{v,n}\)
\[H_{QD}=\sum_{n}\left(\varepsilon_{c,n}c^{\dagger}_{n}c_{n}+\varepsilon_{v,n}v^ {\dagger}_{n}v_{n}\right), \tag{4}\]
where \(c_{n},c^{\dagger}_{n}\) and \(v_{n},v^{\dagger}_{n}\) are the annihilation and creation operators, respectively, for conduction and valence electrons of the \(n\)-th quantum dot.
The two-particle light-matter interaction is described by
\[\begin{split} H_{int}=-i\hbar\sum_{n,q}\left[g_{nq}\left(b_{q}c^{ \dagger}_{n}v_{n}+b_{q}v^{\dagger}_{n}c_{n}\right)\right.\\ \left.-g^{*}_{nq}\left(b^{\dagger}_{q}v^{\dagger}_{n}c_{n}+b^{ \dagger}_{q}c^{\dagger}_{n}v_{n}\right)\right],\end{split} \tag{5}\]
where \(g_{nq}\) is the light-matter coupling strength between a photon in the \(q\)-th mode and the \(n\)-th quantum dot.
In writing the quantum Hamiltonian we have made the standard assumption of neglecting contributions from phonon and Coulomb interactions between charge carriers. This is equivalent to assuming that the quantum dots are operating at cryogenic temperatures (\(\approx 4\)K) [10].
One final remark on the structure of the system Hamiltonian concerns the two-particle operators coming from the interaction part of the Hamiltonian: \(b^{\dagger}_{q}c^{\dagger}_{n}v_{n}\) and \(b_{q}v^{\dagger}_{n}c_{n}\). Quantum mechanically, operators \(b^{\dagger}_{q}c^{\dagger}_{n}v_{n}\) describe a process where a photon in mode \(q\) is created
in conjunction with the excitation of an electron from the valence to the conduction band of the \(n\)-th quantum dot. Operators \(b_{q}v_{n}^{\dagger}c_{n}\) describe its symmetric counterpart, where a photon in mode \(q\) is absorbed in conjunction with the de-excitation of an electron from the conduction to the valence band of the \(n\)-th quantum dot. These two processes do not individually conserve energy, even if their sum is conservative, and oscillate at a frequency approximately double that of the laser, thus in the following we eliminate them from the interaction Hamiltonian (Rotating Wave Approximation).
## III Cluster expansion and nonlinear QED models
To derive the model equations we work in the interaction picture to obtain Heisenberg's equations of motion for the operators appearing in the system Hamiltonian. The variables that appear in the CI models are the quantum operator expectation values and correlations. The dynamics of an \(M\) particle expectation value is directly coupled to an \(M+1\) expectation value through equations of the form
\[i\hbar d_{t}\langle M\rangle=Lo[\langle 1\rangle,\cdots,\langle M\rangle]+Hi[ \langle M+1\rangle], \tag{6}\]
where \(d_{t}\) is the first order derivative with respect to time, \(\langle 1\rangle,\cdots,\langle M+1\rangle\) indicate the sets of the \(1,\cdots,M+1\) particle operators and \(Lo\) and \(Hi\) are matrices that describe coupling to terms of order \(1,\cdots,M\) and \(M+1\), respectively. As a result, there is an infinite hierarchy in which each order \(M\) depends on the higher order \(M+1\). We must therefore find a way of systematically breaking the infinite hierarchy to obtain a closed set of solvable equations at any order \(M\). This is achieved through expressing the expectation values in terms of all possible combinations of products of correlations of lower order operators and introducing approximation schemes for the correlations to truncate the infinite hierarchy [8].
The expectation values of photon number and photon-assisted polarization, central to this work, are
\[\langle b^{\dagger}b\rangle =\delta\langle b^{\dagger}b\rangle+\langle b^{\dagger}\rangle \langle b\rangle, \tag{7}\] \[\langle bc^{\dagger}v\rangle =\delta\langle bc^{\dagger}v\rangle+\langle b\rangle\langle c^{ \dagger}v\rangle, \tag{8}\]
where \(\delta\langle b^{\dagger}b\rangle\) is the two particle correlation between emission and absorption and \(\delta\langle bc^{\dagger}v\rangle\) is the two particle correlation between photon absorption and electron jump from lower to upper energy level.
We find a closed set of equations by including in the model the expectation values \(\langle b\rangle\) and \(\langle c^{\dagger}v\rangle\). These correspond to the complex amplitudes of the coherent field and of the medium polarization and have been neglected in previous microscopic models [10; 12; 19]. They display fast oscillations with frequencies of the order of that of the laser mode. On the contrary, the two-particle quantum correlations Eqs. (7-8) that appear in the cluster expansion of the Hamiltonian oscillate slowly. We call the former variables "coherent" and the latter "incoherent". Coherent quantum correlations such as the correlation between population and field, \(\langle bc^{\dagger}c\rangle\), are neglected in the same way as they are in standard semi-classical models. Models that include all possible two particle correlations have also been shown to display laser thresholds [16] and will be studied in detail in a future paper. We consider all quantum dots identical. This is not a restrictive hypothesis: we have verified numerically that variations in detuning and light-matter coupling strength up to 10% have negligible effects on the system. Thus, we drop the subscript for the Fermi operators and replace the sum over \(n\) with \(N\). The resulting system of equations is
\[d_{t}\langle b\rangle= -(\gamma_{c}+i\nu)\langle b\rangle+Ng^{*}\langle v^{\dagger}c\rangle \tag{9a}\] \[d_{t}\langle c^{\dagger}v\rangle= -(\gamma-i\nu_{\epsilon})\langle c^{\dagger}v\rangle+g^{*}\langle b ^{\dagger}\rangle(2\langle c^{\dagger}c\rangle-1)\] (9b) \[d_{t}\langle c^{\dagger}c\rangle= r(1-\langle c^{\dagger}c\rangle)-(\gamma_{nr}+\gamma_{nl}) \langle c^{\dagger}c\rangle\] (9c) \[-2\Re\{g(\delta\langle b_{q}c^{\dagger}v\rangle+\langle b_{q} \rangle\langle v^{\dagger}c\rangle)\}\] \[d_{t}\delta\langle bc^{\dagger}v\rangle= -(\gamma_{c}+\gamma-i\Delta\nu)\delta\langle bc^{\dagger}v\rangle\] (9d) \[+g^{*}\left[\langle c^{\dagger}c\rangle+\delta\langle b^{\dagger }b\rangle\left(2\langle c^{\dagger}c\rangle-1\right)-|\langle c^{\dagger}v \rangle|^{2}\right]\] \[d_{t}\delta\langle b^{\dagger}b\rangle= -2\gamma_{c}\delta\langle b^{\dagger}b\rangle+2N\Re\left(g \delta\langle bc^{\dagger}v\rangle\right), \tag{9e}\]
where \(\nu_{\epsilon}\) is the frequency of the inter-band energy, \(h\nu_{\epsilon}=\epsilon_{c}-\epsilon_{v}\), \(\Delta\nu\equiv\nu_{\epsilon}-\nu\) is the detuning, \(\Re(\cdot)\) stands for real part of its argument and a superscript \({}^{*}\) indicates the complex conjugate. The equations for \(\langle b^{\dagger}\rangle\), \(\langle v^{\dagger}c\rangle\) and \(\langle b^{\dagger}v^{\dagger}c\rangle\) can be obtained from Eqs. (9) through complex conjugation. The expectation value of the lower-level population has been eliminated using \(\langle c^{\dagger}c\rangle+\langle v^{\dagger}v\rangle=1\). The dissipative part of the equations is obtained by considering Lindblad terms describing the coupling to a Markovian heat bath under the constraint that random excitations into the excited state are neglected [20; 21], a condition which is fulfilled under the assumed cryogenic temperatures. The cavity decay rate, \(\gamma_{c}\), the polarization dephasing rate, \(\gamma\), and the non-radiative losses, \(\gamma_{nr}\), describe the dissipative channels. The coherent field amplitude, \(\langle b\rangle\), and standard polarization, \(\langle v^{\dagger}c\rangle\), are analogous to their amplitudes in semi-classical theories. They describe coherent inter-band processes and therefore need to be externally driven to be sustained. In terms of operators the population density of the excited state, \(\langle c^{\dagger}c\rangle\), describes an intra-band process and does not require any externally driven source to exist; it is the probability of an electron being in the excited state. The photon-assisted polarization describes a correlated event between the annihilation of a photon with an inter-band transition or the opposite scenario for its hermitian conjugate. Finally, the intensity correlation describes the correlation between photon absorption and emission.
The model derived above assumes identical emitters, each with two discrete energy levels and a single electron. We want to show that the existence of coherent laser solutions is not specific to the single-electron nature of this model, thus we include the coherent variables \(\langle b\rangle\) and \(\langle v^{\dagger}c\rangle\) in the model given in Ref. [19], where
the authors relax the single-electron assumption. To ensure that radiative decays can take place only if the upper level is occupied and the lower level empty, the radiative decay terms are now proportional to the product of the probability that an electron is in the excited level with the probability that the lower level is empty. This gives rise to nonlinear terms in the equation for the photon-assisted polarization and the population density, Eqs. (9c,9d), which in the multi-electron model read
\[d_{t}\langle c^{\dagger}c\rangle= r(1-\langle c^{\dagger}c\rangle)-(\gamma_{nr}+\gamma_{nl} \langle c^{\dagger}c\rangle)\langle c^{\dagger}c\rangle\] \[-2\Re\{g(g\langle b_{q}c^{\dagger}v\rangle+\langle b_{q}\rangle \langle v^{\dagger}c\rangle)\} \tag{10c}\] \[d_{t}\delta\langle bc^{\dagger}v\rangle= -(\gamma_{c}+\gamma-i\Delta\nu)\delta\langle bc^{\dagger}v \rangle+g^{*}\left[\langle c^{\dagger}c\rangle^{2}\right. \tag{10d}\]
while all the other equations remain the same. Note that the population of the lower level, \(\langle v^{\dagger}v\rangle\), can be eliminated because \(\langle c^{\dagger}c\rangle+\langle v^{\dagger}v\rangle\to 1\) exponentially in time (see difference between electron and holes densities in Ref. [19, Eq. (3)].
The two sets of equations, (9) and (10), constitute the single-electron and the multi-electron CI models respectively. They both contain coherent and incoherent variables and differ only in the number of electrons in each emitter. With a quantum theory containing variables that can describe both coherent and incoherent processes, we can now investigate how coherence emerges in nanolasers for single- and multi-electron systems.
## IV Linear stability analysis
In analogy with the semi-classical theory of macroscopic lasers [7], we identify the laser threshold as the instability threshold of a non lasing solution where the incoherent variables are different from zero but the amplitudes of the coherent field are \(\langle b\rangle=\langle v^{\dagger}c\rangle=0\). If perturbations of \(\langle b\rangle,\langle v^{\dagger}c\rangle\) grow, the non-lasing solution is unstable. If instead they decay asymptotically to zero, then the solution is stable. The bifurcation point, where the solution with zero values for coherent variables becomes unstable, is the laser threshold. To study its existence as a function of the parameter values, we perform a linear stability analysis of Eqs. (9) and (10). We collect coherent and incoherent variables into two groups, \(\mathbf{c}=\{\langle b\rangle,\langle b^{\dagger}\rangle,\langle v^{\dagger}c \rangle,\langle c^{\dagger}v\rangle\}\) and \(\mathbf{i}=\{\langle c^{\dagger}c\rangle,\delta\langle b^{\dagger}b\rangle, \delta\langle bc^{\dagger}v\rangle,\delta\langle b^{\dagger}v^{\dagger}c\rangle\}\), respectively, and write the two CI models in a more compact form
\[d_{t}\mathbf{i} =\mathbf{F}(\mathbf{i},\mathbf{c}), \tag{11}\] \[d_{t}\mathbf{c} =\mathbf{G}(\mathbf{i},\mathbf{c}), \tag{12}\]
where \(\mathbf{G}(\mathbf{i},\mathbf{c})\) and \(\mathbf{F}(\mathbf{i},\mathbf{c})\) are non-linear vector functions of \(\mathbf{i}\) and \(\mathbf{c}\) whose components are the right-hand side of the two models. With this notation \(G_{\langle b\rangle}(\mathbf{i},\mathbf{c})\) and \(G_{\langle v^{\dagger}c\rangle}(\mathbf{i},\mathbf{c})\) are, for example, the right-hand side of Eq. (9a) and of the complex conjugate of Eq. (9b), respectively, evaluated at \(\mathbf{i},\mathbf{c}\). The linearized dynamics of small perturbations \((\mathbf{\eta}_{\mathbf{i}},\mathbf{\eta}_{\mathbf{c}})\) of a fixed point solution \((\mathbf{i},\mathbf{c})\) is given by
\[d_{t}\left[\begin{array}{c}\mathbf{\eta}_{\mathbf{i}}\\ \mathbf{\eta}_{\mathbf{c}}\end{array}\right]=\left[\begin{array}{cc}\mathbf{\nabla}_{i} \otimes\mathbf{F}(\mathbf{i},\mathbf{c})&\mathbf{\nabla}_{\mathbf{c}}\otimes\mathbf{F}(\mathbf{i},\mathbf{c}) \\ \mathbf{\nabla}_{\mathbf{i}}\otimes\mathbf{G}(\mathbf{i},\mathbf{c})&\mathbf{\nabla}_{\mathbf{c}}\otimes\bm {G}(\mathbf{i},\mathbf{c})\end{array}\right]\left[\begin{array}{c}\mathbf{\eta}_{\mathbf{i} }\\ \mathbf{\eta}_{\mathbf{c}}\end{array}\right]. \tag{13}\]
Each block in the matrix on the right-hand side of Eq. (13) is of dimension \(4\times 4\) (both sets \(\mathbf{i}\) and \(\mathbf{c}\) contain four variables) and corresponds to the Jacobian with respect to the \(\mathbf{i}\) and \(\mathbf{c}\) variables. \(\otimes\) denotes the outer product. For any solution with \(\mathbf{c}=\mathbf{0}\) one has \(\mathbf{\nabla}_{i}\otimes\mathbf{G}(\mathbf{i},\mathbf{0})=\mathbf{0}\) and \(\mathbf{\nabla}_{\mathbf{c}}\otimes\mathbf{F}(\mathbf{i},\mathbf{0})=\mathbf{0}\), so that coherent and incoherent perturbations of the \((\mathbf{i},\mathbf{0})\) solutions decouple.
This is a general feature of _all_ models derived under the rotating wave approximation independently of the order of the quantum correlations considered. Its origin is the separation of time scale between the coherent and the incoherent variables. The (fast) coherent variables oscillate at the lasing frequency \(\nu\), i.e., proportionally to \(\sim e^{-i\nu t}\). They can therefore only appear in complex conjugate quadratic pairs in the equations for the (slow) incoherent variables. As the derivative of a quadratic term at zero is zero, we have that \(\mathbf{\nabla}_{\mathbf{c}}\otimes\mathbf{F}(\mathbf{i},\mathbf{0})=\mathbf{0}\). Conversely, the (slow) incoherent variables can only appear in the equations for the (fast) coherent variables if they are multiplied by a coherent variable. Therefore, \(\mathbf{\nabla}_{\mathbf{i}}\otimes\mathbf{G}(\mathbf{i},\mathbf{0})=\mathbf{0}\).
While these results are generic, in the specific case of the CI models the incoherent perturbations of the incoherent solution always decay to zero. Therefore, the existence of a laser threshold is determined solely by the dynamics of the coherent perturbations, given by
\[d_{t}\mathbf{\eta}_{\mathbf{c}}=\mathbf{\nabla}_{\mathbf{c}}\otimes\mathbf{G}(\mathbf{i},\mathbf{0})\mathbf{ \eta}_{\mathbf{c}}=\left[\begin{array}{cc}J&0\\ 0&J^{*}\end{array}\right]\mathbf{\eta}_{\mathbf{c}}, \tag{14}\]
where
\[J =\left[\begin{array}{cc}\partial_{(b)}G_{(b)}(\mathbf{i},\mathbf{0})& \partial_{\langle v^{\dagger}c\rangle}G_{(b)}(\mathbf{i},\mathbf{0})\\ \partial_{(b)}G_{\langle v^{\dagger}c\rangle}(\mathbf{i},\mathbf{0})&\partial_{ \langle v^{\dagger}c\rangle}G_{\langle v^{\dagger}c\rangle}(\mathbf{i},\mathbf{0}) \end{array}\right]\] \[=\left[\begin{array}{cc}-\gamma_{c}&g^{*}N\\ g(2\langle c^{\dagger}c\rangle-1)&-(\gamma+i\Delta\nu)\end{array}\right], \tag{15}\]
and \(J^{*}\) is the complex conjugate of \(J\). For ease of notation and without loss of generality, we have written \(J\) in a frame rotating with \(\langle b\rangle\). This matrix depends on the system parameters and on the population of the excited state. It is important to note that the structure of the stability matrix is the same for both single- and multi-electron CI models. However, since \(J\) depends on the excited state population the eigenvalues of these two models differ.
The lasing threshold condition is that there is at least one eigenvalue \(\lambda\) of \(J\) such that \(\Re(\lambda)>0\). Since \(0\leq\langle c^{\dagger}c\rangle\leq 1\), this can be satisfied only if
\[N>\frac{\gamma\gamma_{c}}{|g|^{2}}\left[1+\left(\frac{\Delta\nu}{\gamma+\gamma_ {c}}\right)^{2}\right], \tag{16}\]
i.e., if the number of quantum dots is greater than a critical number given by the right hand side of Eq. (16). This applies to both the single- and multi-electron models, is independent of \(\beta\), and increases with losses and detuning. We conclude this section with two observations. The first is that the CI models have been derived assuming weak light-matter coupling. Therefore Eq. (16) does not apply to the strong coupling regime. The second, is that the instability condition on the number of emitters in Eq. (16) is only a necessary one: a sufficiently large pump rate is also necessary to cross the laser threshold, as discussed in the following.
## V Laser threshold: dependence on N
We now investigate photon emission processes in these single- and multi-electron lasers below and above the instability threshold. Since \(\langle b\rangle=0\) when the device is not lasing, we see from the cluster expansion in Eq. (7) that the photon number is given exclusively by the correlation term, which dominates the spontaneous emission regime. Fig. 1 a) illustrates the effect of including the fast variables for the single-electron (solid lines) and the multi-electron (dashed lines with circles) CI models, Eqs. (9) and (10). We compare three devices containing \(N=\{20,21,40\}\) emitters (blue, red and green lines respectively). For the parameter values of this illustration (see caption of Fig. 1), an instability exists if \(N>20\). Below the critical number of quantum dots (blue lines), as the pump increases the photon number saturates and the coherent field amplitude remains zero, confirming the absence of laser emission. For a number of quantum dots just above the minimum number required for lasing, i.e., \(N=21\) (red line), there is a clear jump in the photon number accompanied by an emerging non-zero coherent field amplitude via a pitchfork bifurcation (Fig. 1a). We can see from the graph of \(\delta\langle b^{\dagger}b\rangle\), Fig. 1b, that the initial growth in photon number is due to spontaneous emission, positively and increasingly correlated to absorption, while the coherent part of the field is zero. Indeed, the growth of \(\delta\langle b^{\dagger}b\rangle\) in Fig. 1b precedes the bifurcation (Fig. 1a) and occurs at pump rates toward the end of the steeper growth in the photon number that is visible in Fig. 1c. This is a characteristic feature which distinguishes small from macroscopic lasers. For the latter, it is known that the inflection point of the steeper photon number growth corresponds to the threshold [5] and, by extension, this points has been taken as a reference also for small lasers with the help of clever techniques [4]. Finite-size effects, instead, profoundly modify not only the nature of threshold (as explained below), but also the pump value for which it occurs. The consequences are important since the identification of coherent emission becomes problematic. The difficulty is pragmatically circumvented, in commercial microdevices, by manufacturers [22] whose laser characteristic sheets give a threshold current which is placed well beyond the actual threshold, identified here through linear stability analysis. A discussion of the various "kinds" of threshold experimentally used is offered in [23] (Supplementary Material available in [24]).
From threshold onward, the increase of the coherent field intensity \(|\langle b^{\dagger}\rangle|^{2}\) coincides with a sharp decrease in the correlation between absorption and emission, \(\delta\langle b^{\dagger}b\rangle\), as expected in the presence of stimulated emission. When the correlation \(\delta\langle b^{\dagger}b\rangle\) becomes negative, stimulated emission dominates and \(\langle b^{\dagger}b\rangle=|\langle b^{\dagger}\rangle|^{2}+\delta\langle b ^{\dagger}b\rangle<|\langle b^{\dagger}\rangle|^{2}\). In summary, Fig. 1b clearly shows two features: \((i)\) during the steep parts of the emission growth the light is entirely incoherent; \((ii)\) immediately above threshold, the emitted field consists of a mixture of coherent and incoherent
Figure 1: (a) The modulus of the coherent field amplitude \(|\langle b\rangle|\), (b) the correlation between photon absorption and emission, \(\delta\langle b^{\dagger}b\rangle\), and (c) the expectation value of the photon number, \(\langle b^{\dagger}b\rangle\) as a function of the pump for the single- (solid line) and the multi-electron (dashed with circles) CI models, for \(N=\{20,21,40\}\) (blue, red and green lines respectively). In this and all other figures time and decay and coupling parameters are scaled with \(\gamma_{nr}\), which is equivalent to setting \(\gamma_{nr}=1\) in Eqs. (9) or (10). The other parameter values are \(g=70\), \(\Delta\nu=0\), \(\gamma=10^{4}\), \(\gamma_{c}=10\) and \(\gamma_{nl}=1400\), equivalent to \(\beta=7\times 10^{-4}\).
photons and complete dominance of the coherent component takes place only (well) beyond threshold. These features are typical of nano- and microlasers.
The smoothness of the lasing transition imposed by the finite size of the (small) devices paves the way towards new applications [25; 26]. In contrast, the sharpness of the transition in macrolasers (e.g., Fig. 2) squeezes the pump interval over which the evolution from entirely incoherent to dominantly coherent emission takes place, explaining why, in macroscopic lasers, the threshold can be considered as an on-off effect that corresponds to a single well defined pump value. It is a strength of the CI models that they provide a description of the continuous transformation in the laser emission features as its size increases.
A device with twice the minimum number of quantum dots (e.g., \(N=40\), dashed lines in Fig. 1) crosses the laser threshold at a pump rate lower than that for \(N=21\) and with a sharper transition, see Fig. 1c.
As \(N\) increases the differences between the single- and multi-electron model become apparent (compare the solid and dashed curves in Fig. 1). The multi-electron model reaches threshold for lower values of the pump rate and, hence, the fraction of incoherent emission contributing to the initial growth in photon number is reduced. This is due the lower losses of the upper level population, \(\langle c^{\dagger}c\rangle\), due to the term \(\gamma_{nl}\langle c^{\dagger}c\rangle^{2}\) in Eq. (10c) compared to the losses due to the term \(\gamma_{nl}\langle c^{\dagger}c\rangle\) in Eq. (9c). Both models have the same critical number of emitters necessary for the instability to exist (\(N=21\) for the chosen parameters). Only the pump power at the laser threshold changes.
These results highlight the contributions of the fast variables, and the necessity of their presence in the models to obtain a consistent description of the emission processes in a laser. The position of the laser bifurcation in the I-O curve shows that a simple visual inspection of the output characteristics leads to an incorrect identification of the laser threshold and fails to identify the true nature of the emission process, e.g., the incoherent nature of the photon number in small lasers in the phase of steep growth.
## VI Laser threshold: dependence on \(\beta\)
We now turn to the dependence of threshold on system size, \(\beta\)[14]. Fig. 2a displays the value of pump at threshold as a function of \(\beta\) for devices with different \(N\). This has been computed numerically by finding the pump value for which the correlation \(\delta\langle b^{\dagger}b\rangle\) is maximum (see Fig. 1b). While the threshold pump rate decreases monotonically as \(N\) increases (for all values of \(\beta\)), the dependence on cavity size shows the existence of two regimes: a rapid threshold decrease within the realm of macroscopic lasers, and the onset of near-saturation (in double logarithmic scale) for \(\beta\gtrapprox 10^{-3}\) (i.e., micro- and nanolasers). This latter feature would appear to contradict the common knowledge according to which the threshold linearly decreases with slope \(\frac{1}{2}\) in double logarithmic scale [5, Eq. (20)]) as \(\beta\) increases; this property is, however, based exclusively on the (incorrect) assumption that threshold is always placed at the inflection point of the I-O curve. Instead, the saturation which emerges from the CI models results from the identification of the true laser threshold (self-sustained stimulated emission, Section IV) which progressively and substantially moves away from the macroscopic definition as the laser size is decreased. The loss in threshold reduction is, however, well compensated by the emergence of a broader and richer transition region between incoherent and coherent emission, whose features promise new applications (Sections IV and VII).
A clear visual illustration of the threshold displacement is provided in Fig. 2b, showing the I-O curves in double logarithmic scale for laser devices with \(N=40\)
Figure 2: (a) Numerical estimate of the pump threshold for the single- (S) and multi-electron (M) CI models as a function of the spontaneous emission factor, \(\beta\), for different numbers of emitters. (b): Photon number \(\langle b^{\dagger}b\rangle\) as a function of the pump \(r\) for the single- (S) and multi-electron (M) CI models for different values of \(\beta\) and \(N=40\) quantum dots. The black crosses identify the numerically established laser thresholds for the two models. All other parameters as in Fig. 1.
emitters and three different values of \(\beta\). The straight, superposed I-O curves correspond, as expected, to \(\beta=1\), while those with a gentle curvature to a microlaser: the respective thresholds are marked by black crosses and appear well on the upper branch. It is only with a macroscopic laser that the threshold appears at the inflection point of the steeply growing photon number, matching the well-known properties of macroscopic lasers [5].
## VII First order correlation function \(g^{(1)}(\tau)\)
Surprisingly not included in the recommendations to identify laser threshold in experiments [27], the autocorrelation functions remain the most sensitive and most reliable way of obtaining pertinent threshold information, as long as a meaningful model can be used for comparison. In this section, we tackle precisely this aspect and examine the first order, time-delayed correlation function. Here we study its properties and successfully compare them to the experimental measurements in Ref. [15]. Due to its more complex experimental implementation, it is more seldom used than its second order counterpart, but it has the advantage of providing direct information on the coherence of the emitted radiation [15]. Once the relationship between the two kinds of correlations is clarified, comparison between the two indicators will facilitate their individual use in the interpretation of experimental results.
In order to calculate the first-order correlation function
\[g^{(1)}(\tau)=\frac{\langle b^{\dagger}(t)b(t+\tau)\rangle}{\langle b^{\dagger }(t)b(t)\rangle}, \tag{17}\]
where \(\tau\) is a delay time, we write the differential equation
\[d_{\tau}g^{(1)}=\frac{1}{\langle b^{\dagger}(t)b(t)\rangle}d_{\tau}\langle b^ {\dagger}(t)b(t+\tau)\rangle \tag{18}\]
which we solve with initial condition \(g^{(1)}(0)=1\). To form a close set of equations, we use the quantum regression formula, see Eqs. (1.105-1.107) of Ref. [28]. In the Heisenberg picture, this reads \(d_{\tau}\langle A(t)B(t+\tau)\rangle=\langle A(t)d_{\tau}B(t+\tau)\rangle\) where \(A\) and \(B\) are operators and \(d_{\tau}B\) is calculated applying the Hamiltonian and Lindblad formalism at time \(t+\tau\). We expand the \(\tau\) derivative on the right hand side of Eq. (18) and make use of Eqs. (9a) and (9d) to obtain
\[d_{\tau}\langle\tilde{b}^{\dagger}b\rangle=-(\gamma_{c}+i\nu) \langle\tilde{b}^{\dagger}b\rangle+Ng^{*}\langle\tilde{b}^{\dagger}v^{\dagger }c\rangle, \tag{19a}\] \[d_{\tau}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle=-(\gamma+ i\nu_{\epsilon})\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle+g\left(2\langle \tilde{b}^{\dagger}bc^{\dagger}c\rangle-\langle\tilde{b}^{\dagger}b\rangle \right), \tag{19b}\]
where \(\tilde{b}^{\dagger}\equiv b^{\dagger}(t)\), and all other operators are at time \(t+\tau\). \(\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle\) is the expectation value of a 3-particle operator. To find a closed set of equations at two-particle level we use Eq. (7) and the cluster expansion
\[\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle= \delta\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle+\langle \tilde{b}^{\dagger}\rangle\delta\langle bc^{\dagger}c\rangle\] \[+\langle b\rangle\delta\langle\tilde{b}^{\dagger}c^{\dagger}c \rangle+\langle c^{\dagger}c\rangle\delta\langle\tilde{b}^{\dagger}b\rangle+ \langle\tilde{b}^{\dagger}\rangle\langle b\rangle\langle c^{\dagger}c\rangle\]
together with the semi-classical approximation used to derive the CI models, which for these equations reduces to \(\delta\langle\tilde{b}^{\dagger}c^{\dagger}c\rangle\sim\delta\langle bc^{ \dagger}c\rangle\sim 0\). With these approximations Eqs. (19) become
\[d_{\tau}\langle\tilde{b}^{\dagger}b\rangle=-(\gamma_{c}+i\nu) \langle\tilde{b}^{\dagger}b\rangle+Ng^{*}\langle\tilde{b}^{\dagger}v^{\dagger }c\rangle, \tag{20a}\] \[d_{\tau}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle=-(\gamma+ i\nu_{\epsilon})\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle+g\langle\tilde{b}^{ \dagger}b\rangle(2\langle c^{\dagger}c\rangle-1). \tag{20b}\]
Eqs. (20) are formally identical for the single- and multi-electron models, the only difference in \(g^{(1)}(\tau)\) coming from the different values of the term \(\langle c^{\dagger}c\rangle\). This is due to the fact that the Heisenberg equations and the dissipative Lindblad terms for the operators at time \(t+\tau\) do not depend on the losses of \(\langle c^{\dagger}c\rangle\) that are proportional to \(\gamma_{nl}\).
With the help of these expressions, we can now plot the first order autocorrelation as a function of the model parameters. We expect that below threshold the correlation function decays exponentially with the delay time,
\[g^{(1)}(\tau)\propto e^{-t/\tau_{c}}, \tag{21}\]
with \(\tau_{c}\) the correlation decay time. This behavior is confirmed by the log plot of \(g^{(1)}(\tau)\) in the inset of Fig. 3, where we have set the pump at 15% of the single-electron threshold value [29, Eq. (20)] for \(N=40\) quantum dots.
Figure 3: The main graph plots the coherence decay time \(\tau_{c}\) as a function of the spontaneous emission factor \(\beta\) for the single-(S) and multi-electron (M) CI models for a pump value equal to 15% of the threshold for \(N=40\) quantum dots. \(\tau_{c}\) has been obtained by fitting with a straight line \(\log(|g^{(1)}(\tau)|)\) as a function of \(\tau\). The inset is a log plot \(|g^{(1)}(\tau)|\) as a function of the delay time \(\tau\) for a sample of the values of \(\beta\) used in the main plot. This confirms the exponential decay of the correlation, Eq. (21). All other parameters as in Fig. 1.
We have computed \(\tau_{c}\) as a function of \(\beta\) by fitting these curves with a straight line. The decay rate has a sigmoidal behavior: it is an increasing function of \(\beta\) that jumps by two orders of magnitude as \(\beta\) changes from \(10^{-4}\) to \(10^{-2}\) and is approximately constant outside this interval. This clearly illustrates a fundamental feature of small lasers, whose coherence grows gradually as threshold is approached, in agreement with the smooth response of their I-O curve. For macroscopic lasers, on the other hand, we obtain results which are consistent with the standard picture of a nearly incoherent output up until threshold, with a sudden conversion to full coherence. The single- and multi-electron CI models have similar behavior, with the multi-electron model having larger decay time. This is an effect of the lower effective losses of the multi-electron with respect to the single-electron model: at equal pump values the former is closer to threshold than the latter (cf. the shift of threshold positions between the two models in Fig. 2).
The evolution of coherence with pump power is examined in Fig. 4 for the single- and multi-electron models at fixed number of emitters, \(N=\{40,1000\}\), and cavity volumes, \(\beta=\{1,7\times 10^{-4},3.4\times 10^{-6}\}\). In order to clearly highlight the pump influence, a delay time \(\tau=60/\gamma_{nr}\) is fixed. Experimental information can be gathered, as in [15], by fixing the difference in the Michelson interferometer arm lengths and measuring the fringe visibility as a function of pump. The laser threshold corresponds to the smallest pump value for which \(g^{(1)}(\tau_{M})=1\); at this point the curve slope is discontinuous. While in an experiments unavoidable fluctuations of the control parameters, not included in the model, will limit the coherence time even beyond the threshold, the behavior of the coherence time as a a function of the pump will provide a clear indication of the threshold value. Irrespective of laser size, there is a continuous growth of coherence, driven by the increase in correlation between absorption and emission properties (as in Fig. 2); however, while in smaller systems coherence evolves steadily over a broad pump range below threshold, in macroscopic lasers the change occurs over a narrow interval of pump values. In other words, as \(\beta\) decreases moving toward the macroscopic limit, it becomes more and more difficult to obtain partially coherent emission. This result does not depend on the choice of \(\tau_{M}\), as long as \(\tau_{M}\gg\lambda_{0}/v\), with \(\lambda_{0}\) and \(v\) the light wavelength and velocity in the interferometer, respectively. Changing \(\tau_{M}\) only changes the shape of the curves of Fig. 4. It is worth stressing again that the deformation of the coherence curves progresses continuously from the nano- to the macroscale.
Increasing the number of emitters from \(N=40\), Fig. 4a, to \(N=10^{3}\), Fig. 4b, while keeping the other parameters constant reduces the threshold values and the range of pump values over which the transition toward \(g^{(1)}(\tau_{M})=1\) occurs for all values of \(\beta\). While nanolasers are typically built with tens of emitters, in macroscopic lasers their number will easily be largely in excess of what we are showing here, thus further enhancing the differences between the two categories of devices.
We conclude this section by highlighting that these analytical and numerical results are supported by independent experimental measurement of \(g^{(1)}(\tau)\)[15]. The first order coherence was experimentally obtained in Ref. [15] by measuring the visibility of interference fringes resulting from Michelson interferometry and plotted as a function of the pump power [15, Fig. 2b]. From these data the authors also computed the coherence decay time as a function of the pump power. It is not possible from the experimental data available in Ref. [15] to obtain unique values for the CI model parameters. However, the parameter values used in the figures in this paper are reasonable estimates. We plot in Fig. 5 the correlation decay time as a function of the pump power, measured in units of the analytical threshold for the single electron CI model [29, Eq. (20)]. The similarities between this figure and its in
Figure 4: \(\left|g^{(1)}\right|\) for delay \(\tau_{M}=60/\gamma_{nr}\) as a function of the pump for \(\beta=\{1,7\times 10^{-4},3.4\times 10^{-6}\}\) (blue, red and green lines respectively) for the single- (solid) and multi- (dashed with circle) CI models. The number of quantum dots is \(N=\{40,1000\}\) in panels (a) and (b) respectively. The points where \(g^{(1)}(\tau_{M})\) reaches unity, and where the slopes of the curves suddenly change, are the laser thresholds. All other parameters as in Fig. 1.
set and figures 2c and 2b respectively of Ref. [15] are uncanny, keeping in mind the uncertainty in the mapping of the experimental parameters. We can therefore conclude that the CI model is capable of clearly and unequivocally identifying the onset of coherence, matching it to the crossing of laser threshold (self-sustained growth of stimulated emission), and to explain experimental observations for which no model, derived from first principles, had been available up until now.
## VIII Conclusions
We have presented the details of a model for (semi-conducting) quantum emitters (with single- or multiple-electrons) coupled to an electromagnetic cavity of arbitrary size to describe the transition from thermal to coherent emission. The joining of a fully quantum treatment, based on the explicit description of incoherent fields and the correspondingly induced dipole moments, and of a coherent field with its accompanying polarization, together with an analysis based on nonlinear dynamical properties permits the clear and unequivocal identification of a threshold for the emergence of a self-sustained stimulated emission, i.e., the lasing onset. The Coherent-Incoherent model marks an entirely new approach in the depiction of laser action, due to the traditional attention brought to macroscopic devices and to the resulting attempts at adjusting the latter to cover small devices through simple modifications. This treatment shows that simple adjustments are not sufficient and that a consistent treatment can be obtained only through fundamentally revisiting the physics to explicitly introduce the two categories of incoherent and coherent variables.
The main result is a proper definition of lasing threshold irrespective of laser size, accompanied by a continuous description of the evolution of the degree of coherence from the macro- to the nano- scale; we further find that the number of coupled emitters contribute to sharpening or softening the more extreme aspects of the system size.
In addition to the definition of threshold based on nonlinear physics concepts, the quantum mechanical approach permits the direct evaluation of the coherence properties of the electromagnetic field through the first order coherence function. Its use shows that full coherence is attained at the bifurcation point (laser threshold), which - at variance with scaling laws established at the macroscale - is placed closer and closer to the upper emission branch (or directly on it) as the finite system size contribution increases through the reduced number of electromagnetic cavity modes. Simultaneously, the quantum-mechanical analysis shows that the rapid growth in photon number originates from an increase in correlation between absorption and emission processes in the absence of self-sustained stimulated emission, which account for the entirety of the transition to the upper emission branch in the smallest devices. In macroscopic lasers, instead, this contribution is limited to the lower portion of the (nearly) vertical growth in photon number.
A remarkable aspect of the CI model rests on its ability to predict features experimentally observed in measurements of fringe visibility [15]; a good qualitative agreement is obtained without any free parameters between observations and the predictions shown in this paper. The topic is of great interest since it allows for an unequivocal quantification - and for general model-based predictions - of the amount of coherence, potentially paving the way to numerous applications ranging from novel uses for micro- and nanolasers, but also permitting better assessment of their performance as sources for data treatment (e.g., interconnects in data centers with ultra-low dissipation and small footprint [30; 31; 32; 33; 34; 35]).
The availability of a complete description of the threshold physics at all scales permits the comparison with other experimental choices. For instance, one can envisage computing the output of a mixing interferometer [36] to interpret its results on the basis of a first-principle model, rather than superposing _ad hoc_ radiation packets with preset features.
The simultaneous availability of first-order and second-order autocorrelations, in addition to the threshold information gathered through the LSA, also permits a careful evaluation of the individual properties of these indicators. This way, second-order autocorrelation measurements, easier to perform and routinely used not only in Quantum-Dot-based devices [37; 38; 19; 39], but also with Quantum-Well emitters [40; 41; 42] and metallic
Figure 5: The main graph plots the coherence decay time \(\tau_{\mathrm{c}}\) as a function of the pump power in units of the single electron CI threshold for \(\beta=7\times 10^{-4}\) and \(N=40\) quantum dots. \(\tau_{\mathrm{c}}\) has been obtained by fitting with a straight line \(\log(|g^{(1)}(\tau)|)\) as a function of \(\tau\). The inset is a log plot \(|g^{(1)}(\tau)|\) as a function of the delay time \(\tau\) for the pump values indicated by square symbols in the main plot. All other parameters as in Fig. 1. This figure is the analogous of figures 2b and 2c of Ref. [15]. For ease of comparison time units are expressed in picoseconds. The dimensional time scale has been fixed by setting \(\gamma_{nr}=1\) ns.
nanolasers [43; 44], can acquire a higher degree of reliability in the determination of the nature of the emitted radiation. This can contribute to reaching an agreement on a definite measurement technique for the determination of laser threshold [27] thus sorting the different practical definitions used over the past decades, which give concordant results only at the macroscopic scale [23].
The CI models conclusively show that the transition from incoherent to coherent radiation occurs in a negligibly small pump interval for macroscopic devices. However, they also prove that the physics of laser threshold remains the same even for large lasers, thus implying that the only obstacle in obtaining information from an experiment is of practical nature. This interpretation is consistent with the results of pioneering work of the 1960's and 70's [45; 46], where statistical ensemble measurements gave evidence for a gradual evolution in the nature of the emitted radiation at threshold crossing. More information could become now available through the realization of a novel system constituted by a broadband semiconducting amplifier, where feedback is provided by a fiber loop (also containing adjustable filters) which permit stroboscopic measurements of the light amplification as a function of round trip [47]. The degree of spatio-temporal resolution gained from this realization, thanks to the long delay time of the fibered cavity, enables the measurement of the radiation properties at each round trip. This scheme could garner detailed information to refine our understanding and mathematical description of laser threshold.
|
2307.04618 | Scalar fields with derivative coupling to curvature in the Palatini and
the metric formulation | We study models where a scalar field has derivative and non-derivative
couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation.
We consider both the metric formulation and the Palatini formulation. In the
Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the
same result regardless of whether the connection is unconstrained or the
non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is
included, the unconstrained case and the zero torsion case are physically
different. We reduce all the actions to the Einstein frame with minimally
coupled matter, and find the leading order differences between the metric case
and the Palatini cases. | Hamed Bouzari Nezhad, Syksy Rasanen | 2023-07-10T15:04:16Z | http://arxiv.org/abs/2307.04618v2 | # Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation
###### Abstract
We study models where a scalar field has derivative and non-derivative couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation. We consider both the metric formulation and the Palatini formulation. In the Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the same result regardless of whether the connection is left general or the non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is included, the general case and the zero torsion case are physically different. We reduce all the actions to the Einstein frame with minimally coupled matter, and find the leading order differences between the metric case and the Palatini cases.
###### Contents
* 1 Introduction
* 2 Non-minimal coupling to kinetic terms
* 2.1 Curvature, non-metricity, and torsion
* 2.2 The action
* 2.3 Disformal transformation
* 2.4 Zero torsion case with \(\alpha_{3}\neq 0\)
* 2.5 Metric case
* 3 Conclusions
* A Details of the solution for the connection in the zero torsion case
* B Disformal transformation in the metric formulation
## 1 Introduction
Inflation is the most successful scenario for the early universe [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15], and its predictions agree well with observations [16]. The simplest candidate for driving inflation is a scalar field. The field may be non-minimally coupled to curvature, as such couplings are generated by loop corrections [17]. Direct coupling to the Ricci scalar is the key feature of Higgs inflation [18; 19; 20; 21]. Derivatives of the field can also couple to curvature [22; 23; 24; 25; 26; 27; 28; 29; 30]. In the Higgs case, inflationary models with such couplings are called New Higgs Inflation [31; 32; 33; 34; 35; 36; 37; 38; 39]. When both derivative and non-derivative non-minimal couplings are present, the theories are sometimes called hybrid models [40; 41; 42; 43; 44; 45; 46].
Generic actions with derivative couplings to the curvature, like generic actions with higher order curvature terms, lead to higher than second order equations of motion, which involve extra degrees of freedom that suffer from the Ostrogradsky instability [47]. The most general scalar-tensor theories with second order equations of motion, called Horndeski theories, are explicitly known [48; 49; 50]. They are, however, not the most general stable scalar-tensor theories, because it is possible that the theory is degenerate and some degrees of freedom are not physical. On the gravity side, the simplest example is \(f(R)\) theory [47]. Degenerate higher order scalar-tensor theories (DHOST) have been explicitly catalogued up to terms cubic in the second derivatives of the field [49; 50]. The only such theories that are phenomenologically viable (with propagating gravitational waves and a Newtonian limit), at least at linear order in perturbation theory, are those that are related to Horndeski theories by an invertible disformal transformation [51]. Beyond DHOST are U-degenerate scalar-tensor theories, which are degenerate only in the unitary gauge, where the gradient of the scalar field has to be timelike [52; 53; 54; 55]. They have also been explicitly catalogued up to third order in second derivatives, and the procedure for determining whether a theory with arbitrary powers of second derivatives is DHOST or U-degenerate or neither is known.
These results are for the metric formulation of gravity. In other formulations that are equivalent for the Einstein-Hilbert action with minimally coupled matter but physically distinct for more complicated actions, the stability properties of non-minimally coupled scalar
fields have not been completely categorised. (For Horndeski theories in teleparallel and symmetric teleparallel gravity, see [56; 57].) One of the most common alternatives to the metric formulation is the Palatini formulation, where the connection is an independent variable [58; 59]. Higgs inflation, where the field couples directly to the Ricci scalar has been much studied in the Palatini formnulation, and the predictions are different than in the metric case [60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82]. Inflation in the case when derivatives of the field couple directly to the curvature has also been studied [83; 84; 85; 86]; in [87], such a theory was used for quintessence (see also [88; 89; 90; 91]). Unlike in the case when only the field couples directly to the curvature, in the derivative coupling case the results of the metric and the Palatini formulation are close to each other. We extend previous work by including the co-Ricci tensor in the cases when the connection is taken to be metric-compatible or torsion-free a priori.
In section 2 we give the geometrical background for the Palatini formulation and present the action. We shift to the Einstein frame with minimally coupled matter by making a disformal transformation followed by solving the remaining pieces of the connection from the equation of motion and inserting them back into the action. We calculate the leading order differences between the Palatini cases when the connection is left general, when non-metricity or torsion is put to zero, and the metric case. In section 3 we summarise our findings and outline open questions. Some technical details are relegated to appendices A and B.
## 2 Non-minimal coupling to kinetic terms
### Curvature, non-metricity, and torsion
In the Palatini formulation the metric \(g_{\alpha\beta}\) and the connection \(\Gamma^{\gamma}_{\alpha\beta}\) are independent variables. The connection, defined with the covariant derivative as \(\nabla_{\beta}A^{\alpha}=\partial_{\beta}A^{\alpha}+\Gamma^{\alpha}_{\beta \gamma}A^{\gamma}\), can be decomposed as
\[\Gamma^{\gamma}_{\alpha\beta} = \mathring{\Gamma}^{\gamma}_{\alpha\beta}+L^{\gamma}{}_{\alpha \beta}=\mathring{\Gamma}^{\gamma}_{\alpha\beta}+J^{\gamma}{}_{\alpha\beta}+K^ {\gamma}{}_{\alpha\beta}\, \tag{1}\]
where \(\mathring{\Gamma}^{\gamma}_{\alpha\beta}\) is the Levi-Civita connection of the metric \(g_{\alpha\beta}\). We denote quantities defined with the Levi-Civita connection by \(\mathring{}\). In the second equality we have decomposed the distortion tensor \(L^{\gamma}{}_{\alpha\beta}\) into the disformation tensor \(J_{\alpha\beta\gamma}\) and the contortion tensor \(K_{\alpha\beta\gamma}\), defined as
\[J_{\alpha\beta\gamma} \equiv \frac{1}{2}\left(Q_{\alpha\beta\gamma}-Q_{\gamma\alpha\beta}-Q_{ \beta\alpha\gamma}\right)\,\qquad K_{\alpha\beta\gamma}\equiv\frac{1}{2}(T_{\alpha\beta\gamma}+T_{ \gamma\alpha\beta}+T_{\beta\alpha\gamma})\, \tag{2}\]
where \(Q_{\alpha\beta\gamma}\) and \(T_{\alpha\beta\gamma}\) are the non-metricity and the torsion, respectively, defined as
\[Q_{\gamma\alpha\beta}\equiv\nabla_{\gamma}g_{\alpha\beta}\,\qquad T^{ \gamma}{}_{\alpha\beta}\equiv 2\Gamma^{\gamma}_{[\alpha\beta]}. \tag{3}\]
We have \(Q_{\gamma\alpha\beta}=Q_{\gamma(\alpha\beta)}\), \(J_{\alpha\beta\gamma}=J_{\alpha(\beta\gamma)}\), and \(K^{\gamma}{}_{\alpha}{}^{\beta}=K^{[\gamma}{}_{\alpha}{}^{\beta]}\).
The Riemann tensor can be decomposed into the Levi-Civita and the distortion contributions as
\[R^{\alpha}{}_{\beta\gamma\delta}=\mathring{R}^{\alpha}{}_{\beta \gamma\delta}+2\mathring{\nabla}_{[\gamma}L^{\alpha}{}_{\delta]\beta}+2L^{ \alpha}{}_{[\gamma|\mu|}L^{\mu}{}_{\delta]\beta}. \tag{4}\]
There are three independent first contractions of the Riemann tensor, called Ricci-type tensors,
\[R_{\alpha\beta}\equiv R^{\gamma}{}_{\alpha\gamma\beta}\,\quad\mathring{R}_{ \alpha\beta}\equiv g_{\alpha\epsilon}g^{\gamma\delta}R^{\epsilon}{}_{\gamma \delta\beta}\,\quad\mathring{R}_{\alpha\beta}\equiv R^{\gamma}{}_{\gamma \alpha\beta}. \tag{5}\]
he first is the Ricci tensor, the second is the co-Ricci tensor, and the third is the homothetic curvature tensor. There is only one independent Ricci scalar, \(R=-\hat{R}\), \(\tilde{R}=0\). Instead of the co-Ricci tensor, it can be convenient to use the average of the co-Ricci tensor and the Ricci tensor. Using the definition (5) and the decompositions (1), (2), and (4), we see that the average vanishes when \(Q_{\alpha\beta\gamma}=0\),
\[\hat{\tilde{R}}_{\alpha\beta}\equiv\frac{1}{2}(\hat{R}_{\alpha\beta}+R_{ \alpha\beta})=g^{\mu\nu}\nabla_{[\beta}Q_{\mu]\nu\alpha}-\frac{1}{2}T^{\mu\nu} {}_{\beta}Q_{\mu\nu\alpha}. \tag{6}\]
The Einstein tensor is
\[G_{\alpha\beta}\,\equiv\,-\frac{1}{4}\epsilon_{\alpha\gamma}{}^{\mu_{1}\nu_{ 1}}\epsilon_{\beta}{}^{\gamma\mu_{2}\nu_{2}}R_{\mu_{2}\nu_{2}\mu_{1}\nu_{1}}= \frac{1}{2}(R_{\alpha\beta}-\hat{R}_{\alpha\beta}-g_{\alpha\beta}R). \tag{7}\]
### The action
We consider a scalar field \(\varphi\) whose kinetic term \(X_{\alpha\beta}\equiv\partial_{\alpha}\varphi\partial_{\beta}\varphi\) couples linearly to the first traces of the Riemann tensor, while \(\varphi\) can appear non-linearly. (General non-linear couplings have been studied in [92].) The homothetic curvature tensor does not appear because it is antisymmetric, so in the Palatini case, we have couplings to \(R_{\alpha\beta}\), \(\hat{R}_{\alpha\beta}\) and \(R\), and the action is
\[S = \int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}\frac{1}{2}F(\varphi)g^{ \alpha\beta}R_{\alpha\beta}-\frac{1}{2}K(\varphi)g^{\alpha\beta}X_{\alpha\beta }+\alpha_{1}(\varphi)g^{\alpha\beta}g^{\gamma\delta}R_{\alpha\beta}X_{\gamma\delta} \tag{8}\] \[+\alpha_{2}(\varphi)g^{\alpha\gamma}g^{\beta\delta}R_{\alpha\beta }X_{\gamma\delta}+\alpha_{3}(\varphi)g^{\beta\gamma}g^{\delta\mu}R^{\alpha}{}_ {\beta\gamma\delta}X_{\alpha\mu}-V(\varphi)+\mathcal{L}_{\text{m}}(\Psi, \varphi,g^{\alpha\beta})\bigg{]}\] \[= \int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}(F+\alpha_{1}X)R- \frac{1}{2}KX+(\alpha_{2}R^{\alpha\beta}+\alpha_{3}\hat{R}^{\alpha\beta})X_{ \alpha\beta}-V+\mathcal{L}_{\text{m}}\right]\,\]
where \(g=\det g_{\alpha\beta}\), \(X\equiv g^{\alpha\beta}X_{\alpha\beta}\), and \(\mathcal{L}_{\text{m}}(\Psi,\varphi,g_{\alpha\beta})\) is a matter action1, with \(\Psi\) denoting any matter degrees of freedom other than \(\varphi\).
Footnote 1: Fermion kinetic terms involve the connection. We neglect them; it is always possible to assume that they couple only to the Levi–Civita connection, and thus do not contribute to the distortion.
In the metric case \(\hat{R}_{\alpha\beta}=-R_{\alpha\beta}\), so we can put \(\alpha_{3}=0\). Then when \(\alpha_{1}=-\frac{1}{2}\alpha_{1}\), the action is of the Horndeski form, and there are no extra degrees of freedom, otherwise there is an extra ghost [48]. If also \(\alpha_{2}>0\), the scalar degree of freedom corresponding to \(\varphi\) is healthy, otherwise it is a ghost [31].
In the Palatini case, the theory is different depending on which, if any, constraints are imposed on the connection. The case when no constraints are imposed has been studied in [88, 89]. Solving the connection equation obtained by varying (8) with respect to \(\Gamma^{\gamma}_{\alpha\beta}\) and inserting the solution into the action gives a metric theory with a modified scalar sector. For an action including (8) but more general, it was shown in [89] that the theory is at least U-degenerate (and can be DHOST or Horndeski). The reason is that it is symmetric under the projective transformation \(\Gamma^{\gamma}_{\alpha\beta}\to\Gamma^{\gamma}_{\alpha\beta}+\delta^{\gamma} {}_{\beta}V_{\alpha}\), where \(V_{\alpha}\) is an arbitrary vector field. When the gradient of the scalar field is timelike, the ghost is subsumed in the unphysical projective mode.2 The results of [89] show that for the action (8), the theory is in the DHOST class. (For the case when \(X_{\alpha\beta}\) couples only to the Ricci scalar and the Einstein tensor (7), i.e. \(\alpha_{3}=-\alpha_{2}\), this was shown already in [88].)
Footnote 2: In [53] it is argued that U-degenerate theories could be healthy. However, it is not clear how the theory behaves when spatial gradients are larger than the time derivatives [54], for example during reheating or close to the vacuum at late times. In general, projective symmetry does not guarantee the absence of ghosts, and whether ghosts appear can depend on the background [92].
We will consider the case when either non-metric
### Disformal transformation
We could solve the connection separately in the cases with zero non-metricity or zero torsion and insert the solution back into the action. However, it is easier to first get rid of all non-minimal couplings except those to \(\hat{\overleftarrow{R}}_{\alpha\beta}\) with a disformal transformation. This will also establish that the result is the same in the case when the connection is general and when the non-metricity is put to zero, and that in the zero torsion case the difference arises only from \(\hat{\overleftarrow{R}}_{\alpha\beta}\). It has been shown that observables such as inflationary power spectra are invariant under disformal transformations at least for Horndeski theories [93; 94; 95; 96; 97] (see also [98]).
We will perform an invertible disformal transformation in the action (8) such that only a coupling to \(\hat{\overleftarrow{R}}_{\alpha\beta}\) remains [92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102]:
\[g_{\alpha\beta} = \gamma_{1}(\varphi,\tilde{X})\tilde{g}_{\alpha\beta}+\gamma_{2}( \varphi,\tilde{X})X_{\alpha\beta}\, \tag{9}\]
where \(\tilde{X}\equiv\tilde{g}^{\alpha\beta}X_{\alpha\beta}\). The inverse transformation is
\[\tilde{g}_{\alpha\beta} = \tilde{\gamma}_{1}(\varphi,X)g_{\alpha\beta}+\tilde{\gamma}_{2}( \varphi,X)X_{\alpha\beta}. \tag{10}\]
The original and tilded transformation functions are related to each other as \(\tilde{\gamma}_{1}=-1/\gamma_{1}\), \(\tilde{\gamma}_{2}=-\gamma_{2}/\gamma_{1}\). The inverse metric is
\[g^{\alpha\beta}=\frac{1}{\gamma_{1}}\tilde{g}^{\alpha\beta}-\frac{\gamma_{2}} {\gamma_{1}(\gamma_{1}+\gamma_{2}\tilde{X})}\tilde{g}^{\alpha\mu}\tilde{g}^{ \beta\nu}X_{\mu\nu}\, \tag{11}\]
and \(\tilde{g}^{\alpha\beta}\) is given by the same expression with the replacements \(\gamma_{i}\to\tilde{\gamma}_{i}\), \(\tilde{X}\to X\), \(\tilde{g}^{\alpha\beta}\to g^{\alpha\beta}\). These equations give us the relation between \(X\) and \(\tilde{X}\)
\[X=\frac{\tilde{X}}{\gamma_{1}+\gamma_{2}\tilde{X}}. \tag{12}\]
As the original and tilded variables are in a symmetric position, \(X\) as a function of \(\tilde{X}\) is, again, given by the same equation with the original and tilded quantities switched. The determinants of the metrics are related by
\[g=\tilde{g}\gamma_{1}^{3}(\gamma_{1}+\gamma_{2}\tilde{X}). \tag{13}\]
Under the disformal transformation (9), the curvature coupling terms in the action (8) transform as follows
\[\sqrt{-g}g^{\alpha\beta}R_{\alpha\beta} = \sqrt{-\tilde{g}}\gamma_{1}(1+\gamma\tilde{X})^{1/2}\left(\tilde {g}^{\alpha\beta}R_{\alpha\beta}-\frac{\gamma}{1+\gamma\tilde{X}}\tilde{g}^{ \alpha\gamma}\tilde{g}^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}\right)\] \[\sqrt{-g}g^{\alpha\gamma}g^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}\] \[\sqrt{-g}g^{\beta\gamma}g^{\delta\mu}R^{\alpha}{}_{\beta\gamma \delta}X_{\alpha\mu} = \sqrt{-\tilde{g}}(1+\gamma\tilde{X})^{-1/2}\tilde{g}^{\beta\gamma }\tilde{g}^{\delta\mu}R^{\alpha}{}_{\beta\gamma\delta}X_{\alpha\mu}\, \tag{14}\]
where \(\gamma\equiv\gamma_{2}/\gamma_{1}\).
Applying the disformal transformation (9) to the action (8), using the above results, writing the co-Ricci tensor \(\hat{R}_{\alpha\beta}\) in terms of \(\hat{\overleftarrow{R}}_{\alpha\beta}\) defined in (6), and dropping the tildes on \(g_{\alpha\beta}\) and \(X\), we get
\[S = \int\mathrm{d}^{4}x\sqrt{-g}\Bigg{\{}\frac{1}{2}(1+\gamma X)^{1/2} \left(\gamma_{1}F+\frac{\alpha_{1}X}{1+\gamma X}\right)R+\frac{\alpha_{3}}{(1+ \gamma X)^{1/2}}\hat{\overleftarrow{R}}^{\alpha\beta}X_{\alpha\beta}\]
\[+\frac{1}{2}\frac{1}{(1+\gamma X)^{1/2}}\left[-F\gamma_{2}+\frac{ \alpha_{2}-\alpha_{3}-(\alpha_{1}+\alpha_{3})\gamma X}{1+\gamma X}\right]R^{ \alpha\beta}X_{\alpha\beta}\] \[-\frac{\gamma_{1}}{2(1+\gamma X)^{1/2}}KX-\gamma_{1}^{2}(1+\gamma X )^{1/2}V\] \[+\gamma_{1}^{2}(1+\gamma X)^{1/2}{\cal L}_{\rm m}\left[\Psi,\varphi,\frac{1}{\gamma_{1}}g^{\alpha\beta}-\frac{\gamma}{\gamma_{1}(1+\gamma X)}g^{ \alpha\mu}g^{\beta\nu}X_{\mu\nu}\right]\Bigg{\}}. \tag{15}\]
The non-minimal couplings to \(R\) and \(R_{\alpha\beta}\) are eliminated by choosing
\[(1+\gamma X)^{1/2}\left(\gamma_{1}F+\frac{\alpha_{1}X}{1+\gamma X }\right)=1\] \[F\gamma_{2}+\frac{\alpha_{2}-\alpha_{3}-(\alpha_{1}+\alpha_{3}) \gamma X}{1+\gamma X}=0. \tag{16}\]
From (16) we can solve for \(\gamma_{1}\) and \(\gamma_{2}\) in closed form. The solutions are not very illuminating, so we do not write them down. For \(\alpha_{3}=0\) they simplify; the case \(\alpha_{1}=-\frac{1}{2}\alpha_{2}\), \(\alpha_{3}=0\) is given in [85].
The disformal transformation is invertible and the original and transformed metric describe the same physics when \(\gamma_{1}>0\), \(\gamma_{2}\geq 0\), \(\gamma_{1}+\tilde{X}\gamma_{2}>0\), \(\tilde{\gamma}_{1}-X\partial_{X}\tilde{\gamma}_{1}-X^{2}\partial_{X}\tilde{ \gamma}_{2}\neq 0\). These conditions set a limit on the values \(X\) can take. This is a limitation of the disformal transformation. Large spatial gradients such as may occur during preheating may mean that the coefficient of the Ricci tensor is not positive, so that even in the case \(\alpha_{3}=0\), the theory cannot be mapped to a minimally coupled Einstein frame with a disformal transformation. For study of slow-roll inflation in the super-Hubble regime, this is not a problem.
Inserting \(\gamma_{1}\) and \(\gamma_{2}\) back into the action (15), we get (dropping the matter Lagrangian)
\[S=\int{\rm d}^{4}x\sqrt{-g}\left(\frac{1}{2}R+{\cal G}_{1}\hat{\not{R}}^{ \alpha\beta}X_{\alpha\beta}-\frac{1}{2}{\cal G}_{2}KX-{\cal G}_{3}V\right)\, \tag{17}\]
where we have defined
\[{\cal G}_{1}\equiv\frac{\alpha_{3}}{(1+\gamma X)^{1/2}}\,\ {\cal G}_{2}\equiv \frac{\gamma_{1}}{(1+\gamma X)^{1/2}}\,\ {\cal G}_{3}\equiv\gamma_{1}^{2}(1+\gamma X)^{1/2}. \tag{18}\]
If \(\alpha_{3}=0\), then \({\cal G}_{1}=0\). Then the Ricci scalar is the only term that contains the connection, so the connection equation of motion gives the Levi-Civita connection. (In the case when there are no a priori constraints on the connection, it is determined only up to a projective transformation.) Inserting it back into the action we obtain a metric theory with a minimally coupled scalar field. The physics related to the distortion has been shifted to the modifications of the scalar field kinetic term and potential (and the matter Lagrangian). In the Einstein frame all matter couples to the scalar field and its kinetic term. We have not assumed anything about the connection, showing that if the co-Ricci tensor does not appear in the action, the physics is the same whether we keep the connection general or put non-metricity or torsion to zero. This is also the case in Palatini \(f(R)\) theory, which can be reduced to the Einstein-Hilbert plus minimally coupled matter form via field transformations [103; 104; 105; 106]. If the non-metricity is put to to zero a priori, (6) shows that \(\hat{R}_{\alpha\beta}=-R_{\alpha\beta}\), so the co-Ricci tensor is not independent, and there is no \(\alpha_{3}\) coupling.
In any case, if \(\alpha_{3}=0\), the action (17) reduces to
\[S = \int{\rm d}^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\alpha\beta}R_{\alpha \beta}-\frac{1}{2}\frac{\gamma_{1}}{(1+\gamma X)^{1/2}}KX-\gamma_{1}^{2}(1+ \gamma X)^{1/2}V\right]\]
\[\simeq \int{\rm d}^{4}x\sqrt{-g}\Bigg{\{}\frac{1}{2}\dot{\hat{R}}-\frac{KX}{ 2F}\left[1-\left(\alpha_{1}+\alpha_{2}\right)X\right] \tag{19}\] \[-\frac{V}{F^{2}}\left[1-\left(2\alpha_{1}+\frac{1}{2}\alpha_{2} \right)X+\left(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+\frac{5}{8}\alpha_{2}^{2} \right)X^{2}\right]\Bigg{\}}\,\]
where in the second equality we have expanded to second order in \(X_{\alpha\beta}\). For \(\alpha_{1}=-\frac{1}{2}\alpha_{2}\), \(\alpha_{3}=0\) the result agrees with [85].
The action (19) is manifestly in the Horndeski class. We noted in section 2.2 that based on the results of [89], the action is of the DHOST form. However, as written in the introduction, the only viable DHOST theories (at least to cubic order in second derivatives) seem to be those that are related to Horndeski theories by an invertible disformal transformation. There is no physical difference between Horndeski and DHOST theories as regards physical degrees of freedom and stability.
### Zero torsion case with \(\alpha_{3}\neq 0\)
When \(\alpha_{3}\neq 0\) and the non-metricity is non-zero, we have to solve the connection equation of motion and insert the solution back into the action. Let us consider the case with zero torsion. (The case with no constraints was considered in [89].) Varying the action (17) with respect to the distortion tensor (taking into account that it is symmetric in the last two indices) gives the equation of motion
\[0 = g_{\beta\gamma}L^{\delta}{}_{\delta\alpha}-L_{(\beta\gamma) \alpha}-L_{(\beta|\alpha|\gamma)}+g_{\alpha(\beta}L_{\gamma)}{}^{\delta} \tag{20}\] \[+{\cal G}_{1}\big{(}L^{\delta}{}_{\delta\alpha}X_{\beta\gamma}- g_{\beta\gamma}L^{\delta}{}_{\alpha}X_{\delta\epsilon}-X_{\alpha}Y_{\beta\gamma}+g_{ \beta\gamma}\mathring{\nabla}_{\delta}X_{\alpha}{}^{\delta}+g_{\alpha(\beta} \mathring{\nabla}^{\delta}X_{\gamma)\delta}+L_{(\beta}{}^{\delta}{}_{\gamma) }X_{\alpha\delta}\] \[-L_{(\beta|\alpha|}{}^{\delta}X_{\gamma)\delta}-L_{(\beta}{}^{ \delta}{}_{|\alpha|}X_{\gamma)\delta}-L_{(\beta}{}^{\delta}{}_{|\delta}X_{ \alpha|\gamma)}+L^{\delta}{}_{(\beta|\alpha|}X_{\gamma)\delta}-\frac{3}{2} \mathring{\nabla}_{\alpha}X_{\beta\gamma}+g_{\alpha(\beta}L_{\gamma)}{}^{ \delta\epsilon}X_{\delta\epsilon}\big{)}\] \[+{\cal G}_{1}^{\prime}\big{(}Xg_{\beta\gamma}X_{\alpha}+Xg_{ \alpha(\beta}X_{\gamma)}-2X_{\alpha}X_{\beta\gamma}\big{)}\] \[+\partial_{X}{\cal G}_{1}\big{(}g_{\beta\gamma}X_{\alpha\delta} \mathring{\nabla}^{\delta}X+g_{\alpha(\beta}X_{\gamma)}{}^{\delta}\mathring{ \nabla}_{\delta}X-X_{\beta\gamma}\mathring{\nabla}_{\alpha}X-X_{\alpha(\beta} \mathring{\nabla}_{\gamma)}X\big{)}\,\]
where \(X_{\alpha}\equiv\partial_{\alpha}\varphi\), \(Y_{\alpha\beta}\equiv\mathring{\nabla}_{\alpha}\mathring{\nabla}_{\beta}\varphi\), and prime denotes partial derivative with respect to \(\varphi\). The general solution has the form
\[L_{\alpha\beta\gamma} = l_{1}g_{\beta\gamma}X_{\alpha}+l_{2}g_{\alpha(\beta}X_{\gamma)}+ l_{3}X_{\alpha}Y_{\beta\gamma}+l_{4}\mathring{\nabla}_{\alpha}X_{\beta\gamma}+l_{5}g_{ \beta\gamma}\mathring{\nabla}_{\alpha}X+l_{6}g_{\beta\gamma}\mathring{\nabla }_{\delta}X_{\alpha}{}^{\delta} \tag{21}\] \[+l_{7}g_{\alpha(\beta}\mathring{\nabla}^{\delta}X_{\gamma)\delta }+l_{8}g_{\alpha(\beta}\mathring{\nabla}_{\gamma)}X+l_{9}X_{\alpha}X_{\beta \gamma}+l_{10}X_{\beta\gamma}\mathring{\nabla}_{\alpha}X+l_{11}X_{\beta\gamma }\mathring{\nabla}_{\delta}X_{\alpha}{}^{\delta}\] \[+l_{12}g_{\beta\gamma}X_{\alpha\delta}\mathring{\nabla}^{\delta}X +l_{13}X_{\alpha(\beta}\mathring{\nabla}_{\gamma)}X+l_{14}g_{\alpha(\beta}X_{ \gamma)}{}^{\delta}\mathring{\nabla}_{\delta}X+l_{15}X_{\alpha\delta}X_{\beta \gamma}\mathring{\nabla}^{\delta}X\,\]
Inserting (21) into (20), we solve for the coefficients \(l_{i}(\varphi,X)\). The result is rather lengthy and is given in appendix A. Inserting \(l_{i}\) into the action (17), we get the minimally coupled action
\[S = \int{\rm d}^{4}x\sqrt{-g}\bigg{(}\frac{1}{2}\dot{\hat{R}}-\frac{1 }{2}{\cal G}_{2}KX-{\cal G}_{3}V+{\cal B}_{1}+{\cal B}_{2}Y+{\cal B}_{3}X^{ \alpha\beta}Y_{\alpha\beta} \tag{22}\] \[+{\cal A}_{1}Y_{\alpha\beta}Y^{\alpha\beta}+{\cal A}_{2}Y^{2}+{ \cal A}_{3}X^{\alpha\beta}Y_{\alpha\beta}Y+{\cal A}_{4}X^{\alpha\beta}Y_{\alpha }{}^{\gamma}Y_{\beta\gamma}+{\cal A}_{5}X^{\alpha\beta}X^{\gamma\delta}Y_{ \alpha\beta}Y_{\gamma\delta}\bigg{)}\,\]
where the coefficients \({\cal B}_{1}(\varphi,X)\) and \({\cal A}_{1}(\varphi,X)\) are again relegated to appendix A. The terms on the second line are non-Horndeski, but the functions \({\cal A}_{i}\) satisfy the conditions for the theory
to be DHOST [49; 50]. In order to obtain a minimally coupled action, it was important to consider coupling to \(\hat{\bar{R}}_{\alpha\beta}\), which vanishes for the Levi-Civita connection, rather than \(\hat{R}_{\alpha\beta}\). In [88; 89] the couplings to \(\hat{R}_{\alpha\beta}\) were instead eliminated by writing them in terms of the commutator of the Levi-Civita covariant derivative (the action was not transformed into the Einstein frame). This leads to a different form of the action; it is well known that a Horndeski or a DHOST theory can take quite different-looking forms.
To second order in \(X\), the action (2.22) reads
\[S = \int\mathrm{d}^{4}x\sqrt{-g}\bigg{\{}\frac{1}{2}\hat{R}-\frac{KX}{ 2F}[1-(\alpha_{1}+\alpha_{2}-\alpha_{3})X]-\frac{V}{F^{2}}\Big{[}1-\big{(}2 \alpha_{1}+\frac{1}{2}[\alpha_{2}-\alpha_{3}]\big{)}X \tag{2.23}\] \[+\big{(}\alpha_{1}^{2}+\frac{5}{8}\alpha_{2}^{2}+2\alpha_{1}[ \alpha_{2}-\alpha_{3}]-\frac{3}{4}\alpha_{2}\alpha_{3}+\frac{1}{8}\alpha_{3}^ {2}\big{)}X^{2}\Big{]}\] \[+\frac{5}{8}\alpha_{3}^{2}XY_{\alpha\beta}Y^{\alpha\beta}-\frac{ 13}{24}\alpha_{3}^{2}XY^{2}-\frac{11}{12}\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha \beta}Y+\frac{5}{6}\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta \gamma}\bigg{\}}\.\]
In [92] it was shown that an action that depends on \(\hat{R}_{\alpha\beta}\) has a ghost around Minkowski space in the zero torsion case, and that in the general connection case there is a ghost around some FLRW backgrounds. This is not in contradiction with our result and the results of [88; 89] that these cases are stable. In [92] it was assumed that the Legendre transformation to the Einstein frame is non-degenerate, which means that all degrees of freedom in \(\hat{R}_{\alpha\beta}\) are included in the Einstein frame action. In our case with a scalar field, there are no vector or tensor modes. In [92] the FLRW ghost was in the tensor sector.
### Metric case
Let us compare the Palatini case result (2.22) to the metric case. We again start with the action (2.8), now assuming that the connection is Levi-Civita. The action is of the Horndeski form when the kinetic term couples only to the Einstein tensor \(G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R\), not to the Ricci tensor and the Ricci scalar separately, otherwise it has a ghost. So we set \(\alpha_{1}=-\frac{1}{2}\alpha_{2}\), \(\alpha_{3}=0\). We again shift to the Einstein frame with the disformal transformation (2.9). Now the calculation is more involved, because the Riemann tensor depends on the metric and its first and second derivative, unlike in the Palatini case. Hence, it is not invariant under the disformal transformation, which now introduces second order derivatives of \(\varphi\). The transformation rules of the connection, the Riemann tensor, the Ricci tensor and the Ricci scalar are somewhat lengthy, and are given in appendix B. Inserting the result of the disformal transformation into the action (2.8), expanding to second order in \(X_{\alpha\beta}\) and choosing the disformal functions \(\gamma_{1}\) and \(\gamma_{2}\) so that the non-minimal couplings vanish, we get the Einstein frame action
\[S = \int\mathrm{d}^{4}x\sqrt{-g}\Bigg{\{}\frac{1}{2}\hat{R}-\frac{KX} {2F}\left(1-\frac{\alpha_{2}X}{2}\right)-\frac{3}{4}\frac{F^{\prime 2}}{F^{2}}X- \frac{V}{F^{2}}\left(1+\frac{\alpha_{2}X}{2}-\frac{\alpha_{2}^{2}X^{2}}{8}\right) \tag{2.24}\] \[+\frac{3}{2}\frac{(\alpha_{2}F)^{\prime}}{2F}\frac{F^{\prime}}{F }X^{2}-\frac{(\alpha_{2}F)^{\prime}}{2F}XY+\frac{(\alpha_{2}F)^{\prime}}{2F} X^{\alpha\beta}Y_{\alpha\beta}\] \[+\frac{1}{2}\alpha_{2}^{2}X^{\alpha\beta}Y_{\alpha\beta}Y-\frac{ 1}{2}\alpha_{2}^{2}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta\gamma}\Bigg{\}}\.\]
To first order in \(X_{\alpha\beta}\) and \(Y_{\alpha\beta}\), we recover the result of the original New Higgs Inflation paper [31], apart from the term involving \(F^{\prime}\). (The hybrid case with both \(F^{\prime}\neq 0\) and \(\alpha_{2}\neq 0\)
has been studied in [46].) Apart from the \(F^{\prime}\) term, this leading order result agrees with the Palatini action (2.19) when \(\alpha_{1}=-\frac{1}{2}\alpha_{2}\), \(\alpha_{3}=0\), as observed in [39]. This is easy to understand: the distortion is sourced by \(F^{\prime}\) and \(X_{\alpha\beta}\), and only appears in the action via the total derivative and quadratic terms in the Riemann tensor (2.4) and the coupling to the kinetic terms. So if \(F^{\prime}=0\), the distortion only enters at second order in \(X_{\alpha\beta}\). The second order terms on the first line of (2.24) also agree with the Palatini result, which is less obvious. It is only the non-Horndeski terms that are different.
However, in the Palatini case we can obtain the same action to first (but not second) order in \(X_{\alpha\beta}\) and \(Y_{\alpha\beta}\) by coupling to just \(R\), i.e. with \(\alpha_{2}=\alpha_{3}=0\). In the metric case such a coupling would lead to a ghost. Also, in the metric case the derivatives of \(F\) and \(\alpha_{2}\) enter, unlike in the Palatini case. The terms involving \(Y_{\alpha\beta}\) are also different: in the Palatini case they appear only if \(\alpha_{3}\neq 0\). If the dynamics are dominated by the derivative coupling, the differences are small in slow-roll, but if the non-derivative coupling is important, the theories can have quite different predictions, as comparison of [46] and [85] shows.
## 3 Conclusions
We have considered a theory where a scalar field kinetic term couples linearly to the Ricci tensor and the co-Ricci tensor, which appear linearly in the action, while the field itself can have non-linear non-minimal couplings. We look at both the Palatini formulation and the metric formulation. Extending previous Palatini work, we consider the case when either the non-metricity or the torsion is taken to vanish a priori. To establish the stability properties of the different cases and compare them side-by-side, we use a disformal transformation, followed by solving for the connection and inserting the solution back into the action. In this way we reduce the different cases to metric gravity with the Einstein-Hilbert action minimally coupled to matter.
We find that all the Palatini cases we consider are ghost-free: they are either in the Horndeski or DHOST class. If there is no coupling to the co-Ricci tensor, the Palatini result is independent of the assumptions about the connection. Otherwise, the case with a general connection and the case with zero torsion are physically different. (If non-metricity is zero, the co-Ricci tensor vanishes.) We expand the actions up to second order in the scalar field kinetic term and compare the differences.
At leading order, the metric case and all the Palatini cases all agree with each other. However, in the Palatini case a much wider range of couplings is stable, for example it is possible to simply couple the Ricci scalar to the trace of the kinetic term, simplifying the model. At second order, the Horndeski terms agree in the Palatini and metric cases, but the beyond Horndeski terms are different. The detailed form of the terms beyond the leading order might appear contrived if written in the metric formulation to begin with, but in the original Palatini formulation they are simple. The Palatini formulation can be seen as a selection principle to determine which complicated derivative couplings should appear in a metric formulation action.
Higgs inflation driven by derivative couplings in the metric formulation does not have a unitarity problem, unlike the metric formulation of the original Higgs inflation scenario with a non-derivative coupling to the Ricci scalar alone [33; 35; 36; 39]. The theory is however sensitive to loop corrections [39]. It would be interesting to see whether these features change in the derivative-driven or hybrid Palatini case. In the case with a non-derivative non-minimal
coupling to the Einstein tensor alone, the unitarity problem is ameliorated in the Palatini formulation [71; 72; 74; 75; 76; 77; 78; 71].
It is an interesting question how to characterise the stability properties of theories in the Palatini formulation without reducing the theory a metric equivalent or calculating propagators. In [89] projective symmetry was used to show that a theory is U-degenerate, as the ghosts appear only in the unphysical projective mode. Projective invariance does not guarantee the absence of ghosts in general [92], only in particular cases [89; 107; 91]. It would be interesting to understand better theories whose structure is tuned to the projective symmetry so that it makes them stable, and in particular whether projective symmetry (which has only a vectorial gauge mode) can prevent terms that would lead to tensor ghosts.
###### Acknowledgements.
We thank Katsuki Aoki and Keigo Shimada for helpful correspondence. HBN acknowledges the Mathematica xAct package [108] used in the calculations.
## Appendix A Details of the solution for the connection in the zero torsion case
We give here details of the connection calculation in section 2.4 in the case when \(\alpha_{3}\neq 0\) and the torsion is zero. The general solution of the connection equation of motion (20) in terms of the coefficients (21) is
\[l_{1} =\frac{\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X^{2}}{1+\mathcal{G }_{1}X}\] \[l_{2} =-\frac{2\mathcal{G}_{1}^{2}\mathcal{G}_{1}^{\prime}X^{3}}{1- \mathcal{G}_{1}^{2}X^{2}}\] \[l_{3} =\frac{2\mathcal{G}_{1}}{2-\mathcal{G}_{1}X}\] \[l_{4} =\frac{6\mathcal{G}_{1}(1-\mathcal{G}_{1}X)}{(2-\mathcal{G}_{1}X )^{2}}\] \[l_{5} =-\frac{12\partial_{X}\mathcal{G}_{1}X+9\mathcal{G}_{1}^{3}X^{2} +5\mathcal{G}_{1}^{4}X^{3}-2\mathcal{G}_{1}^{2}X(19-6\partial_{X}\mathcal{G}_ {1}X^{2})+6\mathcal{G}_{1}(2-5\partial_{X}\mathcal{G}_{1}X^{2})}{3(2-\mathcal{ G}_{1}X)(1+\mathcal{G}_{1}X)(6-5\mathcal{G}_{1}X)}\] \[l_{6} =\frac{2\mathcal{G}_{1}(1-\mathcal{G}_{1}X)}{6-3\mathcal{G}_{1}X}\] \[l_{7} =\frac{4\mathcal{G}_{1}\big{[}1-\mathcal{G}_{1}X(1-\mathcal{G}_{ 1}X)\big{]}}{3(2-\mathcal{G}_{1}X)^{2}}\] \[l_{8} =-\frac{2}{3(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_{1}X)(6-5 \mathcal{G}_{1}X)}\big{[}12\partial_{X}\mathcal{G}_{1}X+6\mathcal{G}_{1}^{4} X^{3}-5\mathcal{G}_{1}^{5}X^{4}+3\mathcal{G}_{1}^{3}\partial_{X}\mathcal{G}_{1}X^{4}\] \[+6\mathcal{G}_{1}(2-\partial_{X}\mathcal{G}_{1}X^{2})-\mathcal{G }_{1}^{2}(8X+6\partial_{X}\mathcal{G}_{1}X^{3})\big{]}\] \[l_{9} =\frac{2(\mathcal{G}_{1}^{\prime}+\mathcal{G}_{1}^{2}\mathcal{G}_ {1}^{\prime}X^{2})}{1-\mathcal{G}_{1}^{2}X^{2}}\] \[l_{10} =\frac{125\mathcal{G}_{1}^{2}+84\partial_{X}\mathcal{G}_{1}}{264- 220\mathcal{G}_{1}X}+\frac{4\mathcal{G}_{1}^{2}}{(2-\mathcal{G}_{1}X)^{2}}- \frac{25\mathcal{G}_{1}^{2}+4\partial_{X}\mathcal{G}_{1}}{12(2-\mathcal{G}_ {1}X)}-\frac{28(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{33(1+ \mathcal{G}_{1}X)}\] \[l_{11} =\frac{2\mathcal{G}_{1}^{2}(1-2\mathcal{G}_{1}X)}{3(2-\mathcal{G }_{1}X)^{2}}\]
\[l_{12}=\partial_{X}\mathcal{G}_{1}+\frac{\mathcal{G}_{1}^{2}}{12-6 \mathcal{G}_{1}X}+\frac{5(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{11(1+ \mathcal{G}_{1}X)}-\frac{125\mathcal{G}_{1}^{2}+84\partial_{X}\mathcal{G}_{1}}{6 6(6-5\mathcal{G}_{1}X)}\] \[l_{13}=\frac{2}{(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_{1}X)^{2 }(6-5\mathcal{G}_{1}X)}\big{\{}\mathcal{G}_{1}^{3}X\big{[}15+\mathcal{G}_{1}X( 19-20\mathcal{G}_{1}X)\big{]}\] \[+\partial_{X}\mathcal{G}_{1}(2-\mathcal{G}_{1}X)(6+10\mathcal{G}_ {1}X-5\mathcal{G}_{1}^{2}X^{2}-\mathcal{G}_{1}^{3}X^{3})\big{\}}\] \[l_{14}=2\partial_{X}\mathcal{G}_{1}+\frac{125\mathcal{G}_{1}^{2} +84\partial_{X}\mathcal{G}_{1}}{792-660\mathcal{G}_{1}X}+\frac{5\mathcal{G}_{ 1}^{2}+4\partial_{X}\mathcal{G}_{1}}{24-12\mathcal{G}_{1}X}-\frac{\mathcal{G} _{1}^{2}+\partial_{X}\mathcal{G}_{1}}{(2-\mathcal{G}_{1}X)^{2}}-\frac{ \mathcal{G}_{1}^{2}+\partial_{X}\mathcal{G}_{1}}{1-\mathcal{G}_{1}X}\] \[+\frac{31(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{33( 1+\mathcal{G}_{1}X)}\] \[l_{15}=-\frac{2\mathcal{G}_{1}}{3(2-\mathcal{G}_{1}X)^{2}(1- \mathcal{G}_{1}X)(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\big{[}12 \partial_{X}\mathcal{G}_{1}-126\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}X+ 10\mathcal{G}_{1}^{6}X^{4}\] \[-12\mathcal{G}_{1}^{4}X^{2}(3+5\partial_{X}\mathcal{G}_{1}X^{2}) -\mathcal{G}_{1}^{5}X^{3}(7-15\partial_{X}\mathcal{G}_{1}X^{2})+\mathcal{G}_{ 1}^{3}X(73+33\partial_{X}\mathcal{G}_{1}X^{2})\] \[-2\mathcal{G}_{1}^{2}(26-57\partial_{X}\mathcal{G}_{1}X^{2}) \big{]}. \tag{15}\]
The coefficients of the final Einstein frame action (22) are
\[\mathcal{B}_{1}=\frac{3\mathcal{G}_{1}^{2}\mathcal{G}_{1}^{\prime 2}X^{ 5}}{4-4\mathcal{G}_{1}^{2}X^{2}}\] \[\mathcal{B}_{2}=-\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X^{2}\] \[\mathcal{B}_{3}=\frac{\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X \big{[}1+\mathcal{G}_{1}X^{2}(2\mathcal{G}_{1}+3\partial_{X}\mathcal{G}_{1}X )\big{]}}{1-\mathcal{G}_{1}^{2}X^{2}}\] \[\mathcal{A}_{1}=\frac{\mathcal{G}_{1}^{2}X(5-4\mathcal{G}_{1}X)} {2(2-\mathcal{G}_{1}X)^{2}}\] \[\mathcal{A}_{2}=-\frac{\mathcal{G}_{1}^{2}X(13-12\mathcal{G}_{1}X +\mathcal{G}_{1}^{2}X^{2})}{6(2-\mathcal{G}_{1}X)^{2}}\] \[\mathcal{A}_{3}=-\frac{1}{3}\mathcal{G}_{1}\Big{[}6\partial_{X} \mathcal{G}_{1}X+\mathcal{G}_{1}\frac{11-12\mathcal{G}_{1}X+4\mathcal{G}_{1}^{ 2}X^{2}}{(2-\mathcal{G}_{1}X)^{2}}\Big{]}\] \[\mathcal{A}_{4}=\frac{4}{(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_ {1}X)^{2}(6-5\mathcal{G}_{1}X)}\Big{\{}2(\partial_{X}\mathcal{G}_{1})^{2}X^{ 2}+6\mathcal{G}_{1}^{5}X^{3}-5\mathcal{G}_{1}^{6}X^{4}\] \[+5\mathcal{G}_{1}^{4}\partial_{X}\mathcal{G}_{1}X^{4}-\mathcal{G} _{1}^{3}\partial_{X}\mathcal{G}_{1}X^{3}(16-\partial_{X}\mathcal{G}_{1}X^{2}) +\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}X(14+3\partial_{X}\mathcal{G}_{1}X ^{2})\] \[+\mathcal{G}_{1}^{2}\big{[}5+\partial_{X}\mathcal{G}_{1}X^{2}(5-4 \partial_{X}\mathcal{G}_{1}X^{2})\big{]}\Big{\}}\] \[\mathcal{A}_{5}=-\frac{1}{3(2-\mathcal{G}_{1}X)^{2}(1-\mathcal{G} _{1}X)(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\Big{\{}24(\partial_{X} \mathcal{G}_{1})^{2}X+20\mathcal{G}_{1}^{8}X^{5}\] \[+12\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}(2+\partial_{X} \mathcal{G}_{1}X^{2})+12\mathcal{G}_{1}^{2}\partial_{X}\mathcal{G}_{1}X(1-25 \partial_{X}\mathcal{G}_{1}X^{2})-\mathcal{G}_{1}^{7}(64X^{4}-60\partial_{X} \mathcal{G}_{1}X^{6})\] \[+\mathcal{G}_{1}^{3}\big{[}2-48\partial_{X}\mathcal{G}_{1}X^{2}(9-5 \partial_{X}\mathcal{G}_{1}X^{2})\big{]}+\mathcal{G}_{1}^{6}X^{3}\big{[}23-9 \partial_{X}\mathcal{G}_{1}X^{2}(28-5\partial_{X}\mathcal{G}_{1}X^{2})\big{]}\] \[-2\mathcal{G}_{1}^{4}X\big{[}62-3\partial_{X}\mathcal{G}_{1}X^{2}(6 1+25\partial_{X}\mathcal{G}_{1}X^{2})\big{]}\] \[+\mathcal{G}_{1}^{5}X^{2}[125+3\partial_{X}\mathcal{G}_{1}X^{2}(6 2-63\partial_{X}\mathcal{G}_{1}X^{2})]\Big{\}}. \tag{16}\]
## Appendix B Disformal transformation in the metric formulation
Under the disformal transformation (11), the Levi-Civita connection transforms as
\[\mathring{\Gamma}^{\gamma}{}_{\alpha\beta}\rightarrow\mathring{\Gamma}^{\gamma}{} _{\alpha\beta}+\frac{-\gamma_{1}^{\prime}g_{\alpha\beta}+2\gamma_{2}Y_{\alpha \beta}}{2(\gamma_{1}+\gamma_{2}X)}X^{\gamma}-\frac{1}{2\gamma_{1}}(\partial_{X} \gamma_{1}g_{\alpha\beta}+\partial_{X}\gamma_{2}X_{\alpha\beta})\mathring{\nabla}^{ \gamma}X+\frac{\gamma_{1}^{\prime}}{2\gamma_{1}}\delta_{(\alpha}{}^{\gamma}X_{\beta)} \tag{17}\]
\[+\frac{\partial_{X}\gamma_{1}}{2\gamma_{1}}\delta_{(\alpha}{}^{\gamma} \hat{\nabla}_{\beta)}X+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)}\Big{\{} \partial_{X}\gamma_{1}\gamma_{2}g_{\alpha\beta}X^{\gamma}{}_{\delta}\hat{ \nabla}^{\delta}X\] \[+\big{[}(-2\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{2}^{ \prime})X^{\gamma}+\gamma_{2}\partial_{X}\gamma_{2}X^{\gamma}{}_{\delta}\hat{ \nabla}^{\delta}X\big{]}X_{\alpha\beta}\] \[+2(-\partial_{X}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma_ {2})X_{(\alpha}{}^{\gamma}\hat{\nabla}_{\beta)}X\Big{\}}\;. \tag{10}\]
For the Riemann tensor we get
\[\hat{R}^{\alpha}{}_{\beta\gamma\delta} \rightarrow \hat{R}^{\alpha}{}_{\beta\gamma\delta}+\frac{\partial_{X}\gamma_{ 1}g_{\beta[\gamma}\dot{\bar{\nabla}}{}^{\alpha}\dot{\bar{\nabla}}{}_{\delta]} X-\partial_{X}\gamma_{1}\delta_{[\gamma}{}^{\alpha}\dot{\bar{\nabla}}{}_{[\beta]} \dot{\bar{\nabla}}{}_{\delta]}X+\partial_{X}\gamma_{2}X_{\beta[\gamma}\dot{ \bar{\nabla}}{}^{\alpha}\dot{\bar{\nabla}}{}_{\delta]}X}{\gamma_{1}}\] (11) \[+\frac{\gamma_{1}^{\prime}g_{\beta[\gamma}Y_{\delta]}{}^{\alpha}- \gamma_{1}^{\prime}\delta_{[\gamma}{}^{\alpha}Y_{[\beta]\delta]}+2\gamma_{2}(X ^{\alpha}\dot{\bar{\nabla}}{}_{[\gamma}Y_{[\beta]\delta]}-Y_{\beta[\gamma}Y_{ \delta]}{}^{\alpha})}{\gamma_{1}+\gamma_{2}X}\] \[+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\{}\big{[} -2\gamma_{1}^{2}\gamma_{1}^{\prime\prime}+2\gamma_{1}^{\prime 2}\gamma_{2}X+ \gamma_{1}(3\gamma_{1}^{\prime 2}-2\gamma_{1}^{\prime\prime}\gamma_{2}X+ \gamma_{1}^{\prime}\gamma_{2}^{\prime}X)\big{]}g_{\beta[\gamma}X_{\delta]}{}^{\alpha}\] \[+\big{[}-2\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}+2 \partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}(3\partial_{X }\gamma_{1}\gamma_{1}^{\prime}+\gamma_{1}^{\prime}\gamma_{2}-2\partial_{X} \gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X) \big{]}\] \[g_{\beta[\gamma}X^{\alpha}\dot{\bar{\nabla}}{}_{\delta]}X\Big{\}} +\frac{1}{2\gamma_{1}^{2}}\big{[}(\partial_{X}\gamma_{1})^{2}\dot{\bar{\nabla} }{}_{\epsilon}X\dot{\bar{\nabla}}{}^{\epsilon}Xg_{\beta[\gamma}{}_{\delta]}{} ^{\alpha}-\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}\dot{\bar{\nabla}}{}_{ \epsilon}X\dot{\bar{\nabla}}{}^{\epsilon}X\delta_{[\gamma}{}^{\alpha}X_{[ \beta]\delta]}\] \[-(3(\partial_{X}\gamma_{1})^{2}-2\gamma_{1}\partial_{X}^{2}\gamma _{1})(g_{\beta[\gamma}\dot{\bar{\nabla}}{}_{\delta]}X\dot{\bar{\nabla}}{}^{ \alpha}X-\delta_{[\gamma}{}^{\alpha}\dot{\bar{\nabla}}{}_{\delta]}X\dot{\bar{ \nabla}}{}_{\beta}X)\big{]}\] \[+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)}\big{[}\gamma_{1}^{ \prime 2}Xg_{\beta[\gamma}\delta_{\delta]}{}^{\alpha}+2\partial_{X}\gamma_{1} \gamma_{1}^{\prime}X^{\epsilon}\dot{\nabla}{}_{\epsilon}Xg_{\beta[\gamma} \delta_{\delta]}{}^{\alpha}\] \[-2\partial_{X}\gamma_{1}\gamma_{2}X^{\epsilon}\dot{\bar{\nabla} }{}_{\epsilon}Xg_{\beta[\gamma}Y_{\delta]}{}^{\alpha}+2\partial_{X}\gamma_{1} \gamma_{2}X^{\epsilon}\dot{\bar{\nabla}}{}_{\epsilon}X\delta_{[\gamma}{}^{ \alpha}Y_{[\beta]\delta]}+(4\gamma_{1}^{\prime}\gamma_{2}-2\gamma_{1}\gamma_{ 2}^{\prime})\] \[X_{\beta[\gamma}Y_{\delta]}{}^{\alpha}-2\gamma_{2}\partial_{X} \gamma_{2}X^{\epsilon}\dot{\bar{\nabla}}{}_{\epsilon}XX_{\beta[\gamma}Y_{ \delta]}{}^{\alpha}+(2\partial_{X}\gamma_{1}\gamma_{2}-2\gamma_{1}\partial_{X} \gamma_{2})(X_{[\gamma}{}^{\alpha}\dot{\bar{\nabla}}{}_{\beta]}\dot{\bar{ \nabla}}{}_{\delta]}X\] \[-X_{[\gamma}Y_{[\beta]\delta]}\dot{\bar{\nabla}}{}^{\alpha}X+X_{[ \gamma}Y_{\delta]}{}^{\alpha}\dot{\bar{\nabla}}{}_{\beta}X-X_{\beta}Y_{[\gamma} {}^{\alpha}\dot{\bar{\nabla}}{}_{\delta]}X)-2\partial_{X}\gamma_{1}\gamma_{2} g_{\beta[\gamma}X^{\alpha\epsilon}\dot{\bar{\nabla}}{}_{[\epsilon]}\dot{\bar{\nabla}}{}_{ \delta]}X\] \[-2\gamma_{2}\partial_{X}\gamma_{2}X_{\beta[\gamma}X^{\alpha\epsilon }\dot{\bar{\nabla}}{}_{[\epsilon]}\dot{\bar{\nabla}}{}_{\delta]}X\Big{]}+ \frac{1}{(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\{}(\gamma_{1}\gamma_{2}^{\prime}- \gamma_{1}^{\prime}\gamma_{2})X_{[\gamma}{}^{\alpha}Y_{[\beta]\delta]}+\big{[} \gamma_{2}(\partial_{X}\gamma_{1}\] \[+\gamma_{2})-\gamma_{1}\partial_{X}\gamma_{2}\big{]}X^{\alpha}Y_{[\beta [\gamma}\dot{\bar{\nabla}}{}_{\delta]}X\Big{\}}-\frac{1}{2\gamma_{1}^{2}( \gamma_{1}+\gamma_{2}X)}\Big{\{}(\partial_{X}\gamma_{1})^{2}\gamma_{2}X_{ \epsilon}\dot{\bar{\nabla}}{}^{\epsilon}X\dot{\bar{\nabla}}{}^{\epsilon}X\dot{ \bar{\nabla}}{}^{\zeta}Xg_{\beta[\gamma}{}^{\alpha}\delta_{]}{}^{\alpha}\] \[+\partial_{X}\gamma_{1}(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1} \partial_{X}\gamma_{2})\dot{\bar{\nabla}}{}_{\epsilon}X\dot{\bar{\nabla}}{}^{ \epsilon}Xg_{\beta[\gamma}X_{\delta]}{}^{\alpha}+\big{[}2\gamma_{1}^{2}\gamma_{1} ^{\prime\prime}-\gamma_{1}^{\prime 2}\gamma_{2}X-\gamma_{1}\] \[(3\gamma_{1}^{\prime 2}-2\gamma_{1}^{\prime\prime}\gamma_{2}X+\gamma_{1}^{ \prime}\gamma_{2}^{\prime}X)\big{]}{}_{[\gamma}{}^{\alpha}X_{[\beta]\delta]}+(2 \partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{1}^{ \prime}\partial_{X}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{1}\gamma_{2}^{ \prime})\] \[X^{\epsilon}\dot{\bar{\nabla}}{}_{\epsilon}X\delta_{[\gamma}{}^{ \alpha}X_{[\beta]\delta]}-\partial_{X}\gamma_{1}\gamma_{2}\partial_{X} \gamma_{2}X_{\epsilon}\dot{\bar{\nabla}}{}^{\epsilon}X\dot{\bar{\nabla}}{}^{ \zeta}X\delta_{[\gamma}{}^{\alpha}X_{[\beta]\delta]}-\big{[}2\gamma_{1}^{2} \partial_{X}\gamma_{1}^{\prime}\] \[-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}(3 \partial_{X}\gamma_{1}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}^{\prime} \gamma_{2}+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}g_{\beta[ \gamma}X_{\delta]}{}^{\alpha}\] \[+\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}-2 \partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}(3\partial_{X} \gamma_{1}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{
\[+4\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}+\partial_{X}\gamma_{1} (\gamma_{2}-2\partial_{X}\gamma_{2}X)]\Big{\}}X_{[\gamma}\dot{\nabla}_{\delta]}X \dot{\nabla}_{\delta}X+\Big{[}2\gamma_{1}^{3}\partial_{X}\gamma_{2}+(\partial_ {X}\gamma_{1})^{2}\gamma_{2}+\partial_{X}\gamma_{1}(-2\gamma_{1}\partial_{X} \gamma_{2})\] \[+\partial_{X}\gamma_{1}\partial_{X}\gamma_{2})-3(\partial_{X} \gamma_{1})^{2}\gamma_{2}^{2}X-\gamma_{1}\gamma_{2}(4(\partial_{X}\gamma_{1}) ^{2}+\partial_{X}\gamma_{1}\gamma_{2}-2\partial_{X}^{2}\gamma_{1}\gamma_{2}X) \Big{]}\] \[g_{\beta[\gamma}X^{\alpha\epsilon}\dot{\nabla}_{\delta]}X\dot{ \nabla}_{\epsilon}X-\gamma_{2}\big{[}2\gamma_{1}^{2}\partial_{X}^{2}\gamma_{2 }-2\partial_{X}\gamma_{1}\gamma_{2}\partial_{X}\gamma_{2}X-\gamma_{1}(3 \partial_{X}\gamma_{1}\partial_{X}\gamma_{2}\] \[+\gamma_{2}\partial_{X}\gamma_{2}+(\partial_{X}\gamma_{2})^{2}X -2\gamma_{2}\partial_{X}^{2}\gamma_{2}X)\big{]}X_{\beta[\gamma}X^{\alpha\epsilon }\dot{\nabla}_{\delta]}X\dot{\nabla}_{\epsilon}X\bigg{\}}. \tag{10}\]
The Ricci tensor transforms as
\[\dot{R}_{\alpha\beta} \rightarrow \dot{R}_{\alpha\beta}+\frac{-2\gamma_{1}\gamma_{1}^{\prime}-3 \gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}\gamma_{2}^{\prime}X}{2(\gamma_{1}+ \gamma_{2}X)^{2}}Y_{\alpha\beta}+\frac{-\gamma_{1}^{\prime}g_{\alpha\beta}Y+2 \gamma_{2}(Y_{\alpha\beta}Y+X^{\gamma}\dot{\nabla}_{\gamma}Y_{\alpha\beta})}{ 2(\gamma_{1}+\gamma_{2}X)} \tag{11}\] \[-\frac{(\partial_{X}\gamma_{1}g_{\alpha\beta}+\partial_{X}\gamma_ {2}X_{\alpha\beta})\dot{\nabla}_{\gamma}\dot{\nabla}^{\gamma}X}{2\gamma_{1}}- \frac{1}{4\gamma_{1}(\gamma_{1}+\gamma_{2}X)^{2}}\bigg{\{}X\big{[}2\gamma_{1}^ {\prime\prime}+\gamma_{1}^{\prime 2}\gamma_{2}X\] \[+\gamma_{1}(2\gamma_{1}^{\prime\prime}\gamma_{2}-\gamma_{1}^{ \prime}\gamma_{2}^{\prime})X\big{]}g_{\alpha\beta}+\Big{\{}4\gamma_{1}^{2} \partial_{X}\gamma_{1}^{\prime}+2\partial_{X}\gamma_{1}\gamma_{1}^{\prime} \gamma_{2}X+\gamma_{1}\big{[}4\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X\] \[-\partial_{X}\gamma_{1}\gamma_{2}^{\prime}X-\gamma_{1}^{\prime}( \gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}\Big{\}}g_{\alpha\beta}X^{\gamma} \dot{\nabla}_{\gamma}X-2\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+ \partial_{X}\gamma_{1}\gamma_{2}^{2}X\] \[+\gamma_{1}\gamma_{2}(-\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]} X^{\gamma}Y_{\alpha\beta}\dot{\nabla}_{\gamma}X\bigg{\}}-\frac{1}{4\gamma_{1}^{2}( \gamma_{1}+\gamma_{2}X)}\Big{\{}\big{[}2\gamma_{1}^{2}\partial_{X}^{2}\gamma_ {1}-(\partial_{X}\gamma_{1})^{2}\gamma_{2}X\] \[+\gamma_{1}(2\partial_{X}^{2}\gamma_{1}\gamma_{2}+\partial_{X} \gamma_{1}\partial_{X}\gamma_{2})X\big{]}g_{\alpha\beta}\dot{\nabla}_{\gamma} X\dot{\nabla}^{\gamma}X+\big{[}2(\partial_{X}\gamma_{1})^{2}\gamma_{2}+ \partial_{X}\gamma_{1}(-2\gamma_{1}\partial_{X}\gamma_{2}\] \[+\gamma_{2}\partial_{X}\gamma_{2}X)+\gamma_{1}(2\gamma_{1} \partial_{X}^{2}\gamma_{2}-(\partial_{X}\gamma_{2})^{2}X+2\gamma_{2}\partial_{X }^{2}\gamma_{2}X)\big{]}X_{\alpha\beta}\dot{\nabla}_{\gamma}X\dot{\nabla}^{ \gamma}X\Big{\}}\] \[+\frac{1}{4\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)^{2}}\bigg{\{} \big{[}-4\gamma_{1}^{3}\gamma_{1}^{\prime\prime}+3\gamma_{1}^{\prime 2}\gamma_{2}^{2}X^{2}+2\gamma_{1}^{2}(3 \gamma_{1}^{\prime 2}-5\gamma_{1}^{\prime\prime}\gamma_{2}X+\gamma_{1}^{\prime} \gamma_{2}^{\prime}X)\] \[+\gamma_{1}\gamma_{2}X(10\gamma_{1}^{\prime 2}-6\gamma_{1}^{\prime\prime} \gamma_{2}X+3\gamma_{1}^{\prime}\gamma_{2}^{\prime}X)\big{]}X_{\alpha\beta}+2 \big{[}-\gamma_{1}^{3}(4\partial_{X}\gamma_{1}^{\prime}+\gamma_{2}^{\prime})\] \[+4\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}^{2}X^{2}+2 \gamma_{1}\gamma_{2}X(5\partial_{X}\gamma_{1}\gamma_{1}^{\prime}-2\partial_{X} \gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)+ \gamma_{1}^{2}(6\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\] \[+\gamma_{1}^{\prime}\gamma_{2}-8\partial_{X}\gamma_{1}^{\prime} \gamma_{2}X+2\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}X_{(\alpha} \dot{\nabla}_{\beta)}X+\Big{\{}3(\partial_{X}\gamma_{1})^{2}_{2}X^{2}+2 \gamma_{1}\gamma_{2}X(4(\partial_{X}\gamma_{1})^{2}\] \[-\partial_{X}^{2}\gamma_{1}\gamma_{2}X+\partial_{X}\gamma_{1} \partial_{X}\gamma_{2}X)-2\gamma_{1}^{3}(2\partial_{X}^{2}\gamma_{1}+\partial_{X }\gamma_{2}+\partial_{X}^{2}\gamma_{2}X)+\gamma_{1}^{2}\big{[}6(\partial_{X} \gamma_{1})^{2}+\gamma_{2}^{2}\] \[+(\partial_{X}\gamma_{2})^{2}X^{2}+2\partial_{X}\gamma_{1}(\gamma_{ 2}+2\partial_{X}\gamma_{2}X)-2\gamma_{2}X(3\partial_{X}^{2}\gamma_{1}+\partial_{X }^{2}\gamma_{2}X)\big{]}\dot{\nabla}_{\alpha}X\dot{\nabla}_{\beta}X\] \[+\big{[}-\gamma_{1}^{2}(4\partial_{X}\gamma_{1}^{\prime}\gamma_{2} +2\gamma_{1}^{\prime}\partial_{X}\gamma_{2}-2\partial_{X}\gamma_{1}\gamma_{2}^{ \prime}+\gamma_{2}\gamma_{2}^{\prime})-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime }\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(2\gamma_{1}^{\prime}\gamma_{2}\] \[-4\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}^{\prime} \partial_{X}\gamma_{2}X+3\partial_{X}\gamma_{1}^{\prime}\gamma_{2}^{\prime}X) \big{]}X_{\alpha\beta}X\dot{\nabla}_{\gamma}X+2\big{[}2\gamma_{1}^{3}\partial_{X }^{2}\gamma_{2}+(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X\] \[+\gamma_{1}\gamma_{2}(2(\partial_{X}\gamma_{1})^{2}+\partial_{X} \gamma_{1}\gamma_{2}-2\partial_{X}^{2}\gamma_{1}\gamma_{2}X)-\gamma_{1}^{2}( 2\partial_{X}^{2}\gamma_{1}\gamma_{2}+2\partial_{X}\gamma_{1}\partial_{X} \gamma_{2}+\gamma_{2}\partial_{X}\gamma_{2}\] \[+(\partial_{X}\gamma_{2})^{2}X-2\gamma_{2}\partial_{X}^{2}\gamma_{2 }X)\big{]}X_{(\alpha|\gamma|}\dot{\nabla}_{\beta)}X\dot{\nabla}^{\gamma}X+ \Big{\{}\gamma_{1}\big{[}2\gamma_{1}\partial_{X}^{2}\gamma_{1}\gamma_{2}- \partial_{X}\gamma_{1}\gamma_{2}(2\partial_{X}\gamma_{1}+\gamma_{2})\]
\[-\gamma_{2}\partial_{X}\gamma_{2}X_{\alpha\beta}X_{\gamma\delta}\mathring{ \nabla}^{\delta}\mathring{\nabla}^{\gamma}X\Big{\}}. \tag{100}\]
Finally, the Ricci scalar transforms as
\[\mathring{R} \rightarrow \frac{\mathring{R}}{\gamma_{1}}+\frac{1}{2\gamma_{1}(\gamma_{1}+ \gamma_{2}X)^{2}}\Big{\{}3X\big{[}\gamma_{1}^{\prime 2}+\gamma_{1}^{\prime} \gamma_{2}^{\prime}X-2\partial_{\varphi}^{2}\gamma_{1}(\gamma_{1}+\gamma_{2}X )\big{]} \tag{101}\] \[-2(3\gamma_{1}\gamma_{1}^{\prime}+4\gamma_{1}^{\prime}\gamma_{2}X -\gamma_{1}\gamma_{2}^{\prime}X)Y+\big{[}6\partial_{X}\gamma_{1}\gamma_{1}^{ \prime}+4\gamma_{1}^{\prime}\gamma_{2}-\gamma_{1}(12\partial_{X}\gamma_{1}^{ \prime}+\gamma_{2}^{\prime})\] \[-12\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+3\gamma_{1}^{ \prime}\partial_{X}\gamma_{2}X+3\partial_{X}\gamma_{1}\gamma_{2}^{\prime}X \big{]}X^{\alpha}\mathring{\nabla}_{\alpha}X\Big{\}}\] \[+\frac{1}{2\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\{} \big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+\partial_{X}\gamma_{1}\gamma_{2}^ {2}X-\gamma_{1}\gamma_{2}(\gamma_{2}-\partial_{X}\gamma_{2}X)\big{]}X^{\alpha} Y\mathring{\nabla}_{\alpha}X\] \[+\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+3\partial_{X}\gamma _{1}\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(2\partial_{X}\gamma_{1}-\gamma_{2}+ \partial_{X}\gamma_{2}X)\big{]}\mathring{\nabla}^{\alpha}X\mathring{\nabla}_{ \beta}X_{\alpha}{}^{\beta}\Big{\}}\] \[-\frac{\gamma_{2}(\mathring{R}^{\alpha\beta}X_{\alpha\beta}- \mathring{\nabla}_{\beta}\mathring{\nabla}_{\alpha}X^{\alpha\beta})}{\gamma_{ 1}(\gamma_{1}+\gamma_{2}X)}\] \[-\frac{1}{4\gamma_{1}^{3}(\gamma_{1}+\gamma_{2}X)^{2}}\mathring{ \nabla}^{\alpha}X\Big{\{}\Big{[}-6(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X^ {2}+\gamma_{1}\gamma_{2}X(-10(\partial_{X}\gamma_{1})^{2}+3\partial_{X}\gamma _{1}\gamma_{2}\] \[+8\partial_{X}^{2}\gamma_{1}\gamma_{2}X+2\partial_{X}\gamma_{1} \partial_{X}\gamma_{2}X)+2\gamma_{1}^{3}(6\partial_{X}^{2}\gamma_{1}+3 \partial_{X}\gamma_{2}+2\partial_{X}^{2}\gamma_{2}X)+\gamma_{1}^{2}\big{[}-6( \partial_{X}\gamma_{1})^{2}\] \[-3\gamma_{2}^{2}-2(\partial_{X}\gamma_{2})^{2}X^{2}-2\partial_{X} \gamma_{1}(\gamma_{2}+\partial_{X}\gamma_{2}X)+\gamma_{2}X(20\partial_{X}^{2} \gamma_{1}+\partial_{X}\gamma_{2}+4\partial_{X}^{2}\gamma_{2}X)\big{]}\mathring {\nabla}_{\alpha}X\] \[+2\big{[}-2\gamma_{1}^{3}\partial_{X}^{2}\gamma_{2}+3(\partial_{X }\gamma_{1})^{2}\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(5(\partial_{X}\gamma_{1 })^{2}+2\partial_{X}\gamma_{1}\gamma_{2}-4\partial_{X}^{2}\gamma_{1}\gamma_{2}X\] \[-\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}X)+\gamma_{1}^{2}(- 4\partial_{X}^{2}\gamma_{1}\gamma_{2}-2\partial_{X}\gamma_{1}\partial_{X} \gamma_{2}+\gamma_{2}\partial_{X}\gamma_{2}+(\partial_{X}\gamma_{2})^{2}X-2 \gamma_{2}\partial_{X}^{2}\gamma_{2}X)\big{]}\] \[X_{\alpha\beta}\mathring{\nabla}^{\beta}X\Big{\}}-\frac{1}{ \gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)}\Big{\{}\big{[}2\partial_{X}\gamma_{1 }\gamma_{2}X+\gamma_{1}(3\partial_{X}\gamma_{1}+\gamma_{2}+\partial_{X}\gamma_{2 }X)\big{]}\mathring{\nabla}_{\alpha}\mathring{\nabla}^{\alpha}X\] \[-(2\partial_{X}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma _{2})X_{\alpha\beta}\mathring{\nabla}^{\beta}\mathring{\nabla}^{\alpha}X \Big{\}}\.\]
|
2304.05056 | Real-Time Character Rise Motions | This paper presents an uncomplicated dynamic controller for generating
physically-plausible three-dimensional full-body biped character rise motions
on-the-fly at run-time. Our low-dimensional controller uses fundamental
reference information (e.g., center-of-mass, hands, and feet locations) to
produce balanced biped get-up poses by means of a real-time physically-based
simulation. The key idea is to use a simple approximate model (i.e., similar to
the inverted-pendulum stepping model) to create continuous reference
trajectories that can be seamlessly tracked by an articulated biped character
to create balanced rise-motions. Our approach does not use any key-framed data
or any computationally expensive processing (e.g., offline-optimization or
search algorithms). We demonstrate the effectiveness and ease of our technique
through example (i.e., a biped character picking itself up from different
laying positions). | Ben Kenwright | 2023-04-11T08:26:11Z | http://arxiv.org/abs/2304.05056v1 | # Real-Time Character Rise Motions
###### Abstract
This paper presents an uncomplicated dynamic controller for generating physically-plausible three-dimensional full-body biped character rise motions on-the-fly at run-time. Our low-dimensional controller uses fundamental reference information (e.g., center-of-mass, hands, and feet locations) to produce balanced biped get-up poses by means of a real-time physically-based simulation. The key idea is to use a simple approximate model (i.e., similar to the inverted-pendulum stepping model) to create continuous reference trajectories that can be seamlessly tracked by an articulated biped character to create balanced rise-motions. Our approach does not use any key-framed data or any computationally expensive processing (e.g., offline-optimization or search algorithms). We demonstrate the effectiveness and ease of our technique through example (i.e., a biped character picking itself up from different laying positions).
R 8 (2013) 8
I.3.7Computer GraphicsThree-Dimensional Graphics and RealismAnimation I.6.8Computer GraphicsEvaluation and EstimationTypes of SimulationAnimation
R 8 (2013) 8
## 1 Introduction
**Motivation:** While a tremendous amount of research has been made over the past decade on controllers and animations that have focused specifically on virtual characters' upright motions (e.g., standing, walking, and dancing) [23, 24, 25, 26]. Less research has addressed the issue of how a character would regain its balance by picking itself-up (e.g., after falling down). We focus on a biped site controller that does not depend on any key-framed motion capture libraries, is simple, easy-to-implement, and remarkably robust; since, an algorithmic approach has the potential of producing a more general solution which is capable of generating unique physically-plausible movements.
**Interest \(\&\) Importance:** We want our animated characters to appear as realistic and life-like as possible. Hence, our characters should be able to fall-down (e.g., due to disturbances from pushes or trips) and mimic the real-world. Therefore, it would be significant if the characters could generate adaptable, physically-plausible, and natural-looking motions to synthesize the characters picking themselves-up. A physics-based approach allows us to create motions that are interactive, dynamic, and customizable, while possessing physically correct-properties to produce visually plausible life-like rise animations; for example, when a biped rises from a laying pose it needs to shift its center-of-mass and move its hands and feet while maintaining its balance to reach its final goal of an upright standing posture. Finally, if we use the physical properties of a character (e.g., feature sizes and mass), we can customize and adapt the get-up motions without constantly needing to search for and edit pre-canned animation libraries to fit the specific situation changes.
**Challenges:** Humans possess a large number of degrees-of-freedom (DOF), and it is difficult to determine joint angles that will achieve the multiple goals with varying priority and in real-time. In addition, there are the physical attributes, whereby the character needs to use its body to maintain balance while picking itself-up. The key challenges of our approach is generating robust poses in real-time without key-framed data that can be used to aimate a full-body, three-dimensional biped to faithfully reproduce a natural realistic rise motion. The motion need to account for ground contacts, swing-hand placement, and the center-of-mass support area balance priority considerations, while ensuring the poses are always physically-plausible and life-like.
**Existing Solutions:** A popular uncomplicated method for representing character rise motion is to switch to a pre-created key-framed sequence that will make the character get-up. The animations can possess life-like properties and provide a repertoire of unique and diverse animation solutions [26]. These animation sequences can further be adapted by means of kinematic techniques, for example, so the hands and feet engage the environment (i.e., in contact with the terrain) [1, 11]. For example, a set of key poses are guided by inverse kinematics to move the character along a realistic get-up motion. However, the kinematic motions might not always be physically-plausible, and a library of animations is needed, while the final motions cannot be easily adapted to different character features (e.g., short, tall, fat) or unique environmental situations (e.g., un-even terrain). However, physics-based models have been used to create balanced key-pose by analyzing, planning and solving full-body
Figure 1: Rise from Laying. The rise controller model could be mapped onto skeletons of different levels of complexity to reduce ambiguity and singularity issues. The figure shows the full-body skeleton represented as a basic 9-link stick man rolling over and getting-up from the front.
problems to create rise motions [10, 11].
**Our Approach:** This paper presents a practical, straightforward, and robust system for producing physically plausible get-up motions on-the-fly at run-time. Our model can be applied to different character feature sizes (e.g., short, tall, fat), and can produce various unique get-up motions by changing control parameters. We focus on time critical systems, such as games, for producing practical fundamental motions _without key-framed data_. The resulting get-up motions are physically-accurate, and the generated poses are based on an uncomplicated approximation model that provides crucial balanced reference trajectory information, which can be used to control a full-body articulated biped character. We use inverse kinematic (IK) techniques to combine our simplified base-controller with an articulated skeleton to produce fluid, physically accurate, and controllable get-up movements.
**Contribution:** The key contribution of this paper is the introduction of a novel controller method for generating physically-plausible rising (i.e., get-up) character motions for real-time environments. In summary, the main contributions of this paper are:
* Real-time approach for generating rising (get-up) motions without key-framed data
* Low-dimensional physics-based model to produce character animations that are self-driven (i.e., the character picks itself-up) while obeying geometric and kinematic constraints (e.g., joint and reach limits) and physical laws (e.g., gravity, non-slip contacts)
* We demonstrate and explain our approaches simple, practical, and straightforward ability for correcting and generating balanced rise poses
## 2 Related Work
There has been a broad range of exciting and interesting approaches across different disciplines (i.e., graphics, robotics, and biomechanics) towards creating and adapting character animation solutions. Whereby, we briefly review some of the most recent and relevant research within these fields that has contributed towards synthesizing biped characters picking themselves up.
### Computer Graphics
Kinematic methods use pre-generated motions, either from being painstakingly created by artists or through motion capture recordings. Whereby, the motions can be blended together using motion graphs to achieve a particular movement [13]. The motion graphs can be extended to generate _paramaterized_ motions that are more flexible and less repetitive by blending kinematic motions with a physically-based systems (e.g., including collision responses) [1, 2]. However, these generated motions depend on the available motion libraries to accomplish the specific action.
Faloutsos et al. [14] illustrated an application of their approach on rising from a supine position. They focused on combining controllers for different types of motion. The generated rise motion were based on a fixed posture with no mention of generating rising motions for different start poses. The controllers used pose tracking and timed state transitions, making it difficult to transfer controllers to new characters or environments. Essentially, using key-framed data to implement the character pushing itself up onto all four, then rising to its feet in a final upright balanced pose.
Liu et al. [15] proposed a sampling-based approach to reconstruct controlled movement from underlying reference motions. Their technique demonstrates excellent robustness while preserving physical correctness on contact-rich motions, including rolling, get-up, and kip-up. This sampling-based approach can produce small motion variations that can be treated as noise. However, reference motions are still needed for producing larger motion variations. Their method uses an expensive offline process, and does not incorporate feedback into the generated controllers.
Lin et al. [15], recently demonstrated life-like rising animations by using motion planning based on an RRT (rapidly-explored random tree) based approach (i.e., picking the most plausible motion from a planned motion path). This approach was also applied to manipulation planning by Yamane et al. [11]. Whereby, the motion planning was used to compute the path of an object being manipulated. For each planned object orientation and position, the pose of the character is computed to satisfy geometric, kinematic, and posture constraints. While both approaches are similar, Yamane et al. [11] focused on object space planning, while Lin et al. [15] used a posture space planning with the RRT-blossom algorithm.
Zordan et al. [17] connected a physically simulated movement to a MOCAP motion, with a focused on generating dynamic responses by tracking a desired trajectory, which is formed by linearly interpolating the intermediate postures from the two motion capture sequences before and after the transition. Their approach synthesizes a trajectory using a posture database. In particular, an arbitrary laying posture or key-pose can be used, but the linear interpolation approach always produces the same trajectories. Similarly, Wrotek et al. [18] exploited a data-driven solution to control a physically accurate model, one of the solutions, including a rising model.
Jones [19] generated key-poses using a physics-based model to produce rising motions, while Nunes et al. [11] controlled an articulated structure using a state machine logic to accomplish show jumping, running, rising motions.
A novel alternative was the "motion doodle" system presented by Thorne et al. [10], which let the user sketch the intended motion path to obtain a desired representation of the movement. This user feedback approach could synthesize appropriate motions that were visually correct; some of the example motions included, jumping and getting-up.
### Robotics
Morimoto and Doya [16, 17] proposed a hierarchical reinforcement learning method to generate standing-up movement on a simplified character.
Hirukawa et al. [18], and Fujiwara et al. [17] divided a rising motion into several contact states and used a contact-state graph to represent them. This approach works well on robots, but is difficult to define a proper contact-state graph for human motions rising from various laying postures.
Kanehiro et al. [19] generated getting up motions by linearly interpolating any given laying posture to its most similar posture in a predefined falling state graph on a HRP-2P robot, which is 1.5m tall and weighs 58kg. A similar controller for a smaller robot 0.5m, 2.3kg has also been developed [12]. Their work focuses on generating a smooth sequence of rise postures and less on the physical plausibility of the rising motion.
Kuniyoshi et al. [12] used an adult sized humanoid robot to analyze the critical aspects of a highly dynamic getting up motion, and used this information to find the parameters to generate successful motions. However, transferring the motion from simulation to the robot was challenging and error prone, due to the dependence of the motion on the difficult to simulate ground contact forces.
Mettin et al. [12] created rise motions for a character
sifting (i.e., a chair sitting-to-standing and vice-versa motion) while keeping balanced, using torques and arm forces. Fujiwara et al. [20] created a humanoid robot with features the same as a human that could lie down and pick itself-up. Kuniyoshi et al. [21] examined roll-and-rise motion capture data to generate robot movements based on temporal localization of features to extract crucial information about the task.
### Biomechanics
Standing up motions are well studied in biomechanics, in particular, is the analysis of sit-to-stand motions, including the dynamics and stability which have been studied in detail [Robert and McCollum 1996].
Muscle activity through the motion can be divided into three distinct phases: a forward lean of the upper body, an upward acceleration due to leg extension, and a deceleration phase [Hirschfeld et al. 1999; Kralj et al. 1990; Roebroeck et al. 1994]. In addition, the contact forces between the buttocks and the feet are controlled by muscle activations to generate the forward and upward acceleration to stand.
McCoy and VanSant [McCoy and VanSant 1993], and Ford-Smith and VanSant [Ford-Smith and VanSant 1993] compared movement patterns of people rising from a bed in different ages. For adolescents, they developed four categories of movement patterns: far upper extremity, near upper extremity, axial region and lower extremities. In the age between 30 to 59, they developed four categories of movement patterns: left upper limb movement patterns, right upper limb movement patterns, head and trunk movement patterns and lower limb movement patterns. They experimented and computed the probability of each movement pattern. The majority of the biomechanics studies are to analyze rather than generate the rising motion.
## 3 System Overview
In our approach, we use a low-dimensional model (i.e., a particle-mass with weightless telescopic arms and legs) for estimating key information that is common with a complex biped character structure to create a fast, robust, and simple solution for generating get-up motions. The final motions enable the character to pick itself up using its feet-hand placement and by shifting its own body position to maintain balance and achieve an upright standing pose. The controller gives initial information on where to place the character's hand and pelvis. Then as the controller iteratively proceeds to stand up, a feedback loop between the low-dimensional controller and the character's model corrects for errors as the character slowly gets up. The systems key focus is a character's motion that is anatomically-correct (i.e., bound to joint limits), physically-plausible (i.e., obeys mechanical laws, such as balancing), realistic motion (i.e., analogous to a real-world human), and which is computationally fast, simple, and robust. The character picks **himself** up by positioning his body in a balanced pose and using his own joint torques.
## 4 Optimization
This section explains the optimization problem for generating approximate poses that accomplish the goal of getting up from laying down. These poses need to enforce imposed constraints (e.g., CoM above the support area) to accomplish the primary task of maintaining continuous balance. Furthermore, the poses should be as natural-looking and as comfortable as possible (i.e., avoiding contorted or overstretched positions). We approach the problem by subdividing the task into numerous stages. Each stage contributes towards the final goal of a vertically upright balanced posture.
Figure 4: **Simple Optimization Problem -** We reduce the complexity of the problem down to a simple point-mass with three-contact points. We iteratively move the CoM so that it is as horizontally close as possible to the feet (i.e., above the feet). Then we move the free-hand closer and make it the new support hand, and repeat the process. Figure shows the simple optimization problem.
Figure 3: **Simplified Point-Mass Biped Model** - The arms and legs of the character are analogous with a spring-damper mechanism and work in synergy to control the overall character’s center-of-mass (CoM).
Figure 2: **System Overview -** An overview of the rising (get-up) motion framework.
### _Simplified Model_
We use a simplified model for the optimization problem. The problem and model focus on essential elements (such as, feet, hands, CoM position, and support region). The simplified model consists of a point-mass (m) representing the character's overall center-of-mass (CoM), two mass-less legs, and two mass-less arms (as shown in Figure 3). The legs, arms, and CoM have constraints imposed upon them (e.g., minimum and maximum lengths, target location) that we find an optimum solution at each iteration to accomplish the goal of maintaining balance while moving towards an upright pose. Hence, we evaluate the contact points (i.e., end-effectors) for each hand and foot by means of an uncomplicated systematic analysis of the balancing situation. The simplified model enables us to reduce the complexity of the problem and find an optimized solution in real-time while retaining crucial characteristics of a character's rise motion. For example, to demonstrate the principal driving logic for our approach, imagine a simple 2D couple-rod, shown in Figure 5, picking itself up by iteratively moving its center-of-mass above the foot support region, then it can raise its upper rod while keeping the CoM above the support region to accomplish a vertical upright pose. Simplifying the knees and elbows to extendable rigid links with minimum and maximum lengths helps reduce ambiguity and singularities when searching for an optimum solution to the low-dimensional constraint problem.
### _Geometric and kinematic optimization steps_
The model has to account for the geometric relationship between the character and the virtual environment as well as the kinematic control; for example, the coupled control between the simplified low-dimensional model and the high-dimensional articulated biped character.
Our model's logic is divided into three phases (as shown in Figure 6).
**Phase 0 (move towards start pose)**
After the character is laying down, we want to have him roll onto his front or side. So we locate comfortable hand and foot positions (i.e., four targets) to the left and right of the character's body (i.e., avoid crossed arms and legs).
We can roll the character and move their arms and legs towards their target start locations. We have now started with both feet and hands' locked in place ready for the next phase.
**Phase 1 (move CoM towards the feet support region)**
We release one hand so the body is balanced above the two feet and single support hand. The two feet and the support hand form a projected triangle on the ground known as the body support region. The CoM should be within this body support region for the body remain balanced (i.e., not fall over).
We then search for an optimal solution for the simple model (i.e., two feet and a single hand). We set constraints, leg and arm minimum and maximum lengths, and the CoM as close to the feet support region as possible.
When we find a solution we interpolate the model towards its optimal goal (i.e., legs and arm lengths).
Is the CoM above the foot support region?
No: Keeping the support hand locked, we move the free hand to the side of the body at the location of the CoM (e.g., the left free hand to the left side of the body). The free hand's location is now made the support hand, and a new body support region is formed (restart Phase 1).
Yes: Balanced by our feet (goto Phase 2).
**Phase 2 (move to an upright pose)** Release both hand constraints, since we are supported by our feet support region, as the CoM is located within it (from Iteration 1).
Move vertically upwards while keeping the CoM above the feet support region area.
Figure 5: _Rigid Rod Picking Itsell Up - A simple rod picking itself up (assuming the bottom of the rod is shorter or of equal in length to the top - and the mass is not biased towards the top)._
Figure 6: _Flow Graph - Our approach uses three phases: initial pose, iteratively shifting balance to feet, and rising to an upright posture._
The constraint optimization conditions for phases 1 and 2 are (with primary and secondary priority for importance):
\[\text{Phase 1:}\left\{\begin{array}{ll}d_{lmin}>d_{l}>d_{lmax}&(primary) \\ d_{amin}>d_{a}>d_{amax}&(primary)\\ d_{c}=0&(secondary)\\ \end{array}\right. \tag{1}\] \[\text{Phase 2:}\left\{\begin{array}{ll}d_{c}=0&(primary)\\ d_{l}=d_{lmax}&(secondary)\\ \end{array}\right.\]
where \(d_{l}\) and \(d_{a}\) are the arm and leg distance from the CoM, \(d_{c}\) is the CoM horizontal distance from the foot support region, and subscript min and max dictate minimum maximum constraint conditions (as shown in Figure 7). Foot-wise balanced poses, such as Figure 8 and Figure 9, are considered part of phase 2. Since they are only concerned with keeping the CoM above the feet support area.
We concentrate on slow (i.e., static) get-up motions whereby the CoM and body support area can be used to classify the balance stability criteria (i.e., for static or slow motions the zero moment point (ZMP) is equal to the projected CoM [12]).
Figure 5 illustrates how our controller would go about picking itself up. Initially, it positions the center-of-mass above the center-of-pressure (CoP), from there on, it lifts its front body up, while compensating with the lower body to maintain the CoM above the foot position. Due to the dynamic feedback from the model, any disturbances which might arise during rising, will be fed back into the base which will attempt to compensate for them.
Figure 8: _Arm-less Getting-Up - The character performs a crouch-to-rise, if the character’s CoM is above the feet support area. We can approximate the mass as a particle-point and the problem reduces to a spherical object extending the support leg length. (a) The leg muscle is analogous to a spring-damper system extending its rest length, and (b) illustrating a crouched character on the ground._
Figure 7: **Rising Phases - (a) Phase 1, which iteratively swaps hand locations to keep moving the CoM towards the feet support region, (b) Phase 2 starts when the CoM is above the feet support region; whereby, the arm constraints are released, and the focus is on extending the legs and keeping the CoM above the feet, and is complete when the legs are fully extended and the arms come to rest at the body’s side.**
Figure 9: **Arm-less Elongated Body Getting-Up - The uncomfortable and uncommon pose of a character stretched out can keep their CoM above their feet support area while getting-up; (a) the CoM stays above the feet support area while the posture rotates, and (b) the legs are connected to the end of the elongated body while gradually rotated and keeping the CoM above the feet and reaching the vertical stance pose.**
### Trajectories (Smooth Interpolated Motions)
The trajectories for the hands and feet are calculated using Bezier spline paths when moving them between old and new position during the get-up sequences (the height they are lifted above the ground and the speed they move all affect the final style of the motion).
### Controller Constraints (Priority)
For the base controller to achieve an upright posture, a number of constraints must be imposed. It must be possible for the controller to place its center-of-mass above its foot position. If both the upper and lower body have the same radial dimensions, this means the upper body must great in length than the lower body, but less than twice the length of the lower body.
The steps the controller goes through while picking itself up, are:
1. Align the center-of-mass above the foot position.
2. Slowly raise the upper body, and while doing so, compensate for center-of-mass moving outside the foot position region using the lower body.
### Biped Model
For our character simulation tests, we used a variety of different models. Primarily, for 3D simulations, we used a 15-link biped model, shown in Figure 10 and Figure 19. The character model possesses 36 degrees-of-freedom (DOF), including 6 for the world root (3 translation and 3 rotation), 1 DOF for each knee and elbow, and 2 DOF for each hip and shoulder.
## 5 Inverse Kinematics (IK)
We use inverse kinematics (IK) to map our low-dimensional model's information onto our articulated biped character skeleton. The IK is responsible for generating the final biped joint angles, it also imposes physical joint angle constraints to ensure the model always produces physically-plausible poses. Since we are interested in real-time applications, we employ a fast and simple analytic solution (as done by Coros et al. [13]) for two-link inverse kinematic (IK) problems given by Kulpa et al. [12]. We compute the unique solution for **the elbow and knee by forcing them to lie on a plane** (e.g., the elbow plane would encumber the shoulder and hand while the knee would encumber the hip and ankle). The rotational degree-of-freedom (DOF) of this embedded plane is specified as an input parameter that allows for bow-legged styles and control over the expressive nature of arm reaching movements.
Mapping the low-dimensional model's center-of-mass (CoM) position onto the full-body biped skeleton, there are two fundamental inverse kinematic (IK) approaches. They are:
1. A computationally fast less accurate approach - e.g., the hip midpoint as the CoM position similar to SIMBICON [23] due to it being fast and simple
2. A more precise globally solution (CoM of all limbs) - e.g., constantly update and track the whole articulated body CoM position synonymous with the approach by Tsai et al. [23].
For 2D cases, we use a global CoM tracking constraint IK solution [10], while for 3D simulations, we opt for the simpler and computationally faster hip midpoint for the CoM position. For example, the mapping of the model onto a 3D biped skeleton is shown in 11. As for the hand orientation, we chose to have the hand initially rotate and align to face comfortably forwards at the start of the get-up motion and neglect any twisting.
### Style (Priority ordered IK)
Incorporating a primary and secondary IK solver allows us to mix in behavioral motions. The second optional constraint condition embeds optional characteristic motions while the crucially primary constraints are enforced to ensure the motion is physically-correct and balanced. For example, we could mix in a tired sluggish movement, looking-around motions, or coherent random life-like movements (e.g., swaying and looking around) to make the movement less robot-like and unique. This is done in two parts, using the primary key elements to keep the character balanced and physically correct (locked with the optimized model's rise solution), and a secondary motion added on top to introduce stylistic control (as shown by [10]).
## 6 Rigid Body Control
The IK solver provides joint angles that we used to calculate joint torques to control the full-body rigid body skeleton structure. This approach is analogous to a _puppet on strings_, since the rigid body structure emulates the IK solution through angular springs (i.e., proportional derivative servos). However, since the final motions are generated using an articulated rigid body structure, the move
Figure 11: **Mapping Simple Model onto Biped Structure - The inverse kinematic problem to reconstruct the biped character pose. For example, in the figure, we use a 9 link biped model, with 18 degree-of-freedom (DOF), 6-DOF root (i.e., position and orientation), 2-DOF each hip and shoulder, 1-DOF for each knee and elbow. It has three fixed ground contact points, one for each foot, and one for the support hand.**
Figure 10: **Articulated Biped Character Model - The 3D biped simulation model is composed of 15-links and 14-joints.**
ments are smoother while still possessing their responsive and interactive properties. The joint torques for the articulated character are generated using a proportional derivative (PD) controller, i.e., \(\tau=k_{p}(\theta_{d}-\theta)-k_{d}\theta^{\prime}\), where \(\theta_{d}\) is the desired joint angle, \(\theta\) and \(\theta^{\prime}\) are the current joint angle and joint angular velocity, and \(k_{p}\) and \(k_{d}\) are the gain and damping constants. While the gain and damping constants are crucial for the character's motions to appear responsive and fluid, calculating reliable, robust coefficients that result in the pose reaching its desired target within a specific time reliably and safely is difficult due to the highly dynamic model. Whereby, we hand-tuned the coefficients to achieve the necessary pleasing results. We used simple convex shapes (i.e., boxes) to represent the character limbs as shown in Figure 17(c) and Figure 14.
## 7 Experimental Results
We applied our approach to different simulation situations to demonstrate the advantages of our method and its potential for creating rising motions without pre-recorded animation libraries (i.e., key-framed data). The simulations were ran at 100 frames per second (fps) and were executed on an Intel Core i7-2600 CPU with 16-GB of memory running Windows-7 64-bit on a desktop PC. The results are shown through a series of experiments to warrant the practicality and robustness of our approach for generating adaptive biped stepping motions without key-framed data. In short, the visual results testify to the robustness and simplicity of our approach for synthesizing balancing get-up actions. The overall computational time for generating the character motions with control, including dynamic simulation overheads (e.g., rigid body constraints and contacts) was on average less than 3 ms, respectively (i.e., better than real-time performance).
The controller generates essential information for the biped character to get-up. This information pertains to the end-effectors' locations and upper body's posture. With inverse kinematics and data-driven approaches, the generated motions are not physically accurate. These approaches usually fail to produce realistic get-up poses, as the character's dimensions are changes, and do not reflect the strength of the character's muscles. However, our simulations use torque and joint forces to move the final rigid body skeleton to an upright pose.
When it has been identified that the character has fallen down and has come to a complete stop. Whereby, we wait for the character's angular and linear velocity to reach a minimum threshold, e.g., in-case he is sliding down stairs, or rolling. Once the character has come to a complete stop, we engage the rise-up controller and monitor its progress at repeated intervals. The generated rise motions where robust against initial posture poses (i.e., see Figure 13 and would roll to a comfortable (i.e., on their front or back) before moving their hand and feet contacts while shifting the CoM towards their feet.
Figure 12 shows the model being applied to the sigattal plane to illustrate how we apply the base controller to our biped character model. This can be compared with simulation results in 3D and 2D as shown in Figure 16 and Figure 14.
The preliminary work shows promising results, with a great deal of flexibility for improvement and adaptation. Our simple model provides a robust, computationally fast, and controllable solution for generating fundamental balancing character pose information.
dress other approaches. For example, rising tangentially (i.e., from the sides), using other contact points (e.g., the elbows and knees), and we do not include any feet or hand slipping. Additionally, we only looked at static slow motions, whereby the projected ground CoM stayed within the support region to remain continuously balanced during movement transitions. We did not address highly dynamic get-up motions where the character could gain momentum by swinging their body. Although our get-up approach is been based on restrictive assumptions (i.e., from the front or back), the created motions proved to be visually plausible and life-like.
## 9 Conclusion and Further Work
We have presented a computationally simple, robust, and flexible approach for generating character rise animations. We do not require any key-framed data (i.e., motion capture libraries) to generate the fundamental movements. The biped character get-up motions are physically-plausible and balanced. We enforce joint limits, non-sliding hand and feet contacts, and geometric environmental considerations.
Our experimental results show that our approach can be customized to produce a wide variety of basic rise motions (e.g., height of CoM, max/min, leg/arm extending, speed, front-back). As the character's features are changed (i.e., support polygon and CoM) our model _automatically_ adapts the posture and contact placement information for the rise animation in real-time; for example, a designer would not need to compensate for any changes when creating the rise motion. Furthermore, our approach can be combined with motion capture data (or random human-like rhythmic movements) to create more captivating and life-like motions that are more unique and possess a character's personality.
While the basic model has been introduced here, further work could be to investigate how the model copes with uneven terrain (e.g. on a slope). Furthermore, when we are pushed over, we rotate and reach out in the direction we are falling. Hence, we believe that our model can be adapted to other situations, for example, if the character loses its balance and is unable to recover, it could switch to the get-up motion logic so that the fall sequence is more natural looking (e.g., rotating in the direction of the fall and placing the arms out); compared with matching pre-canned animations to fit the unique fall situation which can look unnatural for that moment.
## 10 Acknowledgments
The author would like to thank the anonymous reviewers for taking time out of their busy schedules to provide helpful comments to make this a more concise, clear, and readable paper.
|
2306.10544 | Magnetic scattering with spin-momentum locking: Single scatterers and
diffraction grating | Simultaneous manipulation of charge and spin density distributions in
materials is the key element required in spintronics applications. Here we
study the formation of coupled spin and charge densities arising in scattering
of electrons by domains of local magnetization producing a position-dependent
Zeeman field in the presence of the spin-momentum locking typical for
topological insulators. Analytically and numerically calculated scattering
pattern is determined by the electron energy, domain magnetization, and size.
The spin-momentum locking produces strong differences with respect to the
spin-diagonal scattering and leads to the scattering asymmetry with nonzero
mean scattering angle as determined by only two parameters characterizing the
system. To extend the variety of possible patterns, we study scattering by
diffraction gratings and propose to design them in modern nanostructures based
on topological insulators to produce desired distributions of the charge and
spin densities. These results can be useful for engineering of magnetic
patterns for electron optics to control coupled charge and spin evolution. | S. Wolski, V. K. Dugaev, E. Ya. Sherman | 2023-06-18T12:41:43Z | http://arxiv.org/abs/2306.10544v1 | # Magnetic scattering with spin-momentum locking: Single scatterers and diffraction grating
###### Abstract
Simultaneous manipulation of charge and spin density distributions in materials is the key element required in spintronics applications. Here we study the formation of coupled spin and charge densities arising in scattering of electrons by domains of local magnetization producing a position-dependent Zeeman field in the presence of the spin-momentum locking typical for topological insulators. Analytically and numerically calculated scattering pattern is determined by the electron energy, domain magnetization, and size. The spin-momentum locking produces strong differences with respect to the spin-diagonal scattering and leads to the scattering asymmetry with nonzero mean scattering angle as determined by only two parameters characterizing the system. To extend the variety of possible patterns, we study scattering by diffraction gratings and propose to design them in modern nanostructures based on topological insulators to produce desired distributions of the charge and spin densities. These results can be useful for engineering of magnetic patterns for electron optics to control coupled charge and spin evolution.
## I Introduction: Electron optics with spin-momentum locking
The ability to manipulate and control electron charge and spin dynamics by external fields is one of the challenges in modern applied physics. This goal can be achieved by electron optics, that is using elements similar to conventional optics based on wave properties of electrons. Another option is related to the electron spin optics, that is to control both coupled electron spins and charge dynamics. A conventional tool to manipulate electron spin is the Zeeman-like coupling either with the external magnetic field or with material magnetization. However, simultaneous control of spin and charge motion requires spin-orbit coupling [1] resulting in spin-momentum locking. This can be achieved by using electrons in low-energy two-dimensional surface states of topological insulators, where this coupling demonstrates itself as a strong spin-momentum locking expressed in the Hamiltonian [2; 3]
\[H=-i\hbar v\mathbf{\sigma}\cdot\mathbf{\nabla}+\Delta_{\rm m}({\bf r})\sigma_{z}, \tag{1}\]
and produces relativistic-like Dirac cones. Here position \({\bf r}=(x,y)\), \(v\) is the electron bandstructure velocity parameter and \(\Delta_{\rm m}({\bf r})\) is the local magnetization assumed to be along the \(z-\)axis. We assume that the electron energy is considerably low and the corresponding momentum is sufficiently small to satisfy the validity of the Hamiltonian (1), including only the linear \(\mathbf{\sigma}\cdot\mathbf{\nabla}\) term and neglecting the band warping [4], with \(\sigma_{i}\) being the Pauli matrices. Here it is convenient to present the electron spin in the form \(\mathbf{\sigma}=(\mathbf{\sigma}_{\perp},\sigma_{z})\), where \(\mathbf{\sigma}_{\perp}=(\sigma_{x},\sigma_{y})\) is the two-dimensional in-plane component. The velocity of the electron defined as \({\bf v}=i[H,{\bf r}]/\hbar=v(\sigma_{x},\sigma_{y})=v\mathbf{\sigma}_{\perp}\) is determined by the electron spin. Thus, by acting at the electron spin by a position- and time-dependent Zeeman field one can modify the electron velocity and, thus, influence the charge transport (see, e.g., Ref. [5] for electron propagation in the presence of one-dimensional stripe-like magnetization). Recently, Ref. [6] analyzed the effects of skew scattering by magnetic monopoles in spin-orbit coupled systems.
On the one hand, random magnetic disorder modifies weak localization [7] in conventional semiconductors and strongly influences the conductivity of topological insulators [8]. On the other hand, the ability to produce magnetization pattern \(\Delta_{\rm m}({\bf r})\)[9; 10] permits design of position-dependent spin dynamics and, correspondingly, the manipulation of electron wavefunction producing the coupled spin-charge transport [11; 12; 13]. The effects of spin-dependent velocity are known for conventional semiconductors for electrons [14] and holes [15] and can manifest itself in electron scattering by impurities in the presence of strong spin-orbit coupling [16], formation of equilibrium spin currents [17; 18] and in the spin-Hall effect [19; 20; 21; 22; 23; 24; 25] kinetics.
Here we explore the possibility to control electron spin and position dynamics by a single- and arrays of magnetized quantum dots with spatially confined magnetization on the surface of topological insulators. One possible application is the variety of spin torques [26; 27; 28; 29; 30; 31; 32; 33] produced on the magnetized quantum dots by scattered electrons to manage magnetization dynamics and coupled spin-charge transport. We study the scattering processes in different regimes and conclude that a net charge current injection can be produced. Based on the results of
a single-dot scattering approach, we propose to design diffraction gratings made by a one-dimensional array of magnetic quantum dots for this purpose. This grating will produce on purpose asymmetric patterns of spin and charge densities and currents.
The rest of the paper is organized as follows. In Sec. II we formulate the scattering problem and present the observables of interest. In Sec. III we describe partial wave summation approach and present general analytical results. Different sets of parameters and scattering domains for single scatterers will be analyzed in Sec. IV while scattering by diffraction grating will be considered in Sec. V. The main numerical results for cross-section and scattering angles will be presented in Sec. VI. Section VII provides the conclusions and outlook of this paper.
## II Scattering process and observables
For a circular magnetized disk on the surface of topological insulator we rewrite Eq. (1) in the form:
\[H=-iv\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}+\theta(R-r)\Delta_{\rm m} \sigma_{z}, \tag{2}\]
where \(\theta(R-r)\Delta_{\rm m}\) is the local magnetization with the Heaviside function \(\theta(R-r)\), as shown in Fig. 1. Here and below we use the system of units with \(\hbar\equiv\,1\).
In the absence of external magnetization (\(\Delta_{\rm m}\equiv 0\)) the free-space plane-wave function for \(\boldsymbol{\psi_{\rm k}(\rm r)}\sim e^{i\mathbf{k}\cdot\mathbf{r}}[\psi_{\rm k }^{\dagger},\psi_{\rm k}^{\dagger}]^{\rm T}\) (T stands for transposition) satisfies the equation:
\[\left[\begin{array}{cc}-\varepsilon&vk_{-}\\ vk_{+}&-\varepsilon\end{array}\right]\left[\begin{array}{c}\psi_{\rm k}^{ \dagger}\\ \psi_{\rm k}^{\dagger}\end{array}\right]=0, \tag{3}\]
where \(k_{\pm}\equiv\,k_{x}\pm ik_{y}\). We obtain two linear branches of spectrum \(\varepsilon=\pm vk\) where \(k=\sqrt{k_{x}^{2}+k_{y}^{2}}\). For the eigenstates one has
\[\boldsymbol{\psi_{\rm k}(\rm r)}=\frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{\sqrt{ 2}}\left[\begin{array}{c}1\\ \varepsilon\,k_{+}/|\varepsilon|\,k\end{array}\right]. \tag{4}\]
and the spin is parallel (\(\varepsilon>0\)) or antiparallel (\(\varepsilon<0\)) to the momentum. At a large distance from the quantum dot the electron wavefunction at \(\varepsilon>0\) is presented for the wave coming from \(x=-\infty\) along the \(x-\)axis with \(\mathbf{k}=(k,0)\)[34]
\[\boldsymbol{\psi}(r,\varphi)=\frac{e^{ikx}}{\sqrt{2}}\left[\begin{array}{c} 1\\ 1\end{array}\right]+\boldsymbol{f}(\varphi)\,\frac{e^{ikr}}{\sqrt{r}}, \tag{5}\]
where
\[\boldsymbol{f}(\varphi)=\left[\begin{array}{c}f^{\uparrow}(\varphi)\\ f^{\downarrow}(\varphi)\end{array}\right] \tag{6}\]
is the two-component spinor scattering amplitude. We note that in two-dimensional systems the cross-section \(l\) has the length units and present the differential cross-section as \(dl/d\varphi=|\boldsymbol{f}(\varphi)|^{2}\) where the total \(l\) is:
\[l=\int_{-\pi}^{\pi}|\boldsymbol{f}(\varphi)|^{2}d\varphi. \tag{7}\]
Since, as we will demonstrate below, the scattering is anisotropic with \(|\boldsymbol{f}(-\varphi)|\neq|\boldsymbol{f}(\varphi)|\), we introduce the mean value of the scattering angle \(\left\langle\varphi\right\rangle\) and of its square \(\left\langle\varphi^{2}\right\rangle\) :
\[\left\langle\varphi^{n}\right\rangle=\frac{1}{l}\int_{-\pi}^{\pi}\varphi^{n}| \boldsymbol{f}(\varphi)|^{2}d\varphi, \tag{8}\]
where \(n=1\) or \(n=2.\) The dispersion \(D_{\varphi}=\sqrt{\left\langle\varphi^{2}\right\rangle-\left\langle\varphi \right\rangle^{2}}\), characterizes the width of the scattering aperture.
In addition, we mention that the asymmetric scattering produces effective charge current along the \(y-\)axis, which can be defined as:
\[\left\langle j\right\rangle=\frac{ev}{l}\int_{-\pi}^{\pi}\sin\varphi| \boldsymbol{f}(\varphi)|^{2}d\varphi \tag{9}\]
where \(e\) is the electron charge. The behavior of this current as a function of system parameters is qualitatively similar to the behavior of \(\left\langle\varphi\right\rangle.\)
## III Partial wave summation: analytical results
### Wave functions and boundary conditions
In polar coordinates with \(r=\sqrt{x^{2}+y^{2}}\), \(\varphi=\arctan(y/x)\), the eigenstates in the form of the circular waves are determined by:
\[\left[\begin{array}{cc}\varepsilon-\Delta_{\rm m}(r)&ve^{-i\varphi}\left(i \,\partial_{r}+\frac{\partial_{\varphi}}{r}\right)\\ ve^{i\varphi}\left(i\,\partial_{r}-\frac{\partial_{\varphi}}{r}\right)& \varepsilon+\Delta_{\rm m}(r)\end{array}\right]\boldsymbol{\psi}(r,\varphi)=0. \tag{10}\]
Figure 1: Asymmetric scattering of electron with the wavevector \(\mathbf{k}\) by a single magnetized nanodot.
To calculate the sum of partial waves attributed to the \(z-\)components of the angular momentum \(m\), we first substitute in Eq. (10) the spinor characterized by given \(m\) in the form:
\[\mathbf{\psi}_{m}(r,\varphi)=\,e^{im\varphi}\left[\begin{array}{c}\psi_{m}^{ \dagger}(r)\\ \psi_{m}^{\downarrow}(r)\,e^{i\varphi}\end{array}\right] \tag{11}\]
and obtain coupled equations for the radial functions (omitting the explicit \(r-\)dependence for brevity):
\[\left[\begin{array}{cc}\Delta_{\rm m}(r)-\varepsilon&-iv\left(\frac{d}{dr}+ \frac{m+1}{r}\right)\\ -iv\left(\frac{d}{dr}-\frac{m}{r}\right)&-\Delta_{\rm m}(r)-\varepsilon \end{array}\right]\left[\begin{array}{c}\psi_{m}^{\dagger}\\ \psi_{m}^{\downarrow}\end{array}\right]=0. \tag{12}\]
Inside the dot, \(r<R\) and \(\Delta_{\rm m}(r)=\Delta_{\rm m}:\)
\[(\Delta_{\rm m}\,-\varepsilon)\,\psi_{m}^{\dagger}-iv\left(\frac{d}{dr}+ \frac{m+1}{r}\right)\psi_{m}^{\downarrow}=0 \tag{13}\]
\[-iv\left(\frac{d}{dr}-\frac{m}{r}\right)\psi_{m}^{\dagger}-\left(\Delta_{\rm m }\,+\varepsilon\right)\psi_{m}^{\downarrow}=0. \tag{14}\]
Extracting \(\psi_{m}^{\downarrow}\) in Eq. (14)
\[\psi_{m}^{\downarrow}=-\frac{iv}{\Delta_{\rm m}\,+\varepsilon}\left(\frac{d}{ dr}-\frac{m}{r}\right)\psi_{m}^{\uparrow} \tag{15}\]
and substituting it into (13) we obtain for \(\psi_{m}^{\uparrow}\)
\[\kappa^{2}\psi_{m}^{\uparrow}-\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d} {dr}-\frac{m^{2}}{r^{2}}\right)\psi_{m}^{\uparrow}=0, \tag{16}\]
with the energy-dependent \(\kappa\equiv\sqrt{|\Delta_{\rm m}^{2}-\varepsilon^{2}|}/v.\) We begin with the realization \(\varepsilon<\Delta_{\rm m}\) where the solution regular at \(r\to 0\) is the modified Bessel function \(\psi_{m}^{\uparrow}(r\kappa)=I_{m}(r\kappa)\)[35] and then consider the \(\varepsilon>\Delta_{\rm m}\) case by analytical continuation. Using (15) we can write:
\[\psi_{m}^{\downarrow}(z)=-is\left(\frac{d}{dz}-\frac{m}{z}\right)\psi_{m}^{ \uparrow}(z)=-isI_{m+1}(z), \tag{17}\]
where \(z\equiv r\kappa\) and \(s\equiv\sqrt{|\Delta_{\rm m}-\varepsilon|/(\Delta_{\rm m}+\varepsilon)}.\)
Thus, the general solution at \(r<R\) is
\[\left[\begin{array}{c}\psi_{m}^{\uparrow}\\ \psi_{m}^{\downarrow}\end{array}\right]=A_{m}\left[\begin{array}{c}I_{m}( \kappa r)\\ -isI_{m+1}(\kappa r)\end{array}\right], \tag{18}\]
where \(A_{m}\) is a constant.
For \(r>R\) with \(\Delta_{\rm m}=0\) introducing for brevity \(\mu\equiv r\varepsilon/v=kr\) we obtain
\[\left[\mu^{2}\frac{d^{2}}{d\mu^{2}}+\mu\frac{d}{d\mu}+(\mu^{2}-m^{2})\right] \psi_{m}^{\uparrow}=0, \tag{19}\]
with the Bessel functions \(J_{m}(kr)\) and \(Y_{m}(kr)\)[35] solutions where \(\psi_{m}^{\downarrow}\) being expressed with \(J_{m+1}(kr)\) and \(Y_{m+1}(kr).\) The resulting general solution at \(r>R\) is the superposition of the waves with harmonics
\[\left[\begin{array}{c}\psi_{m}^{\uparrow}\\ \psi_{m}^{\downarrow}\end{array}\right]=B_{m}\left[\begin{array}{c}J_{m}(kr) \\ iJ_{m+1}(kr)\end{array}\right]+C_{m}\left[\begin{array}{c}Y_{m}(kr)\\ iY_{m+1}(kr)\end{array}\right]. \tag{20}\]
These equations are supplemented by the two continuity conditions at \(r=R\) which can be reduced to a single equation as:
\[B_{m}J_{m}(kR)+C_{m}Y_{m}(kR)\] \[= -\frac{I_{m}(\kappa R)}{sI_{m+1}(\kappa R)}\left(B_{m}J_{m+1}(kR )+C_{m}Y_{m+1}(kR)\right).\]
### Scattering amplitude: summed partial waves
To perform summation over harmonics \(m\), we begin with the plane wave resolution \(e^{ikr\cos\varphi}=\sum_{m=-\infty}^{\infty}i^{m}e^{im\varphi}\,J_{m}(kr)\)[36] resulting in
\[\mathbf{\psi}_{\rm k}(r,\varphi)=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right]\sum_{m=-\infty}^{\infty}i^{m}e^{im\varphi}\,J_{m}(kr). \tag{22}\]
At large distances, \(kr\gg|m^{2}-1/4|\), we use asymptotics of the Bessel functions [35]:
\[J_{m}\left(kr\right) \simeq \sqrt{\frac{2}{\pi kr}}\cos\left(kr-\frac{m\pi}{2}-\frac{\pi}{4 }\right), \tag{23}\] \[Y_{m}\left(kr\right) \simeq \sqrt{\frac{2}{\pi kr}}\sin\left(kr-\frac{m\pi}{2}-\frac{\pi}{4 }\right). \tag{24}\]
By using condition that the wave function contains only the outgoing \(\exp(ikr)\) and no ingoing \(\exp(-ikr)\) waves [34], that is the ingoing wave terms mutually cancel each other, we obtain
\[f^{\uparrow}(\varphi) = -ie^{-i\pi/4}\sqrt{\frac{1}{\pi k}}\sum_{-\infty}^{\infty}\frac{ \gamma_{m}}{1+i\gamma_{m}}e^{im\varphi}, \tag{25}\] \[f^{\downarrow}(\varphi) = f^{\uparrow}\left(\varphi\right)e^{i\varphi}. \tag{26}\]
Here \(\gamma_{m}\equiv\,C_{m}/B_{m}\) is obtained with the boundary conditions in the form of Eq. (21). The spin component expectation values for the scattered wave defined as
\[\sigma_{i}(\varphi)=\langle\mathbf{f}(\varphi)|\sigma_{i}|\mathbf{f}(\varphi)\rangle \tag{27}\]
are the same as those of the free state since \(k_{+}=ke^{i\varphi}\) with \(\sigma_{z}(\varphi)=0.\)
In the low-energy domain \(\varepsilon<\Delta_{\rm m}\) we obtain
\[\gamma_{m}=-\frac{sJ_{m}(kR)I_{m+1}(\kappa R)+I_{m}(\kappa R)J_{m+1}(kR)}{sY_{m}( kR)I_{m+1}(\kappa R)+I_{m}(\kappa R)Y_{m+1}(kR)}. \tag{28}\]
Similar calculation for the high-energy domain \(\varepsilon>\Delta_{\rm m}\) using relation \(I_{m}(i|z|)=(-i)^{m}\,J_{m}\left(-|z|\right)\) and \(J_{m}(-x)=(-1)^{m}J_{m}(x)\) yields
\[\gamma_{m}=-\frac{sJ_{m}(kR)J_{m+1}(\kappa R)-J_{m}(\kappa R)J_{m+1}(kR)}{sY_{m}( kR)J_{m+1}(\kappa R)-J_{m}(\kappa R)Y_{m+1}(kR)}. \tag{29}\]
Various scattering regimes described by these equations will be discussed below analytically and numerically.
### Scattering cross-section and asymmetry
It is convenient to introduce even \(\Gamma_{g}\left(m_{1},m_{2}\right)=\Gamma_{g}\left(m_{2},m_{1}\right)\) and odd \(\Gamma_{u}\left(m_{1},m_{2}\right)=-\Gamma_{u}\left(m_{2},m_{1}\right)\) matrices as
\[\frac{\gamma_{m_{1}}}{1-i\gamma_{m_{1}}}\frac{\gamma_{m_{2}}}{1+ i\gamma_{m_{2}}}=\Gamma_{g}\left(m_{1},m_{2}\right)+\Gamma_{u}\left(m_{1},m_{2}\right) \tag{30}\] \[\Gamma_{g}\left(m_{1},m_{2}\right)=\frac{\gamma_{m_{1}}\gamma_{m _{2}}}{\left(1+\gamma_{m_{1}}^{2}\right)\left(1+\gamma_{m_{2}}^{2}\right)} \left(1+\gamma_{m_{1}}\gamma_{m_{2}}\right)\] \[\Gamma_{u}\left(m_{1},m_{2}\right)=i\frac{\gamma_{m_{1}}\gamma_{m _{2}}}{\left(1+\gamma_{m_{1}}^{2}\right)\left(1+\gamma_{m_{2}}^{2}\right)} \left(\gamma_{m_{1}}-\gamma_{m_{2}}\right),\]
and use them to define \(l/R\), \(\left\langle\varphi\right\rangle\) and \(\left\langle\varphi^{2}\right\rangle.\) Using Eqs. (8), (25), (26), and (30), the \(l/R\) and the mean value \(\left\langle\varphi\right\rangle\) can be expressed with these matrices as:
\[\frac{l}{R} = \frac{4}{kR}\sum_{m}\frac{\gamma_{m}^{2}}{1+\gamma_{m}^{2}}=\frac {4}{kR}\text{tr}\,\Gamma_{g}\left(m_{1},m_{2}\right), \tag{31}\] \[\left\langle\varphi\right\rangle = -\frac{4i}{kl}\sum_{m_{1},m_{2}}\Gamma_{u}\left(m_{1},m_{2}\right) \frac{\left(-1\right)^{m_{2}-m_{1}}}{m_{2}-m_{1}} \tag{32}\]
making the scattering asymmetric with nonzero \(\left\langle\varphi\right\rangle\) solely due to the imaginary terms \(i\gamma_{m}\) in the denominators of Eqs. (25), (26), which appear due to the phase shift between the spin components in Eq. (11). This effect is qualitatively different from the spin-diagonal scattering by a radially-symmetric potential, which is always \(\varphi\leftrightarrow-\varphi\) symmetric, and is similar to the scattering mechanisms producing the anomalous Hall effect [37]. For the \(\left\langle\varphi^{2}\right\rangle\) we obtain similarly:
\[\left\langle\varphi^{2}\right\rangle=\frac{8}{kl}\sum_{m_{1},m_{2}}\Gamma_{g} \left(m_{1},m_{2}\right)\frac{\left(-1\right)^{m_{2}-m_{1}}}{\left(m_{2}-m_{1 }\right)^{2}}+\frac{\pi^{2}}{3}. \tag{33}\]
## IV Sets of parameters and scattering domains
We introduce two parameters which fully describe the scattering process \(M\equiv\,R\Delta_{\text{m}}/v,\epsilon\equiv kv/\Delta_{\text{m}}\) and express the scattering amplitudes with
\[s=\sqrt{\frac{|1-\epsilon|}{1+\epsilon}};\qquad\kappa R=M\sqrt{|1-\epsilon^{2 }|};\qquad kR=M\epsilon, \tag{34}\]
where \(\kappa/k=\sqrt{|1-\epsilon^{2}|}/\epsilon.\) Parameter \(M\) corresponds to the angular momentum of the electrons with the resonant energy \(\Delta_{\text{m}}\) and can be seen as \(\tau_{p}\Delta_{\text{m}},\) where \(\tau_{p}=R/v\) is the typical passing time through the magnetic domain while the limit \(M\ll\,1\) corresponds to the Born approximation of the scattering theory [38].
### Low-energy domain \(\epsilon<1\)
We consider first the low-energy domain \(\epsilon<1.\) To demonstrate the main properties of the scattering, we begin with the small-radius, large wavelength limit \(kR\ll 1,\) where spin-independent scattering theory predicts angle-independent probability with \(|\mathbf{f}(\varphi)|^{2}=\text{const}.\) As we will show, however, it is not the case in the presence of spin-momentum locking. For this purpose we use small-\(x\) behavior of the Bessel functions:
\[J_{m}(x) \simeq I_{m}(x)\simeq\frac{1}{m!}\left(\frac{x}{2}\right)^{m}; \tag{35}\] \[Y_{m}(x) \simeq -\frac{(m-1)!}{\pi}\left(\frac{x}{2}\right)^{-m}\]
and their index-parity transformations:
\[J_{-m}(x) = \left(-1\right)^{m}J_{m}(x);\quad I_{-m}(x)=I_{m}(x), \tag{36}\] \[Y_{-m}(x) = \left(-1\right)^{m}Y_{m}(x).\]
We consider first a nonresonant scattering with \(kR\ll 1\) and \(\kappa R\gg\,kR.\) Thus, we select terms by the lowest powers of \(kR\) in the numerator [34; 39] and highest powers of \(\left(kR\right)^{-1}\) in the dominator and obtain for \(m\geq 0\)
\[\gamma_{m\geq 0} = -\frac{J_{m}(kR)I_{m+1}(\kappa R)}{I_{m}(\kappa R)Y_{m+1}(kR)}\] \[= \frac{I_{m+1}(\kappa R)}{I_{m}(\kappa R)}\frac{\pi}{(m!)^{2}} \left(\frac{kR}{2}\right)^{2m+1}.\]
Making similar \(kR-\)powers selection for \(m<0\) we obtain:
\[\gamma_{m<0}=-\frac{I_{m}(\kappa R)}{I_{m+1}(\kappa R)}\frac{\pi}{\left(|m+1| |\right)^{2}}\left(\frac{kR}{2}\right)^{2|m|-1} \tag{38}\]
and see fast decrease with \(|m|\) both for positive and negative \(m.\) Therefore, at \(kR\ll 1\) and \(M\sim\,1\) one obtains the resulting angular dependence \(|\mathbf{f}(\varphi)|^{2}\sim\sin^{2}\varphi/2\) with predominant backscattering, qualitatively different from the spin-diagonal scattering [39]. The ratio \(l/R\sim kR\) is linear in the energy and a weak asymmetry \(\left\langle\varphi\right\rangle\sim\gamma_{0}-\gamma_{-1}\sim kR\sim\,l/R\)[38]. In this limit \(\left\langle\varphi^{2}\right\rangle=2+\pi^{2}/3\) and, therefore, \(D_{\varphi}=\sqrt{2+\pi^{2}/3}.\)
Next, we turn to small wavelength, large radius limit \(kR\gg|m^{2}-1/4|\) away from resonance with \(\kappa R\gtrsim\,1\) but \(\kappa R<|m^{2}-1/4|.\) Then, by using asymptotics for the functions of \(kR\) and exact expressions for the functions of \(\kappa R\) and noticing that no power selection is required here, we obtain after a straightforward calculation:
\[\gamma_{m}=-\tan\left(kR-\frac{\pi m}{2}-\frac{\pi}{4}+\xi_{m}\right) \tag{39}\]
with \(\xi_{m}=\arctan\left(sI_{m+1}(\kappa R)/I_{m}(\kappa R)\right).\)
### Resonant scattering \(\epsilon\to 1\)
Next, consider resonant scattering as the energy of electron is close to \(\Delta_{\text{m}}\) with \(kR=M\) and \(\kappa R\ll\,kR\) at \(\epsilon\to 1.\) Here \(s=\sqrt{(1-\epsilon)/(1+\epsilon)}\approx\sqrt{(1-\epsilon)/2}\) and
\(\kappa R=M\sqrt{2}\sqrt{1-\epsilon}\) yield \(s\approx\kappa R/2M\ll 1.\) Making expansions in Eq. (28), we obtain
\[\gamma_{m}=-\frac{\kappa RJ_{m}(M)I_{m+1}(\kappa R)+2MI_{m}(\kappa R)J_{m+1}(M)}{ \kappa RY_{m}(M)I_{m+1}(\kappa R)+2MI_{m}(\kappa R)Y_{m+1}(M)}. \tag{40}\]
Here we perform selection by power counting of small \(\kappa R\) and obtain
\[\gamma_{m\geq\,0}=-\frac{J_{m+1}(M)}{Y_{m+1}(M)}, \tag{41}\]
in the limit \(M\ll 1\) this yields:
\[\gamma_{m\geq\,0}\approx\frac{\pi}{(m+1)!m!}\left(\frac{M}{2}\right)^{2(m+1)}. \tag{42}\]
For \(m<0\) we take into account that:
\[I_{m+1}(\kappa R)\approx\frac{1}{|m+1|!}\left(\frac{\kappa R}{2}\right)^{|m|-1} \tag{43}\]
and obtain:
\[\gamma_{m<0}=-\frac{|m|J_{m}(M)+MJ_{m+1}(M)}{|m|Y_{m}(M)+MY_{m+1}(M)}, \tag{44}\]
yielding in the limit \(M\ll\,1:\)
\[\gamma_{m<0}=\frac{(-1)^{m}\pi}{|m|!(|m|-1)!}\left(\frac{M}{2}\right)^{2|m|}. \tag{45}\]
Since in this limit \(\gamma_{0}=-\gamma_{-1}\) with \(|\gamma_{|m|>1}|\ll|\gamma_{0}|,\) the scattering behavior remains the same as in the \(\epsilon\ll\,1\) case.
### High-energy domain \(\epsilon>1\)
We use in Eq. (29) known asymptotics in Eqs. (23) and (24) for the realization \(kR\approx\kappa R\gg 1\) and where both the effective angular momentum and energy are large. Summing the terms and taking into account that: \(\kappa R\pm\,kR=M\left(\sqrt{\epsilon^{2}-1}\pm\epsilon\right),\) we obtain for the realization \(kR\approx\kappa R\gg|m^{2}-1/4|:\)
\[\gamma_{m}=-\frac{(-1)^{m}(1-s)\cos\left(M\epsilon_{+}\right)+(1+s)\sin\left( M\epsilon_{-}\right)}{(1+s)\cos\left(M\epsilon_{-}\right)+(-1)^{m}(1-s)\sin \left(M\epsilon_{+}\right)}, \tag{46}\]
where \(\epsilon_{\pm}\equiv\,\sqrt{\epsilon^{2}-1}\pm\,\)\(\epsilon.\)
In the case \(\epsilon\gg 1\) the leading terms in the expansion by \(1/\epsilon\) result in:
\[\gamma_{m}=(-1)^{m}\frac{\cos(2M\epsilon)}{2\epsilon} \tag{47}\]
rapidly decreasing with the energy. The limit \(\epsilon\rightarrow\infty\) evidently yields \(\gamma_{m}=0.\)
## V Scattering by diffraction grating
We consider now a diffraction grating formed by the linear chain of magnetic dots (nanodiscs) at the surface of topological insulator as shown schematically in Fig. 2. The array contains \(N\) identical dot scatterers separated by the distance \(d\) such that position of the center of domain is given by: \(y_{i}=d(1-N)/2+d(i-1),\)\(i=1,\ldots,N.\)
In this geometry, we assume that each dot is an independent scatterer of the incoming plane wave with spin polarization along axis \(x\). The distance \(d\) between neighboring dots is of the order of electron wavelength \(\lambda,\) and we are observing the diffraction pattern at a relatively large distance \(L\gg\lambda\). To consider the grating as a chain of independent scatterers, we first formulate the scattering independence condition:
\[\frac{R}{d}\left|\mathbf{f}\left(\pi/2\right)\right|^{2}\ll\frac{l}{4} \tag{48}\]
meaning that the wave scattered by one dot cannot be re-scattered by its neighbors.
The scattering pattern produced at points \((L,y)\) on the screen is given by [40]:
\[\mathbf{F}(y)=\sum_{i=1}^{N}\mathbf{f}(\varphi_{i})\,\frac{e^{ikri}}{\sqrt{r_{i}}}, \tag{49}\]
where \(r_{i}=\sqrt{\left(y-y_{i}\right)^{2}+L^{2}}\) and \(\varphi_{i}=\arctan(\left(y-y_{i}\right)/L).\) Then, we obtain the scattering density \(\left|\mathbf{F}(y)\right|^{2}\) and density of spin components \(\sigma_{j}(y)=\mathbf{F}^{\dagger}(y)\sigma_{j}\mathbf{F}(y).\)
The points, where the scattered waves produce constructive interference are determined by the constructive interference condition, \(d\sin\varphi=n\lambda.\) For asymmetric scattering this relation also determines the spin orientation in the diffraction spots. The whole diffraction picture is asymmetric as the "brightness" of spots is more
Figure 2: Schematic plot of electron scattering by a diffraction grating. Only the principal peaks are shown. The presented asymmetry corresponds to asymmetric scattering in Fig. 1.
pronounced in one of \(\varphi\)-directions (this is shown in Fig. 5 as a larger peak for \(\varphi<0\) than for \(\varphi>0\), in accordance with Fig. 1). Thus, we obtain scattering profile corresponding to \(|f\left(\varphi\right)|^{2}\) with asymmetric scattering pattern. This asymmetric profile corresponds to formation of the spin current also.
Now we turn to the diffraction picture for the spin polarization where the qualitative feature is the emergence of nonzero \(z-\)axis spin polarization. To clarify this effect we consider two-dots realization with \(N=2\), \(y_{1}=-d/2\), and \(y_{2}=d/2\), where
\[\mathbf{F}(y)=\mathbf{f}(\varphi_{1})\frac{e^{ikr_{1}}}{\sqrt{r_{1}}}+\mathbf{f}( \varphi_{2})\frac{e^{ikr_{2}}}{\sqrt{r_{2}}}, \tag{50}\]
with
\[r_{1,2} = \overline{r}+\Delta r_{1,2}=\overline{r}\pm\frac{d}{2}\frac{y}{ \sqrt{L^{2}+y^{2}}}, \tag{51}\] \[\varphi_{1,2} = \overline{\varphi}+\Delta\varphi_{1,2}=\overline{\varphi}\pm \frac{d}{2}\frac{L}{L^{2}+y^{2}}, \tag{52}\]
where \(\overline{r}=\sqrt{L^{2}+y^{2}}\) and \(\overline{\varphi}=\arctan(y/L).\) Expansions in Eq. (50) with small \(|\Delta r|\ll L\) and \(|\Delta\varphi|\ll|\overline{\varphi}|\sim 1\), show for the scattering density:
\[|\mathbf{F}(y)|^{2}\approx\frac{4}{\overline{r}}\cos^{2}\left(k\Delta r\right) |\mathbf{f}(\overline{\varphi})|^{2} \tag{53}\]
and for the \(z-\)component of spin:
\[\mathbf{F}^{\dagger}(y)\sigma_{z}\mathbf{F}(y)=\left|F^{\dagger} (y)\right|^{2}-\left|F^{\dagger}(y)\right|^{2}\] \[\approx -\frac{1}{\overline{r}}\sin\left(k\Delta r\right)\Delta\varphi| \mathbf{f}(\overline{\varphi})|^{2},\]
where \(\Delta r=\Delta r_{2}-\Delta r_{1}\) and \(\Delta\varphi=\Delta\varphi_{2}-\Delta\varphi_{1}.\) Since \(k\Delta r=2\pi(d/\lambda)\times y/\sqrt{L^{2}+y^{2}}\) at \(d\ll L\) is much larger than \(\Delta\varphi,\) the scattering intensity weakly depends on \(\Delta\varphi.\) The resulting \(\mathbf{F}^{\dagger}(y)\sigma_{z}\mathbf{F}(y)\) is not zero but rapidly decreases with \(y.\)
## VI Numerical results: cross-section and scattering angles
### Single scatterers
The results of numerical calculations of the scattering cross-section length \(l/R\), mean angle \(\langle\varphi\rangle\) and dispersion \(D_{\varphi}\) based on Eqs. (28) and (29) are presented in Fig. 3 as the universal functions of parameters \(\epsilon\) and \(M\).
The upper panel shows that the ratio \(l/R\) is small both for small \(M,\) corresponding to the Born approximation and for relatively large \(M\) and \(\epsilon,\) where electron energy is sufficient to ensure a relatively weak effect of the nanosize dot on the electron propagation.
The mean scattering angle in the middle panel is typically small since the scattering is still close to \(\varphi\leftrightarrow-\varphi\) symmetric in the domain of \(\epsilon\sim 1\) and \(M>1/2.\) Also, at large energies the forward scattering dominates leading to a small mean \(|\langle\varphi\rangle|\ll 1.\)
On the contrary, \(D_{\varphi}\) is relatively large at \(M<1\) being of the order of one and then decreases since the forward scattering with \(|\langle\varphi\rangle|\ll 1\) and \(\langle\varphi^{2}\rangle\ll 1\) becomes dominating. Notice hyperbolic structure clearly seen at \(M\epsilon>1\) in the mean scattering angle and its dispersion demonstrating a periodic dependence on \(M\epsilon\) product with the \(\pi/2\) period, corresponding to Eq. (47).
Figure 3: Numerical results for \(l/R\), \(\langle\varphi\rangle,\) and \(D_{\varphi},\) as marked above the plots. In all numerical calculations we use \(\hbar v=2.5\times 10^{2}\) meVnm with \(v=4\times 10^{7}\) cm/s typical for Bi-based topological insulators [41], \(\Delta_{\rm m}=25\) meV and consider \(-10\leq m\leq 10\) harmonics. We emphasize that the results in terms of \(M\) and \(\epsilon\) parameters are universal and do not depend on the choice of these numerical values.
To illustrate this behavior of the cross-section and scattering angle, we plot in Fig. 4 the angular dependence of the differential scattering cross-section. Figure 4 shows that at small energies this function behaves as \(\sin^{2}(\varphi/2)\) and with the increase in the energy the scattering becomes less symmetric till it becomes mainly forward at high energies. At higher energies and larger \(M\), one obtains forward scattering with a relatively weak asymmetry \(|\langle\varphi\rangle|\ll\,1\) and small aperture \(D_{\varphi}\ll\,1\).
### Diffraction gratings
Having discussed single-dot scattering, here we present in Fig. 5 the numerical results for the density probability and spin density produced by a diffraction grating.
As shown in Fig. 5, the diffraction pattern consists of strong principal scattering peaks and of weak secondary intermediate peaks, as predicted by the diffraction theory [40]. As expected, the diffraction pattern is strongly asymmetric with \(\langle\varphi\rangle<0\), as can be understood from Figs. 3 and 4. In addition, we see that the \(\sigma_{z}(y)\) spin density is small but not zero, as expected from the discussion above when for a single scatterer \(\sigma_{z}(y)=0.\) Figure 5 shows that with the given gratings geometry one can achieve the spin polarization \(\sigma_{z}(y)/\left|\mathbf{F}(y)\right|^{2}\leq 0.1.\) This is a result of interference of the waves scattered by different angles \(\varphi_{i}\), similar to the effects observed in the scattering of bunches of ultrafast electrons in solids [42].
## VII Conclusions
We studied cross-section and diffraction patterns of electron scattering by magnetic nanodots and their diffraction gratings on the surface of a topological insulator with spin-momentum locking. For a single nanomagnet, we considered analytically and numerically var
Figure 4: Differential cross-section length of electron scattering of magnetic quantum dot for different electron energies (as marked near the plots). Upper panel corresponds to \(R=2\) nm (\(M=0.2\)) and lower panel corresponds to \(R=15\) nm (\(M=1.5\)). The system parameters are the same as in Fig. 3. Weak forward scattering at \(M\ll\,1\) corresponds to the perturbation theory in the Born approximation [38].
Figure 5: Scattering probability and spin density produced by nanodots-formed diffraction grating with \(N=10\) dots, the interdot distance \(d=150\) nm, and distance to the screen \(L=2000\) nm. Upper panel corresponds to \(R=5\) nm, lower panel corresponds to \(R=15\) nm. Electron energy \(\varepsilon=\Delta_{\rm m}/2=12.5\) meV, wavevector \(k=5\times 10^{-2}\) nm\({}^{-1}\), and the wavelength \(\lambda=1.26\times 10^{2}\) nm. Scattering angle \(\varphi\) for the first principal maxima obtained with \(d|\sin\varphi|=\lambda\) yields \(|\sin\varphi|=\lambda/d=0.84\) and corresponds to position on the screen \(|y|=L|\sin\varphi|/\sqrt{1-\sin^{2}\varphi}\approx\,3100\) nm, in agreement with the presented plots. Although Eqs. (53) and (54) are not directly applicable here, they agree with these results demonstrating small \(\sigma_{z}(y)\) near the principal peaks maxima.
ious scattering regimes in terms of the electron energy and nanodott size and magnetization and demonstrated that they can be universally described by two dimensionless system parameters. The scattering probability is usually angle-asymmetric, presenting its qualitative feature due to the spin-momentum locking as can occur in a broad interval of the scattering angles. It becomes angle-symmetric (i) at high energies, where it is concentrated in a narrow angle and (ii) in the energy-independent Born approximation leading to the universal broad scattering probability distribution. We demonstrated that the spins of scattered electrons remain parallel to the surface of the topological insulator. Next, we obtained the corresponding patterns of the scattering by diffraction gratings. In qualitative contrast to single scatterers, diffraction gratings produce nonzero perpendicular to the surface spin component of the scattered electrons.
These results can be applied for the design of magnetization patterns such as arrays of magnetic quantum dots or magnetization lattices of nanomagnets of the size between 10 and 100 nm [43] to produce in a controllable way spin and charge currents and densities at the surfaces of topological insulators. This approach can be used for studies of spin torques [26; 27; 28; 29; 30; 31; 32; 33] produced on the magnetized quantum dots by scattered electrons. Another application can be related to electron interferometry and holography of magnetic structures and nonuniform magnetic fields [44; 45] providing detailed information about magnetization patterns by visualization of the phase of the electron wavefunction.
## Acknowledgements
This work was supported by the National Science Center in Poland as a research project No. DEC-2017/27/B/ST3/02881. The work of E.S. is financially supported through Grants No. PGC2018-101355-B-I00 and No. PID2021-126273NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ERDF "A way of making Europe," and by the Basque Government through Grants No. IT986-16 and No. IT1470-22.
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2306.07038 | Hilbert Bundles and Holographic Space-time Models | We reformulate Holographic Space-time (HST) Models as Hilbert bundles over
the space of time-like geodesics on a background manifold. The background,
following Jacobson, is viewed as a hydrodynamic flow, which the quantum model
must reproduce. Work of many authors, but particularly the Verlindes, Carlip
and Solodukhin, suggest that the relevant quantum model is a sequence of 1+1
dimensional conformal field theories (CFT) on the boundaries of nested causal
diamonds. Finiteness of the entropy suggests the CFTs be built from cutoff
fermion fields, and the spin/statistics connection, combined with Connes'
demonstration that Riemannian geometry is encoded in the Dirac equation,
suggests that these fields are labelled by the cutoff eigenspectrum of the
Dirac operator on the holoscreen of each diamond. This leads to a natural
conjecture for the density matrix of arbitrary diamonds and,in a subclass of
space-times, for the time evolution operator between them. We conjecture that
the 't Hooft commutation relations on diamond boundaries are related to
Schwinger terms in U(1) currents constructed from the fermion fields. We review
the notion of "locality as constraints on holographic variables" discovered by
Fiol, Fischler and the present author and, in an appendix, explain how it
differs from the notion of locality arising from tensor network constructions
in AdS/CFT. | T. Banks | 2023-06-12T11:37:37Z | http://arxiv.org/abs/2306.07038v1 | # Hilbert Bundles and Holographic Space-time Models
###### Abstract
We reformulate Holographic Space-time Models in terms of Hilbert bundles over the space of time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite dimensional Hilbert spaces. Following Jacobson[1] we view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein-Hawking relation between area and entropy, generalized to arbitrary causal diamonds. Time-like geodesics are equivalent to nested sequences of causal diamonds and the area of the holoscreena encodes the entropy of a certain density matrix on a finite dimensional Hilbert space. We review arguments[2][3][4] that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator \(L_{0}\) of a \(1+1\) dimensional CFT of large central charge, living on an interval in longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is \(\rm{lnD}_{\diamond}\), up to subleading corrections, in the limit of large dimension diamond Hilbert space. We also connect those arguments to the derivation[5] of the 't Hooft commutation relations for horizon fluctuations. We present a tentative connection between the 't Hooft relations and \(U(1)\) currents in the CFT's on the past and future diamond boundaries. The 't Hooft relations are related to the Schwinger term in the commutator of vector and axial currents. The paper[5] can be read as a proof that near horizon dynamics for causal diamonds much larger than Planck scale is equivalent to a topological field theory of the 't Hooft CR, plus small fluctuations of transverse geometry. Connes'[6] demonstration that Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy, and makes the geometry both "fuzzy"[7] and quantum. Following the analysis of Carlip and Solodukhin, we model the expansion coefficients as two dimensional fermionic fields. We argue that local excitations in the interior of a diamond are constrained states where the spinor variables vanish in regions of small area on the holoscreen. This leads to an argument that quantum gravity in asymptotically flat space must be exactly supersymmetric.
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###### Contents
* I Introduction
* I.1.1 A \(Verlinde^{2}\to Carlip+Solodukhin\) in a General Causal Diamond
* II Transverse Geometry in Terms of Fluctuating Fuzzy Spinors
* II.1 Time Evolution
* III The Quantum Principle of Relativity
* III.1 Space-time Locality and the Origin of Quantum Field Theory
* IV Representation of Non-compact Isometry Groups
* IV.1 Summary
* V The 't Hooft Commutation Relations
* VI Conclusions, Speculations and (Ugh) Philosophy
* VII Appendix The Curious Case of Anti de Sitter Space
* VIII Appendix II Some Remarks about Black Holes in Minkowski Space
## I Introduction
Quantum Field Theory (QFT) is our most successful model of the physics of the universe and enjoys a remarkable degree of agreement with experiment. Every experiment ever done, or likely to be done in the forseeable future, is performed by a detector moving for a finite amount of proper time along a near geodesic time-like trajectory in a Lorentzian manifold with curvature that is small in Planck units. Such an experiment defines a causal diamond. The authors of[8] showed that the agreement of QFT with current experiments was unaffected if we omitted from the QFT Hilbert space all states whose semi-classical gravitational back reaction would create a black hole larger than the causal diamond. With the UV/IR cutoffs imposed in[8] current experiments were near the limits implied by these constraints, but[9] showed that other forms of cut off put experimental tests of these _a priori_ constraints on QFT off into the distant future. The total entropy allowed by these constraints is \(\sim S_{BH}^{\frac{d-1}{d}}\), in \(d\) dimensional space-time. \(S_{BH}\) is the area (\(d-2\) volume) of the maximal area \(d-2\)
surface on the null boundary of the diamond, divided by \(4G_{N}\). We will call this the area of the holographic screen.
Although the experimental success of QFT does not require us to believe this, the theoretical structure of QFT gives us a compelling reason to suspect that there is something special about an area law for the entropy of a causal diamond. In the 1960s, algebraic quantum field theorists[10] discovered that the algebra of operators localized in a diamond was always Type III in the Murray von Neumann classification. This means that the algebra has no density matrices of finite entropy. In 1983, Sorkin[11] computed the vacuum entanglement entropy of a diamond and found that it was proportional to the area of the holographic screen, with a UV divergent proportionality constant. This calculation languished in obscurity until it was repeated by more sophisticated methods in the 1990s[12][13]. It is clear that these calculations are unchanged in any state obtained by acting with a finite number of smeared local operators on the vacuum, as long as their support is spacelike separated from the boundary of the diamond. The infinite entanglement comes from divergent light cone commutators of fields on either side of the diamond boundary.
Entanglement entropy of one system with another is a lower bound on the maximal entropy of the system. The fact that the finite temperature entropy of the portion of a Cauchy slice whose causal boundary is the diamond, is finite after vacuum entropy subtraction, combined with the Cohen Kaplan Nelson (CKN) argument, is a strong hint that this divergent boundary entropy has fundamental significance. Susskind and Uglum[14] and Jacobson[15] suggested that this divergence should be viewed as a renormalization of Newton's constant in the Bekenstein Hawking entropy law. None of these authors commented on the revolutionary nature of this suggestion. The Bekenstein-Hawking law was a proposal about black holes. The calculations of[11][12][13] were all done in empty flat space. At the time, the present author found this confusing, but was not clever enough to follow up on the hint.
The most convincing evidence that we should take the area law proposal for the entropy of a causal diamond seriously came one year later in the work of Jacobson[1]. This paper was phrased in terms of local changes of entropy and so did not explicitly invoke the _Covariant Entropy Principle_(CEP): **In models of quantum gravity, the entropy of a causal diamond is equal to the area of its holographic screen divided by \(4G_{N}\).** Entropy should be thought of as the expectation value of the modular Hamiltonian associated with the "empty diamond" state. The CEP is meant as the first term in an asymptotic expansion of the entropy as a function of the area. The term "area" makes sense only in the context of this expansion. It was not until 1998 that the CEP was formulated explicitly, first in the special case of FRW cosmology by Fischler and Susskind[16] and then in the masterful
follow up work by Bousso[17].
Implicit in the CEP is the central notion proven in[1]: the equations
\[k^{m}(x)k^{n}(x)(R_{mn}(x)-\frac{1}{2}g_{mn}(x)R(x)-8\pi G_{N}T_{mn}(x))=0, \tag{1}\]
are the hydrodynamic equations of the CEP. Here \(k^{m}(x)\) is an arbitrary null vector in space-time. That is, these equations follow from the definition of Lorentzian geometry (which includes the Raychaudhuri equation for the local spreading of a congruence of geodesics) and the covariant conservation of the stress tensor, applied to the thermodynamic equation \(dE=TdS\), assuming that entropy is given by the CEP. One also uses the local relation \(E=k^{m}k^{n}T_{mn}\) and chooses a congruence of trajectories centered around a maximally accelerated Unruh trajectory, which grazes a point on the holographic screen of the diamond. The fact that the CEP is only an asymptotic relation follows from the fact that the derivation assumes that everything can be treated as locally flat.
A few comments are in order. The CEP can be "derived" from Euclidean path integrals[22] just like black hole or dS entropy[23]. The relation between Euclidean path integrals and hydrodynamics has been clarified recently in[25]. For a system consisting of large independent subsystems \(X_{i}\) connected by small "interfaces" with many fewer degrees of freedom, hydrodynamics can be derived from quantum mechanics as a Markov equation for the diagonal matrix elements of the density matrix in a basis of states in which the local values \(C(X_{i})\) of conserved quantities are simultaneously diagonal[26]. Kac's path integral solution[27] of this Markov equation is the Euclidean path integral for hydrodynamics.
The hydrodynamic view of Einstein's equations means that they are valid in contexts where there is no systematic weak coupling or large N expansion. Strongly coupled condensed matter systems have high entropy states whose coarse grained properties are described by the Navier-Stokes equations, even though the underlying quantum mechanics is not well approximated by perturbative quasi-particle physics. As a bonus, Jacobson's derivation makes it clear that the cosmological constant in Einstein's equations should not be thought of as an energy density since it does not appear in the equations of hydrodynamics. String theorists should have known this since the invention of the AdS/CFT correspondence, where the c.c. plays the role of a parameter in the high energy density of states in a CFT and is determined by discrete parameters like the gauge group \(SU(N)\).
Further evidence for the CEP, and for Jacobson's view of GR as hydrodynamics, comes from remarkable work done by Carlip[2] and Solodukhin[3] in 19981. These authors proposed a state counting interpretation of black hole entropy for quite general black holes. More recently, K. Zurek
and the present author realized that their derivation applied to a general causal diamond[4]. We will explore these ideas using an argument introduced by the Verlindes[5] in order to provide a formal derivation of 't Hooft's commutation relations between light ray operators in high energy scattering. The same argument is precisely suited for studying the near horizon limit of any causal diamond. In a later section, we will return to provide a microscopic model of the 't Hooft commutation relations.
### \(Verlinde^{2}\to Carlip+Solodukhin\) in a General Causal Diamond
The observation of the Verlindes is that one can analyze high energy scattering at large transverse distance by taking length scales in a two dimensional subspace of Minkowski signature to be Planck scale, while those in the transverse dimensions are of order \(L\gg L_{P}\). The same is true in the near horizon limit of any causal diamond, since the two dimensional length scales are nearly null and the transverse length scale \(L\), will generally be much larger than \(L_{P}\). The Einstein Hilbert action splits into three terms of very different size.
\[I_{EH}=(L/L_{P})^{d-4}I_{tH}+(L/L_{P})^{d-2}I_{\perp}+I_{3}. \tag{2}\]
\[I_{\perp}=\int\sqrt{-g}\sqrt{h}[-R(g)-g^{ab}\partial_{a}h_{mn}\partial_{b}h_{ kl}(h^{ml}h^{nk}-h^{hm}h^{nl})]. \tag{3}\]
\[I_{tH}=\int\sqrt{-g}\sqrt{h}[-R(h)-h^{mn}\partial_{m}g_{ab}\partial_{n}g_{cd }(g^{ad}g^{bc}-g^{ac}g^{bd})]. \tag{4}\]
We will not need the formula for the third term. Here \(g_{ab}=\hat{g}_{ab}-\hat{g}_{ai}h^{ij}\hat{g}_{jb}\), where \(\hat{g}\) is the original metric. The equations of motion of \(I_{\perp}\) say that \(g_{ab}\) is flat, and that \(h_{mn}\) is independent of \(y\). The authors of[5] then show that \(I_{tH}\) becomes a purely topological action for the boundary values of the coordinate transformation that takes \(g_{ab}\) to \(\eta_{ab}\). Those boundary values satisfy the 't Hooft commutation relations. We will return to a microscopic model of those commutation relations below.
A quick and dirty way to understand the argument of[5] and to see that it is valid in any dimension, is to note that we can view the near horizon limit of any causal diamond as an asymmetric rescaling of coordinates in which the transverse coordinates are taken much larger than those in the two dimensional Minkowksi signature directions. Under this rescaling, the covariant components of the Ricci tensor transform as a tensor and its clear that the dominant term for large \((L/L_{P})\) is \(R_{ab}\). The leading terms in Einstein's equations, for a stress tensor that consists of pure traceless boundary waves (as suggested by the CEP) are precisely those following from \(I_{\perp}\).
The fact that the fluctuations of the transverse metric become small whenever length scales are large compared to the Planck scale is another indication that Jacobson's hydrodynamic view of gravity
is correct. One can construct a very general derivation of hydrodynamics for lattice quantum systems, using only the assumption that large blocks of the lattice have Hamiltonians that commute with each other up to subleading surface terms[26]. The Schrodinger equation leads directly to classical stochastic equations for the hydrodynamic variables, which have a standard Euclidean path integral solution. Fluctuations of transverse geometry are similarly suppressed simply by the size of the region they describe. No assumptions of "weak string coupling" are necessary.
Solodukhin's analysis of the linearized fluctuations around the classical solution is particularly transparent. One defines a scalar field \(\phi\) in the Lorentzian dimensions in terms of the fluctuations of the conformal factor of the metric \(h\) around its classical value. After a \(\phi\) dependent Weyl transformation of the two dimensional metric (which, to linear order in \(\phi\) is just a constant rescaling) the action takes the form
\[I[\phi]=-\int d^{2}y\sqrt{-g}[\frac{1}{2}g^{ab}\partial_{a}\phi \partial_{b}\phi+q\phi\sqrt{\frac{(d-3)S}{(d-2)8\pi}}R(g)+U(\phi)]. \tag{5}\]
The metric is set to be Minkowski by the equations of motion of \(I_{\perp}\), but we have kept it general to show that the field \(\phi\) has a stress tensor with a large classical central charge. The first two terms define a classical conformal field theory, which was shown in[28] to have a stress tensor whose correlation functions satisfy all the Ward identities of a general CFT with the same value of the central charge, because stress tensor correlators in two dimensional CFT are completely determined by the central charge. This is often called the Liouville theory, though it does not have the usual two dimensional cosmological constant term that gives Liouville's equation for constant curvature metrics. An illuminating way to state the result of[28] is that the "Liouville" theory is the solution of the hydrodynamic equations of two dimensional CFT with a given value of \(c\). Solodukhin shows that the potential term \(U(\phi)\) contains only "classically irrelevant" perturbations of this CFT, in the near horizon limit.
In[20], Strominger argued that the boundary "Liouville" theory discovered by Brown and Henneaux[21] in \(2+1\) dimensional AdS gravity, was an avatar of the AdS/CFT correspondence. In our Jacobson language: asymptotic 3d gravity in AdS space is the hydrodynamics of \(1+1\) CFT. Strominger argued, as a consequence, that quantum gravity should be quantum CFT. This was an after the fact justification for the \(AdS_{3}/CFT_{2}\) correspondence. Carlip and Solodukhin advocate using the same logic for all black holes and show that Cardy's entropy formula reproduces the Bekenstein-Hawking formula in every case. In[4] we argued that these arguments applied to every causal diamond, giving further justification for the CEP. In addition we argued that they implied the universal fluctuation
formula \((\Delta K)^{2}=\langle K\rangle\) for the modular Hamiltonian2. Solodukhin remarks that there is no apparent reason for neglecting fluctuations of the unimodular part of the transverse metric \(h_{mn}\). In the next section we will use insights from the Holographic Space-time formalism to incorporate these into the Carlip-Solodukhin picture.
Footnote 2: The sole known exceptions are the horizons of large stable black holes in AdS space. These are also the only horizons that propagate sound modes. Both facts can be understood in terms of the tensor network model of AdS/CFT. The universal fluctuation formula is valid in nodes of the tensor network but the entropy of a large black hole is dominated by sound modes propagating between the nodes, which have a different capacity of entanglement. See the Appendix for more detail.
When combined with the CKN argument, the work of Carlip and Solodukhin leads one to a remarkable conclusion. Most of the quantum degrees of freedom of a model of quantum gravity are inaccessible to local experiments done on near geodesic trajectories through the diamond. They reside on the boundaries of causal diamonds and we will only be able to get very indirect experimental information about them. What is more, so far the CS picture does not give us a clue about the nature of the local excitations, to which the usual rules of QFT apply. These are the excitations explored by experiments done along geodesics inside a causal diamond. Fortunately, the answer to that question was found by Fiol, Fischler and the present author some time ago, and we will return to it below.
## II Transverse geometry in terms of fluctuating fuzzy spinors
The Holographic Space-time (HST) formalism was introduced in order to construct models of quantum gravity with a closer connection to local physics than that achieved in perturbative string theory or the AdS/CFT correspondence. Its most basic postulate is the existence of a net of local subalgebras of the operator algebra describing an entire space-time. These subalgebras are finite dimensional and are in one to one correspondence with finite area causal diamonds in the space-time. When the area is much larger than any microscopic scale the dimension of the Hilbert space associated with a diamond is approximately3
Footnote 3: A more precise statement is that the area is the expectation value of the modular Hamiltonian of the empty diamond state in the Hilbert space. However, in models constructed so far, this coincides with the log of the dimension up to subleading corrections.
\[d(\mathcal{H}_{\diamond})\approx e^{\frac{A}{4G_{N}}}. \tag{6}\]
A nested sequence of intervals along any time-like trajectory defines a nested sequence of causal diamonds. Every diamond contains a unique timelike geodesic that connects its past and future tips. Given a choice of nested intervals on a timelike geodesic we have a sequence of causal diamonds. The relations between proper time interval and area for all diamonds along all geodesics completely determines the space-time metric \(g_{mn}\). The relation between area and Hilbert space dimension thus
allows us to construct a map from quantum mechanics to geometry: we consider a _Hilbert bundle_ over the space of time-like geodesics that connect two Cauchy slices in the space-time4. We can make a synchronized choice of proper time intervals,starting from the past Cauchy surface or going backwards from the future surface. In time symmetric space-times we can nest outwards from a small diamond surrounding a point of time symmetry (Figure 1).
Footnote 4: For space-times that are asymptotically AdS, we must restrict the maximal proper time between points on the two Cauchy surfaces, so that the maximal causal diamonds have finite area, in order to treat the infinite area limit with proper care. Note that for non-negative c.c. we either have no restriction on the proper time between past and future surfaces or only the restriction that it be finite.
Specifying the dimensions of Hilbert spaces for each time interval is equivalent to specifying the space-time metric, in the limit of exponentially large dimensions. The basic hypothesis of HST is that Hilbert space dimension is the correct extension of the concept of area of a causal diamond's holographic screen, to short distances. This idea lines up perfectly with Jacobson's[1] claim that Einstein's Equations are the hydrodynamic equations of the area law, and provides an immediate resolution of space-like singularities in solutions of GR. Space-like singularities[29][30] are places where causal diamonds shrink to zero area and it's precisely in low entropy situations that we can expect the hydrodynamic approximation to a quantum system to break down.
Figure 1: Future directed, time symmetric, and past directed nested coverings of a causal diamond.
The hydrodynamic view of Einstein's equations casts doubt on a number of assumptions common among researchers in all branches of quantum gravity. The first of these is that there should be a single "background independent" formulation of quantum gravity, with all particular instances of the theory some sort of "vacuum states". Among string theorists, this myth should have been dispelled long ago by the success of the AdS/CFT correspondence. From the hydrodynamic perspective it is tantamount to saying that because almost all systems have states that satisfy the Navier-Stokes equation, then they are described by the same microscopic quantum Hamiltonian.
The second erroneous idea, most common among string theorists, is that the Einstein equations should always be viewed as the first term in some kind of semi-classical expansion, which is in some way related to a "path integral over metrics". For condensed matter systems with a gapless ground state, it is often the case that one can view the low lying excitations as quantized fields obeying the linearized hydrodynamic equations, with interactions that can be understood by Wilsonian renormalization group analysis and symmetry considerations. The classical hydrodynamic equations have a much broader validity than this and apply to states far removed from the ground state, in which it is inappropriate to quantize them. Instead, one searches for a microscopic system that reproduces the same coarse grained hydrodynamic flow. This is the principle that we will be following. Our models of quantum gravity always begin with a fixed classical background, which we view as a hydrodynamic flow to which we must match a quantum theory, following the clues outlined in the previous section. Hydrodynamic equations have to be supplemented by stochastic "stirring" terms, in order to agree with observation. There is a long history of representing the solutions of these stochastic equations by Euclidean path integrals, following Kac' original treatment of the diffusion equation. This is the way one should view Euclidean quantum gravity.
As noted, the big lacuna in the discussion of the previous section was a model for the fluctuations in the transverse geometry. Here we follow the analysis of HST. Finite entropy implies that the algebra of operators in a diamond be Type II in the Murray-von Neumann classification, but it is essentially impossible for experimental physics to distinguish between infinite Type II algebras and finite dimensional algebras, so HST has always been restricted to finite dimensional Hilbert spaces. Any such Hilbert space is a representation of a fermionically generated superalgebra, or equivalently of a system of canonical fermionic oscillators with constraints. Alain Connes pointed out long ago[6] that all of Riemannian geometry was encoded in the Dirac operator. Indeed, on page 544 of[6] one can find a (rather abstract) formula for the geodesic distance between two points in a manifold in terms of properties of the Dirac operator. Physicists are more familiar with the fact that the short time expansion of the heat kernel of the square of the Dirac operator can be written in terms of
curvature invariants, and that topological properties of the manifold are also related to the index of the Dirac operator. This means that a cut off on the spectrum of the Dirac operator provides a method of "fuzzifying" a Riemannian geometry even if the space does not have a symplectic structure[7]. Furthermore, if we postulate that the expansion coefficients of a generic section of the spinor bundle into Dirac eigenfunctions of a _fixed_ background metric, are \(1+1\) dimensional (cut-off) quantum fields, then each quantum field theory state will define a probability distribution for the curvature invariants of the transverse geometry. More precisely, the quantum field
\[\Psi(x,y)=\sum_{E}e^{-iEt}\psi_{E}(y)\chi_{E}(x), \tag{7}\]
satisfies the Dirac equation, with Hamiltonian \(H=[L_{0},]+D\), where \(D\) is the transverse Dirac operator in the background geometry, and \(E\) its cutoff spectrum. \([L_{0},]\) is the action of the \(L_{0}\) generator on operators in the CFT. Its spectrum is determined by that of \(L_{0}\). Thus, we can view the fluctuations in \(L_{0}\) as being fluctuations in the transverse Dirac spectrum, and thus of the transverse geometry.
This is not the only possible definition of a fluctuating transverse geometry. The fermion bilinears
\[\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\ldots A_{k})}e^{A_{1}}_{m(1)}\ldots e^{A_{ k}}_{m(k)}\psi(\sigma,x), \tag{8}\]
are rank \(k\) differential forms, and the bundle of all these forms satisfies the Kahler-Dirac equation
\[(d-d^{*})F=0, \tag{9}\]
where \(d\) is the exterior derivative on the transverse manifold and \(d*\) its Hodge dual. This equation implies that there is a linear combination of the bilinear
\[\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\ldots A_{4})}e^{A_{1}}_{m(1)}e^{A_{2}}_{m( 2)}\psi(\sigma,x), \tag{10}\]
and the divergence of the rank 5 bilinear, which has the symmetry properties and satisfies the Bianchi identity of the field strength of an \(O(d-2)\) connection on the spinor bundle. This is an operator quantum field and we can view it as a fluctuating curvature tensor for the holoscreen. It's not clear which, if any of these definitions is correct, or whether there is any way of testing either hypothesis. We've emphasized that the kinds of measurements available to a local observer use detectors with far fewer q-bits than the number that actually describe the holoscreen. We should not be surprised to discover that our experiments will never be able to probe the details of "quantum geometry".
If we arrange that the quantum density matrix for each causal diamond satisfies
\[\langle K\rangle=\frac{A}{4G_{N}}, \tag{11}\]
\[\langle(K-\langle K\rangle)^{2}\rangle=\frac{A}{4G_{N}}, \tag{12}\]
then we will have a quantum model whose hydrodynamic equations coincide with the solution of the Einstein equations from which we began, according to the analysis of Carlip and Solodukhin.
Another immediate benefit of adopting the Connes-Dirac formulation of Riemannian geometry and the quantization rules of HST is that it builds in the spin statistics connection. The connection between spin and statistics is a theorem in local quantum field theory, but even more importantly, it is a property of the real world, and _local quantum field theory is not_. We'll see below that in asymptotically flat space-time it also (almost) proves the necessity for Supersymmetry (SUSY) in asymptotically flat space: a "phenomenological" fact about string theory with no obvious explanation.
We begin to see the outlines of a quantum theory of gravity, applicable on scales much larger than the Planck scale. One starts from a classical space-time, obeying Einstein's equations with a stress tensor satisfying the null energy condition5. For each time-like geodesic in the space-time one introduces a nested sequence of proper time intervals and constructs a corresponding Hilbert bundle of cutoff \(1+1\) dimensional fermion fields. The UV and IR cutoffs in two dimensions are related to the minimal size of causal diamond (in Planck units) for which one believes the Carlip/Solodukhin arguments are valid. At the moment, we have no principled way of deciding how large the minimal diamond is. This choice of course removes any Big Bang singularities from the background space-time. Once this choice is made, one increases the number of fermion fields according to the Bekenstein-Hawking-Carlip-Solodukhin area/entropy law. The quantum state of an empty diamond has modular Hamiltonian equal to the \(L_{0}\) generator of a \(1+1\) CFT constructed from the fermions, with the UV/IR cutoffs described above. The cutoff is imposed on the spectrum of \(L_{0}\), with a central value chosen by the solution of the CS Liouville equation. In general (see below) there appears to be a conformal manifold of CFTs to choose from and it is not clear whether these correspond to different consistent models for quantum fluctuations around the same classical geometry.
Footnote 5: The null energy condition follows from the second law of thermodynamics in Jacobson’s derivation of the Einstein equations.
### Time Evolution
For causal diamonds in maximally symmetric spaces, the \(L_{0}\) generator used by Carlip and Solodukhin showed up in more recent work of Casini, Huerta and Myers[31] and Jacobson and Visser[32] on the modular Hamiltonian of conformal field theory on these space-times. \(L_{0}\) is a quantum operator,
which implements the action of a certain space-time vector field on the time-slice through the holographic screen of the diamond. The vector field preserves the diamond and the flows associated with it are time-like inside it and define a set of inextensible Diamond Universe (DU) coordinates inside the diamond. In fact this is a conformal Killing vector of the maximally symmetric space-time, so the same statements will be true for any geometry conformal to the maximally symmetric space-times. The DU coordinates will be different for different conformal factors. The Virasoro algebra fixes the normalization of the CKV, which is of course ambiguous as a geometrical object.
Now let's consider two causal diamonds in a future directed nesting, with future tips separated by a single Planck unit. We have argued that each should be described by a \(1+1\) dimensional CFT built from fermion fields, with the larger diamond containing more fermion fields than the smaller one. The density matrix for the empty diamond state is \(e^{-L_{0}(\tau)}\) for the small diamond and \(e^{-L_{0}(\tau+L_{P})}\) for the larger one. Geometrically, at least for geometries conformal to maximally symmetric ones, it would seem that the time evolution operator for the Planck scale time slice between the two diamonds, in DU coordinates, should be \(e^{-iL_{0}(\tau+L_{P})}\). By Planck scale time slice, we mean one Planck time along the geodesic. Since this operator is a unitary in the larger Hilbert space it will, in general, entangle the smaller diamond's fermions with those of the larger diamond.
On the other hand, if we think about time evolution between time \(\tau-L_{P}\) and \(\tau\) and time \(\tau\), then the evolution operator in that time slice is \(e^{iL_{0}(\tau)}\), but the extra degrees of freedom that are added between \(\tau\) and \(\tau+L_{P}\) must remain decoupled during evolution from \(\tau-L_{P}\) to \(\tau\). Combined with conformal invariance this motivates us strongly to add the new fermions as free fermions. That is, between time \(\tau\) and \(\tau+L_{P}\) the full evolution operator on the entire Hilbert space is
\[e^{-iL_{0}(\tau+L_{P})}\otimes e^{-iL_{0}^{out}}. \tag{13}\]
\(L_{0}^{out}\) acts on the tensor complement of the diamond Hilbert space of the interval \([-T,\tau+L_{P}]\) in the Hilbert space of \([-T,T]\). The tensor complement can be generated by fermions that are added in each of the intervals \(\tau+nL_{p}\) to \(\tau+(n+1)L_{P}\). Choosing their Hamiltonian to be that of free fermions in \(L_{0}^{out}\) guarantees that the time evolution preserves causality and also "prepares" them for the interaction with the expanding nest of prior diamonds.
The interaction, to leading order in \(L_{P}/L\), is largely determined by two constraints: \(1+1\) dimensional conformal invariance and _fast scrambling[33]_. The evidence for fast scrambling comes predominantly from the rapid homogenization of perturbations on the horizon of a causal diamond, as viewed from an accelerated trajectory that avoids penetrating the horizon. For non-negative c.c. or
for boundary anchored RT diamonds in AdS space, perturbations become homogeneous in times of order the horizon curvature scale. In[34] we argued that this could be explained if the dynamics was invariant under area preserving diffeomorphisms in 2 or more dimensions. This is a global, rather than a gauge symmetry. It implies that the dynamics has no respect for geodesic distance in the transverse space. A small spherical cap is equivalent to a many fingered amoeba of the same area, which touches points that are arbitrarily far away on the manifold.
Fast scrambling is often defined by the statement that a perturbation of a single q-bit operator at time zero, will fail to commute with every other q-bit in a time that scales like the number of q-bits. It is characteristic of models that involve "all to all" couplings of a set of q-bits. The homogenization of mass and charge in logarithmic time certainly does not require non-locality which is quite so drastic. A paradox pointed out by Hayden and Preskill[35] is the strongest argument for the logarithmic time scale. It's not clear, at least to this author, whether it implies the kind of "all to all" coupling that is often assumed.
Figure 2: Holoscreen dynamics invariant under area preserving diffeomorphisms can propagate quantum information without regard to geometric distance. Consistent with exponentially fast homogenization of charge and mass on horizons.
Since our fundamental variables are spinors, we can easily construct differential forms
\[J^{a}_{(m(1)\dots m(k))}=\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\dots A_{k})}\gamma^{a }e^{A_{1}}_{m(1)}\dots e^{A_{k}}_{m(k)}\psi(\sigma,x), \tag{14}\]
where \(e\) is the veilbein on the transverse manifold, the \(\Gamma\) matrices are \(d-2\) dimensional Euclidean signature and the \(\gamma\) matrices 2 dimensional Lorentzian signature. Recalling that our manifold has fixed volume, we can make area preserving diffeomorphism invariant Hamiltonians by integrating products of differential forms of total rank \(d-2\). If the fermions at each \(x\) were independent canonical variables then we could construct independent \(U(1)\) currents for differential forms of each rank \(p\)
\[J^{p}_{a}(x,\sigma)=\bar{\psi}(x,\sigma)\gamma_{a}\Gamma_{(A(1)\dots A(p))}e^{ A(1)}_{m(1)}(x)\dots e^{A(p)}_{m(p)}(x)\psi(x,\sigma). \tag{15}\]
If \(d\) is even the currents with rank \(p\) and \(d-2-p\) commute with each other and we can construct a fixed line of conformally invariant Thirring interactions from products of these currents. For \(p\) odd, we would have to use the reducible representation of the \(d-2\) dimensional Clifford algebra and insert the extra anti-commuting element \(\Gamma\) into either the \(p\) or \(d-p-2\) rank current to construct a similar interaction. These interactions are formally invariant under volume preserving diffeomorphisms. However, if the \(\psi(x,\sigma)\) for different \(x\) were truly independent two dimensional fermion fields, the interactions would be ultra-local in \(x\) and could not propagate information along the horizon at all. It's the fixed cutoff on transverse Dirac eigenvalues that leads to fast scrambling.
This same cutoff will violate \(1+1\) dimension conformal invariance, but recall that the arguments of Carlip and Solodukhin are only valid to leading order in an expansion in \(L_{P}/L\). The finiteness of the entropy of the causal diamond already told us that the CFT had to be cut off, violating conformal invariance. A central conjecture of the current paper is that for large \(L/L_{P}\) the system is fast scrambling, but also close enough to a CFT that the Cardy formula for the spectral density is valid in the vicinity of the value of \(L_{0}\) picked out by the classical argument of Carlip and Solodukhin.
Our proposal now is that we choose the CFT cutoff once and for all by the criterion that for each causal diamond along a nested sequence of proper time intervals the modular Hamiltonian is the \(L_{0}\) generator of the cutoff CFT, where the cutoff is imposed by choosing an \(L_{0}\) value closest to the classical "Liouville" value computed by Carlip and Solodukhin's methods, and one chooses an interval of discrete eigenvalues around that value such that
\[\langle L_{0}\rangle=\langle(L_{0}-\langle L_{0}\rangle)^{2}\rangle=\frac{A}{4 G_{N}}. \tag{16}\]
This condition is enforced for some minimal area diamond, the smallest area for which we believe the CS arguments are valid. It determines the cutoff on the width of the band of \(L_{0}\) eigenvalues once and
for all. Only the number of fermion fields is allowed to vary as we change the proper time interval along the geodesic, and that changes with the area of the transverse manifold in order to continue obeying the above equation. The change is implemented by changing the eigenvalue cutoff on the transverse Dirac operator _for the appropriate transverse manifold at each time slice_. Time evolution in the nested sequence of diamonds is given by \(e^{iL_{0}(\tau+L_{P})}\) in the wedge between the diamond whose future tip is at \(\tau\) and the next one in the sequence. \(L_{0}(\tau+L_{P})\) is the modular Hamiltonian of the empty diamond state of that large diamond.
This procedure produces a finite quantum model for a given choice of a solution of Einstein's equations with a stress tensor satisfying the null energy condition, at least if the geometry is conformal to a maximally symmetric space-time. In higher dimension there seems to be a bit of ambiguity in the choice of Thirring couplings for the various allowed values of \(p\). These models are certainly well defined and have the hydrodynamic behavior that Jacobson's argument tells us to expect from Einstein's equations. By construction they obey a primitive notion of causality: the (even functions of the) degrees of freedom in a given causal diamond form a factor in the algebra of all operators in the Hilbert space of a large finite proper time interval6, and time evolution preserves that factor throughout the proper time in the diamond. To generalize these models to arbitrary geometries, one would have to find the analog of DU coordinates for each space-time. That is, the geometric transformation that acts as \(L_{0}=\partial_{t}\) on the holographic screen of each diamond, should be the boundary value of a vector field whose flow lines inside the diamond define a set of time-like trajectories (including the geodesic) which start and end at the tips of the diamond.
Footnote 6: As always, we inserted a proper time cutoff in order to deal only with finite dimensional algebras. We know how to take the limit for AdS/CFT, and it involves a drastic change in the formalism in which we introduce a tensor network of the systems discussed here in order to obtain a QFT limit. This is discussed in the appendix. Below we’ll discuss the subtleties of the infinite entropy limit in asymptotically flat space.
The formalism so far is missing at least three different key elements: it is tied to a particular geodesic and does not obey a "principle of relativity". The correspondence with local quantum field theory is obscure. The connection to the AdS/CFT correspondence, our best understood model of quantum gravity, is also absent. We will deal with the last of these issues in an appendix. The other two will be discussed in the following sections.
## III The quantum principle of relativity
Given two proper time intervals along geodesics \(G_{1,2}\), the corresponding causal diamonds have an intersection, which might be empty. The intersection is not itself a diamond, but contains a maximal area diamond, \(D_{12}\), unique up to symmetries. The Quantum Principle of Relativity (QPR) is the
statement that \(D_{12}\) corresponds to an isomorphism between a tensor factor in the Hilbert space of \(D_{1}\) and that of \(D_{2}\). Given initial states on the past boundaries of \(D_{1,2}\), there are density matrices \(\rho_{1,2}\) on this tensor factor. The QPR further states that these density matrices have the same entanglement spectra. The QPR is the dynamical principle that makes the Hilbert bundle over the space of geodesics into a quantum version of Einstein's equations. It has, unfortunately, proven extremely difficult to find efficient machinery for implementing this principle. What has been done is, I believe, non-trivial, but is mostly at the level of pictures and stories.
The simplest complete HST model is one in which space-time is a flat Friedmann-Robertson-Walker (FRW) universe with scale factor
\[a(t)=\sinh^{1/3}(3t/R). \tag{17}\]
This is a universe that begins its life with equation of state
\[p=\rho, \tag{18}\]
and asymptotes to dS space with Hubble radius \(R\). The quantum model consists of fermions fields \(\psi(\Omega,\sigma)\) on an interval \(\times S^{2}\) with an angular momentum cutoff that increases linearly with time. The fermion Hamiltonian is the sum of a free \(1+1\) dimensional Dirac Hamiltonian for every transverse angular momentum mode and a term
\[g\int d\sigma\ J_{a}^{0}(\sigma,x)J^{a2}(\sigma,x), \tag{19}\]
built from the product of the 0-form and 2-form currents. The integral includes an integral of the two form over the transverse two sphere. When the expectation value of \(L_{0}(t_{max})\) reaches the dS entropy \(\pi R^{2}\), we stop increasing the number of fermion fields and continue the time evolution of the system with \(L_{0}(t_{max})\) as the Hamiltonian.
Now consider another geodesic in the same space-time and the causal diamond corresponding to the interval \([0,\tau]\) in proper time along that diamond. The intersection with the diamond \([0,\tau]\) along the original geodesic contains a maximal diamond of area \(A(\tau,D)\) where \(D\) is the space-like distance between the two geodesics on some fiducial surface of fixed proper time. As long as the rule for the dynamics in any diamond is that for the empty diamond initial state, the modular Hamiltonian in any subdiamond is unitarily equivalent to \(L_{0}(A)\) for the same CFT described above, with the appropriate value of the central charge. Then the QPR will be satisfied, if the initial state is chosen to be such that, for each choice of geodesic, the initial state is a tensor product of \(e^{-L_{0}^{min}}\), where \(L_{0}^{min}\) is the Virasoro generator of the CFT corresponding to the minimal diamond for which the Carlip-Solodukhin
analysis is valid. This is close7 to the minimal diamond for which a description of the space-time as a flat FRW model with equation of state \(p=\rho\) makes sense. Once the total entropy gets small enough, the description of the density of states by Cardy's formula will become less accurate. In our model of the system as fermion fields with Thirring interactions, the detailed values of the couplings and the precise nature of the cutoff will become more important. Although the quantum model that Carlip and Solodukhin guessed from hydrodynamics is well defined, we have no particular reason to consider it to be the only correct answer when the diamond becomes small. If one were really probing Planck scale distances in the laboratory, only experiment would be the arbiter of what the right microscopic theory was.
Footnote 7: The CS analysis provides a finite quantum model of the space-time down to diamond areas of Planck scale. The classical equations clearly break down in the vicinity of the Planck scale, while the CS _model_ does not. The question is whether there is any reason to believe this particular finite quantum mechanical model. A physicist’s answer should be that only experiment could decide.
From the point of view of a space-time picture, we can think of this as a restriction on how useful it is to think of two different FRW geodesics as being spatially close to each other on a fixed time slice. The initial state above, with a given choice of the minimal entropy, prescribes a minimal initial distance between FRW geodesics. If evolution along each geodesic is chosen to be identical (this is the definition of our Hilbert bundle model, not a fine tuning of the initial state), then the QPR will be satisfied. We have described elsewhere[36] how to construct more interesting, inflationary, cosmologies based on these ideas. In those models the QPR implies that inflationary horizon volumes along distant geodesics appear in the DU coordinates of a given geodesic as a dilute gas of black holes. Some mild assumptions about the properties of those black holes produce a very economical theory of early universe cosmology, which accounts for CMB fluctuations8, the Hot Big Bang, baryogenesis, and dark matter. The model has no Transplanckian problem, and is completely consistent with quantum mechanics, causality and unitarity.
Footnote 8: and makes predictions that are in principle distinguishable from those of QFT models of inflation.
The QPR is also essential for establishing that our Hilbert bundle formulation of quantum gravity actually corresponds to unitary evolution on a Hilbert space. The prescription that the DU evolution operator on the time interval \([\tau,\tau+L_{P}]\) is \(e^{-iL_{0}(\tau+L_{P})}\) defines time evolution in terms of embedding maps of the Hilbert space of a smaller diamond into that of a larger one. When we compare two diamonds along the same geodesic, whose future tips differ by one Planck unit, we add new degrees of freedom only along the new section of null boundary. These degrees of freedom are however already included in the diamonds of smaller proper time, along other geodesics. So the QPR puts constraints on the evolution of these "out of diamond" degrees of freedom. Applying the QPR along every pair of geodesics we conclude that on the full Hilbert space along each geodesic there is a unitary operator
\(U(t)=U_{in}(t)\otimes U_{out}(t)\), where \(U_{in}(t)\) operators only on the Hilbert space of the diamond at time \(t\), and \(U_{out}\) only on its tensor complement. Strictly speaking, this argument only applies to the dS case, where the Hilbert space of a complete geodesic is finite dimensional. For non-positive c.c. we must be more careful about infinite dimensional limits. We'll discuss some aspects of this in the next section and in the appendix. Above we have proposed that \(U_{out}\) be defined in terms of the \(L_{0}\) of a collection of free fermion fields. An important question that we have not yet studied is whether this proposal is consistent with the QPR. The real force of the QPR comes into play for states describing localized excitations in a diamond, to which we now turn.
### Space-time Locality and the Origin of Quantum Field Theory
The formalism described so far allows us to localize quantum information on the boundaries of causal diamonds in space-time but has not given us any clue about the description of the quintessential object of experimental physics: a localized excitation traveling on a time-like or null trajectory through space-time, particularly a timelike geodesic. The essential hint about the description of localized objects came from the study of black holes in de Sitter (dS) space. Before explaining that, we should comment on the unfortunate fact that localization in AdS space is quite a different story, which we will mostly relegate to an appendix. In brief, AdS space is best described by the Error Correcting Code/Tensor Network formalism, in which a cutoff \(1+d-2\) dimensional conformal field theory is described as a sequence of lattice field theories arranged as shells in \(d-1\) dimensional space, with embedding maps (called a Tensor Network Renormalization Group) connecting consecutive shells in the sequence[37]. This is a manifestly local presentation of an approximation to \(AdS_{d}\) geometry, which also manifests Maldacena's scale radius duality. When the radius of AdS space is much larger than all microscopic scales, the number of degrees of freedom inside each node of the tensor network is very large and space-time locality has to do with what is going on inside the node, rather than with communication between the nodes. The ECC/TN picture becomes irrelevant. Thus, our claim is that locality on scales smaller than the AdS radius is a property of the cutoff lattice models of the tensor network, and cannot be addressed easily by computations in the continuum boundary field theory.
In the year 2000, Fischler and the present author independently conjectured[38] that the finite Gibbons-Hawking entropy of dS space implied that the Hilbert space of a quantum model of dS space was finite dimensional. An essential part of that conjecture was the fact that there is a maximal mass black hole in dS space, so that it is extremely plausible that a quantum model of dS space has a UV cutoff. In discussing this at the Festschrift for L. Susskind at Stanford, Fischler and I were struck by
the fact that the Bekenstein-Hawking area law for black holes much smaller than the maximal size in dS space showed that the total entropy of the space-time with a black hole in it was _less_ than the entropy of empty dS space! Moreover, this gave an almost classical derivation of the Gibbons-Hawking temperature of dS space, which Gibbons and Hawking had derived from quantum field theory in a de Sitter background[23]. We conjectured that the density matrix of empty dS space was maximally uncertain (and that the same was true for any empty causal diamond of finite area9) and that energy in the static patch was _defined_ by saying the number of "constraints" on the states was of order \(ER\), where \(R\) was the dS radius. We did not have a very clear model of this in mind but in the summer of 2006 Tomeu Fiol walked into my office in Santa Cruz and presented me with such a model.
Footnote 9: A similar claim was made by R. Bousso[17] at about the same time.
Suppose one has a matrix, with matrix elements that are fermionic oscillators
\[[\psi_{i}^{j},\psi_{l}^{\dagger\ k}]_{+}=\delta_{i}^{k}\delta_{l}^{j}. \tag{20}\]
Then if we insist on being in a subspace of the Hilbert space of fermions where we set the off diagonal blocks between a block of size \(n\) and one of size \(N\) to zero, the statistical probability of being in a quantum state that has order one probability of being in that subspace is \(P\sim e^{-nN}\), which looks like a Boltzmann factor, with \(N^{-1}\) proportional to the temperature and \(n\) proportional to the energy. Something we did not realize at the time, was that if we only require this when the probability is small, then it remains true for a wide variety of density matrices which are not maximally uncertain[24]. In particular, even if dS space follows the Carlip-Solodukhin law, the explanation of dS temperature can be valid for the states in dS space that, according to CKN, are well described by QFT.
Fiol's matrix model idea needs a little tweaking to make it compatible with the ideas presented in the present paper. Some of that tweaking was already done in 2006[39]. First of all, the fermions should be spinors. In four dimensions, we can think of the cut-off spinor bundle as \(N\times N+1\) matrices, since these can be thought of as transforming in all half integer spin representations of \(SU(2)\) up to \(J=N-\frac{1}{2}\). We can then construct square \(N\times N\) matrices \(M_{i}^{j}=\psi_{i}^{A}\psi_{A}^{\dagger\ j}\) and impose the constraints on states in terms of these. These bilinears in spinors can be thought of as "fuzzy differential forms on the two sphere". The trace of products of these matrices is the fuzzy analog of the integral of two forms over the sphere. When a matrix is block diagonal its trace is the sum of the traces of the blocks. We can think of this as writing the sphere as a line bundle over an interval and doing a one dimensional sum over slices of two volume which are then discretized into matrices with matrix elements that are bounded operators in a fermionic Hilbert space. Block diagonal matrices correspond to bands of vanishing volume form. This does not quite capture the full structure of a trace however,
since we can do unitary transformations, which change the order of the blocks. A better picture then is regions of finite area surrounded by bands where the variables vanish. 3 is a cartoon of what we mean.
Readers will note the large black region on the figure, the white bands of vanishing variables, and the smaller red regions. The latter represent small blocks of the matrix of size \(n_{i}\) while the black region has size \(N-\sum n_{i}\gg 1\). Constraints setting the off diagonal blocks between different small blocks to zero can easily be removed by finite time dynamics, while it's plausible that with an appropriate scaling law for the Hamiltonian, the \(o(N)\) constraints between the large block and each of the small ones will never be removed. In the strict \(N\rightarrow\infty\) limit, \(\sum n_{i}\) would become an asymptotic conservation law if the proper time was also scaling like \(N\).
On the other hand, in our model for dS space, \(N\) is kept finite while the proper time goes to infinity, so the quantity \(\sum n_{i}\) is not conserved, but controls the Boltzmann factor of low probability states in the density matrix. \(\sum n_{i}\) is thus proportional to the static energy, which will become the conserved Minkowski energy if we let \(N\) go to infinity with proper time. The measure theoretic cartoon of 3 tells us how to generalize these ideas to higher dimensions, for large \(N\). Isolated localized excitations
Figure 3: The holoscreen of a diamond with localized excitations penetrating it. Red are regions where jet momentum flows into or out of the diamond. White regions have strictly zero momentum, while black regions have momentum flows of magnitude less than the inverse diamond radius.
entering a causal diamond with area large in microscopic units, but small compared to the scale defined by the c.c,, are defined by red regions of finite area \(A_{i}\) on the holographic screen, whose area scales like \(N^{d-2}\). The energy carried by those excitations is proportional to \(A_{i}^{\frac{d-3}{d-2}}\), _i.e._ the volume of its bounding surface. The empty region surrounding the red region has an area \(\sim NA_{i}^{\frac{d-3}{d-2}}\). To implement this picture in a matrix model, we need to write the matrices as
\[M_{i}^{j}=\psi_{i}^{A}\psi_{A}^{\dagger\ j}, \tag{21}\]
where the index \(A\) runs over of order \(N^{d-3}\) values. Fischler and the author[40], using the fact[7] that the number of solutions of the Dirac equation with angular momentum cutoff \(N\) is equal to the number of components of a \(d-2\) form in \(N\) dimensions, tried to write \(\psi_{i}^{A}\) as a \(d-2\) form. However, one would also like to be able to impose constraints on of order \(n_{i}^{d-3}N\) fermion number operators in order to make all the off diagonal matrix elements between \(n_{i}\) and \(N-n_{i}\) indices vanish. It is not yet clear how to do this. Alternatively, we could try to implement the constraints directly from the figure in terms of the "local" spinor field expanded in a basis of solutions of the Dirac equation with an eigenvalue cutoff. Fuzzily localized fields cannot reproduce exact characteristic functions of regions but for large values of the cutoff the approximation should be good enough for statements about non-interaction of the localized objects with the horizon to be valid. Indeed, a Hamiltonian written in terms of integrals of products of differential forms constructed from truly local fermions would be ultra-local and could not propagate information around the horizon. Fuzzy localization is essential if we are to reproduce the fast scrambling properties of horizons. In four dimensions there is a well known connection between the matrix model formulation and area preserving diffeomorphisms, with unitary conjugations of the matrices playing the role of a regularization of the infinite dimensional group of area preserving diffeomorphisms. We need a d dimensional generalization of that formalism.
It is unlikely that there is an interesting quantum theory of dS space in dimension greater than 4. Since 2001[41], Fischler and the present author have argued that dS space should be thought of as a regulated version of a scattering theory of "particles" in Minkowski space10. There is no S-matrix in dS space: every state returns to a maximal entropy equilibrium with no local excitations in a time in any static coordinate system which is not much longer than the dS radius11. However, if the \(R_{dS}/L_{P}\rightarrow\infty\) limit of finite time transition operators in dS space defines a Minkowski scattering operator, then we can view this as a mathematical definition of dS space that is independent of particular complex localized objects (see previous footnote). However, we have an enormous amount of evidence from
string theory that Poincare invariance implies exact super Poincare invariance in models of quantum gravity. Rigorous theorems in classical SUGRA show us that there are no dS solutions of SUGRA for \(d\geq 5\).
The real reason to be interested in this description of localized excitations for \(d>4\) is that it applies equally well to the Minkowski limit. Indeed it provides the crucial link between the formalism we have been discussing and QFT. Let us begin by noting the analogy between 3 and a layer in a modern particle detector. We can think of the red regions as the pixels in the detector that have been "lit up" by a particular scattering event and the white regions as "jet isolation zones" that are used to define particle jets. In a real detector these are defined by including all soft particles that we think should be included in the jet, leaving an empty region around it. The dark regions are pixels that don't light up, which means that anything that flowed through them was below the detector threshold. Here we are insisting that not even "asymptotically zero energy" flows through the annular regions surrounding the jets. The dark region has energy of order \(1/N\) as measured along the geodesic in a finite area diamond of area \(N^{d-2}\) in Planck units, Now imagine a time symmetric nesting of diamonds as a very very finely layered particle detector. Then following the red regions from layer to layer, we get a picture of the trajectory of the jet through the bulk of space-time4. In the description of this trajectory from the point of view of the Hilbert fiber over a given geodesic, all we learn about this trajectory is how much area it takes up on each holoscreen in the nest of diamonds, but the QPR relates this to the Hilbert fibers over all other trajectories and consistency between them determines the geometric shape of the jet trajectory through space-time.
Thus, given an initial state on a very large diamond, containing some number of well separated jets on its past boundary, we can compute the amplitudes for Feynman like diagrams describing their evolution into a different collection of well separated jets on the future boundary (5).
These are not exactly Feynman diagrams because each picture in which energy \(E\) propagates between two points contains amplitudes for this energy to split up into distinct pieces carrying fractions of the total energy. The reason for this is that each jet contains all fermion operators contributing to the matrix \(M_{i}^{j}\) with indices running from 1 to \(n\) with \(n^{d-3}\propto E\).
It is also very easy to see where the particle/jet picture characteristic of QFT breaks down in these models. Suppose that for some macroscopically large \(N\) we find that \(\sum n_{i}^{d-3}\sim N^{d-3}\). Then clearly the approximation that we have a small deviation from the equilibrium state has broken down, and we can expect a rapid evolution back to equilibrium. This is precisely the criterion that the Schwarzschild radius is of order the size of the diamond, here derived from a quantum model, rather than general relativity.
There are other ways in which the diagrams derived from the Hilbert bundle formalism differ from Feynman diagrams. They are definitely causally ordered. The meaning of a diagram like that of figure 5 is that a jet that exits the future boundary of the diamond 1 enters the past boundary of diamond 2 and interacts with other jets entering that diamond, producing some final state, for which the diagram computes the amplitude12. The resemblance to time ordered Feynman diagrams then is that these pictures give a bulk space-time decomposition of an amplitude into processes that occur in different causal diamonds. Unlike Feynman diagrams they include both small diamonds, which resemble localized vertices in which a number of particles come into a diamond and a generally different number emerge and "black hole formation processes" in which particles enter a diamond, which then retains its identity as a meta-stable object of fixed energy, emitting jets of energy in a random thermal manner for an extended period. As far as scaling laws are concerned, these objects behave like the black holes of general relativity, but are excitations of a unitary quantum mechanical system. In the next section, we will indicate how these models can lead to a Lorentz invariant scattering theory in Minkowski space.
Footnote 12: Of course another big difference is that we do not have a nice closed formula for the amplitude, just a sort of Trotter product formula in terms of amplitudes in \(1+1\) dimensional CFTs with time varying central charge.
Figure 4: The holoscreens of two consecutive nested causal diamonds, showing how following the constraints leads to a picture of the trajectory of a jet of particles through space-time.
## IV Representation of non-compact isometry groups
All of the maximally symmetric space-times have non-compact groups of isometries, which we will denote collectively by \(G\). Finite causal diamonds in these space-times are invariant only under an \(SO(d-1)\) subgroup and geodesics only under \(SO(d-1)\otimes R\). Transformations in \(G/(SO(d-1)\otimes R)\) map one fiber of the Hilbert bundle onto another and so are not in general represented on any fixed Hilbert space. The static time translation \(U_{S}\) maps one interval along a geodesic to another one. It is _not_ the same as the time dependent Hamiltonian that describes propagation in a diamond. In terms of conventional coordinate systems \(U_{S}\) uses the global time slicing, while the Hamiltonian \(H(t)=iU(t+1)U^{\dagger}(t)-1\) describes propagation between slices within the diamond, as in Figure 1. For maximally symmetric space-times these are the _diamond universe coordinates_ of[31][32]. In the case of dS space, \(H(t)\) approaches the global Hamiltonian \(i\partial_{t^{s}}\) as the static time goes to infinity. The asymptotic dimension of the Hilbert space is \(e^{A_{d-2}R_{dS}^{d-2}/4G_{N}}\) up to subleading corrections in the exponential. \(A_{d-2}\) is the area of the unit sphere. Strictly speaking, the area is just the expectation value of the modular Hamiltonian, but in[24] we've shown that the conditions \(\langle K\rangle\rightarrow\ln\,\mathrm{dim}\mathcal{H}\) and
Figure 5: Decomposition of amplitudes in HST models into time ordered Feynman-like diagrams, describing jets of particles propagating between causal diamonds in the background space-time.
\((\Delta K)^{2}=\langle K\rangle\) are compatible in the limit of a Hilbert space of large dimension.
For AdS space, as \(t\) approaches \(\pi R_{AdS}/2\) the Hilbert space dimension goes to infinity and the Hamiltonian approaches the generator of the AdS isometry group that leaves a particular time-like geodesic invariant. The AdS/CFT correspondence tells us that this is the Hamiltonian of a CFT, and the c.c. is determined by the asymptotic density of states in that CFT. The transformations that do not leave that particular geodesic invariant act as unitary transformations on the CFT Hilbert space. So in this case, the full isometry group acts on a single Hilbert space. We can think of the CFT Hilbert space as being a trivial Hilbert bundle over the space of time-like geodesics in AdS space. A choice of the Hamiltonian \(K_{0}+P_{0}\) within its conjugacy class in the conformal group is equivalent to a choice of timelike geodesic and we can think of the fiber over that geodesic as the Hilbert space expressed in the basis where \(K_{0}+P_{0}\) is diagonal. There is no holonomy when we traverse a closed loop in the conjugacy class, so the bundle is trivial. We'll discuss the case of negative c.c. in more detail in the appendix.
The case of vanishing c.c. is more subtle and is intertwined with the existence (or not) of the Bondi-Metzner-Sachs algebra on the Hilbert space on which the S-matrix acts. There is also considerable confusion about whether that Hilbert space is a traditional Fock space. It is clear that it is not in four dimensions, and the present author has argued for some time[46] that Fock space is not the correct arena for asymptotically flat quantum gravity in any dimension. That conviction grew out of the picture of particle/jets as constrained states of holographic degrees of freedom on the boundaries of a diamond, which we described in the previous section. In the appendix, we'll see that the same conclusion follows if we follow the approach of Polchinski and Susskind[42] and derive the Minkowski S matrix as a limit of CFT correlators.
Let us then consider the variables in a finite causal diamond in \(d\) dimensional Minkowski space times a compact \(11-d\) manifold \(\mathcal{K}\) of fixed size much larger than the 11 dimensional Planck scale. The magic number 11 appears because we anticipate the result of our investigation, namely that the limiting theory must be super-Poincare invariant. Decades of investigation into string theory make it fairly clear that every supersymmetric model of quantum gravity can be viewed as a compactification plus duality transform of eleven dimensional SUGRA.
The variables are fields \(\psi_{a}^{I,M}(\mathbf{\Omega},\sigma)\). \(a\) is a two dimensional Dirac index and \(I\) a \(d-2\) dimensional Dirac index, while \(M\) labels a finite set of eigenfunctions of the Dirac operator on \(\mathcal{K}\). We will omit the indices \(I,M\) in future equations. \(\mathbf{\Omega}\) is a coordinate on the \(d-2\) sphere. The spherical harmonic expansion on this sphere is cut off, but we're considering the limit of an infinite diamond where the cutoff goes to infinity. Recall that the volume of the sphere scales like \(N^{d-2}\), where \(\pm N\) is the scaling
of the cutoff on the Dirac eigenvalue.
Now we want to define in(out) scattering states. We consider a sequence of diamonds with \(N_{i}\) going to infinity, in a future(past) oriented nested covering of the Penrose diagram of Minkowski space (Figure 1). Pick a finite set of points \(\mathbf{\Omega}_{k}\) on the sphere. Around each point we impose a constraint that the variables \(\psi_{a}(\mathbf{\Omega},\sigma)\) "vanish" in a shell of volume \(N_{i}A(\mathbf{\Omega}_{k})^{\frac{d-3}{d-2}}\equiv N_{i}E_{k}\) surrounding an area \(A(\mathbf{\Omega}_{k})\), which includes that point. By independent area preserving maps on each diamond boundary, we can turn each of these regions into a spherical cap surrounded by an annulus of "vanishing" variables. Of course, with only a finite number of spinor spherical harmonics, we cannot make a function that vanishes exactly in an annulus. We mean the closest approximation to such a half-measure with the available basis set.
For finite \(N_{i}\) the "areas" are discrete numbers, but in the limit they become continuous and the \(E_{k}\) are continuous variables, which we associate with the magnitudes of null momenta in Planck units13. In other words, as we let \(N_{i}\rightarrow\infty\) we are free to rescale the \(E_{k}\) at will since we can absorb the rescaling into \(N\). As usual, in order to obtain finite results in the limit, one must impose some kind of scaling symmetry, and the obvious one is the conformal group of the \(d-2\) sphere, which is the Lorentz group of the null cone \(P^{2}=0\) in \(d\) dimensions. Our spinor variables thus become spinors \(Q_{\alpha}(P)\) on the null cone and they satisfy the Lorentz covariant condition
Footnote 13: Stable massive particles are always associated with BPS states or bound states of them and require a generalization of the SUSY algebras we will write below.
\[P_{m}(\Gamma^{m})_{\alpha\beta}Q_{\beta}(P)=0. \tag{22}\]
which reduces the number of components to the transverse ones that existed before we took the limit. In the limit, the \(Q_{\alpha}(P)\), are generalized functions on the null cone. For non-zero values of \(P\), these generalized functions only exist in a finite number of spherical caps, surrounded by annuli in which \(Q_{\alpha}(0)=0\).
Note that we have dropped the two dimensional structure that was a crucial part of the description of the boundary of finite causal diamonds, which we've emphasized by also changing the names of the operators. That structure was used by Carlip and Solodukhin to understand the finite boundary entropy and by[4] to understand the finite fluctuations of that entropy. In the limit, both of these quantities become infinite and must be "renormalized away" if there is to be a finite limiting scattering operator. We have interpreted the fluctuations implied by this two dimensional structure as fluctuations in the transverse geometry, so its disappearance in the asymptotic limit is the statement that the geometry at null infinity is frozen. This implies an asymptotic decoupling of the dynamics of
the emergent zero energy "topological" sector of the theory in the infinite diamond limit. The various "soft theorems"[43] in the extensive literature on BMS symmetries prove this kind of decoupling, but there is as yet no clear picture of what the Hilbert space of asymptotically flat quantum gravity looks like. Above four dimensions, most authors would just claim that it is Fock space, but this is an incredibly ambitious claim. Initial states consisting of a few particles with large sub energies, can create large black holes which orbit around each other for extended periods of time, merge, emit Hawking radiation and so on. To claim that the final state of any such process is a normalizable state in Fock space, even in 11 dimensional SUGRA where there are no apparent infrared problems in perturbation theory, is the height of hubris. In the BFSS matrix model[44], which is a manifestly unitary scattering theory with the particle content of 11 D SUGRA (in the large N limit), the claim is equivalent to having control over processes in which large block diagonal matrices come in from remote regions of the moduli space and go out into regions in which other large blocks plus an arbitrary number of small blocks with transverse momenta of order \(N^{-1/2}\) are emitted. In four dimensions we _know_ that Fock space is not the right answer and there is recent evidence[45] that the Fadeev-Kulish proposal does not work either.
I have proposed[46] that the proper framework for scattering theory in quantum gravity is a map between representations of the algebras of the \(Q_{\alpha}^{M}(P)\) operators on the positive and negative energy parts of the null cone. These algebras are
\[[Q_{\alpha}^{M}\ {}^{\pm}(P),Q_{\beta}^{N}\ {}^{\pm}(P^{\prime}]_{+}=\pm \delta(P\cdot P^{\prime})\delta^{MN}P_{m}(\Gamma^{m})_{\alpha\beta}. \tag{23}\]
The angular delta function follows from the fact that the fermion fields corresponding to individual angular momenta anti-commute. The delta function in the "internal" indices is achieved by taking appropriate linear combinations. The rest follows from Lorentz invariance and the fact that the objects we are describing are localized in angle. They cannot carry tensor charges at infinity.
The delicate question of what the Hilbert space of asymptotically flat quantum gravity is, has no experimental relevance. For practical purposes, the inclusive cross sections defined by Weinberg are all we could ever hope to measure if we lived in asymptotically flat space, even in four dimensions, where IR effects are most important. In eleven dimensions these IR questions could effect scattering amplitudes only via terms exponential in ratios of Mandelstam invariants to the Planck mass. In ten dimensional superstring theory they might be of order \(e^{-g_{S}^{-2}}\), exponentially subdominant compared to the leading non-perturbative \(e^{-1/g_{S}}\) contributions.
We should note that while Lorentz invariance seems necessary in order to get any kind of sensible infinite diamond limit, it also follows from the QPR. Our formalism has a built in requirement that
everything must be invariant under passage to the fiber over any geodesic of the background space-time. We've described the infinite diamond limit along a particular geodesic, and discovered an apparent need for Lorentz invariance in the infinite diamond limit, which is consistent with the requirement that the scattering operator should be identical to that obtained by working over a different geodesic.
Something similar happens for space and time translation invariance. Our Hamiltonian description of time evolution in a causal diamond uses a time dependent Hamiltonian because it is evolving in DU coordinates. However, when we define asymptotic energy in terms of constraints on initial conditions on the past boundary of the diamond we can see that it is asymptotically conserved. The argument has two parts and its form depends on whether we use the past or future directed nest of diamonds to cover the Penrose diagram. Let us consider the future directed nesting. Then, at early times it is unlikely that the constraints are imposed on the degrees of freedom in the small nested diamond. Those outside the diamond are described as free fermions, so the constraints are preserved by time evolution. However, even after the constrained DOF are engulfed by the growing diamond, the size of the Hamiltonian is of order \(1/R_{diamond}\), the Hamiltonian is \(4-local\) in the fermion variables, and the number of constraints is of order the radius of the Penrose sphere, which we are taking to infinity. The asymptotic energy is proportional to the coefficient of that infinite number of constraints, so it cannot be changed. The same conclusion follows from the QPR applied to causal diamonds that are shifted by a rigid time translation along the same geodesic.
The situation for conservation of spatial momentum is a bit different. Here we use the QPR to impose constraints on the tensor complement of the Hilbert space corresponding to proper time interval \([-T,\tau]\) along a particular geodesic \(G_{1}\) in Minkowski space in terms of the constraints on the Hilbert spaces corresponding to that same time interval along geodesics related to \(G_{1}\) by a rigid space translation. In words, these relations tell us when and where a certain amount of energy enters into the causal diamond of a detector traveling along \(G_{1}\). Thus, although the constraints themselves are defined only in terms of areas, the QPR and the background geometry tell us about the trajectories followed by the centers of jets of particles through the background space-time. One must recall that the jets are not particles, but actually systems that have (asymptotically) infinite dimensional Hilbert spaces14. The jet trajectory is a decoherent semi-classical observable of these large quantum systems.
Footnote 14: For finite size causal diamond the jet Hilbert spaces always have dimension exponentially smaller than that of the diamond, because of the bound that says that the jet energies must be much smaller than the mass of a black hole that would fill the diamond. But as the diamond approaches infinite size, this allows the jet Hilbert spaces to grow without bound.
Using the area preserving diffeomorphism invariance of the large \(N\) Hamiltonians, we can always view the jets as living in spherical caps, surrounded by symmetric annuli where the spinor variables
vanish. The map from the past boundary to the future boundary is invariant under relative rotations of the centers of the jets, since this is a residual symmetry of the area preserving group after we have fixed the jet profiles to be symmetric. Translation invariance then follows from Lorentz invariance and energy conservation. It can be derived independently by insisting that the scattering operator obtained at one fiber of the Hilbert bundle be independent of the fiber. As in the case of energy conservation and boost invariance, the first derivation involves dynamics of the time evolution operator for a given geodesic, while the second is a symmetry constraint on the Hilbert bundle as a whole.
While these arguments are not rigorous, their conclusion is remarkable, and explains an empirical fact. What we have found is that the zero c.c. limit of our Hilbert bundle theory of dS space, has stable localized excitations. If we enforce the Poincare symmetry of Minkowski space, the asymptotic algebra automatically has the form of the supersymmetric generalization of the Bondi-Metzner-Sachs algebra (actually its Fourier transform) derived from SUGRA by Awada, Gibbons and Shaw[47]. Since the invention of string theory in the late 1960s, every attempt to formulate a non-supersymmetric theory of quantum gravity in asymptotically flat space has met with failure. If we accept the premise of this paper, that quantum gravity is a Hilbert bundle over the space of timelike geodesics on its hydrodynamic space-time manifold, and the the variables of quantum gravity are Connes-Carlip-Solodukhin conformal fields on causal diamond boundaries, then we understand that failure.
### Summary
The Hilbert bundle formulation of HST resolves all problems with representations of non-compact isometry groups on finite dimensional Hilbert spaces. In AdS space, the proper time \(\rightarrow\frac{\pi R}{2}\) limit, leads to a trivial Hilbert bundle and a unitary representation of the isometry group on the Hilbert space of the CFT on \(R\times S^{d-2}\). In asymptotically flat space we have suggested that one obtains a unitary representation of the semi-direct product of the Lorentz group and the AGS superalgebra (or its generalization to include massive BPS states), but the nature of the asymptotic Hilbert space remains unresolved. In dS space, which probably only makes sense in dimensions below \(5\)15, the isometry group acts only on the Hilbert bundle as a whole, not on any single Hilbert space.
Footnote 15: See[48] for a model of \(dS_{3}\).
The 't Hooft commutation relations
The basic postulate of our approach to quantum gravity is that the Einstein equations are the hydrodynamic equations of a collection of quantum systems living on the boundaries of causal diamonds in space-time, whose equilibrium density matrix \(\rho=e^{-K}\) satisfies \(\langle K\rangle=\langle(K-\langle K\rangle)^{2}\rangle=\frac{A}{4G_{N}}\). When we use the Verlinde analysis and examine the near boundary limit of a causal diamond, the EH action splits into two parts. Our discussion so far has dealt with the large part of the action, when the transverse size of the diamond \(L\) satisfies \(L\gg L_{P}\). We have argued that it is given by a cut off quantum field theory of \(1+1\) dimensional fermions, which is a particular kind of abelian Thirring model. The fermion fields encode fluctuations of the transverse geometry around a classical background metric \(h_{mn}\).
The authors of[5] showed that the small term in the action was purely topological (once one imposed the classical equations of motion of the large term), and that its entire content was the 't Hooft commutation relations
\[[X^{+}(\Omega),X^{-}(\Omega^{\prime})]=(\frac{L_{P}}{L})^{d-4}(\triangle_{h}- R_{h})^{-1}(\Omega,\Omega^{\prime}). \tag{24}\]
Our purpose in this section is to provide a model for these commutation relations in terms of the \(1+1\) CFT that we have constructed to describe the transverse fluctuations16 The variables \(X^{\pm}\) are defined on the future/past boundary of the causal diamond, and the equal time surface on which the commutation relations are evaluated is the bifurcation surface of the diamond. The reason that there is such a relation at every point along the past/future boundary is because every point is on the bifurcation surface of some diamond in a future or past directed nested cover of the diamond.
Footnote 16: For another approach to the ’t Hooft commutation relations, see[51]. The first of these papers gives a derivation of \((\Delta K)^{2}=\langle K\rangle\) directly from the ’t Hooft commutation relations.
Our model of transverse dynamics has separate CFTs on the past and future boundaries of a diamond, related by the time evolution operator. Note that in going from past to future boundary, the role of the time and space coordinates of the CFT are switched. Thus, the space and time components of a \(U(1)\) current are also exchanged. The space and time components of a conserved \(U(1)\) current in \(1+1\) CFT have a c-number Schwinger term in their commutator17. Thus we postulate
Footnote 17: A previous relation between the ’t Hooft commutation relations and Schwinger terms in 1+1 dimensional electrodynamics was pointed out in[59].
\[X^{+}(\mathbf{\Omega})=\int d\sigma f^{+}(\sigma)J^{0}(\mathbf{\Omega},\sigma). \tag{25}\]
\[X^{-}(\mathbf{\Omega})=\int d\sigma f^{-}(\sigma)J^{1}(\mathbf{\Omega},\sigma). \tag{26}\]
Here
\[\int d\sigma f^{+}\partial_{\sigma}f^{-}=1, \tag{27}\]
and the Schwinger term is
\[[J^{0}(\mathbf{\Omega},\sigma),J^{1}(\mathbf{\Omega}^{\prime},\sigma^{\prime})]=(\triangle _{h}-R_{h})^{-1}(\mathbf{\Omega},\mathbf{\Omega}^{\prime})\partial_{\sigma}\delta( \sigma-\sigma^{\prime}). \tag{28}\]
\(\triangle_{h}\) is the scalar Laplacian of the transverse manifold, and \(R_{h}\) its scalar curvature. To understand how to obtain the last equation, we consider a set of \(1+1\) dimensional fermion fields labelled by the eigenspinors of the transverse Dirac operator \(D\) and write
\[J^{\mu}=\bar{\psi}\gamma^{\mu}D^{-1}\psi. \tag{29}\]
There are two ways in which this formula deviates from that of 't Hooft. First, the currents, and thus the light ray operators \(X^{\pm}\) are bilocal rather than local operators on the transverse manifold. The second is that the transverse integral operator we obtain from this ansatz is
\[D^{-2}=(\triangle_{h}-\frac{1}{4}R_{h})^{-1}. \tag{30}\]
The discrepancy in the coefficient of \(R_{h}\) can perhaps be fixed by choosing a non-Riemannian connection for the Dirac operator, though one would clearly like to have a deep physical justification for such a replacement. Another possibility, which should be explored is to replace the Dirac operator by the Rarita-Schwinger operator as the proper representation of transverse geometry. If we do choose such a connection, or if the Rarita-Schwinger idea works, then the equations of motion[5] show that the null energy fluxes, which enter into actual scattering amplitudes are still local operators. The very preliminary nature of this analysis should be evident.
## VI Conclusions, Speculations and (UGH) Philosophy
For many researchers in quantum gravity there is much in the present work that will appear negative. Quite frankly, it is probably because some of the conclusions I've been drawn to were so distasteful that it has taken a long time to come to the present formulation of HST, despite the fact that they were implied by the basic postulates of the formalism. I've long believed that the deepest insight into the problem of quantum gravity was Jacobson's demonstration that Einstein's equations, with an arbitrary stress tensor satisfying the null energy condition, were the hydrodynamic equations of the Bekenstein-Hawking entropy law, applied to the boundary of an arbitrary diamond in space-time. If
one thinks about the role the Navier-Stokes equations play in the physics of ordinary matter, Jacobson's paper should have made us all extremely suspicious of the idea that quantum gravity was all about quantizing Einstein's equations. Extremely suspicious as well of the idea that there was some kind of "background independent" formulation of QG, just because Einstein's equations were universal. Thinking about this in condensed matter language, this is like saying that there's a "substance independent" theory of condensed matter, just because the NS equations are so universal. The real clue in Jacobson's work, in a phrase that J.A. Wheeler invented, but clearly did not understand, was that space-time was the IT in "IT from BIT". The areas of the holoscreens of all causal diamonds determine space-time geometry and these are determined by the von Neumann entropies of quantum density matrices associated with those diamonds.
It follows that one should investigate QG by choosing a background geometry satisfying Einstein's equations with a stress tensor obeying the null energy condition (the second law of thermodynamics), and try to find a quantum system obeying Jacobson's principle in that geometry. This leads directly to the Hilbert bundle formulation of QG because there is a one to one correspondence between the set of causal diamonds, and the set of choices of nested proper time intervals along time-like geodesics. The QPR is the natural consistency condition to impose on this system of Hilbert spaces and density matrices.
We next face the question of what to choose for the density matrix of a diamond. Obviously there can't be a unique choice for this because we know from experiment that there can be many different states inside a causal diamond. Here we get two clues from quantum field theory and another big boost from the hydrodynamic interpretation of Einstein's equations. Paradoxically, one of the most important clues is the demonstration by CKN that all of our experiments probe an entropy that scales at most like \((A/G_{N})^{3/4}\) (in four dimensions), and cannot account for the entropy used by Jacobson to derive Einstein's equations. On the other hand, QFT, the _theoretical framework_ we use to explain our experiments, predicts area law entropy for an _empty_ diamond, roughly independent of the state in the QFT Hilbert space. The QFT prediction for the coefficient of area is UV divergent and was conjectured to be "absorbed into Newton's constant" in the Bekenstein-Hawking law. So we should be looking for a universal prediction for the density matrix of an empty diamond and try to understand the tiny corrections due to excitations localized inside the diamond after we've understood "nothing".
That step was taken by Carlip and Solodukhin in 1998, whose analysis is best understood in an analysis done by the Verlindes, where to leading order in \(L/L_{P}\) a redefined metric is block diagonal between a two dimensional Lorentzian and a \(d-2\) dimensional Riemannian block, with the Riemannian length scales \(L\gg L_{P}^{(d)}\) while those of the Lorentzian block are of order the Planck length in the
near horizon limit. The equations of motion of the large part of the EH action imply that the two dimensional metric is flat and the Riemannian metric is independent of the two dimensional coordinates. The two dimensional stress tensor fluctuations of the conformal factor of the Riemannian metric satisfy the Ward identities of a stress tensor of a CFT of large central charge. That is, they are the hydrodynamic equations of such a CFT. Carlip and Solodukhin thus postulate that the actual near horizon quantum theory is such a CFT, and show that that postulate reproduces most known black hole entropy formulae. Zurek and the present author generalized their hypotheses to an arbitrary causal diamond and pointed out that it implied the universal fluctuation formula \(\langle(K-\langle K\rangle)^{2}\rangle=\langle K\rangle\).
In the present paper we have taken this analysis one step further, using the insight gained from my work with Fischler on HST. Carlip and Solodukhin do not deal with fluctuations of the unimodular part of the transverse geometry, and Solodukhin comments explicitly that he does not know why it is less important than the conformal factor. The resolution of this is to imagine that the transverse geometry is encoded in the target space of the CFT. The fact that the finite diamond has finite entropy suggests immediately that the target space fields be fermions18. Connes' work on Riemannian geometry and the Dirac equation tells us that there is a natural set of fermion variables lying around, once we remember the spin statistics connection. This relation was long ago incorporated into HST. We update it here by turning the spinor fermionic oscillators into \(1+1\) dimensional spinor fields. They're actually Dirac spinors both in the Lorentzian and Riemannian geometries, and so can in a sense be thought of as fields on the full \(d\) dimensional space-time. However, the UV cutoff in the transverse space depends on the proper time in two dimensions in a way that a \(d\) dimensional field theorist would find bizarre. The UV cutoff on the \(L_{0}\) generator is also a bit bizarre. One takes a central value determined by the classical formula of Carlip and Solodukhin and chooses a width in discrete \(L_{0}\) eigenvalue space such that the Carlip-Solodukhin formula for the entropy is satisfied exactly, for "the smallest causal diamond for which one believes the Carlip-Solodukhin analysis". That last phrase has unfortunate philosophical implications, to which we will return at the conclusion of these conclusions.
Footnote 18: It’s already implicit in the Carlip/Solodukhin work that the CFT lives on an interval and has a cut off spectrum of the Virasoro generator \(L_{0}\).
We are almost done with empty causal diamonds. To proceed we have to switch our focus from the density matrices of empty diamonds to the time evolution operator between two successive diamonds in a nested covering of a given diamond. Let us choose a future oriented one 1 for definiteness. Here we will restrict attention to geometries that are conformal to maximally symmetric spaces. For such geometries, the modular Hamiltonian \(L_{0}\) is related[31][32] to a conformal Killing transformation which leaves the diamond invariant. More precisely, if we consider a conformal quantum field theory on the
same background space-time, then the modular Hamilton of that field theory is proportional to the quantum generator of the action of the conformal Killing vector (CKV) acting on the bifurcation surface of the diamond. The proportionality constant is fixed by the Virasoro algebra. The vector field associated with that CKV defines a set of inextensible coordinates inside the diamond, and the flows associated with the vector field are timelike lines, one of which is the geodesic in the diamond. For these space-times at least, it seems plausible that also in QG we should associate time evolution between two diamonds whose future tips are \(\tau\) and \(\tau+L_{P}\) to be
\[U(\tau+L_{P},\tau)=e^{-iL_{0}(\tau+L_{P})}. \tag{31}\]
This unitary operator acts in the Hilbert space of the diamond corresponding to the time interval \([-T,\tau+L_{P}]\) along the geodesic and we expect it to entangle degrees of freedom in the Hilbert space corresponding to \([-T,\tau]\) with the new degrees of freedom that are added in the larger diamond. It should be tensored with a unitary that acts in the tensor complement of \([-T,\tau+L_{P}]\) in the full Hilbert space \([-T,T]\)19
Footnote 19: \(T\) is a proper time cutoff inserted to make sure that we are always dealing with finite dimensional Hilbert spaces. Its value depends on the background space-time. The limiting cases where the Hilbert space dimension goes to infinity have to be treated with great care.
We have argued in the previous section that the twin requirements of approximate two dimensional conformal invariance and fast transverse scrambling of information restrict the form of \(L_{0}\) to be that of free fermions plus a limited set of abelian Thirring interactions constructed from quartic fermion operators that would be invariant under transverse area preserving diffeomorphisms if the transverse cutoff were removed. More work is required to see whether those arguments are indeed correct, but if they are, we have constructed a consistent set of models of quantum gravity for a certain class of space-times.
The most disturbing thing about these models is the fact that we have had to insert an arbitrary restriction to a "smallest causal diamond for which we believe the Carlip/Solodukhin analysis" into our construction. This is inevitable in an approach that tries to bootstrap quantum theory from hydrodynamics. By definition, hydrodynamics contains only coarse grained information about an underlying quantum theory. In principle one should not expect to find the correct description of the microscopic theory applying to the real world, without comparing to microscopic experiments. This raises the question about models in asymptotically flat or AdS space, where we have a mathematically precise description of asymptotic observables, at least in principle. In the case of AdS space there seems to be a pretty definitive negative answer to the question of whether we could have an absolutely precise and unambiguous description of the physics inside a causal diamond whose radius was much
smaller than the cosmological radius of curvature. According to the description we have given of that physics in the appendix to this paper, that physics is described by a single node in a tensor network. But a tensor network is a cutoff quantum field theory and we have known since the work of Wilson that many different cutoff theories converge to the same CFT. Which one gives the "correct" physics in a small causal diamond, with arbitrary precision? One can try to impose symmetries to refine the allowed set of cutoff theories, but it seems doubtful that one could come up with a unique definition of time evolution in a sequence of small causal diamonds in terms of unambiguous CFT data.
The same issue arises in asymptotically flat space. In quantum field theory, we're familiar with the fact that the S-matrix does not determine local correlation functions. Even in integrable theories in two dimensions one has to supply extra information to determine them, and we can do so only because we know their mathematical definition and properties. The problem for asymptotically flat quantum gravity is even worse. For the most part we only know it as a perturbation expansion which is not Borel summable. There are a few cases where it can be treated by BFSS Matrix Theory, but the large \(N\) limit is not under control. In no case do we have a rigorous understanding of the Hilbert space in which the scattering operator acts. It seems unlikely that this will lead us to an unambiguous definition of finite time physics in a local region of space-time.
There is a "philosophical" issue at stake here, in which the nasty questions of the interpretation of quantum mechanics raise their heads. The present author adheres to the modern version of the Copenhagen interpretation of QM. Certain quantum systems have collective variables whose dynamics takes place on a time scale slower than typical micro-scales. Hydrodynamic flows are prime examples of such variables. For collective variables the uncertainties implied by QM are small, down by powers of some ratio of a small length scale to a large one. More importantly, the violations of Bayes' sum over histories rule for probabilities in the stochastic evolution equations for collective variables are _exponentially_ small in the ratio of the large length scale over the small one. This means that we can treat the quantum fluctuations in these quantities like measurement errors or errors caused by random disturbances. They form the basis for what Bohr and Heisenberg called the Classical World. In our models, space-time geometry is part of this classical world.
In contemplating the physics that takes place inside a causal diamond, we're imagining a measuring apparatus traveling along some time-like trajectory and "doing experiments". The meaning of a mathematically precise theory of some set of physical phenomena is that we can account for everything that measuring apparatus can measure, with arbitrary precision. _If the total Hilbert space of the causal diamond is finite dimensional this is simply impossible._ There is some point at which the quantum fluctuations in the measuring device will limit the precision. Furthermore, in a theory of quantum
gravity, when we make the measuring device have more q-bits, in order to endow it with more robust collective observables, it uses up more of the entropy of the diamond, eventually creating a black hole. So the claim is that there is no such thing as a mathematically precise theory of the physics of a local region of space-time in quantum gravity, and the smaller the region we want to study, the more ill defined the theory becomes. At a certain point the physics in a "causal diamond" will depend on the nature of the device one uses to probe it. In language that is probably too classical we can say that the device's gravitational field will distort the local space-time geometry, so that one no longer knows what one was probing.
For this reason, it seems to me that the phrase "the smallest causal diamond for which we can trust the approximation of Carlip and Solodukhin" represents a fundamental barrier to the construction of local theories of quantum gravity, which can only be removed, if at all, by experimental probes. That is, the models described in the present paper might one day be tested, and found to apply at large enough scales. Experiments at smaller scale might then provide guidance about how to construct a more fine grained quantum theory. Hydrodynamics, as embodied in Einstein's equations and the QPR, can only take us so far.
I want to end this rather ambitious paper with an outline of how a general theory of quantum gravity looks from the perspective advanced here.
* Einstein's equations with a smooth stress tensor satisfying the null energy condition are the hydrodynamic equations of the CEP for a general causal diamond. The c.c. is not included in these equations but is an asymptotic boundary condition which controls the high entropy limit.
* Generically, causal diamonds have only surface excitations. Bulk excitations are constrained states. The empty diamond state has maximal entropy. In space-times which are asymptotically AdS, these statements are all true on scales below the AdS radius. The full space-time is a tensor network made up of locally coupled subsystems satisfying these principles.
* Given a background space-time/hydrodynamic flow, we construct a Hilbert bundle over the space of time-like geodesics on that space-time. A choice of nested proper time intervals along each geodesic corresponds to a set of nested causal diamonds. For large enough causal diamonds the modular Hamiltonian of each diamond is the \(L_{0}\) generator of a model of \(1+1\) dimensional fermion fields, labeled by the eigenspinors of the Dirac operator on the holoscreen of the diamond. The interactions between these fields is determined by a small number of abelian Thirring couplings, described above. For space-times conformal to maximally symmetric ones, \(L_{0}\) is the quantum realization of the action of a smooth vector field, restricted to the holographic screen of the
diamond. The flows of that field inside the diamond are time-like and define a set of inextensible coordinates on the interior. This leads to the natural conjecture that the time evolution operator along those flows, between proper times \(\tau\) and \(\tau+L_{P}\) along the geodesic, is just \(e^{-iL_{0}(\tau+L_{P})}\). The proper times along other time-like trajectories in the coordinate system are shorter than those along the geodesic, so this ansatz properly incorporates the "redshift" between different observers. A full unitary operator on the Hilbert space over the geodesic is the tensor product of this operator with the \(L_{0}\) of free fermions for those degrees of freedom that are not causally connected to the diamond \([-T,\tau+L_{P}]\). For empty diamond states, this ansatz satisfies the QPR. For constrained states, with localized excitations, the QPR leads, at the level of pictures, to the Feynman-like diagrams we have discussed in the text. A more quantitative verification of the QPR for localized states is an important unsolved problem.
* In the limit of asymptotically flat space-time, all of these models are exactly supersymmetric. That is, their spectrum contains supergravitons and their unitary S-matrix is non-trivial and Poincare invariant. This implies well known results. The maximum space-time dimension is 11. There are a variety of exactly stable BPS branes, and in particular, various compactifications have stable strings with tension much smaller than the Planck scale, so that a systematic perturbation expansion is possible. Thus, the Hilbert bundle formulation implies conventional string perturbation theory.
* The most important features implied by this formalism are those not covered in this paper: the connection with experiment. The fluctuation formula \(\langle(K-\langle K\rangle)^{2}\rangle=\langle K\rangle\), confirms the arguments of Verlinde and Zurek[49] that there should be observable quantum gravity fluctuations in interferometer experiments. The crucial calculation that remains to be done is the power spectral density (PSD), for which only an _ad hoc_ model[50] exists at present. I have also argued for years that the connection between the finite horizon of dS space and the breaking of Supersymmetry should lead to insight into the SUSY particle spectrum. An update on those ideas will appear soon[52].
* In the appendix I emphasize that the tensor network construction is applicable to quite general CFTs, and that the local picture of AdS space that it provides is not a good guide to the way that bulk locality arises at scales small compared to the AdS radius. \(AdS_{d}/CFT_{d-1}\) models with EH duals have tensor networks whose nodes are Hilbert spaces with entropy that scales like \(R^{d+p}\) where \(p\geq 2\). The reduced density matrices in the node Hilbert spaces, in typical
states obtained by acting on the ground state with local lattice operators near the boundary of the (cut off) network, also have large entropy, and Polchinski-Susskind scattering states in the "arena" are constrained, atypical states of the network field theory. Locality on scales small compared to the AdS radius arises in much the same way as it does in the HST formalism.
I want to end this on a personal/philosophical note. Like most working physicists, I have spent most of my career regarding the problems of the foundations of quantum mechanics in the "shut up and calculate" mode. We know how to use the theory to make stunningly accurate predictions about observations. That is all we know on earth and all we need to know. Controversies that arose during my tenure at UCSC forced me to take a closer look at these issues, as a consequence of which I came away convinced that Bohr and Heisenberg had it essentially right but needed the more modern work on decoherence to justify their somewhat mystical language. I've explained this in great detail in my textbook, but in brief: certain quantum systems with many DOF and "local" interactions, have a large number of collective variables. These are operators whose uncertainties scale like inverse powers of a ratio of microscopic and macroscopic length scale, and whose quantum probability distributions satisfy linear stochastic equations up to corrections that are exponentially small in the ratio of the macro to micro length scale. These variables are the "classical measuring devices" to which the Copenhagen interpretation refers.
The CKN bound on the validity of local field theory in a causal diamond, combined with the apparent inability of black hole dynamics to produce long lived complex collective variables (fast scrambling/incompressible hydrodynamics), suggests that the Copenhagen interpretation is not applicable to the detailed quantum mechanics of gravitational systems. A local detector can be a complex measuring device, but does not have enough q-bits to store information about the diamond in which it resides. This implies that mathematically precise descriptions of the physics measureable by that detector are in principle impossible to construct, because they are impossible to test. For those of us living in an asymptotically de Sitter universe this is somewhat disappointing.
Our mathematically precise theories of AdS space do not help, as discussed in the appendix. For asymptotically flat space the picture is less clear because we do not have a definition, outside of perturbation theory or the finite N BFSS matrix models of the Hilbert space in which the scattering operator is supposed to be unitary. Conservatively, one might assume that ambiguities could show up in non-perturbative corrections to the string perturbation series, which is not Borel summable. As a dramatic example, in [58] I proposed modifications of a model of N copies of the minimal Type 0B string theory in \(1+1\) dimensions. These are non-interacting, integrable systems, which
have a string perturbation series that is not Borel summable. Despite the fact that the low energy scattering resembles linear dilaton gravity, there are no linear dilaton black holes in the exact fermionic field theory that gives rise to the perturbation expansion. The modified models do not effect the perturbation series, but all have meta-stable excitations with the qualitative properties of black holes. There are a large number of such models (for large \(N\)), whose detailed predictions are all different.
## VII Appendix the curious case of anti de Sitter space
For researchers whose interest in quantum gravity started in String Theory, the focus of interest for 25 years has been the AdS/CFT correspondence. There are many excellent reasons for this, the principle one being that quantum field theory is our best understood theoretical tool in physics, and this correspondence reduces quantum gravity to the study of a particular limiting case of quantum field theory. I have nothing but admiration for the work in this field and the insights it has brought us, but I believe it has also led to a number of serious misconceptions.
As indicated in the main text, the most serious of these is a completely incorrect notion of the emergence of locality in models of quantum gravity. This error is both a technical one, and also obscures a fundamental conceptual ambiguity of the notion of local physics in any mathematical theory of quantum gravity. We will leave the conceptual issue to the end of this appendix and focus on the technical. The CFT of the AdS/CFT correspondence lives on the boundary \(R\times S^{d-2}\) of the universal cover of \(AdS_{d}\), and enjoys the usual locality properties of field theory on that boundary. One of the earliest and most celebrated properties of the correspondence is Maldacena's _scale radius duality[53]_. This implies in particular that if we consider any finite region of \(AdS_{d}\), below some radius \(R*\) in a fixed global coordinate system, we are dealing with some kind of cut off version of the CFT[54].
A very explicit version of the cutoff, which displays the local structure of \(AdS_{d}\) is given by the tensor network (TN) or error correcting code (ECC) models of the AdS/CFT correspondence[37]. Commonly referred to as "toy models" these should instead be thought of as sequences of cut off lattice field theories, which converge on the CFT, in a manner that explicitly displays scale/radius duality. The _tensor network renormalization group_ (TNRG) of Evenbly and Vidal[37] makes this even more explicit. These authors construct unitary embedding maps using a variational principle, which map the lattice model on each shell of the tensor network into that on the next shell. Remarkably, the eigenvalues of the "Hamiltonians" for small shells, are, in simple soluble CFTs, close to the dimensions of low dimension operators in the CFT on the boundary.
In[55] Fischler and the present author interpreted the embedding maps of the TNRG as the maps
between successive nested causal diamonds along the geodesic that runs through the center of the tensor network. The shells of the network are the holographic screens of successive nested diamonds. The TN/ECC picture of CFT thus gives a rather remarkable local picture of the structure of AdS space. The observant reader will have noticed however that we have never really used the limiting behavior of CFTs that is required in order to make holographic duality work. This is particularly evident in the work of Evenbly and Vidal, whose numerical studies are done for soluble models with low central charge. The TN/ECC formalism is a property of QFT, independent of the particular limits which guarantee that a QFT has an "Einstein-Hilbert (EH) dual".
There is a universal feature of all known explicit examples of CFTs with an EH dual, namely that one always finds that when the AdS radius \(R\) is much larger than all microscopic scales, there are also at least 2 compact dimensions, whose Kaluza-Klein modes have masses that scale like \(R^{-1}\). Note that it's important here that we say _all microscopic scales_. The free energy of a CFT at temperature \(T\) on a \(d-2\) sphere of radius \(R\) is
\[F=cT(TR)^{d-2}. \tag{32}\]
Comparing this to the Bekenstein-Hawking formula for the free energy of a stable black hole in \(AdS_{d}\) we conclude that the AdS radius is large compared to the Planck scale whenever the constant \(c\) is large. A model has an EH dual only if, in addition to large \(c\) there is a large gap in dimension between the energy momentum tensor and the typical exponentially growing density of states of a CFT. In all known examples, the density of states grows too rapidly to be accounted for by a finite number of fields in \(AdS_{d}\times S^{1}\).
It has been pointed out in various places, by the present author, as well as by Susskind, that the way to account for this in the tensor network picture is that the nodes of the tensor network must be large quantum systems, and that local physics on scales below the AdS radius, _is connected to the behavior of the Hamiltonian within a single node, rather than the local lattice physics of the network_. There are a number of examples of AdS/CFT with tunable parameters, such that for small values of the parameter a Lagrangian picture of the CFT is valid. Although this is the opposite limit from the one in which the model has an EH dual, one can examine the "single node" physics in this limit. It is always a highly non-local "matrix model" (in \(1+1\) dimensions this means a permutation orbifold) in which the AdS and compact dimensions appear on a roughly equal footing. The model exhibits fast scrambling. On larger scales in the tensor network we find ballistic scrambling on the \(d-2\) sphere, obeying a Lieb-Robinson bound, while scrambling in the compact dimensions is still fast. These properties are mirrored in the hydrodynamics of large black holes in AdS space, in the
opposite limit where the system _is_ well described by an EH dual. On scales short compared to the AdS radius, hydrodynamics is incompressible and identical to the hydrodynamics of flat space black hole horizons. There is no hydrodynamic propagation of entropy because information is scrambled at a much faster rate than transverse momentum. On scales large compared to the AdS radius we find sound modes, which propagate information ballistically. These dominate the modular fluctuations of large black holes.
These distinctions also show up in the relation between temperature and infall time to the singularity for large and small black holes. For small black holes, the Schwarzschild radius determines the time scale of infall and also the change in entropy when a particle falls into the black hole. For a large black hole the particle is falling into one particular node of the tensor network, so the infall time is determined by the timescale of that subsystem, which is the AdS radius. The temperature on the other hand is a measure of the equilibrium response of the entire network, after the ballistic scrambling process has taken place, and has a very different value.
A final piece of evidence for the same fact is the difference between the behavior of infinite RT diamonds (AdS Rindler space) and large black holes, both with regard to the coefficient in their modular fluctuation formulae, and the question of the existence of sound modes. From the bulk point of view this might appear a little puzzling, because the infinite RT diamond has a horizon much larger than the AdS radius, yet it has no sound modes and its modular fluctuations are like those of a flat space diamond. The resolution of this puzzle is again found by appealing to scale/radius duality and the tensor network construction. The volume divergence in the entropy of an RT diamond is a UV divergence in the CFT, and UV divergences have to do with modes of the field theory that are very short wavelength on \(S^{d-2}\). In the tensor network picture, these are modes restricted to a single node, and therefore it is no surprise that they behave like flat space horizon modes. This example also makes it clear that _the sphere at infinity in the CFT is not the same as the sphere seen by a local observer in a diamond of radius smaller than the AdS radius._ The latter sphere, like the compact dimensions, appears in the target space of the field theory, rather than the base space over which the fields are defined.
_Thus, the emergence of locality at sub AdS radius scales in AdS models with EH duals is not connected with the tensor network and entanglement structure of the boundary CFT._ Instead, we claim the picture that emerges is the same as the one we abstracted from the zero c.c. limit of dS space. The key observation is the "arena" picture of the emergence of the flat space S-matrix from CFT correlators proposed by Polchinski and by Susskind[42]. These authors focus on a single causal diamond called the arena whose size is larger than all microscopic scales but smaller than the AdS
radius, and construct CFT operators which, when a few of them act on the vacuum, create Witten diagrams that are approximately non-interacting until all the lines enter into the arena. The resulting CFT correlators are argued to be directly related to flat space S-matrix elements.
There are two properties that follow from this picture. The first assumes the CEP, namely that the arena is a finite dimensional quantum subsystem of the CFT Hilbert space. By Page's theorem[57] this would seem to imply that for most states of the CFT, the density matrix of the subsystem is maximally uncertain. It's unclear whether the CEP is true in the exact CFT, but if we identify the arena with the central node in a tensor network approximation to the cut off CFT then it is certainly true and the tensor network shows us that Page's argument is correct for this system. For a finite tensor network the probability that the density matrix is maximally uncertain is less than one. It's clear that a lot of care has to be taken when applying Page's theorem to infinite dimensional systems.
The second property was first pointed out in[56] and has to do with the origin of Minkowski amplitudes with large numbers of soft massless particles. Because of the gap in the AdS energy spectrum for massless particles, one cannot construct Polchinski-Susskind operators that insert an arbitrary number of massless particles into the arena, without creating black holes inside the arena. Instead, amplitudes with large numbers of Minkowski massless particles arise from AdS correlators of the form
\[\langle PS(1)PS(2)PS(3)PS(4)O(y_{1})\ldots O(y_{n})\rangle_{c}, \tag{33}\]
where the 4 Polchinski-Susskind operators prepare a hard \(2\to 2\) scattering process in the arena, and the other operators create particles whose amplitude to be in the arena (as computed from Witten diagrams) is very small because they are equally probable to be anywhere in AdS space. We conjecture that it is linear combinations of such amplitudes that converge to S-matrix amplitudes for emission and absorption of arbitrary numbers of soft particles with arbitrarily low energy in the \(R\rightarrow\infty\), Minkowski limit. It's the non-perturbative behavior of those amplitudes for large \(n\), which determines whether the resulting states are normalizable in Fock space. At the moment, it does not appear that the AdS/CFT correspondence sheds additional light on these questions, but the fact that it leads to the same questions as the approach from dS space suggests that the issues raised in the text are real.
We conclude that the issue of local physics on scales smaller than the AdS radius involves a cut-off version of the CFT. This means that it is not encoded unambiguously in CFT correlators, for many different cut-off systems converge to the same CFT. In the text we have pointed out a possible deep reason for this ambiguity. The phrase "physics in a causal diamond of finite area" is shorthand for "a mathematical theory that explains the results of measurements performed by a detector traveling
along a time-like trajectory between the tips of the diamond". However, because the information gathering capacity of a detector is limited by the CKN bound, no mathematically precise theory is testable, so some ambiguity in the definition of local physics is inevitable. In the real world we could (in principle, though probably not in practice) probe this ambiguity experimentally by showing that the results of experiments depended on the type of detector used. In the AdS/CFT paradigm the ambiguity seems to be related to the well know insensitivity of continuum field theory to UV details. This means that the question of precisely what goes on in finite area causal diamonds is not well defined in AdS/CFT. Different theorists can choose different sequences of cut off models, all of which converge to the same CFT, but differ in their predictions for local physics in finite area diamonds.
## VIII Appendix II Some remarks about black holes in Minkowski space
Large black holes in AdS space are conventional thermal ensembles. Their modular Hamiltonian is just the Hamiltonian of the CFT. Black holes in Minkowski space have negative specific heat. One cannot relate their thermal properties to their modular Hamiltonian. Indeed, the black hole entropy formula would predict a energy level spacing of order \(e^{-S}\), while the width of black hole levels (the inverse lifetime) scales like a power of the entropy. The level spacing of the modular Hamiltonian has nothing to do with the energy of the black hole as measured by an observer at infinity. We have conjectured instead that it's related to the spacing of levels seen by a near horizon detector that measures the operator \(L_{0}\).
We should note that the classical properties of black holes beg for an interpretation as some kind of universal equilibrium ensemble. They are independent of the black hole's formation history and small perturbations of a black hole just lead us to another black hole differing in a small number of macroscopic parameters. Black holes also admit hydrodynamic flows, which, in the Minkowski case, do not allow for entropy transport. Our conjecture is a model for this equilibrium state.
Finally, we note again that negative specific heat gives the simplest resolution of the so called "firewall paradox". It implies that a large amount of entropy is created when a localized, low entropy, object is dropped onto a large black hole. The explanation for this is that, in the Hilbert space of the final state black hole, there must have been a large number of frozen q-bits, which must be activated in order for the object to come into equilibrium with the rest of the black hole q-bits. If the time scale (in proper time of the infalling object) for activation of these q-bits is the Schwarzschild radius, then the infalling object will not begin to feel the effect of being inside the black hole for that amount of time. Furthermore, the consequence of equilibration is that more and more q-bits with which the
object could have interacted if it had been in empty space, will instead be coming into equilibrium. Using the connection between entropy and area, the effective area "felt" by the object shrinks.
**Acknowledgments**
I would like to thank W. Fischler for collaboration over more than 20 years on material that led to this paper. Thanks also to B. Fiol for the clue to understanding localized states in quantum gravity. Special thanks to K. Zurek for showing me the work of the Verlindes and of Solodukhin, which were so important to the formulation of this paper, and for insisting on the universality of the modular fluctuation formula. Thanks also to E. Witten and E. and H. Verlinde for pointing out an error in my description of the Verlindes' work and the way to correct it. This work was supported in part by the U.S. Department of Energy under grant DE-SC0010008
|
2302.03971 | Communicative Robot Signals: Presenting a New Typology for Human-Robot
Interaction | We present a new typology for classifying signals from robots when they
communicate with humans. For inspiration, we use ethology, the study of animal
behaviour and previous efforts from literature as guides in defining the
typology. The typology is based on communicative signals that consist of five
properties: the origin where the signal comes from, the deliberateness of the
signal, the signal's reference, the genuineness of the signal, and its clarity
(i.e., how implicit or explicit it is). Using the accompanying worksheet, the
typology is straightforward to use to examine communicative signals from
previous human-robot interactions and provides guidance for designers to use
the typology when designing new robot behaviours. | Patrick Holthaus, Trenton Schulz, Gabriella Lakatos, Rebekka Soma | 2023-02-08T10:08:18Z | http://arxiv.org/abs/2302.03971v1 | # Communicative Robot Signals: Presenting a New Typology for Human-Robot Interaction
###### Abstract.
We present a new typology for classifying signals from robots when they communicate with humans. For inspiration, we use ethology, the study of animal behaviour and previous efforts from literature as guides in defining the typology. The typology is based on communicative signals that consist of five properties: the origin where the signal comes from, the deliberateness of the signal, the signal's reference, the genuineness of the signal, and its clarity (i.e., how implicit or explicit it is). Using the accompanying worksheet, the typology is straightforward to use to examine communicative signals from previous human-robot interactions and provides guidance for designers to use the typology when designing new robot behaviours.
human-robot interaction, typology, model, communicative signals +
Footnote †: journal: Computer Vision and Pattern Recognition
## 2. An Ethological Approach
Communication between humans and robots can be understood by looking at it from an evolutionary perspective (Sundundar et al., 2016; Sundar et al., 2017). Ethology helps us to find a feasible way of explaining and examining the quality of signals by (_a_) looking at how communicative signals evolved and are used in the animal kingdom, and (_b_) by looking at asymmetric communication between humans and animals and drawing parallels to asymmetric communication between humans and robots. Let us examine concepts and theories commonly employed in ethology and see how they can be useful for HRI and thus for the typology of communicative robot signals we introduce in Sect. 4.
### Communication definitions
Communication in ethology is defined as the behavioural act an animal performs to change the probability another animal will behave in a certain way (Sundar et al., 2016). Often, this act has adaptive value, at least for the _sender_ animal. In addition, signals should have an evolutionary history and should be evolved specifically for a communicative function.
Communication, in general, is the process or act of transmitting a message from a sender to a receiver through a channel with the interference of noise (Bordes and Rafter, 2005). Some broaden the definition to include that the message transmission is intentional and conveys meaning to bring about change. Others restrict the definition to (_a_) the receiver must respond to the message and (_b_) the interacting individuals must be members of the same species (Sundar et al., 2016). A communicational act includes at least two individuals, a _sender_ and a _receiver_. The sender emits a signal that informs the receiver about the sender's inner state or about elements of the environment. The signal's function in the receiver's mental model can change by ontogenetic learning and, thus, one signal can have different functions in different contexts. In this sense, the message of a signal is the function the signal fills in the receiver's mental model.
Csanyi and Kampis split communication based on how well the mental models of the participants correspond (Csanyi and Kampis, 2016). When a component of the sender's mental model (the signal) has the same function in the receiver's mental model, the correspondence between the two mental models is 100%. This is _type I communication_. This type of communication rarely occurs, even among individuals of the same species. In _type II communication_, the correspondence is below 100% and cannot be exactly determined. In the inter-species (interspecific) situation, one can only talk about type II communication. The main challenge of such is that the sender's and receiver's mental models require a common set of signals to communicate.
Based on the signals' nature, researchers differentiate between _referential_ and _non-referential_ (motivational) signals (Sundar et al., 2016). Non-referential signals contain information about the actual motivational state of the animals and are independent of the quality of any external stimuli. Referential signals refer to certain elements of the environment or environmental events independent of the inner state of the sender. Referential communication assumes the signals referring to the elements of the environment have the same role as words in human language, but theoretical and experimental research has shown that function and mechanism cannot be separated since the inner state of the animals cannot be determined. For this reason, researchers developed _functional reference_(Garrett, 2001; Garrett, 2002; Sundar et al., 2017). This concept implies that communicational signals have the same function in the animals' communicational system as words in the human language, but it is not assumed that there are similar mechanisms behind the signal production and signal utilisation (Garrett, 2001). Some of these definitions reference interspecific communication, which is relatively rare among animals. The function of communication (both in intra- and interspecific cases) can be to share information or manipulate others. The most commonly investigated examples of interspecific communication are alarm calls, which are classic examples of functionally referential signals. For example, Vervet monkeys (_Cercopithecus aethiop_) can comprehend some signals of a starling species (_Sproo superbus_) living close to them (Bordes and Rafter, 2005). The starling produces specific alarm calls for different predators (terrestrial or air). Vervet monkeys show adequate reactions to a call, avoiding the type of predator that it refers to. Hence, it can be supposed that the monkeys can comprehend the meaning of these calls. It is, however, debatable if such communication is completely bi-directional or simply eavesdropping from the receiver's side.
An extraordinary case of interspecific communication is when a communicational event develops between an animal species and humans. Greater honeyguides (_Indicator indicator_) living in East-Africa have assisted humans living close to them in finding honey for about 20,000 years, guiding them to the hives of wild bees (_Apis mellifera_), which are well-protected and out of the honeyguide's reach. The bird's vocal signals and the height of their flight predict the distance to the bees' nest. In addition, communication between the honeyguide and humans is not unidirectional; members of the Borna tribe also call the honeyguide using specific vocal signals (Sundar et al., 2017) using a dedicated whistle. The honeyguide responds by flying towards the campsite of the Borna, calls for the human's attention and then guides them to the hive. The bird uses specific vocal signals and movements, knowing that when the Borna obtain the honey, they always leave some honey behind for the honeyguide.
### How can ethology help robotics?
The etho-robotic approach suggests that HRI should be considered a specific form of inter-specific interaction. Applying animal models in robot behaviour design thus provides an important alternative in the development of social robots (Sundar et al., 2016). Human-animal interaction is a valid model for HRI as both are asymmetric, may start at any age, are much simpler than human-human interactions, and can develop using only nonverbal communicational behaviour on the animal (or robot) side (Sundar et al., 2017). Hence, much like in type II communication between animals, a common set of communicative signals is required for successful interaction between humans and robots.
Dogs provide an excellent biological model since they adapted to the human social environment exceptionally well, developing specific interspecific communicational skills towards humans, enabling them to participate in numerous complex social interactions with us on a daily basis (Sundar et al., 2016). Dogs have not only evolved exceptional interspecific communicational skills but also develop a strong attachment bond with humans, making them life-long companions (Sundar et al., 2016). This approach argues that the implementation of dog-analogue behaviours in social robots could lead to more believable and acceptable robotic companions (Sundar et al., 2016; Sundar et al., 2016; Sundar et al., 2017).
## 3. Communication Theories and Hri
Let us examine some HRI studies that have used behaviour drawn from animals, humans, and machines for specific interactions. Then, we present how general theory and the arts can also be used for designing communication. We also present some earlier attempts at characterising communicative robot signals in general.
### Studies inspired by etho-robotics, human, and artificial behaviour
There are several experimental studies that have used an etho-robotics approach, and several have been inspired by behaviours from dogs. One study used dogs' motivational (non-referential) signals as a model to design emotionally expressive behaviour (happiness, fear, and guilt) for robots and investigated whether participants could recognise the emotions expressed by the robot and interact with it accordingly (Han et al., 2017). The results suggested people attribute emotions to a social robot and interact with it according to the expressed emotional behaviour. A recent study designed guidelines for 11 affective expressions for the Miro robot and applied behaviour design inspired by dog behaviour among others, and evaluated the expressions through an online video study. All expressions were recognised significantly above the chance level (Han et al., 2017). So, the dog-inspired behaviour proved a suitable medium for making people attribute emotional states to a non-humanoid robot.
Another study used the behaviour of specially trained hearing dogs to design behaviour for an assistive robot. In the study, the robot successfully led participants to two sound sources using visual communication signals inspired by the referential behaviour of the hearing dogs. The findings suggested that participants could correctly interpret the robot's intentions (Han et al., 2017). This study provided further evidence for dog-inspired social behaviour being a suitable medium for communicating with human participants.
Designing robot behaviours using biologically-inspired behaviour is not restricted to only dog behaviour. A recent study used a Roomba and a LED strip to imitate aquatic animals' use of light, examining how bio-luminescence, colour science, and colour psychology can be used in HRI, with a focus on appearing attractive or hostile (Rosen and Seth, 2017). The study found that low-intensity blue lights in a circle or split pattern were more attractive to participants, while high-intensity red lights in a blinking or breathing pattern were perceived as hostile (Rosen and Seth, 2017). Note that aquatic animals' communicative signals are _not_ used in interspecific interactions with humans and hence, have no biological relevance to human communication. This suggests that aquatic animals may not provide a suitable model for robot behaviour design in HRI, and explanations for these findings may be more complex even though the signals were perceived as expected by the participants.
Human behaviour can be a source for creating robot signals, but the result may not always be perceived as human-like. One study used a Nao with three presenting-behaviours: (_a) leader_ where gaze and nod were not derived from human response (machine-like), (_b_) _follower_ version where the robot follows the person's gaze and nodding behaviour (machine-like), and (_c_) _semi-follower_ where the robot would sometimes follow the person and sometimes try to lead (human-like) (Rosen and Seth, 2017). Participants watched and were asked to pick the most human-like behaviour. The follower and semi-follower behaviour were picked as more human-like over the leader behaviour, but some participants preferred the nodding behaviour of the follower behaviour, while others preferred the gaze and face-tracking in the semi-follower behaviour (Rosen and Seth, 2017). The study showed that, given a certain context, pursuing human-likeness may not be the most important aspect to design a robot's behaviour.
Human-like behaviour carries the danger that the robot may appear more socially competent than it is. In some situations, it may be better the robot behaves in a artificial, machine-like manner. For example, one study (Han et al., 2017) had a robot with two behaviours to wait for an elevator in the lobby of a building along with people. In the "machine-like" behaviour, the robot waited at a designated spot and only entered the elevator after everyone else had a spot. For the "natural" behaviour, the robot joined the rear of the cluster of waiting people and used a (weak) first come, first served policy that was similar to what was observed when humans wait for elevators. People in the study felt the robot's "machine-like" behaviour caused less confusion than the robot's "natural," human-like behaviour (Han et al., 2017).
### Describing human-robot communication
These examples in Section 3.1 show robot behaviours can draw inspiration from animal, human, or machine behaviour. If we use some theory of communication or perhaps adopt methods from the arts, we may design clearer and more understandable signals for communication or notice potential problems.
For example, the concept of _speech acts_(Han et al., 2017) and the concept of intentional and consequential sounds (Han et al., 2017) was applied to suggest a robot's movements are functional and communicative at the same time (_movement acts_(Han et al., 2017)). In addition, the communicative part of the movement has an intentional (explicit) component, i.e. designed by the roboticist, and a consequential (implicit) component, i.e. how the person observing a robot's motion interprets that motion (Han et al., 2017). The explicit and implicit components are split between the roboticists and the people interacting with the robot respectively. For example, a robot's motion could be designed to be purely functional, but an observer can implicitly draw unintended meanings out of it (e.g. a Fetch robot's movement to reset its navigation stack leads to people interpreting its movement as confusion or being extra careful).
Designers have deliberately used explicit and implicit elements in the movement of a robotic Ottoman to indicate it wanted people to interact with it (Han et al., 2017). In another study, the researchers used artistic techniques from improvisational acting to have the robot ottonan's motion interpreted by observers as having different levels of dominance (Rosen and Seth, 2017). Designers' attempts may not always be successful and a designed behaviour can be misinterpreted, (e.g. a Cozzo robot's behaviour indicating it was done waiting for a fist bump was misinterpreted as the robot being sad for other reasons (Han et al., 2017)). It is ultimately the person interacting with the robot who interprets the implicitly sent message, but there seems to be a split between an _intended_ implicit signal versus an _unintended_ implicit signal.
Turning to the arts, such as Laban notation for dance (Han et al., 2017) and animation principles and techniques (Han et al., 2017; Rosen and Seth, 2017), can provide inspiration for an intentional, implicit interaction. Bianchini et al. (Bianchini et al., 2017) used Laban notation as inspiration for creating a notation system for
non-anthropomorphic robot movement, focusing on the _qualia_ of gesture: from movement to gesture, and from gesture to behaviour. They noted it was difficult to determine what is or is not a gesture: a gesture cannot be separated from its "... relation to the postural, dynamic, and contextual organisation of the body" (Bartos et al., 2016, p. 16).
Movement acts are valid, important classifications that contribute to the understanding and design of robot communication with humans. Using methods from the arts, such as animation and dance, can be a good way to implement some of these signals. But there is a gap in distinguishing what is intentionally implicit (that is, created by the designer to be understood implicitly) and unintentionally implicit (that is, not created intentionally by the designer, but still communicates something to the human). This split needs to be better developed and presented to minimise misinterpretation.
There have been attempts at describing how communication between humans and robots works. Much of the foundation is from two models. This first is Shannon (Shannon, 1958) that creates a probabilistic communication model that can model signal noise and modulate emphasis in communication. The second is Watzlawick et al. (Watzlawick et al., 2016) and the concept of _non-action_ summarised succinctly as "... no matter how one might try, one cannot _not_ communicate. Activity or inactivity, words or silence all have message value" (Watzlawick et al., 2016, p. 30). One model by Bonarini (Bonarini, 2016) examines the channels of hearing, sight, and touch that are available to humans for communicating with robots and suggests to use machine-learning techniques to teach robot behaviours and then benchmark their performance (Bonarini, 2016). This can provide a starting point to explore communication channels.
A more in-depth review of communication theories in HRI argued that an asymmetrical model for communication is a better fit for human-robot communication since robots are not human and interact with the world differently than humans (Bartos et al., 2016), which is compatible with the notion of type II communication (c.f. Sect. 2.1). The review presents the AMODAL-HRI model, which focuses on actions to build a common ground between a human and a robot (Bartos et al., 2016). It also includes the processes in the human and robot for interpreting and acting on the situation (Bartos et al., 2016). The model includes guidelines based on Norman's design principles for interaction design (Noman, 2017).
Another way to describe the communication is to create a typology for communication for HRI. Similar to our approach, the typology created by Hegel et al. (Hegel et al., 2016) was strongly inspired by communication theories between animals. _Signs_ are the basic term for an entity that transmits information to another entity (Hegel et al., 2016). Signs could be divided into _signals_, the signs acted on to alter the behaviour of another organism, and _cues_, any feature of the world that a receiver could use as a guide to get information about the signaller (Hegel et al., 2016). These signals and cues could be further classified as human--resembling human behaviours and features--or artificial--non-human-like signs (e.g. blinking lights or jerky movement) that could communicate information (Hegel et al., 2016).
In their typology, Hegel et al. emphasised that signals are _explicitly_ designed by the roboticist, while cues may be explicitly or implicitly interpreted by the receiver. That is, there is a distinction between the active nature of a signal and the passive (evolved) nature of a cue (Hegel et al., 2016). For example, the active nature of deliberately sending a signal by an animal has a cost (e.g. a predator may also see the signal), and the passive nature of _cues_ are usually evolved features like colour or patterns that can indicate toxicity.
We argue that distinguishing between signals and cues in the design of robots adds complexity that is not helpful since a robot's appearance (passive cues) and behaviour (active signals) are created by roboticists for communicative purposes in a similar time frame. That is, the signals and cues are not necessarily "evolved" in the traditional sense. Moreover, distinguishing between the explicitness or implicitness in the _design of signals_ by the roboticist may be mistaken for the explicitness of the _signal itself_ (i.e. whether the signal needs interpretation by the receiver or not). In the suggested new typology, we address both these problems by focusing on communicative signals that might be intended or unintended and at the same time contain an explicit or implicit component.
## 4. A new typology of communicative robot signals
In this section, we introduce a new typology for communicative robot signals in HRI that aims to support roboticists in designing and evaluating robot behaviour. Since the suggested typology heavily relies on ethological communication theories and draws inspiration from some earlier attempts at applying communication theories to HRI, we will first describe how the typology in general relates to existing theories and frameworks that consider communication as introduced in Sections 2 and 3. We will then present how the typology looks in detail, proposing a selection of core properties to describe communicative signals and discuss how our suggested worksheet can be used to apply the typology to characterise an example behaviour. The remainder of the section will argue how the typology can guide new robotic behaviour design and the analysis of already conducted interaction studies.
### Scoping considerations
We adopt Slater's view that communication is a behaviour that alters the behaviour of an interaction partner (Steintein and Stein, 1996). Contrarily to some definitions in ethology (see Sect. 2.1), we neither require communicative signals to be evolved to be regarded as communicative nor do we require intentionality to consider a robot as a communicator. Moreover, we consider communication between robots and humans to be asymmetrical, similar to type II communication between different species (Bartos et al., 2016).
We want to stress that our notion of communicative signals also includes non-actions (Watzlawick et al., 2016) like interruptions of robot movements, which sometimes happen deliberately, for example, to allow for human conversation, object recognition, or position estimation. Moreover, the typology also covers and directly addresses non-obvious (implicit) signals and overloaded signals carrying multiple messages. We would further like to emphasise that we do not see communicative signals in isolation but rather embedded in an interaction (Stein and Stein, 1996), potentially involving multiple parties, and complex turn-taking behaviours and feedback channels, as described by DeVito (Deueuf et al., 2017). However, to individually characterise communicative signals, we focus on a specific view considering a single robot as the sender and a single human as the receiver. This allows us to relate the typology to communication models introduced in Section 3.2, e.g. Shannon (Shannon, 1958), while preserving the potential to consider multiple receivers or swapping roles.
While the proposed typology might be applicable to either the robot or the human as a sender, for simplicity here, we take a unidimensional perspective to discuss communicative signals in the context of particular robot behaviours. With that, we aim to provide a structured approach to interaction-centric robot behaviour design and analysis in light of the asymmetric nature of communication between humans and robots. We further acknowledge--but do not further discuss--that communicative signals can be affected by (external) noise, which might distort the message content.
### Characterisation of robot signals
We define a robot behaviour as a higher level function aimed to fulfil a range of functional goals, which in turn might require certain robot actions that are perceptible by a human user (Fig. 1). Each of the actions might have a number of components that the robot uses to reach the functional goal, potentially perceptible via a range of modalities. Such a robot action always elicits one or more communicative signals as long as a human user perceives it.
Like other animals, humans perceive communicative signals using multiple modalities including seeing, hearing, and touch (Bradbury et al., 2016; Bradbury et al., 2016). The typology focuses on structuring the content and properties of such non-verbal messages, i.e. describing what a signal might reveal about the robot or the environment and how the signal might be interpreted by the human (Sect. 3.2). We therefore propose to individually characterise communicative robot signals in the context of HRI, addressing qualities of the sender and how it produces the signal, the signal's message content, and how much interpretation is required at the receiver's end. While we focus on five core properties, the typology is meant to be extensible and welcomes others to define and discuss additional properties. Some of the propositions have proven to be useful when characterising human (or animal) signals, others are novel suggestions that arise from the specific nature of HRI where roboticists have control over the design and expression of all robot behaviour, including communicative signals.
#### 4.2.1. Origin
Designing social behaviours is complex with formal generation still requiring manual fine-tuning of signals (Rasmussen, 2015). However, solutions for manually signalling an intended meaning with a robot can often be derived from biology (Sect. 2.2). It might be worthwhile to consult ethology when assessing whether a signal is expressed correctly or can efficiently transport the intended meaning. On the other hand, imitating biology when artificial solutions are proven to be effective might lead to misunderstandings or increase design efforts unnecessarily. Hence, evaluating a signal from both perspectives allows roboticists to easily identify potential for improvement or reveal unintended side effects.
With the new typology, we thus suggest to determine the **origin** of a signal, i.e. whether the signal mimics communication between humans--or animals--or whether it has been designed otherwise to understand and model communication from the sender's (robot) perspective. We thereby encourage roboticists to reason about a signal's origin and explicitly think along both alternatives and what implications they entail with regard to signal design and evaluation.
Signals might have be derived from human communication but adapted to suit the needs of human-robot interaction, allowing this property to have three values: _biological_, _artificial_, or _hybrid_.
#### 4.2.2. Deliberateness
Consequential signals are a common source of misunderstandings (Bradbury et al., 2016; Bradbury et al., 2016; Bradbury et al., 2016; Bradbury et al., 2016) because the designer cannot control what information is being sent and is often _unaware_ that information is being sent at all. We therefore propose to capture whether a signal has been produced deliberately or whether it is merely the accidental byproduct of an action to allow roboticists to identify and reduce unwanted effects in the interaction partner. The identification of unintended signals might guide the design process of robots or the examination of unexpected user behaviours. One possible aim could be to eliminate unintended signals; another one would be to identify the signal's nature and use them deliberately.
Assessing the signal's **deliberateness** allows roboticists to further investigate a signal from the sender's perspective. With this typology, we encourage roboticists to carefully consider whether perceptible actions only elicit intentional signals or whether they might result in a signal that is unintended. A single action may elicit more than one signal, possibly resulting in unintended secondary meanings, depending on robot state, context, and receiver. This property can have two values: _intentional_ or _consequential_.
#### 4.2.3. Reference
Robot signals with a primarily social function are often designed to reveal some piece of information about the robot's inner state to its human users (Bradbury et al., 2016; Bradbury et al., 2016; Bradbury et al., 2016). We argue that assessing the motivational reference of a signal, similar to ethological approaches (Bradbury et al., 2016), helps roboticists to determine whether the signal appropriately reveals or perhaps fails to reveal any of the robot's inner states. Moreover, by considering the reference of a signal, we aim to foster the identification of additional information that might potentially be carried by a signal. Referential signals like deictic gestures and functional behaviour, e.g. delays or pauses, also have a (possibly consequential) non-referential component that provides human users with information about the robot and shapes their expectations. Sophisticated voice or advanced arm movements when giving directions might suggest other parts of the robot, such as its perception modules, are advanced while in reality, they might not be.
Figure 1. Typology of communicative robot signals in the context of robot behaviours where signals are elicited by perceptible actions required to fulfil a functional goal. Communicative signals can be characterised using five properties.
When looking at the signal's message content we suggest determining a signal's **reference**, i.e. whether it considers information about the sender and its inner state or alternatively any outside entity. This allows us to distinguish between _non-referential_ and _referential_ communicative signals in a similar way that ethology does (Hohmann et al., 2017) for assessing the effectiveness of a signal.
#### 4.2.4. Genuineness
Many communicative robot signals are deceptive in their nature (S
and receiver used different reference systems, the gesture would become ambiguous and hence less explicit. Likewise, a squeaking noise can explicitly signal that robot parts are grinding against each other whereas other noise might be more difficult to interpret and thus be more implicit.
Assessing the properties of the communicative signal allows us to estimate the information that is being perceived by the human user of the robot. In this example, the user notices that the robot explains directions, which is facilitated by all three actions. However, excessive noise while presenting the gesture might cause uncertainty about the capabilities of the robot and the accurateness of its actions. An expected (or observed) human behaviour may therefore include signs of hesitation or requests for confirmation.
### Using the typology in behaviour design
Perceptible robot behaviour results from design choices at different levels of abstraction, ranging from low-level hardware design to the definition of high-level behavioural goals, which may involve a sequence of actions or a combination of actuators. At all levels, roboticists have the opportunity to influence the emerging robot behaviour so that it is, to some degree, shaped by a designer to perform a confined set of functions. Robot behaviour, however, is also dependent on perception and embedded in environment and interaction; not every robot action is predictable. For example, robot behaviour might be generated from a learning component adapting to user preferences. Still, the robot's interaction architecture, training pipelines for machine learning, and individual robot actions are frequently hand-crafted and can be dictated by a roboticist.
The proposed typology aims to be applicable at all abstraction levels of robotic behaviour design by looking at the final robot behaviour from an interactive perspective, i.e. by focusing on perceptible actions. At the same time, it is good practice to always consider the functional goals when designing any robotic behaviour (Stein
the rotation's genuineness, we can classify it as _honest_; The robot was trying to reset its navigation stack and not being deceptive. As to the rotation's clarity, it was clearly _implicit_. People observing the robot would have no idea what it was doing unless they were familiar with the robot's navigation routine. Applying the new typology shows there is potential for this motion to be misunderstood even though the behaviour serves its function. It seems that the implicitness of the signal makes people confuse the non-referential repair mechanism with a referential signal that aims to draw their attention to something in the robot's surroundings.
For the Cozmo robot signal that it did not get a fist bump in time (described in Pelikan et al. (2018)), the origin of the action is _biological_ (human). Here, the deliberateness of the action was _intentional_, as it signalled that the time had expired for giving a fist bump. This is a _non-referential_ signal as well since Cozmo signalled its "desire" for a fist bump on its front loader. The signal was also _honest_, as a desire for this action has been modelled on the robot. The clarity, at least as thought by the designers, was explicit. The typology shows that the understanding of the signal should be straightforward, yet the behaviour in isolation could still be misinterpreted (Pelikan et al., 2018).
The typology can also be used to see differences in behaviour between conditions in a study (and possibly improve the internal validity of an experiment). For example, in the study of the robot waiting for an elevator from Gallo et al. (2018), the origins of the "machine-like" behaviour is _artificial_, and the "human-like" behaviour origin is _biological_, but the other properties of both behaviours are _intentional_, _explicit_, _honest_, and _referential_. Using the typography can show how changing one component of the communicative signal can have an effect on the behaviour participants prefer in a given situation.
## 5. Discussion and Limitations
The typology provides an accessible starting point for designing behaviours or examining how robot behaviour could be interpreted. The typology should work for robot signals in other modalities such as virtual or augmented reality (Sund |
2308.07594 | Effective Continued Fraction Dimension versus Effective Hausdorff
Dimension of Reals | We establish that constructive continued fraction dimension originally
defined using $s$-gales is robust, but surprisingly, that the effective
continued fraction dimension and effective (base-$b$) Hausdorff dimension of
the same real can be unequal in general.
We initially provide an equivalent characterization of continued fraction
dimension using Kolmogorov complexity. In the process, we construct an optimal
lower semi-computable $s$-gale for continued fractions. We also prove new
bounds on the Lebesgue measure of continued fraction cylinders, which may be of
independent interest.
We apply these bounds to reveal an unexpected behavior of continued fraction
dimension. It is known that feasible dimension is invariant with respect to
base conversion. We also know that Martin-L\"of randomness and computable
randomness are invariant not only with respect to base conversion, but also
with respect to the continued fraction representation. In contrast, for any $0
< \varepsilon < 0.5$, we prove the existence of a real whose effective
Hausdorff dimension is less than $\varepsilon$, but whose effective continued
fraction dimension is greater than or equal to $0.5$. This phenomenon is
related to the ``non-faithfulness'' of certain families of covers, investigated
by Peres and Torbin and by Albeverio, Ivanenko, Lebid and Torbin.
We also establish that for any real, the constructive Hausdorff dimension is
at most its effective continued fraction dimension. | Satyadev Nandakumar, Akhil S, Prateek Vishnoi | 2023-08-15T06:40:18Z | http://arxiv.org/abs/2308.07594v1 | # Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals
###### Abstract
We establish that constructive continued fraction dimension originally defined using \(s\)-gales [23] is robust, but surprisingly, that the effective continued fraction dimension and effective (base-\(b\)) Hausdorff dimension of the same real can be unequal in general.
We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable \(s\)-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest.
We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion [8]. We also know that Martin-Lof randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [23]. In contrast, for any \(0<\varepsilon<0.5\), we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\), but whose effective continued fraction dimension is greater than or equal to \(0.5\). This phenomenon is related to the "non-faithfulness" of certain families of covers, investigated by Peres and Torbin [25] and by Albeverio, Ivanenko, Lebid and Torbin [1].
We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.
## 1 Introduction
The concept of an individual random sequence, first defined by Martin-Lof using constructive measure [18], is well-established and mathematically robust - very different approaches towards the definition identify precisely the same sequences as random. These include Kolmogorov incompressibility (Levin [10], Chaitin [3]) and unpredictability by martingales [27]. While the theory of Martin-Lof randomness _classifies_ sequences into random and non-random, it does not _quantify_ the information rate in a non-random sequence. Lutz effectivized the classical notions of Hausdorff and packing dimensions [15], surprisingly extending it to individual infinite binary sequences [16], yielding a notion of information density in sequences. This definition also has several equivalent definitions in terms of Kolmogorov compression rates [19], unpredictability by \(s\)-gales, and using covers [15], [16]. These definitions have led to a rich variety of applications in various domains of computability and complexity theory (see for example, Downey and Hirschfeldt [5], Nies [24]).
Recently, settings more general than the Cantor space of infinite binary (or in general, infinite sequences from a finite alphabet) have been studied by Lutz and Mayordomo [17], and Mayordomo [20], [21]. Prominent among them is Mayordomo's definition of effective Hausdorff dimension for a very general class of metric spaces [20], [21]. Nandakumar and Vishnoi [23] and Vishnoi [31] define the notion of effective dimension of continued fractions, which involves a countably infinite alphabet, and is thus a setting which cannot be studied using Mayordomo's framework. This latter setting is interesting topologically since the space of continued fractions is non-compact, and interesting measure-theoretically since the natural shift invariant measure, the Gauss measure, is a non-product measure.
Nandakumar and Vishnoi [23] use the notion of an \(s\)-gale on the space of continued fractions to define effective dimension. Vishnoi [31] introduced the notion of Kolmogorov complexity of finite continued fraction strings using a one to one binary encoding. Vishnoi [31] also shows that the notion of Kolmogorov complexity is invariant under computable 1-1 encodings, upto an additive constant.
In this work, we first establish the mathematical robustness of the notion of effective dimension, by proving an equivalent characterization using Kolmogorov complexity of continued fractions. The characterization achieves the necessary equivalence by choosing a binary encoding of continued fractions which has a compelling geometric intuition, and then applying Mayordomo's characterization of effective (binary) Hausdorff dimension using Kolmogorov complexity [19]. In the process, analogous to the notion of an optimal constructive supergale on the Cantor space defined by Lutz [16], we provide the construction of a lower semi-computable \(s\)-gale that is optimal for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders using the digits of the continued fraction expansion, a result which may be of independent interest.
The topological and measure-theoretic intricacies involved in this setting imply that some, but not all, "natural" properties of randomness and dimension carry over from the binary setting. For example, while Martin-Lof and computable randomness are invariant with respect to the conversion between the base-\(b\) and continued fraction expansion of the same real [22], [23], Vandehey [29] and Scheerer [26] show that other notions of randomness like absolute normality and normality for continued fractions are not identical.
Staiger [28] showed that the Kolmogorov complexity of a base \(b\) expansion of a real \(\alpha,0\leq\alpha\leq 1\), is independent of the chosen base \(b\). Aligning with this, Hitchcock and Mayordomo [8] establish that feasible dimension of a real is the same when converting between one base to another. Hitherto, it was unknown whether effective dimension is invariant with respect to conversion between base-\(b\) and continued fraction representations. Since we can convert between the representations efficiently, it is possible that these are equal. We show this is true in one direction, that the effective base \(b\) dimension is a lower bound for effective continued fraction dimension.
However, using the technique of diagonalization against the optimal lower semicomputable continued fraction \(s\)-gale and using set covering techniques used in recent works by Peres and Torbin [25], Albeverio, Ivanenko, Lebid and Torbin [1] and Albeverio, Kondratiev, Nikiforov and Torbin [2] to show the "non-faithfulness" of certain families of covers, we show that the reverse direction does not hold, in general. We prove the following result: for every \(0<\varepsilon<0.5\), there is a real whose effective (binary) Hausdorff dimension is less than \(\varepsilon\) while its effective continued fraction dimension is at least 0.5. By the result of Hitchcock and Mayordomo [8], this also implies that the effective base-\(b\) dimension of this real is less than \(\varepsilon\) in every base-\(b\), \(b\geq 2\). Thus, surprisingly, there is a sharp gap between the effective (base-\(b\)) dimension of a real and its effective continued fraction dimension, highlighting another significant difference in this setting.
Preliminaries
We denote the binary alphabet by \(\Sigma\). The set of strings of a particular length \(n\) is denoted \(\Sigma^{n}\). The set of all finite binary strings is denoted \(\Sigma^{*}\) and infinite binary sequences is denoted \(\Sigma^{\infty}\). For a binary string \(v\in\Sigma^{n}\setminus\{0^{n}\cup 1^{n}\}\), \(v-1\) denotes the string occurring just before \(v\) lexicographically, and \(v+1\) the string occurring just after \(v\) lexicographically. We use \(\mathbb{N}\) to denote the set of positive integers. The set of finite continued fractions is denoted \(\mathbb{N}^{*}\) and the set of all infinite continued fractions, as \(\mathbb{N}^{\infty}\).
We adopt the notation \([a_{1},a_{2},\dots]\) for the continued fraction
\[\frac{1}{a_{1}+\frac{1}{a_{2}+\cdots}}\]
and similarly, \([a_{1},a_{2},\dots,a_{n}]\) for finite continued fractions.
If a finite binary string \(x\) is a prefix of a finite string \(z\) or an infinite binary sequence \(Z\), then we denote this by \(x\sqsubseteq z\) or \(x\sqsubseteq Z\) respectively. If \(x\) is a proper prefix of a finite string \(z\), we denote it by \(x\sqsubset z\). We adopt the same notation for denoting that a finite continued fraction \(v\) is a prefix of another continued fraction. For a \(v\in\mathbb{N}^{*}\), the _cylinder set_ of \(v\), denoted \(C_{v}\), is defined by \(C_{v}=\{Y\in\mathbb{N}^{\infty}\mid v\sqsubset Y\}\). For a \(w\in\Sigma^{*}\), \(C_{w}\) is defined similarly. For a continued fraction string \(v=[a_{1}\dots a_{n}]\), \(P(v)\) denotes the string \([a_{1},\dots a_{n-1}]\). \(\lambda\) denotes the empty string and we define \(P(\lambda)=\lambda\).
For \(v\in\mathbb{N}^{*}\), \(\mu(v)\) refers to the Lebesgue measure of the continued fraction cylinder \(C_{v}\). \(\gamma(v)\) refers to the Gauss measure of the continued fraction cylinder \(C_{v}\), defined by \(\gamma(v)=\int_{C_{v}}\frac{1}{1+x}\;dx\). We use the same notation for a binary cylinder \(w\in\Sigma^{*}\). It is well-known that the Gauss measure is absolutely continuous with respect to the Lebesgue measure, and is invariant with respect to the left-shift transformation on continued fractions (see for example, [4], or [6]). Wherever there is no scope for confusion, for a \(v\in\mathbb{N}^{*}\), we use \(\mu(v)\) and \(\gamma(v)\) to represent \(\mu(C_{v})\) and \(\gamma(C_{v})\) respectively. The same holds for a \(v\in\Sigma^{*}\). We also use the notation \(\mu^{s}(v)\) and \(\gamma^{s}(v)\) to denote \((\mu(v))^{s}\) and \((\gamma(v))^{s}\) respectively. For a continued fraction string \(v=[a_{1},\dots,a_{n}]\), we call \(n\) as the rank of \(v\), and we denote it using \(rank(v)\). \([v,i]\) denotes the continued fraction \([a_{1},\dots a_{n},i]\). For an infinite continued fraction string \(Y=[a_{1},a_{2},\dots]\), \(Y\upharpoonright n\) denotes the continued fraction string corresponding to the first n entries of \(Y\), that is \(Y\upharpoonright n=[a_{1},a_{2}\dots a_{n}]\). For \(k\in\mathbb{N}\), \(\mathbb{N}^{\leq k}\) refers to the set of continued fraction strings having rank less than or equal to \(k\). All logarithms in the work have base \(2\), unless specified otherwise. For any sets \(A\) and \(B\), \(A\Delta B\) denotes the symmetric set difference operator, defined by \((A\setminus B)\cup(B\setminus A)\). In this work, for ease of notation, \(Y\in\mathbb{N}^{*}\) denotes an infinite continued fraction and \(X\in\Sigma^{\infty}\) denotes an infinite binary sequence.
### Constructive dimension of binary sequences
Lutz [16] defines the notion of effective (equivalently, constructive) dimension of an individual infinite binary sequence using the notion of the success of \(s\)-gales.
**Definition 1** (Lutz [16]).: _For \(s\in[0,\infty)\), a binary \(s\)-gale is a function \(d:\Sigma^{*}\to[0,\infty)\) such that \(d(\lambda)<\infty\) and for all \(w\in\Sigma^{*}\), \(d(w)[\mu(C_{w})]^{s}=\sum_{i\in\{0,1\}}d(wi)[\mu(C_{wi})]^{s}\)._
_The success set of \(d\) is \(S^{\infty}(d)=\bigg{\{}X\in\mathbb{N}^{\infty}\mid\limsup_{n\to\infty}d(X \upharpoonright n)=\infty\bigg{\}}\)._
_For \(\mathcal{F}\subseteq[0,1]\), \(\mathcal{G}(\mathcal{F})\) denotes the set of all \(s\in[0,\infty)\) such that there exists a lower semicomputable binary \(s\)-gale \(d\) with \(\mathcal{F}\subseteq S^{\infty}(d)\)._
_The constructive dimension or effective Hausdorff dimension of \(\mathcal{F}\subseteq[0,1]\) is \(\operatorname{cdim}(\mathcal{F})=\inf\mathcal{G}(\mathcal{F})\) and the constructive dimension of a sequence \(X\in\Sigma^{\infty}\) is \(\operatorname{cdim}(X)=\operatorname{cdim}(\{X\})\)._
## 3 Effective Continued Fraction Dimension using \(s\)-gales
Nandakumar and Vishnoi [23] formulate the notion of effective dimension of continued fractions using the notion of lower semicomputable continued fraction \(s\)-gales. Whereas a binary \(s\)-gate bets on the digits of the binary expansion of a number, a continued fraction \(s\)-gales places bets on the digits of its continued fraction expansion.
**Definition 2** (Nandakumar, Vishnoi [23]).: _For \(s\in[0,\infty)\), a continued fraction \(s\)-gale is a function \(d:\mathbb{N}^{*}\to[0,\infty)\) such that \(d(\lambda)<\infty\) and for all \(w\in\mathbb{N}^{*}\), the following holds._
\[d(w)[\gamma(C_{w})]^{s}=\sum_{i\in\mathbb{N}}d(wi)[\gamma(C_{wi})]^{s}.\]
_The success set of \(d\) is \(S^{\infty}(d)=\bigg{\{}Y\in\mathbb{N}^{\infty}\mid\limsup_{n\to\infty}d(Y \upharpoonright n)=\infty\bigg{\}}\)._
In this paper, we deal with the notion of effective or equivalently, constructive dimension. In order to effectivize the notion of \(s\)-gales, we require them to be _lower semicomputable_.
**Definition 3**.: _A function \(d:\mathbb{N}^{*}\longrightarrow[0,\infty)\) is called lower semicomputable if there exists a total computable function \(\hat{d}:\mathbb{N}^{*}\times\mathbb{N}\longrightarrow\mathbb{Q}\cap[0,\infty)\) such that the following two conditions hold._
* _Monotonicity : For all_ \(w\in\mathbb{N}^{*}\) _and for all_ \(n\in\mathbb{N}\)_, we have_ \(\hat{d}(w,n)\leq\hat{d}(w,n+1)\leq d(w)\)_._
* _Convergence : For all_ \(w\in\mathbb{N}^{*}\)_,_ \(\lim_{n\to\infty}\hat{d}(w,n)=d(w)\)_._
For \(\mathcal{F}\subseteq[0,1]\), \(\mathcal{G}_{CF}(\mathcal{F})\) denotes the set of all \(s\in[0,\infty)\) such that there exists a lower semicomputable continued fraction \(s\)-gale \(d\) with \(\mathcal{F}\subseteq S^{\infty}(d)\).
**Definition 4** (Nandakumar, Vishnoi [23]).: _The effective continued fraction dimension of \(\mathcal{F}\subseteq[0,1]\) is_
\[\operatorname{cdim}_{CF}(\mathcal{F})=\inf\mathcal{G}_{CF}(\mathcal{F}).\]
_The effective continued fraction dimension of a sequence \(Y\in\mathbb{N}^{\infty}\) is defined by \(\operatorname{cdim}_{CF}(\{Y\})\), the effective continued fraction dimension of the singleton set containing \(Y\)._
### Conversion of binary \(s\)-gales into continued fraction \(s\)-gales
In this subsection, from a continued fraction \(s^{\prime}\)-gale \(d:\mathbb{N}^{*}\to[0,\infty)\), for any \(s>s^{\prime}\), we construct a binary \(s\)-gale \(h:\Sigma^{*}\to[0,\infty)\) which succeeds on all the reals on which \(d\) succeeds. The construction proceeds in multiple steps. We first mention some technical lemmas which we use in the proof.
The following lemma is an easy consequence of the fact that the Gauss measure is absolutely continuous with respect to the Lebesgue measure (see for example, Nandakumar and Vishnoi [23]).
**Lemma 1**.: _For any interval \(B\subseteq(0,1)\), we have_
\[\frac{1}{2\ln 2}\mu(B)\leq\gamma(B)\leq\frac{1}{\ln 2}\mu(B).\]
In the construction that follows, we formulate betting strategies on binary cylinders based on continued fraction cylinders. In order to do this conversion, we require the following bounds on the relationships between the lengths of continued fraction cylinders and binary cylinders.
**Lemma 2** (Nandakumar, Vishnoi [23]).: _For any \(0\leq a<b\leq 1\), let \(\left[\frac{m}{2^{k}},\frac{m+1}{2^{k}}\right)\), where \(0\leq m\leq 2^{k}-1\) be one of the largest dyadic intervals which is a subset of \([a,b)\), then \(\frac{1}{2^{k}}\geq\frac{1}{4}(b-a)\)._
**Lemma 3** (Falconer [7]).: _For any \(0\leq a<b\leq 1\), let \(\left[\frac{m}{2^{k}},\frac{m+1}{2^{k}}\right)\), \(\left[\frac{m+1}{2^{k}},\frac{m+2}{2^{k}}\right)\), where \(0\leq m\leq 2^{k}-2\), be the smallest consecutive dyadic intervals whose union covers \([a,b)\). Then \(\frac{1}{2^{k}}\leq 2(b-a)\)._
Proof.: Let \(j=\lfloor-\log_{2}(b-a)\rfloor\). Since \((b-a)/2<2^{-j}\), it follows that at most two dyadic rationals of the form \(m/2^{j}\), \(0\leq m<2^{j}\) is in \((a,b)\). Thus, three dyadic intervals of length \(\frac{1}{2^{j}}\) cover the interval \([a,b]\). Hence two dyadic intervals of length \(\frac{2}{2^{j}}\) cover \([a,b]\). Since \(2^{-j}\leq(b-a)\), it follows that \(\frac{2}{2^{j}}\leq 2(b-a)\).
The following lemma is a generalization of the Kolmogorov inequality for continued fraction martingales (Vishnoi [30]) to \(s\)-gales. The lemma states that an equality holds in the case of decompositions using prefix-free subcylinder sets upto a finite depth.
**Lemma 4**.: _Let \(d:\mathbb{N}^{*}\rightarrow[0,\infty)\) be a continued fraction \(s\)-gale. Let \(v\in\mathbb{N}^{*}\) and for some \(k\in\mathbb{N}\), let \(A\) be a prefix free set of elements in \(\mathbb{N}^{\leq k}\) such that \(\cup_{w\in A}C_{w}=C_{v}\). Then, we have \(d(v)\gamma^{s}(v)=\sum_{w\in A}d(w)\gamma^{s}(w)\)._
Proof.: We prove this result by induction on \(k\), the maximum rank of an element in \(A\). We first observe that for any \(v\in\mathbb{N}^{*}\), if \(C_{v}=\cup_{w\in A}C_{w}\), then for all \(w\in A\), \(v\sqsubseteq w\). Therefore, we have that \(k\geq rank(v)\).
Let's consider the case when \(k=rank(v)\). In this case, the only possibility is that \(A=\{v\}\) and therefore the lemma holds trivially.
Assume that the lemma holds for any \(k\in\mathbb{N}\) such that \(k\geq rank(v)\). Let \(A\) be a prefix free set of elements in \(\mathbb{N}^{\leq k+1}\) such that \(\cup_{w\in A}C_{w}=C_{v}\).
Consider the sets, \(P=\{w\in A\mid rank(w)=k+1\}\) and \(P^{\prime}=\{u\in\mathbb{N}^{k}\mid[u,i]\in P\text{ for some }i\in\mathbb{N}\}\).
Now since \(\cup_{w\in A}\)\(C_{w}=C_{v}\) and \(A\) is prefix free, we have that for all \(u\in P^{\prime}\), and for all \(i\in\mathbb{N}\), \([u,i]\in P\).
Since \(d\) is a continued fraction \(s\)-gale, it follows that,
\[\sum_{u\in P^{\prime}}d(u)\gamma^{s}(u)=\sum_{w\in P}d(w)\gamma^{s}(w)\]
Also, note that since for any \(u\in\mathbb{N}^{*}\), \(C_{u}=\cup_{i\in\mathbb{N}}\)\(C_{[u,i]}\), we have
\[\cup_{u\in P^{\prime}}\)\(C_{u}=\cup_{w\in P}\)\(C_{w}.\]
Construct the \(A^{\prime}=P^{\prime}\cup(A-P)\). We have that the elements in \(A^{\prime}\) are prefix free and \(\cup_{w\in A^{\prime}}C_{w}=C_{v}\). Since the maximum rank of an element in \(A^{\prime}\) is \(k\), we have that,
\[d(v)\gamma^{s}(v)=\sum_{w\in A^{\prime}}d(w)\gamma^{s}(w)\]
Since \(P^{\prime}\cap(A-P)=\phi\), it follows that
\[d(v)\gamma^{s}(v)=\sum_{w\in A-P}d(w)\gamma^{s}(w)+\sum_{u\in P^{\prime}}d(u) \gamma^{s}(u).\]
Using \(\sum_{u\in P^{\prime}}d(u)\gamma^{s}(u)=\sum_{w\in P}d(w)\gamma^{s}(w)\), we have
\[d(v)\gamma^{s}(v)=\sum_{w\in A-P}d(w)\gamma^{s}(w)+\sum_{w\in P}d(w)\gamma^{s}(w).\]
From which we have
\[d(v)\gamma^{s}(v)=\sum_{w\in A}d(w)\gamma^{s}(w).\]
In the construction of a binary \(s\)-gale from continued fraction gales, The first step is the following decomposition of a binary cylinder into a set of prefix free continued fraction cylinders.
**Lemma 5** (Vishnoi [30]).: _For every \(w\in\Sigma^{*}\), there exists a set \(I(w)\subseteq\mathbb{N}^{*}\) and a constant \(k\in\mathbb{N}\) such that,_
1. \(y\in\mathbb{N}^{\leq k}\) _for every_ \(y\in I(w)\)_._
2. \((\cup_{y\in I(w)}C_{y})\Delta C_{w}\subseteq\{\inf(C_{w}),\sup(C_{w})\}\)__
3. \(I(w0)\cup I(w1)=I(w)\)__
4. \(I(w0)\cap I(w1)=\phi\)__
Moreover, given \(w\in\Sigma^{*}\), Vishnoi [30] gives a division algorithm to compute \(I(w)\). It is also clear from the division algorithm that for all \(w\in\Sigma^{*}\), there exists a \(u\in I(w)\) such that for all \(v\in(I(w0)\cup I(w1))\setminus I(w)\), we have \(u\sqsubset v\). This \(u\in I(w)\) is the continued fraction cylinder for which the mid point of \(w\), \(m(w)\) is an interior point in \(C_{u}\) and therefore gets divided.
From a continued fraction martingale, Vishnoi [30] uses the decomposition \(I(w)\) to construct a binary martingale that places the same aggregate bets on an interval. We generalize this construction to the setting of \(s\)-gales. Given a continued fraction \(s^{\prime}\)-gale \(d:\mathbb{N}^{*}\to[0,\infty)\), using the decomposition \(I(w)\), we construct a binary \(s^{\prime}\)-gale \(H_{d}\) from \(d\).
**Definition 5**.: _Given any continued fraction \(s^{\prime}\)-gale \(d:\mathbb{N}^{*}\to[0,\infty)\), define the Proportional binary \(s^{\prime}\)-gale of \(d\), \(H_{d}:\Sigma^{*}\to[0,\infty)\) as follows:_
\[H_{d}(w)=\sum_{y\in I(w)}d(y)\left(\frac{\gamma(y)}{\mu(w)}\right)^{s^{\prime }}.\]
For a \(w\in\Sigma^{*}\), let \(I^{\prime}(w)=I(w0)\cup I(w1)\). Then we have,
\[H_{d}(w0)+H_{d}(w1)=2^{s^{\prime}}\sum_{y\in I^{\prime}(w)}d(y)\Big{(}\frac{ \gamma(y)}{\mu(w)}\Big{)}^{s^{\prime}}.\]
Let \(u\in I(w)\) such that for all \(v\in I^{\prime}(w)\setminus I(w)\), \(u\sqsubset v\). Hence, by Lemma 4, it follows that \(\sum_{y\in I^{\prime}(w)}d(y)\gamma^{s^{\prime}}(y)=\sum_{y\in I(w)}d(y)\gamma ^{s^{\prime}}(y)\). Therefore, we have \(H_{d}(w0)+H_{d}(w1)=2^{s^{\prime}}H_{d}(w)\), so \(H_{d}\) is an \(s^{\prime}\)-gale. Also as \(\gamma(\lambda)=1\), we have that \(H_{d}(\lambda)=d(\lambda)\).
As \(I(w)\) is computably enumerable, it follows that \(H_{d}\) is lower semicomputable if \(d\) is lower semicomputable.
The construction by Vishnoi [30] proceeds using the _savings-account trick_ for martingales. In the setting of \(s\)-gales, however, the concept of a savings account does not work directly. Therefore, we require additional constructions in this setting.
Using ideas from the construction given in Lemma 3.1 in Hitchcock and Mayordomo [8], we construct a "smoothed" \(s\)-gale \(H_{h}:\Sigma^{*}\to[0,\infty)\) from the proportional \(s^{\prime}\)-gale constructed in Definition 5.
**Definition 6**.: _For a \(w\in\Sigma^{*}\), and an \(n>|w|\), we define_
\[F_{n}(w) =\{u\in\{0^{n}\cup 1^{n}\}\mid w\sqsubseteq u\}\;\cup\;\{u\in \Sigma^{n}\setminus\{0^{n}\cup 1^{n}\}\mid w\sqsubseteq u+1\text{ and }w\sqsubseteq u-1\},\] \[H_{n}(w) =\{u\in\Sigma^{n}\mid w\sqsubseteq u\text{ or }w\sqsubseteq u+1 \text{ or }w\sqsubseteq u-1\}\setminus F_{n}.\]
**Definition 7**.: _Given an \(s^{\prime}\)-gale \(h:\Sigma^{*}\to[0,\infty)\), for any \(s>s^{\prime}\) and for each \(n\in\mathbb{N}\), define:_
\[h_{n}(w)=\left\{\begin{array}{ll}2^{s|w|}\left(\;\sum\limits_{u\in H_{n}(w)} \tfrac{1}{2}\;h(u)\;+\;\sum\limits_{u\in F_{n}(w)}h(u)\right)&\text{if }|w|<n\\ \\ 2^{(s-1)(|w|-n+1)}\;\;h_{n}(w[0\ldots n-2])&\text{otherwise}.\end{array}\right.\]
_Define \(S_{h}:\Sigma^{*}\to[0,\infty)\) by_
\[S_{h}(w)=\sum\limits_{n=0}^{\infty}2^{-sn}\;h_{n}(w).\]
_We call \(S_{h}\) as the smoothed \(s\)-gale of \(h\)._
Consider a string \(w\in\Sigma^{n}\) other than \(0^{n}\) and \(1^{n}\). In \(h_{n}\), a factor of half the capital of \(w\) gets assigned to it's immediate parent \(w^{\prime}\). The other half is assigned to the neighbor of \(w^{\prime}\) to which \(w\) is adjacent to.
It is straightforward to verify that each \(h_{n}\) is an \(s\)-gale. \(S_{h}\) is a combination of \(s\)-gales, and hence is a valid \(s\)-gale. Note that \(h_{n}(\lambda)=\sum_{u\in\Sigma^{n}}h(u)=2^{s^{\prime}n}\). Therefore as \(s>s^{\prime}\), \(S_{h}(\lambda)=\sum_{n\in\mathbb{N}}2^{(s^{\prime}-s)n}\) is finite. If \(h\) is lower semicomputable, it follows that \(S_{h}\) is lower semicomputable.
Combining the constructions given in the section, for any \(s>s^{\prime}\), we show the construction of a binary \(s\)-gale from a continued fraction \(s^{\prime}\)-gale, satisfying certain bounds on the capital acquired.
This construction helps to establish a lower bound on effective continued fraction dimension using effective binary dimension. It is also central in formulating a Kolmogorov complexity characterization for continued fraction dimension.
**Lemma 6**.: _For \(s^{\prime}\in(0,\infty)\), let \(d:\mathbb{N}^{*}\to[0,\infty)\) be a continued fraction \(s^{\prime}\)- gale. Then, for any \(s>s^{\prime}\), there exists a binary \(s\)-gale \(h:\Sigma^{*}\to[0,\infty)\) such that for any \(v\in\mathbb{N}^{*}\) and for any \(b\in\Sigma^{*}\) such that \(C_{b}\cap C_{v}\neq\phi\) and \(\frac{1}{16}\mu(v)\leq\mu(b)\leq 2\mu(v)\), we have_
\[h(b)\geq c_{s}d(v),\]
_where \(c_{s}\) is a constant that depends on \(s\). Moreover, if \(d\) is lower semicomputable, then \(h\) is lower semicomputable._
Proof.: Given an \(s^{\prime}\)-gale \(d:\mathbb{N}^{*}\to[0,\infty)\), let \(h^{\prime}=H_{d}\), the proportional binary \(s^{\prime}\)-gale of \(d\) given in Definition 5.
For a \(v\in\mathbb{N}^{*}\), consider the smallest \(w_{1},w_{2}\in\Sigma^{*}\) such that \(C_{v}\subseteq C_{w_{1}}\cup C_{w_{2}}\). Also we can see that there exists a \(S\subseteq I(w_{1})\cup I(w_{2})\) such that \(C_{v}=\cup_{u\in S}C_{u}\).
Therefore from Lemma 4, we get \(h^{\prime}(w_{1})+h^{\prime}(w_{2})\geq d(v)\frac{\gamma^{\prime}(v)}{\mu^{s^{ \prime}}(w_{1})}\). From Lemma 3, we get that \(\mu(w_{1})=\mu(w_{2})\leq 2\mu(v)\). Also from Lemma 1, we have that \(\gamma(v)\leq(ln2)^{-1}\mu(v)\). Therefore, \(h^{\prime}(w_{1})+h^{\prime}(w_{2})\geq(2ln2)^{-s^{\prime}}d(v)\). Now for any \(s>s^{\prime}\), we have that \(h^{\prime}(w_{1})+h^{\prime}(w_{2})\geq c_{1}.d(v)\), where \(c_{1}=1/(2ln2)^{s}\).
Now for any \(s>s^{\prime}\), consider the smoothed \(s\)-gale \(h=S_{H_{d}}\) of the \(s^{\prime}\)-gale \(H_{d}\) given in Definition 6.
Let \(|w_{1}|=n\), and let \(W_{1}=P(P(w_{1}))\) be the parent cylinder of parent of \(w_{1}\). Similarly let \(W_{2}=P(P(w_{2}))\). We see that for any \(W\in\{W_{1},W_{2}\}\), \(h_{n}(W)\geq 2^{s(n-2)}\frac{h^{\prime}(w_{1})+h^{\prime}(w_{2})}{2}\geq c_{2}.2 ^{sn}.d(v)\), where \(c_{2}=2^{-(2s+1)}c_{1}\).
Take any any \(b\in\Sigma^{*}\) such that \(C_{b}\cap C_{v}\neq\phi\) and \(2.\mu(v)\geq\mu(b)\geq\frac{1}{16}\mu(v)\). Since \(\mu(b)\leq 2\mu(v)\) and \(\mu(v)\leq 2.\mu(w_{1})\), it follows that for some \(W\in\{W_{1},W_{2}\}\), \(W\sqsubseteq b\). Also since \(\mu(b)\geq\frac{1}{16}\mu(v)\), we have that, \(\mu(b)\geq\frac{1}{32}\mu(w_{1})\).
Therefore, we have that \(h_{n}(b)\geq 2^{5(s-1)}h_{n}(W)\geq c_{3}.2^{sn}.d(v)\), where \(c_{3}=c_{2}.2^{5(s-1)}\).
Since \(h(b)\geq 2^{-sn}h_{n}(b)\), we have that \(h(b)\geq c_{3}.d(v)\).
## 4 Kolmogorov Complexity characterization of Continued Fraction Dimension
Mayordomo [19] extended the result by Lutz [14] to show that effective dimension of a binary sequence \(X\in\Sigma^{\infty}\) can be characterized in terms of the Kolmogorov complexity of the finite prefixes of \(X\).
**Theorem 1** (Mayordomo [19] and Lutz [14]).: _For every \(X\in\Sigma^{\infty}\),_
\[\operatorname{cdim}(X)=\liminf_{n\to\infty}\frac{K(X\upharpoonright n)}{n}.\]
We provide a similar characterization for effective continued fraction dimension. To obtain the Kolmogorov complexity of a continued fraction string, we use the Kolmogorov complexity of one of its binary encodings.
The idea of encoding a finite continued fraction using a 1-1 binary encoding is present in Vishnoi [31]. The author presents an invariance theorem stating that every computable binary 1-1 encoding of continued fractions defines the same Kolmogorov complexity, up to an additive constant. Hence in this work, we use a new binary encoding to define Kolmogorov complexity of continued fractions, which helps us establish the characterization of effective dimension of continued fractions in a fairly simple manner while having intuitive geometric meaning.
**Definition 8** (Many-one binary encoding).: _For a continued fraction string \(v\in\mathbb{N}^{*}\), let \(b_{v}\) be the leftmost maximal binary cylinder which is enclosed by \(C_{v}\). We define \(E(v)=b_{v}\)._
**Lemma 7**.: _For any \(b\in\Sigma^{*}\), there exists at most three \(v\in\mathbb{N}^{*}\) such that \(E(v)=b\)._
Proof.: For \(b\in\{0,1\}^{*}\) assume there exists distinct \(v_{1},v_{2},v_{3},v_{4}\in\mathbb{N}^{*}\) such that \(E(v_{1})=E(v_{2})=E(v_{3})=E(v_{4})=b\). Since \(b\) is enclosed by all of these continued fraction cylinders, it follows that these continued fraction strings are extensions of each other. Assume without loss of generality that \(v_{1}\sqsubset v_{2}\sqsubset v_{3}\sqsubset v_{4}\). We can also assume that these cylinders are one length extensions of each other. From Lemma 11, using the fact that \(s_{n}>0\) and \(i\geq 1\), it follows that for any \(v\in\mathbb{N}^{*}\) and \(i\in\mathbb{N}\), \(\frac{\mu[v,i]}{\mu[v]}\leq 1/2\). Therefore we have, \(\mu(v_{4})\leq\frac{1}{8}\mu(v_{1})\). But from Lemma 2, we get \(\mu(b)\geq\frac{1}{4}\mu(v_{1})\). Combining these two, we get that \(\mu(b)\geq 2\mu(v_{4})\), which leads to a contradiction, since \(b\) is enclosed by \(v_{4}\)
Therefore for any \(b\in\Sigma^{*}\), at most three continued fraction cylinders, say \([v]\), \([v,i]\) and \([v,i,j]\) get mapped to \(b\). Therefore we pad two additional bits of information to \(E(v)\), say \(b_{1}(v).b_{2}(v)\) to identify the continued fraction cylinder that \(E(v)\) corresponds to.
**Definition 9** (One-one binary encoding).: _For \(v\in\mathbb{N}^{*}\), let \(\mathcal{E}(v)=E(v).b_{1}(v).b_{2}(v)\). This forms a one to one binary encoding of \(v\)._
We define Kolmogorov complexity of continued fraction string \(v\in\mathbb{N}^{*}\) as the Kolmogorov complexity of \(\mathcal{E}(v)\).
**Definition 10** (Kolmogorov complexity of continued fraction strings).: _For any \(v\in\mathbb{N}^{*}\), define \(K_{\mathcal{E}}(v)=K(\mathcal{E}(v))\)._
**Notation.** By the invariance theorem of Vishnoi [31], for any \(v\in\Sigma^{*}\), \(K_{\mathcal{E}}\) is at most an additive constant more than the complexity of \(v\) as defined in [31]. Hence, we drop the suffix and denote the above complexity as \(K(v)\).
In the proof of Theorem 1, Mayordomo [19] provides the construction of an \(s\)-gale that succeeds on all \(X\) for which \(s>s^{\prime}>\liminf_{n\to\infty}\frac{K(X\upharpoonright n)}{n}\). We extend the construction to the setting of continued fractions.
Additionally, we take a convex combination of gates to remove the dependence of the \(s\)-gale on the parameter \(s^{\prime}\). Due to this, we obtain the notion of an optimal lower semicomputable continued fraction \(s\)-gale. This notion is crucial in the proofs we use in the upcoming sections.
**Definition 11**.: _Given \(0<s^{\prime}<s\leq 1\) let_
\[G_{s^{\prime}}=\{w\in\mathbb{N}^{*}\mid K(w)\leq-s^{\prime}\log(\mu(w))\}.\]
_Consider the following function \(d_{s^{\prime}}:\mathbb{N}^{*}\to[0,\infty)\) defined by_
\[d_{s^{\prime}}(v)=\frac{1}{\gamma^{s}(v)}\left(\sum_{w\in G_{s^{\prime}};v \sqsubseteq w}\gamma^{s^{\prime}}(w)+\sum_{w\in G_{s^{\prime}};w\sqsubseteq v} \gamma^{s^{\prime}}(w)\frac{\gamma(v)}{\gamma(w)}\right).\]
_Now for each \(i\in\mathbb{N}\), let \(s_{i}=s(1-2^{-i})\). Finally, define \(d^{*}:\mathbb{N}^{*}\to[0,\infty)\) by_
\[d^{*}(v)=\sum_{i=1}^{\infty}2^{-i}d_{s_{i}}(v).\]
We now go on to show that the function \(d^{*}\) given in Definition 11 is a lower semicomputable \(s\)-gale. Additionally, it succeeds on all continued fraction sequences \(Y\) for which the Kolmogorov complexity of its prefixes, \(K(Y\upharpoonright n)\) dips below \(s\times-\log(\mu(Y\upharpoonright n))\) infinitely often.
**Lemma 8**.: _For any \(0<s\leq 1\), there exists a lower semicomputable continued fraction \(s\)-gale \(d^{*}:\mathbb{N}^{*}\to[0,\infty)\) that succeeds on all \(Y\in\mathbb{N}^{\infty}\) such that \(\liminf_{n\to\infty}\frac{K(Y\upharpoonright n)}{-\log(\mu(Y\upharpoonright n))}<s\)._
Proof.: Consider any \(s^{\prime}<s\leq 1\). Let \(G_{s^{\prime}}=\{w\in\mathbb{N}^{*}\mid K(w)\leq-s^{\prime}.\log(\mu(w))\}\). Consider the following continued fraction \(s\)-gale \(d_{s^{\prime}}:\mathbb{N}^{*}\to[0,\infty)\)
\[d_{s^{\prime}}(v)=\frac{1}{\gamma^{s}(v)}\ \left(\sum_{w\in G_{s^{\prime}};v \sqsubseteq w}\gamma^{s^{\prime}}(w)+\sum_{w\in G_{s^{\prime}};w\sqsubseteq v }\gamma^{s^{\prime}}(w)\frac{\gamma(v)}{\gamma(w)}\right).\]
We can see that for any \(s^{\prime}\), since \(G_{s^{\prime}}\) is computably enumerable, \(d_{s^{\prime}}\) is lower semicomputable.
For each \(w\in G\), let \(\mathcal{E}(w)=b\). By definition \(K(w)=K(b)\). Therefore, \(K(b)\leq-s^{\prime}\log(\mu(w))\). From the Definition of \(\mathcal{E}\) and Lemma 2, we see that \(\mu(b)\geq\frac{1}{16}\mu(w)\). The extra factor of \(\frac{1}{4}\) comes from the two additional bits that are padded in \(b\). Therefore, we have that \(K(b)\leq-s^{\prime}\log(16.\mu(b))\leq s^{\prime}|b|-4s^{\prime}\leq s^{\prime }|b|\).
Define \(B=\{x\in\{0,1\}^{*}\ \mid\ K(x)\leq s^{\prime}|x|\}\). Hence if \(w\in G\), then \(\mathcal{E}(w)\in B\).
Now, \(d_{s^{\prime}}(\lambda)=\sum_{w\in G}\gamma^{s^{\prime}}(w)\). Using Lemma 1, we get that \(d_{s^{\prime}}(\lambda)\leq c\sum_{w\in G}\mu^{s^{\prime}}(w)\) where \(c=(\ln 2)^{-s^{\prime}}\). Since \(s^{\prime}<1\), we have that \(c<\ln 2\).
From Lemma 2, it follows that \(\mu^{s}(\mathcal{E}(w))\geq\frac{1}{16}.\mu^{s}(w)\). Using the fact that \(\mathcal{E}\) is a one to one encoding, we get that \(d_{s^{\prime}}(\lambda)\leq 16\ln 2\sum_{x\in B}2^{-s^{\prime}|x|}\).
In the proof of the Kolmogorov complexity characterization of constructive dimension [19], Mayordomo shows that \(\sum_{x\in B}2^{-s^{\prime}|x|}\leq 2^{c}\) for some constant \(c\). The proof uses the fact that \(|B^{=n}|\leq 2^{s^{\prime}n-K(n)+c}\) along with the Kraft inequality \(\sum_{n}2^{-K(n)+c}\leq 2^{c}\). Therefore \(d_{s^{\prime}}(\lambda)\) is finite.
Note that for each \(w\in G_{s^{\prime}}\) we see that \(d_{s^{\prime}}(w)\geq(\frac{1}{\gamma(w)})^{s-s^{\prime}}\).
Now for each \(i\in\mathbb{N}\), let \(s_{i}=s(1-2^{-i})\). Finally, consider the following continued fraction \(s\)-gale.
\[d^{*}(v)=\sum_{i=1}^{\infty}2^{-i}d_{s_{i}}(v).\]
Consider any \(Y\in\mathbb{N}^{\infty}\) such that \(\liminf\limits_{n\to\infty}\frac{K(Y\mid n)}{-\log(\mu(Y\mid n))}<s\). Let \(i\in\mathbb{N}\) be the smallest number such that \(\liminf\limits_{n\to\infty}\frac{K(Y\mid n)}{-\log(\mu(Y\mid n))}<s_{i}\).
For each \(w\in G_{s_{i}}\) we see that \(d^{*}(w)\geq 2^{-i}(\frac{1}{\gamma(w)})^{s-s_{i}}\). We see that there are infinitely many prefixes \(w\) of \(Y\) for which \(w\in G_{s_{i}}\). Hence it follows that \(d^{*}\) succeeds on Y.
We refer to Downey and Hirschfeldt's (Theorem 13.3.4 [5]) proof of the lower bound on constructive dimension using Kolmogorov complexity. The proof fundamentally uses properties of the universal lower semicomputable super-martingale.
For any real having continued fraction dimension less than \(s\), we obtain a lower semicomputable binary \(s\)-gale that succeeds on it from Lemma 6. We use the success of this binary \(s\)-gale along with the same properties of the universal lower semicomputable super-martingale, to prove the following lemma.
**Lemma 9**.: _For any \(Y\in\mathbb{N}^{\infty}\) and any \(s>\operatorname{cdim}_{CF}(Y)\), we have \(\liminf\limits_{n\to\infty}\frac{K(Y\mid n)}{-\log(\mu(Y\mid n))}\leq s\)._
Proof.: Given any \(s>dim_{CF}(Y)\), take an \(s^{\prime}\) such that \(s>s^{\prime}>dim_{CF}(Y)\). By definition, there exists a continued fraction \(s^{\prime}\)-gale, say \(d:\mathbb{N}^{*}\to[0,\infty)\) that succeeds on \(Y\). For each prefix \(v_{i}\sqsubseteq Y\), let \(\mathcal{E}(v_{i})=b_{i}\). From the definition of \(\mathcal{E}\) and Lemma 2, we get that \(\mu(v)\geq\mu(b_{i})\geq\frac{1}{16}\mu(v_{i})\). The extra factor of \(\frac{1}{4}\) comes from the two additional bits that gets added in \(\mathcal{E}\).
Using Lemma 6, we get a binary \(s\)-gale \(h:\Sigma^{*}\to[0,\infty)\) such that forall \(i\), \(h(b_{i})\geq c_{s}d(v_{i})\) for some constant \(c_{s}\). Therefore \(\limsup_{i\to\infty}h(b_{i})=\infty\).
Let \(f\) be the universal lower semicomputable super-martingale [11]. As \(2^{(1-s)|b_{i}|}h(b_{i})\) is a martingale, it follows that \(f(b_{i})\geq c_{h}2^{(1-s)|b_{i}|}h(b_{i})\) for some constant \(c_{h}\) that depends only on \(h\). Therefore, we have that \(\limsup_{i\to\infty}\frac{f(b_{i})}{2^{(1-s)|b_{i}|}}=\infty\).
Let \(|b_{i}|=n\). As noted in the proof of Theorem 13.3.4 in [5], \(f(b_{i})=2^{n-KM(b_{i})\pm O(1)}\) where \(K(b_{i})\leq KM(b_{i})+O(\log n)\). Hence it follows that \(\limsup_{i\to\infty}2^{sn-K(b_{i})+O(\log n)}=\infty\). Therefore for infinitely many \(i\in\mathbb{N}\), we have \(K(b_{i})<sn+O(\log n)\).
By definition \(K(v_{i})=K(b_{i})\). Also, we have that \(|b_{i}|=-\log(\mu(b_{i}))\leq-\log(\mu(v_{i}))+4\). Therefore for infinitely many \(i\in\mathbb{N}\), we have \(K(v_{i})<-s\log(\mu(v_{i}))+4s+O(\log(-\log(\mu(v_{i}))))\). From this, it follows that \(\liminf\limits_{i\to\infty}\frac{K(v_{i})}{-\log(\mu(v_{i}))}\leq s\).
Therefore, we have the following Kolmogorov complexity based characterization of effective continued fraction dimension.
**Theorem 2**.: _For any \(Y\in\mathbb{N}^{\infty}\),_
\[dim_{CF}(Y)=\liminf\limits_{n\to\infty}\frac{K(Y\upharpoonright n)}{-\log( \mu(Y\upharpoonright n))}.\]
Proof.: For any \(Y\in\mathbb{N}^{\infty}\), let \(s^{*}=\liminf\limits_{n\to\infty}\frac{K(Y\upharpoonright n)}{-\log(\mu(Y \upharpoonright n))}\).
For any \(s>s^{*}\), from Lemma 8, it follows that there exists a lower semicomputable \(s\)-gale \(\mathcal{D}\) that succeeds on \(Y\). Hence \(\dim_{CF}(Y)\leq s^{*}\).
For any \(s>\dim_{CF}(Y)\), from Lemma 9, we have that \(s^{*}\leq s\). Therefore, we have \(s^{*}\leq\dim_{CF}(Y)\).
### Optimal gases and effective continued fraction dimension of a set
Lutz [16] utilizes the notion of the optimal constructive subprobability supermeasure \(\mathbf{M}\) on the Cantor space [32] to provide a notion of an optimal constructive supergale.
We note that using Theorem 2, the gale that we obtain from Lemma 8 leads to an analogous notion in the continued fraction setting. We call the continued fraction \(s\)-gale \(d^{*}\) thus obtained as the _optimal lower semicomputable continued fraction \(s\)-gale_.
**Lemma 10**.: _For any \(s>0\), there exists a lower semicomputable continued fraction \(s\)-gale \(d^{*}:\mathbb{N}^{*}\to[0,\infty)\) such that for all \(Y\in\mathbb{N}^{\infty}\) with \(\dim_{CF}(Y)<s\), \(d^{*}\) succeeds on \(Y\)._
Proof.: For all \(Y\in\mathbb{N}^{*}\) such that \(\dim_{CF}(Y)<s\), from Theorem 2, it follows that \(\liminf\limits_{n\to\infty}\frac{K(Y\upharpoonright n)}{-\log(\mu(Y\upharpoonright n ))}<s\). Now applying Lemma 8, we see that the given lemma holds.
Lutz (Theorem 4.1 in [16]) shows that the effective dimension of a set is precisely the supremum of effective dimensions of individual elements in the set, that is for all \(X\subseteq[0,1]\), \(\dim(X)=\sup_{S\in X}\dim(S)\). Using the notion of the optimal lower semicomputable continued fraction \(s\)-gale from Lemma 10, we extend this result to continued fraction dimension.
**Theorem 3**.: _For all \(\mathcal{F}\subseteq[0,1]\), \(\dim_{CF}(\mathcal{F})=\sup_{Y\in\mathcal{F}}\cdim_{CF}(Y)\)._
Proof.: For any \(s>\dim_{CF}(\mathcal{F})\), for all \(Y\in\mathcal{F}\) there exists a lower semicomputable continued fraction \(s\)-gale that succeeds on \(Y\). Thus we have \(\sup_{Y\in\mathcal{F}}\cdim_{CF}(Y)\leq s\).
Take any any \(s>\sup_{Y\in\mathcal{F}}\cdim_{CF}(Y)\). It follows that for all \(Y\in\mathcal{F}\), \(\dim_{CF}(Y)<s\). Therefore from Lemma 10, we have that there exists a lower semicomputable continued fraction \(s\)-gale \(d^{*}:\mathbb{N}^{*}\to[0,\infty)\) that succeeds on all \(Y\in\mathcal{F}\). Therefore, \(\cdim_{CF}(\mathcal{F})\leq s\).
## 5 Reals with unequal Effective Dimension and Effective Continued Fraction Dimension
In this section, we show that for any set of reals \(\mathcal{F}\subseteq[0,1]\), the effective Hausdorff effective dimension of \(\mathcal{F}\) cannot exceed its effective continued fraction dimension. We show that this cannot
be improved to an equality. Hence, this inequality is strict in general. We show this by proving the existence of a real such that its effective continued fraction dimension is strictly greater its effective dimension.
### Effective Hausdorff dimension is at most the effective continued fraction dimension
**Theorem 4**.: _For any \(\mathcal{F}\subseteq[0,1]\), \(\operatorname{cdim}(\mathcal{F})\leq\operatorname{cdim}_{CF}(\mathcal{F})\)._
Proof.: Let \(s>s^{\prime}>\operatorname{cdim}_{CF}(\mathcal{F})\). By definition, there exists a lower semicomputable continued fraction \(s^{\prime}\)-gale \(d:\mathbb{N}^{*}\to[0,\infty)\) such that \(\mathcal{F}\subseteq S^{\infty}[d]\).
Take any \(Y\in S^{\infty}[d]\). Let \(X\in\Sigma^{\infty}\) be the corresponding binary representation of \(Y\). By definition, for any \(m\in\mathbb{N}\), there exists an \(n\in\mathbb{N}\) such that \(d(Y\upharpoonright n)>m\). Let \(v=Y\upharpoonright n\).
Using Lemma 3, we get two binary cylinders \(w_{1}\) and \(w_{2}\) such that \(C_{v}\subseteq C_{w_{1}}\cup C_{w_{2}}\) such that \(\mu(w_{1})=\mu(w_{2})\leq 2\mu(v)\). We have that since \(v\sqsubseteq Y\), \(w1\sqsubseteq X\) or \(w2\sqsubseteq X\). Without loss of generality assume that \(w_{1}\sqsubseteq X\).
From Lemma 6, we obtain a lower semicomputable \(s\)-gale \(h\) such that \(h(w_{1})\geq c_{s}.d(v)\geq c_{s}.m\) for some positive constant \(c_{s}\).
Since \(m\) is arbitrary, we see that \(h\) succeeds on \(X\).
### Reals with unequal effective Hausdorff and effective continued fraction dimensions
We now provide the main construction of the paper, utilizing the results we show in previous sections.
We first require some technical lemmas about the estimation of Lebesgue measure of a continued fraction cylinder in terms of digits of the continued fraction. Some of the bounds derived in this section may be of independent interest.
In combinatorial arguments, the Gauss measure is often difficult to deal with directly, and it is convenient to use the Lebesgue measure, and derive inequalities.
The following equation, Proposition 1.2.7 in Kraaikamp and Iosifescu [9], is extremely useful in deriving an estimate for the Lebesgue measure of continued fraction cylinders. We derive consequences of this Lemma below, and these are crucial in estimating the dimension of the continued fraction we construct in Section 5. Note that the bounds for Gauss measure are not simple to derive directly.
**Lemma 11** (Kraaikamp, Iosifescu [9]).: _For any \(v=[a_{1},\ldots a_{n}]\) and \(i\in\mathbb{N}\),_
\[\frac{\mu([v,i])}{\mu([v])}=\frac{s_{n}+1}{(s_{n}+i)(s_{n}+i+1)}\]
_where \(s_{n}=[a_{n},\ldots a_{1}]\) is the rational corresponding to the reverse of string \(v\)._
The lemma given above gives the following bounds on the Lebesgue measure of a continued fraction cylinder in terms of the digits of the continued fraction.
**Lemma 12**.: _For any \(v=[a_{1},\ldots a_{k}]\in\mathbb{N}^{k}\) we have_
\[\prod_{i=1}^{k}\frac{1}{(a_{i}+1)(a_{i}+2)}\ \leq\ \mu(v)\ \leq\ \prod_{i=1}^{k} \frac{2}{a_{i}\;(a_{i}+1)}\]
Proof.: If \(k=1\), then \(\mu(v)=\frac{1}{a_{1}(a_{1}+1)}\), therefore the lemma holds. For \(k>1\), let \(v=[a_{1},\ldots a_{k-1}]\) and consider \([v,a_{k}]\). From Lemma 11, it follows that
\[\mu([v,a_{k}])=\frac{s_{k-1}+1}{(s_{k-1}+a_{k})(s_{k-1}+a_{k}+1)}\ \ \mu(v),\]
where \(s_{k-1}=[a_{k-1},\ldots a_{1}]\). Since \(s_{k-1}\in[0,1]\), it follows that
\[\frac{1}{(a_{k}+1)(a_{k}+2)}\ \mu(v)\leq\mu([v,a_{k}])\leq\frac{2}{(a_{k})(a_{k }+1)}\ \mu(v)\]
Therefore, by induction on \(k\), the lemma holds.
**Lemma 13**.: _Let \(v=[a_{1}\ldots a_{k}]\in\mathbb{N}^{*}\). Then for any \(a,b\in\mathbb{N}\) such that \(b>a\),_
\[\mu\left(\ \bigcup_{i=a}^{b}\,[v,i]\ \right)\leq\frac{2}{a}\ \prod_{i=1}^{k} \frac{2}{a_{i}\ (a_{i}+1)}\]
Proof.: From Lemma 11, it follows that
\[\frac{\mu([v,i])}{\mu(v)}=\frac{s_{k}+1}{(s_{k}+i)(s_{k}+i+1)}\]
where \(s_{k}\in[0,1]\). Therefore
\[\mu\Big{(}\ \bigcup_{i=a}^{b}\,[v,i]\ \Big{)}\leq\sum_{i=a}^{b}\frac{2}{i(i+1)} \mu(v)\leq\frac{2}{a}\ \mu(v).\]
From Lemma 12, \(\mu(v)\leq\prod\limits_{i=1}^{k}\frac{2}{a_{i}\ (a_{i}+1)}\). Therefore this lemma holds.
The following lemma is a direct constructive extension of the proof by Lutz [15]. Using this technique, we convert a set of computably enumerable prefix free binary covers into a lower semi-computable binary \(s\)-gale.
**Lemma 14** (Lutz [15]).: _For all \(n\in\mathbb{N}\), and \(\mathcal{F}\subseteq[0,1]\), if there is a computably enumerable prefix free binary cover \(\{B_{n}^{n}\}\) of \(\mathcal{F}\), such that \(\sum_{i}|B_{i}^{n}|^{s}<2^{-n}\), then there exists a lower semicomputable binary \(s\)-gale that succeeds on \(\mathcal{F}\)._
Proof.: For an \(n\in N\), define an \(s\)-gale \(d_{n}:\{0,1\}^{*}\to[0,\infty)\) as follows:
If \(\exists\ v\sqsubseteq w\) such that \(v\in\{B_{n}^{i}\}\), then
\[d_{n}(w)=\frac{\mu^{s}(v)}{\mu^{s}(w)}\frac{\mu(w)}{\mu(v)}\]
Otherwise,
\[d_{n}(w)=\frac{1}{\mu^{s}(w)}\sum_{v\in A_{r}\ ;\ w\sqsubseteq v}\mu^{s}(v)\]
Finally define
\[d(w)=\sum_{n=0}^{\infty}2^{n}d_{2n}(w).\]
It is straightforward to verify that \(d_{n}\) is an \(s\)-gate. It follows that \(d_{n}(\lambda)\leq 2^{-n}\), hence \(d(\lambda)\leq 1\). Since \(\{B_{n}^{i}\}\) is computably enumerable, \(d\) is lower semicomputable. Now, for all \(X\in F\) and \(n\in\mathbb{N}\), there exists a prefix \(w\sqsubseteq X\) such that \(w\in\{B_{n}^{i}\}\). From the definition of \(d\), we can see that \(\forall w\in\{B_{n}^{i}\}\), \(d_{n}(w)=1\). Therefore, \(d(w)\geq 2^{n}d_{2n}(w)=2^{n}.\) Since \(n\) is arbitrary, the \(s\)-gale d succeeds on \(F\).
We now proceed to show the construction of the set \(\mathcal{F}\). The definition uses a parameter \(\mathbf{s}\). We later go on to show that for all such \(\mathcal{F}\), there exists an element \(Y\in\mathcal{F}\) such that the effective continued fraction dimension of \(Y\) is greater than \(0.5\). We also go on to show that \(cdim(F)\leq\mathbf{s}\).
We first provide the stage-wise construction of a set \(\mathcal{F}_{k}\subseteq[0,1]\), such that for each \(k\in\mathbb{N}\)\(\mathcal{F}_{k+1}\subseteq\mathcal{F}_{k}\). We then define the set \(\mathcal{F}\) using an infinite intersection of the sets \(\mathcal{F}_{k}\).
**Definition 12**.: _Let \(0<\mathbf{s}<0.5\). Define \(a_{1}=1\). For any \(k\in\mathbb{N}\), such that \(k>1\), define \(a_{k}\) inductively as:_
\[a_{k}=2\bigg{(}k\prod\limits_{i=1}^{k-1}100a_{i}\bigg{)}^{1/\mathbf{s}}.\]
_For any \(k\in\mathbb{N}\), define \(b_{k}=50.a_{k}\). Take \(\mathcal{F}_{0}=\lambda\)._
_Let \(\mathcal{F}_{k}=\{[v_{1}\dots v_{k}]\in\mathbb{N}^{k}\text{ such that }v_{i}\in[a_{i},b_{i}]\text{ for }i\in 1 \text{ to }k\}\). Finally define_
\[\mathcal{F}=\bigcap\limits_{k=1}^{\infty}\mathcal{F}_{k}.\]
We use the bounds obtained from Lemma 11, along with basic properties of harmonic numbers to prove the following property of measures of continued fraction sub cylinders.
**Lemma 15**.: _For any \(x\in\mathbb{N}^{*}\), \(s\leq 0.5\) and \(a_{k},b_{k}\in N\) such that \(b_{k}=50.a_{k}\), \(\sum\limits_{i=a_{k}}^{b_{k}}\gamma^{s}([x,i])>c\gamma^{s}([x])\) for some \(c>1\)._
Proof.: From Lemma 11, it follows that
\[\sum\limits_{a_{n}}^{b_{k}}\frac{\mu^{s}([x,i])}{\mu^{s}([x])} =\sum\limits_{a_{n}}^{b_{k}}\bigg{(}\frac{s_{k}+1}{(s_{k}+i)(s_{ k}+i+1)}\bigg{)}^{s}\] \[\geq\sum\limits_{a_{k}}^{b_{k}}\bigg{(}\frac{1}{(i+1)(i+2)}\bigg{)} ^{s}\]
The second inequality follows from the fact that \(s_{k}\in[0,1]\).
Using Lemma 1, we get that
\[\sum\limits_{a_{n}}^{b_{k}}\frac{\gamma^{s}([x,i])}{\gamma^{s}([x])}\geq\sum \limits_{a_{k}}^{b_{k}}\bigg{(}\frac{1}{2(i+1)(i+2)}\bigg{)}^{s}\]
Putting \(b_{k}=50a_{k}\) and \(s\leq 0.5\), we get
\[\sum\limits_{a_{n}}^{b_{k}}\frac{\gamma^{s}([x,i])}{\gamma^{s}([x ])} \geq\frac{1}{2}\sum\limits_{a_{k}}^{50a_{k}}\frac{1}{i+2}\] \[=0.5(H(50a_{k}+2)-H(a_{k}+1)).\]
where \(H_{n}\) is the \(n^{th}\) Harmonic number. From the fact that \(\ln n\leq H_{n}\leq\ln n+1\), we have
\[H(50a_{k}+2)-H(a_{k}+1) \geq\ln(50.a_{k})-\ln(2.a_{k})-1\] \[=\ln(25)-1.\]
The lemma holds as \(0.5(\ln 25-1)\) is greater than \(1\).
Using the bound derived above, we show that for \(s=0.5\), the optimal \(s\)-gale \(d^{*}\) formulated in Section 4.1 does not succeed on some sequence in \(Y\in\mathcal{F}\). Using this we establish that \(\operatorname{cdim}_{CF}(Y)\geq 0.5\).
**Lemma 16**.: _There exists a \(Y\in\mathcal{F}\) such that \(\operatorname{cdim}_{CF}(Y)\geq 0.5\)._
Proof.: Let \(s=0.5\). Consider the continued fraction \(s\)-gale \(d^{*}\) from Lemma 10. It follows that for all \(Y\in\mathbb{N}^{*}\), if \(d^{*}\) does not succeed on any \(Y\in\mathbb{N}^{*}\), then \(\operatorname{cdim}_{CF}(Y)\geq s\).
Consider any \(v\in\mathbb{N}^{*}\), let \(rank(v)=k\). From lemma 15, we have that for some \(c>1\), \(\sum\limits_{i=a_{k}}^{b_{k}}\gamma^{s}([v,i])>c.\gamma^{s}([v])\).
Now if \(\forall i\in[a_{k},b_{k}]\), \(d^{*}([v,i])\geq\frac{1}{c}.d^{*}(v)\), from the \(s\)-gale condition it follows that \(d^{*}(v)\gamma^{s}(v)\geq\frac{d^{*}(v)}{c}\sum\limits_{i=a_{k}}^{b_{k}}\gamma ^{s}([v,i])>d^{*}(v)\gamma^{s}(v)\), which is a contradiction.
Therefore, it follows that for all \(v\in\mathbb{N}^{*}\), there exists some \(n_{v}\in[a_{k},b_{k}]\) such that \(d^{*}([v,i])<\frac{1}{c}.d^{*}([v])\).
Let \(v_{0}=\lambda\), for each \(i\in\mathbb{N}\), we define \(v_{i}=[v_{i-1},n_{v_{i-1}}]\). Take \(Y=\cap_{i=1}^{\infty}C_{v_{i}}\), it follows that \(Y\in F\). Taking \(d^{*}(\lambda)=k\) we get \(d^{*}(Y\upharpoonright n)<\frac{k}{c^{n}}\). Therefore the continued fraction \(s\)-gale \(d^{*}\) does not succeed on \(Y\). Hence \(\operatorname{cdim}_{CF}(Y)\geq 0.5\).
From this, it follows that the constructive dimension of the entire set \(\mathcal{F}\) is also greater than or equal to \(0.5\).
**Lemma 17**.: \(\operatorname{cdim}_{CF}(\mathcal{F})\geq 0.5\)_._
Proof.: From Theorem 3, we get that \(\operatorname{cdim}_{CF}(\mathcal{F})=\sup_{Y\in\mathcal{F}}\operatorname{cdim }_{CF}(Y)\). From Lemma 16, it follows that there exists a \(Y\in F\) such that \(\operatorname{cdim}_{CF}(Y)\geq 0.5\). Combining these two, we get that \(\operatorname{cdim}_{CF}(\mathcal{F})\geq 0.5\).
Now we show that the effective Hausdorff dimension of all points in the set \(\mathcal{F}\) is less than \(\mathbf{s}\). Using ideas from [2], we devise a set of covers \(S_{k}\) for \(\mathcal{F}\), by combining adjacent continued fraction cylinders into a single cover.
Using the bounds derived on Lebesgue measure of continued fraction cylinders, we show that for the set of covers \(S_{k}\) for \(\mathcal{F}\), the \(\mathbf{s}\)-dimensional Hausdorff measure shrinks arbitrarily small.
**Lemma 18**.: _For \(k\in\mathbb{N}\), let \(S_{k}=\{\ \bigcup\limits_{i=a_{k}}^{b_{k}}[v,i]\text{ for }v\in\mathcal{F}_{k-1}\}\). Then, \(\sum\limits_{S\in S_{k}}\mu^{\mathbf{s}}(S)\leq 1/k\)._
Proof.: The largest element in \(S_{k}\) is \(I=[(a_{1}\dots a_{k-1},a_{k}),[(a_{1}\dots a_{k-1},b_{k})]\). The number of elements in \(S_{k}\) equals \(\prod_{i=1}^{k-1}(b_{i}-a_{i})\). Additionally, we have \(b_{i}=50a_{i}\) for all \(i\in\mathbb{N}\). Therefore,
\[\sum\limits_{S\in S_{k}}\mu^{\mathbf{s}}(S)\leq\mu^{\mathbf{s}}(I)\prod_{i=1} ^{k-1}50a_{i}.\]
From Lemma 13, it follows that \(\mu(I)\leq\frac{2}{a_{k}}\ \prod\limits_{i=1}^{k-1}\frac{2}{a_{i}\ (a_{i}+1)}\). Therefore,
\[\sum\limits_{S\in S_{k}}\mu^{\mathbf{s}}(S) \leq\Big{(}\frac{2}{a_{k}}\ \prod\limits_{i=1}^{k-1}\frac{2}{a_{i}^{2}}\Big{)}^{ \mathbf{s}}\ \prod\limits_{i=1}^{k-1}(50a_{i})\] \[\leq\frac{2^{\mathbf{s}}}{a_{k}^{\mathbf{s}}}\ \prod\limits_{i=1}^{k-1}100a_{i}\]
Since \(a_{k}=2(k\prod\limits_{i=1}^{k-1}100a_{i})^{1/\mathbf{s}}\), this value is less than \(1/k\).
To show that the constructive dimension of \(\mathcal{F}\) is less than \(\mathbf{s}\), we construct a lower semicomputable binary \(\mathbf{s}\)-gale that succeeds on \(\mathcal{F}\). Using standard techniques, we first convert the covers obtained in Lemma 18 to a set of binary covers of \(\mathcal{F}\). Finally applying Lemma 14, we convert the binary covers into a semicomputable \(\mathbf{s}\)-gale that succeeds on \(\mathcal{F}\).
**Lemma 19**.: \(\mathrm{cdim}(\mathcal{F})\leq\mathbf{s}\)_._
Proof.: Given \(k\in\mathbb{N}\), from Lemma 18, we have that for \(S_{k}=\{\ \bigcup_{i=a_{k}}^{b_{k}}\ [v,i]\text{ for }v\in\mathcal{F}_{k-1}\}\), \(\sum_{S\in S_{k}}\mu^{\mathbf{s}}(S)\leq 1/k\). For each \(S\in S_{k}\), using Lemma 3, we get that for the two smallest consecutive binary cylinders say \(b_{1}(S)\) and \(b_{2}(S)\) that cover \(S\), we have that \(\mu(b_{1})=\mu(b_{2})\leq 2\mu(C)\).
Hence the set \(B_{k}=\{\{b_{1}(S)\}\cup\{b_{2}(S)\}\) such that \(S\in S_{k}\}\) forms a binary cover of \(S_{k}\). Also from Lemma 3, we have that \(\sum_{b\in B_{k}}\mu^{\mathbf{s}}(b)\leq 2^{1+\mathbf{s}}\sum_{S\in S_{k}}\mu^{ \mathbf{s}}(S)\leq 2^{1+\mathbf{s}}/k\).
Note that the set \(S_{k}\) is computable as \(a_{k}\) and \(b_{k}\) are computable for all \(k\). Given any interval \(S\), \(b_{1}(S)\) and \(b_{2}(S)\) are also computable. Hence the set \(B_{k}\) is computable.
Since \(B_{k}\) is a finite set, we can remove all \(v\in B_{k}\) such that \(u\sqsubset v\) for some \(u\in B_{k}\), to make \(B_{k}\) prefix free.
For an \(n\in\mathbb{N}\), taking \(k=\lceil 2^{1+\mathbf{s}}.2^{n}\rceil\), the set \(B_{k}\) forms a computably enumerable prefix free binary cover of \(\mathcal{F}\) such that \(\sum_{b\in B_{k}}\mu^{\mathbf{s}}(b)\leq 2^{-n}\).
Applying Lemma 14, we get that there exists a lower semicomputable \(\mathbf{s}\)-gale that succeeds on \(\mathcal{F}\). Hence the lemma holds.
We sum up the results from Lemma 16 and Lemma 19 into the following theorem.
**Theorem 5**.: _Given any \(0<\varepsilon<0.5\), there exists a \(Y\in\mathbb{N}^{\infty}\) such that \(\mathrm{cdim}_{CF}(Y)\geq 0.5\) and \(\mathrm{cdim}(Y)\leq\varepsilon\)._
Proof.: Given \(0<\varepsilon<0.5\), taking \(\mathbf{s}=\varepsilon\), construct the set \(\mathcal{F}\) given in Definition 12.
From Lemma 16, it follows that there exists a \(Y\in\mathcal{F}\) such that \(\mathrm{cdim}_{CF}(Y)\geq 0.5\).
From Lemma 19, it follows that for all \(X\in\mathcal{F}\), \(cdim(X)\leq\varepsilon\). Hence \(cdim(Y)\leq\varepsilon\).
|
2301.04596 | Interplay of structure and magnetism in LuFe4Ge2 tuned by hydrostatic
pressure | LuFe$_4$Ge$_2$ crystallizes in the ZrFe$_4$Si$_2$-type structure, hosting
chains of Fe-tetrahedra giving rise to geometric frustration and
low-dimensionality. The compound orders antiferromagnetically at around 36 K
accompanied by a simultaneous structural transition from a tetragonal to an
orthorhombic phase. The hydrostatic pressure dependence of the magnetic and
structural transitions is investigated using electrical-transport, ac
magnetic-susceptibility, ac calorimetry, M$\ddot{\rm o}$ssbauer, muon-spin
relaxation ($\mu$SR), and x-ray diffraction measurements. External pressure
suppresses the first-order transition to the antiferromagnetic phase (AFM1)
around 1.8 GPa. The structural transition is largely unaffected by pressure and
remains between 30 to 35 K for pressures up to 2 GPa. A second
antiferromagnetic phase (AFM2) is observed at higher pressures. The transition
from the paramagnetic to the AFM2 phase is of second-order nature and appears
to be connected to the structural transition. The magnetic volume fraction
obtained from $\mu$SR and M$\ddot{\rm o}$ssbauer measurements reveal that the
entire sample undergoes magnetic ordering in both magnetic phases. In addition,
similar low-temperature muon-precession frequencies in AFM1 and AFM2 phases
point at similar ordered moments and magnetic structures in both phases. Our
results further indicate enhanced magnetic fluctuations in the pressure induced
AFM2 phase. The experimental observations together with density functional
theory calculations suggest that the magnetic and structural order parameters
in LuFe$_4$Ge$_2$ are linked by magnetic frustration, causing the simultaneous
magneto-structural transition. | M. O. Ajeesh, P. Materne, R. D. dos Reis, K. Weber, S. Dengre, R. Sarkar, R. Khasanov, I. Kraft, A. M. León, W. Bi, J. Zhao, E. E. Alp, S. Medvedev, V. Ksenofontov, H. Rosner, H. -H. Klauss, C. Geibel, M. Nicklas | 2023-01-11T17:44:42Z | http://arxiv.org/abs/2301.04596v1 | # Interplay of structure and magnetism in LuFe\({}_{4}\)Ge\({}_{2}\) tuned by hydrostatic pressure
###### Abstract
LuFe\({}_{4}\)Ge\({}_{2}\) crystallizes in the ZrFe\({}_{4}\)Si\({}_{2}\)-type structure, hosting chains of Fe-tetrahedra giving rise to geometric frustration and low-dimensionality. The compound orders antiferromagnetically at around 36 K accompanied by a simultaneous structural transition from a tetragonal to an orthorhombic phase. The hydrostatic pressure dependence of the magnetic and structural transitions is investigated using electrical-transport, ac magnetic-susceptibility, ac calorimetry, Mossbauer, muon-spin relaxation (\(\mu\)SR), and x-ray diffraction measurements. External pressure suppresses the first-order transition to the antiferromagnetic phase (AFM1) around 1.8 GPa. The structural transition is largely unaffected by pressure and remains between 30 to 35 K for pressures up to 2 GPa. A second antiferromagnetic phase (AFM2) is observed at higher pressures. The transition from the paramagnetic to the AFM2 phase is of second-order nature and appears to be connected to the structural transition. The magnetic volume fraction obtained from \(\mu\)SR and Mossbauer measurements reveal that the entire sample undergoes magnetic ordering in both magnetic phases. In addition, similar low-temperature muon-precession frequencies in AFM1 and AFM2 phases point at similar ordered moments and magnetic structures in both phases. Our results further indicate enhanced magnetic fluctuations in the pressure induced AFM2 phase. The experimental observations together with density functional theory calculations suggest that the magnetic and structural order parameters in LuFe\({}_{4}\)Ge\({}_{2}\) are linked by magnetic frustration, causing the simultaneous magneto-structural transition.
+
Footnote †: preprint: APS/123-QED
## I Introduction
Compounds with competing ground states have attracted tremendous attention in condensed matter research. This is because novel properties such as quantum criticality and unconventional superconductivity are often observed in regions of competing energy scales in a variety of material classes [1; 2; 3; 4; 5; 6]. In such compounds the interplay of magnetic, electronic, and structural degrees of freedom dictates the emerging phenomena. A prime example is the unconventional superconductivity observed in iron pnictides, where the role of magnetic and nematic fluctuations are crucially debated [7; 8; 9; 10; 11]. In this regard, compounds with low-dimensionality and magnetic frustration are of particular interest due to enhanced quantum fluctuations. Therefore identifying new systems and detailed investigations by tuning their ground-state properties are important for improving our understanding of unconventional phenomena.
Intermetallic \(A\)Fe\({}_{4}\)\(X_{2}\) (\(A=\) rare-earth, \(X=\) Si, Ge) compounds are ideal candidates for studying unconventional phases and the effect of magnetic frustration on their properties. These compounds crystallize in the ZrFe\({}_{4}\)Si\({}_{2}\)-type structure (space group \(P4_{2}/mnm\)) consisting of a slightly distorted tetrahedral arrangement of Fe atoms, a geometry well-known for exhibiting magnetic frustration [12]. Moreover, the Fe tetrahedra are edge-shared to form chains along the crystallographic \(c\)-axis, resulting in a quasi-one-dimensional structure (see Fig. 1). These quasi-one-dimensional chains of geometrically frustrated Fe tetrahedra form a very different type of magnetic lattice compared to the 122 Fe-pnictides where the magnetic frustration is caused by competing exchange interactions. In that way the intermetallic \(A\)Fe\({}_{4}\)\(X_{2}\) materials offer a new perspective on the entanglement of crystal structure and magnetism.
Previous investigations on the isostructural compound ZrFe\({}_{4}\)Si\({}_{2}\) using chemical substitution and external pressure revealed that this compound is close to a lattice-volume tuned quantum critical point [13]. Furthermore, significantly large electronic heat capacity observed at low temperatures has been ascribed to the effect of magnetic frustration. The combination of magnetic frustration and low-dimensionality in these compounds draws
the attention for a detailed investigation on other materials in the family.
Earlier studies on LuFe\({}_{4}\)Ge\({}_{2}\) showed an antiferromagnetic (AFM) transition at 32 K with first-order character accompanied by a structural transition from tetragonal \(P4_{2}/mnm\) to orthorhombic \(Pnnm\)[14]. The results pointed at a canted arrangement of Fe moments in the \(ab\)-plane yielding a commensurate antiferromagnetic phase with propagation vector \(q=0\). Moreover, the size of the ordered Fe moment of 0.44 \(\mu_{\rm B}\) appeared to be highly reduced, which is attributed to the presence of magnetic frustration. However, a detailed study on the physical properties and the interplay of structure and magnetism is lacking.
In this article, we report a detailed investigation on the pressure evolution of the magnetic and structural transitions in LuFe\({}_{4}\)Ge\({}_{2}\), carried out using electrical-transport, magnetic-susceptibility, ac calorimetry, Mossbauer spectroscopy, muon-spin relaxation, and x-ray diffraction experiments under external pressure. The nature of the phase transitions and the details of the various magnetic phases are discussed. Furthermore, our experimental findings together with theoretical calculations based on density functional theory elucidate the role of magnetic frustration in the interplay of magnetic and structural degrees of freedom in LuFe\({}_{4}\)Ge\({}_{2}\).
## II Results and discussion
### Ambient Pressure Characterization
The temperature dependence of the dc magnetic susceptibility \(\chi=M/H\) of LuFe\({}_{4}\)Ge\({}_{2}\) at ambient pressure is shown in Fig. 2a. Here, the ferromagnetic contribution in the magnetization from a small amount of Fe\({}_{3}\)Ge phase (\(<4\%\)) in the sample is estimated by measuring magnetization at different fields, which is then subtracted from the measured data to obtain the intrinsic magnetic susceptibility of LuFe\({}_{4}\)Ge\({}_{2}\). High-temperature \(\chi(T)\) follows a Curie-Weiss (CW) behavior \(\chi(T)=C/(T-\theta_{\rm W})\) with an effective moment \(\mu_{\rm eff}(=\sqrt{3K_{\rm B}C/N_{\rm A}\mu_{0}\mu_{\rm B}^{2}})\) of 2.2 \(\mu_{\rm B}\)/Fe and a Weiss temperature \(\theta_{\rm W}=-90\) K. These CW parameters indicate relatively large Fe moments in the paramagnetic phase with dominant antiferromagnetic interaction among them. At low temperature, a sharp drop in susceptibility is observed at \(T_{N}=36\) K corresponding to the antiferromagnetic transition. It should be noted that the observed \(T_{N}\) differs from the earlier reported value of 32 K [14]. This may be due to a difference in sample quality, where a large amount of impurity phases could have affected the precise determination of the ordering temperature using neutron diffraction studies.
Heat capacity \(C_{p}\) of LuFe\({}_{4}\)Ge\({}_{2}\) as a function of temperature is presented in Fig. 2b. A very sharp peak in \(C_{p}(T)\) is observed at \(T=36\) K, consistent with \(T_{N}\) obtained from magnetic-susceptibility data. The low-temperature part of \(C_{p}(T)\) is fitted with \(C_{p}(T)=\gamma T+\beta T^{3}\) in the interval between 2 and 25 K. The fit yields an enhanced value for the Sommerfeld coefficient \(\gamma=96\) mJ/molK\({}^{2}\) which indicates strong electron correlation effects. \(C_{p}(T)\) upon heating and cooling, obtained by analyzing the thermal-relaxation curves recorded in a standard relaxation-type measurement setup following a method outlined by Lashley _et al._[15], is plotted in the inset of Fig. 2b. A thermal hysteresis observed between heating and cooling curves confirms the first-order nature of the magneto-structural transition.
Figure 1: (a) Crystal structure of LuFe\({}_{4}\)Ge\({}_{2}\) viewed along the \(c\) axis. In the low-temperature orthorhombic phase (\(Pnnm\)), the Fe sites split in to two sites marked as Fe1 and Fe2 [14]. (b) The chainlike arrangement of edge-shared Fe tetrahedra viewed along the \(b\) axis.
Figure 2: (a) Temperature dependence of the dc magnetic susceptibility of LuFe\({}_{4}\)Ge\({}_{2}\). The red curve is a Curie-Weiss fit to the data in the temperature interval between 100 and 300 K. (b) Heat capacity \(C_{p}\) of LuFe\({}_{4}\)Ge\({}_{2}\) as a function of temperature. The line is a fit to the data for 2 K\(\leq T\leq 25\) K using \(C_{p}(T)=\gamma T+\beta T^{3}\). The inset displays the thermal hysteresis in \(C_{p}(T)\) obtained from the analysis of heating and cooling parts of the thermal-relaxation cycle.
Experiments under hydrostatic pressure
#### ii.2.2 Electrical Resistivity
To investigate the evolution of the magneto-structural transition upon application of hydrostatic pressure we first turn to electrical-resistivity measurements. Figure 3a presents the evolution of the temperature dependence of the electrical resistivity \(\rho(T)\) under external pressure. At ambient pressure, LuFe\({}_{4}\)Ge\({}_{2}\) shows metallic behavior upon cooling followed by a sudden drop in the resistivity at 36 K coinciding with the magneto-structural transition. The decrease in the resistivity at the transition temperature can be understood as the reduction in the scattering contribution due to the transition from the disordered paramagnetic to the antiferromagnetic phase. Upon further cooling, the resistivity decreases with a higher slope, indicating a further reduction of the scattering in the antiferromagnetic phase. The resistivity ratio (RR) at ambient pressure is determined as \(\rho_{\rm 300K}/\rho_{\rm 1.8K}\approx 11\) for the investigated sample. Under pressure, initially, the anomaly in the resistivity shifts to lower temperatures with increasing pressure while the sharp nature of the anomaly changes to a gradual decrease. The anomaly becomes much weaker and moves to around 12 K at a pressure of 1.7 GPa. Above 1.7 GPa, the feature becomes too small and not traceable within the resolution of our data. However, the resistivity isotherms plotted as a function of pressure (see the inset of Fig. 3a) display clear jumps at the phase boundary. These data suggest that the first-order-like antiferromagnetic transition (AFM1) is suppressed to zero temperature by the application of a pressure of \(p_{c}\approx 1.8\) GPa. In addition, a weak shoulder-like anomaly in \(\rho(T)\) develops around \(T=35\) K which becomes more prominent at higher pressures. This anomaly is better visible in the temperature derivative of the electrical resistivity \(d\rho/dT\) presented in Fig. 3b. With increasing pressure this feature slowly shifts to higher temperatures, reaching about 40 K at \(p=2.52\) GPa, suggesting a phase boundary from the PM phase to a pressure-induced phase. Furthermore, for \(p\geq 1.86\) GPa, two additional features are clearly seen in \(d\rho/dT\) at low temperatures around 20 K and 10 K (marked by \(*\) and \(\#\) symbols, respectively). We note that the residual resistance above \(p_{c}\), in the pressure-induced phase, is considerably larger than that in the AFM1 phase at low pressures.
#### ii.2.3 Magnetic Susceptibility
The results from \(\rho(T)\) measurements are corroborated by ac magnetic susceptibility which further confirm their magnetic origin. The results of the ac magnetic-susceptibility measurements on LuFe\({}_{4}\)Ge\({}_{2}\) for several applied pressures are presented in Fig. 3c. The susceptibility data at each pressure were normalized by the jump height of the superconducting transition of the Pb manometer in order to compare the data taken at different pressures. At \(p=0.1\) GPa, the real part of ac susceptibility (\(\chi^{\prime}\)) shows a sudden decrease at the antiferromagnetic transition. Upon increasing pressure the drop in \(\chi^{\prime}\) shifts to lower temperatures while the sharpness of the jump is reduced. A broad humplike feature develops around 40 K and shows a slight shift to higher temperatures with further increase in pressure. These observations are in excellent agreement with the resistivity data. Above 1.8 GPa, the nature of the \(\chi^{\prime}(T)\) curve is significantly different from that in the low pressure region. In the high pressure region (\(p>1.8\) GPa), an additional humplike feature and a sharp decrease in the susceptibility are observed at about 20 K and 10 K coinciding with the low-temperature anomalies observed in \(d\rho/dT\). The origin and the nature of these anomalies are not fully understood, however, their implication in other physical properties are discussed later.
Figure 3: (a) Electrical resistivity of LuFe\({}_{4}\)Ge\({}_{2}\) as a function of temperature for several applied pressures. Inset: Resistivity isotherms as a function of pressure. The arrows indicate the phase boundary to a pressure-induced phase. (b) Temperature derivative of electrical resistivity \(d\rho/dT\) vs. T for several pressures. Various anomalies are marked by symbols. (c) Temperature dependence of the real part of the ac magnetic susceptibility (\(\chi^{\prime}\)) of LuFe\({}_{4}\)Ge\({}_{2}\) for selected applied pressures. The pressure dependence of the anomalies are traced by the dashed lines and additional low-temperature anomalies at higher pressures are marked by \(*\) and \(\#\) symbols
In order to get a microscopic understanding of the nature of the different magnetic phases of LuFe\({}_{4}\)Ge\({}_{2}\), we have carried out muon-spin relaxation (\(\mu\)SR) and \({}^{57}\)Fe Mossbauer spectroscopy measurements under several pressures below and above \(p_{c}\approx 1.8\) GPa. \(\mu\)SR measurements in a weak transverse-field of \(B_{\rm{TF}}=50\) Oe were performed to determine the magnetic-volume fraction (\(V_{\rm{mag}}\)) and the transition temperatures. In Fig. 4a, \(V_{\rm{mag}}\) of LuFe\({}_{4}\)Ge\({}_{2}\) as a function of temperature for three different pressures is shown. \(V_{\rm{mag}}(T)\) shows a sharp step-like increase at the magnetic ordering temperature for all pressures, a characteristic feature of long-range magnetic order. It is also important to note that the entire sample volume undergoes magnetic ordering at all pressures. The data confirm that the pressure-induced phase is also long-range magnetically ordered. The transition temperature \(T_{N}\) is extracted by fitting the magnetic-volume fraction with the phenomenological equation \(V_{\rm{mag}}=\frac{1}{1+e^{(T-T_{N})/w}}\) where \(w\) is the transition width. Here, the magnetic ordering temperature seems to be nearly independent of pressure and coincide with the weak \(p\)-independent anomalies in the resistivity and susceptibility data. A 100% magnetic volume fraction in both phases is also confirmed by Mossbauer data (see Fig. 4c). Furthermore, the Mossbauer spectrum obtained at \(p=2.9\) GPa, \(T=3\) K, and an external magnetic field of 6 T (see Fig. 4d) suggests that the pressure-induced phase is antiferromagnetically ordered (AFM2). The fit to the spectrum by _Exact Lineshape Site Analysis_ using RECOIL software revealed contributions from Fe moments parallel and antiparallel to the external magnetic field. The fraction of magnetic moments on the Fe parallel to external field is about 66% with a hyperfine magnetic field of \(B_{\rm{hf}}=11.8\) T. This value is higher than that of the 34% of magnetic moments on Fe antiparallel to external field with \(B_{\rm{hf}}=3.6\) T. This is presumably because of the rotation of some part of the moments to the direction of the applied magnetic field. Such an observation is consistent with the metamagnetic transition seen in the magnetoresistance data (see Appendix). In the magnetoresistance (MR) data, the spin reorientation appears to occur at external fields as low as 5 T at high pressures.
Further information regarding the strength of the local magnetic field \(B_{\rm{loc}}\) at the muon site in the magnetically ordered phases is obtained from zero-field (ZF) \(\mu\)SR measurements. The temperature dependence of the muon-precession frequency \(\omega_{\mu}\) for three different pressures is displayed in Fig. 4b. At ambient pressure, the spontaneous precession occurs at \(T\approx 36\) K with a slight enhancement in \(\omega_{\mu}\) upon further decreasing temperature. This sharp step-like behavior of \(\omega_{\mu}(T)\) is consistent with the first-order nature of the phase transition at ambient pressure. At \(p=1.39\) GPa, the precession starts at \(T\approx 34\) K. The frequency gradually increases upon cooling followed by a sudden jump at \(T\approx 26\) K. These features can be well ascribed to the two consecutive phase transitions upon lowering temperature: the first from the paramagnetic to the pressure-induced AFM2 phase at 34 K and the second from the AFM2 to the AFM1
Figure 5: Representative peaks in the X-ray diffraction patterns of LuFe\({}_{4}\)Ge\({}_{2}\) at different temperatures and applied pressures. The structural transition from the tetragonal to the orthorhombic phase is evidenced by the splitting of the diffraction peak around \(2\theta\approx 15.5^{0}\).
Figure 4: (a) Temperature dependence of the magnetic-volume fraction \(V_{\rm{mag}}\) of LuFe\({}_{4}\)Ge\({}_{2}\) obtained using \(\mu\)SR at different pressures. The solid lines are fits to the data using a phenomenological equation provided in the text. (b) Temperature dependence of the muon-spin precession frequency under different applied pressures. (c) \(V_{\rm{mag}}\) obtained from time-domain \({}^{57}\)Fe Mossbauer spectroscopy under several pressures. (b) Energy-domain \({}^{57}\)Fe Mossbauer spectra measured at \(p=2.9\) GPa, \(T=3\) K, and an external magnetic field of 6 T. The solid lines are fits to the spectra by _Exact Lineshape Site Analysis_ using RECOIL software.
phase at 26 K. The gradual increase in \(\omega_{\mu}\) at the first phase transition implies that this transition is of second order-type. The sharp jump in \(\omega_{\mu}\) around 26 K arises from the transition from AFM2 to AFM1 phase and the sharpness points at a first-order-type phase transition. At \(p=2.26\) GPa, the precession starts around 35 K, where the compound undergoes the transition from the PM to the AFM2 phase. Upon further cooling, \(\omega_{\mu}\) increases gradually until the lowest temperature in our experiment. We note that the \(\omega_{\mu}(T)\) curve for \(p=2.26\) GPa shows noticeable slope changes at temperatures around 20 K and 10 K. These features occur at the same temperatures as the low-\(T\) anomalies in the \(d\rho(T)/dT\) and \(\chi^{\prime}(T)\) data discussed earlier and suggest the possibility of multiple order-parameters associated with the phase transition from the paramagnetic to the high pressure AFM2 phase. It is worth noting that, at the lowest temperature, the muon-precession frequencies for the three pressures have similar values. This indicates that, at low temperature, both the AFM1 and the pressure-induced AFM2 phase have similar local magnetic fields at the muon sites.
#### X-ray Diffraction
We now turn to the evolution of the structural transition in LuFe\({}_{4}\)Ge\({}_{2}\) under external pressure. The structural transition from tetragonal to orthorhombic symmetry is characterized by the splitting of certain peaks in the diffraction pattern. For example, the [310] peak splits in the orthorhombic phase while the [200] peak does not. Therefore, the evolution of these peaks with varying conditions of temperature and pressure provides a straight forward determination of the structural transition in the phase diagram. Figure 5 presents the [200] and [310] diffraction peaks obtained at various temperatures and pressures. Remarkably, the splitting of the [310] peak occurs between 30 and 35 K in the entire pressure range. This confirms that the structural transition temperature does not significantly change with pressure and it remains between 30 and 35 K. The lack of a more precise determination of the structural transition temperature is due to the limited temperature sampling available in our experiment.
### Phase Diagram
The temperature-pressure phase diagram of LuFe\({}_{4}\)Ge\({}_{2}\) is established using the results from the different high-pressure studies, see Fig. 6a. The results from bulk measurements as well as from microscopic probes are in excellent agreement. \(T_{N1}\) corresponding to the transition from the PM to the AFM1 phase is continuously suppressed toward zero temperature upon increasing pressure at a critical pressure of \(p_{c}\approx 1.8\) GPa. A second antiferromagnetic phase (AFM2) is confirmed by Mossbauer and \(\mu\)SR measurements. The structural transition temperature which coincide with the AFM1 ordering temperature at ambient pressures does not significantly change with application of pressure and remains at about 35 K, i.e. the AFM1 transition decouples from the structural transition. Moreover, the onset of the magnetic ordering from the paramagnetic (PM) to the AFM2 phase appears to be connected to the structural transition, i.e. its transition temperature \(T_{N2}\) is also almost independent of pressure.
The nature of the transitions between the various phases have been studied in detail. As described earlier, the magneto-structural transition at ambient pressure is of first-order type. The application of pressure decouples the \(T_{N1}\) and the structural transition. Figure 6b depicts heat-capacity data (measured using an ac-calorimetry technique) taken upon heating and cooling cycles under selected pressures of 0.29 and 0.76 GPa. The peak in \(C(T)\) associated with the transition from the AFM2 to the AFM1 phase shows a thermal hysteresis, confirming the first-order character of the transition. We note that the thermal hysteresis was measured with a very slow temperature sweep-rate in order to minimize the effect of the relatively slow thermalization of the pres
Figure 6: (a) \(T-p\) phase diagram of LuFe\({}_{4}\)Ge\({}_{2}\). The transition temperatures obtained from \(\rho(T)\) (circle), \(\chi^{\prime}(T)\) (square), \(\mu\)SR (star), and Mössbauer (sphere) data are indicated. The \(*\) and \(\#\) symbols stand for the low-temperature anomalies observed in \(\rho(T)\) and \(\chi^{\prime}(T)\) data. (b) Thermal hysteresis in \(C(T)\), measured using an ac-calorimetry technique under pressure, confirms the first-order nature of the transition at \(T_{N1}\). The arrow points at a very weak feature at \(T_{N2}\) (c) Magnetoresistance MR(\(B)=[\rho(B)-\rho(B=0)]/\rho(B=0)\) of LuFe\({}_{4}\)Ge\({}_{2}\) measured at \(T=2\) K.
sure cell. A similar hysteresis is also observed in resistivity and ac magnetic-susceptibility data (not shown). The first-order nature of the transition is also consistent with the sharp jump in \(\omega_{\mu}\) at this phase boundary. The transition from the PM to the AFM2 phase appears as very weak features in \(C(T)\), \(\rho(T)\), and \(\chi^{\prime}(T)\) without any observable thermal hysteresis. Moreover, \(\omega_{\mu}\) shows a gradual increase at this phase boundary, confirming that the transition from the PM to the AFM2 phase is of second order.
The magnetoresistance and the magnetic-susceptibility data provide insights into the relative strength of magnetic fluctuations in the two antiferromagnetic phases of LuFe\({}_{4}\)Ge\({}_{2}\). In the AFM1 phase, the MR = \([\rho(B)-\rho(B=0)]/\rho(B=0)\) exhibits a monotonous increase with a quadratic dependence on the magnetic field (see Fig. 6c). This is the typical behavior expected for metallic systems and is due to the cyclotron motion of the conduction electrons in the transverse magnetic field. However, in the AFM2 phase, MR(\(B\)) shows an opposite behavior displaying negative values. These observations can be taken as indication for stronger magnetic fluctuations in the AFM2 phase compared to the AFM1 phase. The negative MR stems from the reduction in the resistivity as the external magnetic field suppresses the magnetic fluctuations. The small reduction in the \(\chi^{\prime}\) at the magnetic ordering at the phase boundary to AFM2 phase compared to the large drop in \(\chi^{\prime}\) at the ordering at the phase boundary to AFM1 phase is also in support of the existence of strong magnetic fluctuations in AFM2 phase. Moreover, the gradual increase in the muon-precession frequency, which is a measure of the local magnetic field, in the AFM2 phase upon deceasing temperature suggests a strong temperature dependence of the magnetic fluctuations.
### Electronic structure calculations
Electronic-structure calculations for LuFe\({}_{4}\)Ge\({}_{2}\) based on density functional theory (DFT) have been performed to understand its magnetic and structural phase transitions. The magnetic properties are determined by the dominating contribution of the Fe \(3d\) related rather narrow bands near the Fermi level \(E_{F}\), resulting in a pronounced peak in the density of states (DOS) near \(E_{F}\) (see Fig. 7a). With a value of about 3.5 states (per Fe and eV) at the Fermi level, the Stoner criterion is fulfilled, evidencing a magnetic instability and the tendency for spontaneous magnetic ordering. The magnetic and structural transition to a collinear AFM state leads to a strong reduction of the DOS at \(E_{F}\) (compare Fig. 7 a and b). To a large part, this drop in the DOS(\(E_{F}\)) is caused by the band splitting related to the localized Fe \(3d\) moments and thus similar for different magnetic structures.
A full relaxation of the crystal structure (at the experimental low-temperature volume) yields that the orthorhombic symmetry is energetically more stable than the tetragonal. Interestingly, the resulting equilibrium distortion \(2(a-b)/(a+b)\) of the lattice is strongly dependent on the size of the magnetic moment, essentially independent from the choice of the exchange correlation potential (see Fig. 7c). This strong and non-linear dependence indicates that the magnetism and the crystal lattice are coupled in a nontrivial way. The energy dependence of the DOS for both the tetragonal and the orthorhombic experimental crystal structures _without_ spin polarization reveals that the magnetic ordering can occur independent of the structural transition since for both symmetries we obtain a similarly large DOS near the Fermi level. Therefore, the reason for the magnetic ordering occurring simultaneously with the structural transition might not be directly related to the electronic band structure but rather due to the change in the strength of the magnetic frustration due to the lattice distortion. High-field magnetization measurements on LuFe\({}_{4}\)Ge\({}_{2}\) showed that the magnetization remains small
Figure 7: Total and partial density of states of LuFe\({}_{4}\)Ge\({}_{2}\) for (a) the non-magnetic tetragonal and (b) the antiferromagnetic orthorhombic case. The Fermi level is at zero energy. Panel (c) shows the dependence of the lattice distortion (in %) on the Fe magnetic moment for different exchange correlation potentials.
up to high fields and the extrapolated saturation field is more than 150 T [16]. This corresponds to strong antiferromagnetic exchange interactions on the scale of more than 100 K. The strong exchange interaction along with a relatively low ordering temperature points at the significance of the magnetic frustration. It is likely that the magnetic frustration, at least to some extent, is released by the distortion of the Fe tetrahedra during the structural transition from the tetragonal to the orthorhombic phase and, thereby, facilitating the magnetic ordering. This scenario is supported by the calculations which predict the structural transition even without involving magnetic polarization. Here the large magnetic entropy connected with a fluctuating frustrated paramagnetic system, which stabilizes the tetragonal phase upon increasing temperature, might explain why the tetragonal phase is stable down to such comparatively low temperature. Our results suggest a mechanism where the magnetic and structural order parameters in LuFe\({}_{4}\)Ge\({}_{2}\) are linked by the magnetic frustration causing the simultaneous magneto-structural transition.
The commensurate magnetic ground state proposed by Papamantellos _et al._[14] is also reproduced by our calculations. Moreover, a collinear spin structure is found to be energetically close to the non-collinear ground state. This points at the possibility that the pressure-induced AFM2 phase could be a spin rearrangement from non-collinear to collinear structure. Such a subtle spin rearrangement is also consistent with the similar values of low-temperature muon-precession frequency found in AFM1 and AFM2 phases.
## III Conclusion
In summary, LuFe\({}_{4}\)Ge\({}_{2}\) presents an interesting interplay of magnetic, structural, and electronic degrees of freedoms. At ambient pressure LuFe\({}_{4}\)Ge\({}_{2}\) undergoes a simultaneous antiferromagnetic and structural transition at 36 K with first-order character. The pressure dependence of the magnetic transition in LuFe\({}_{4}\)Ge\({}_{2}\) has been investigated using electrical-transport, ac magnetic-susceptibility, ac calorimetry, Mossbauer, \(\mu\)SR, and PXRD measurements. External pressure suppresses the first-order magnetic transition (AFM1) to zero temperature around 1.8 GPa. The structural transition is largely unaffected by pressure. A new antiferromagnetic phase (AFM2) is observed at higher pressures, confirmed by Mossbauer and \(\mu\)SR measurements. The transition from the paramagnetic to the AFM2 phase is of second order and appears to be connected to the structural transition. \(\mu\)SR and Mossbauer data revealed 100% magnetic volume fraction in both the magnetic phases. In addition, similar values of muon-precession frequency at low temperatures in the AFM1 and AFM2 phases point at similar ordered moments and closely related magnetic structures in the two phases. Our results also indicate enhanced magnetic fluctuations in the pressure-induced AFM2 phase. The experimental observations are supported by DFT band-structure calculations, suggesting a scenario where the magnetic and structural order parameters in LuFe\({}_{4}\)Ge\({}_{2}\) are linked by magnetic frustration, causing the simultaneous magneto-structural transition. Our results reveal an interesting and unusual interplay of structure and magnetism in LuFe\({}_{4}\)Ge\({}_{2}\), which differ from the situation observed in the \(A\)Fe\({}_{2}\)As\({}_{2}\) pnictides. Therefore, LuFe\({}_{4}\)Ge\({}_{2}\) is an attractive system where further in-depth studies could provide deeper insight into the interaction between frustrated magnetism and structural instability, a topic of general interest which is also relevant for other classes of quantum materials.
## IV Methods
Polycrystalline samples of LuFe\({}_{4}\)Ge\({}_{2}\) were synthesized by a standard arc-melting technique on a copper hearth. Constituent elements (at least 99.9% purity), taken in the stoichiometric ratio, were melted in an arc furnace under argon atmosphere, followed by several flipping and remelting of the resulting ingot to ensure homogeneity. Then the as-cast samples were annealed at 1150\({}^{0}\)C under a static argon atmosphere for a week. The phase purity of the annealed samples was checked by powder x-ray diffraction (PXRD) using Cu K\(\alpha\) radiation and a scanning electron micrograph (SEM), which revealed only a small amount (up to 4%) of eutectic phase Fe\({}_{3}\)Ge in our samples. The stoichiometry of the samples was verified using energy dispersive x-ray (EDX) analysis.
DC magnetic susceptibility was measured using a superconducting quantum interference device (SQUID) magnetometer (magnetic properties measurement system, Quantum Design). The heat capacity at ambient pressure was recorded by a thermal-relaxation method using a physical property measurement system (PPMS, Quantum Design). Electrical-transport, ac magnetic-susceptibility, and ac-calorimetry measurements under hydrostatic pressure were performed using a double-layered piston-cylinder-type pressure cell with silicon oil as the pressure-transmitting medium. The pressure inside the sample space was determined at low temperatures by the shift of the superconducting transition temperature of a piece of Pb. The electrical resistivity was measured using a standard four-terminal method, where electrical contacts to the sample were made using 25 \(\mu\)m gold wires and silver paint. Resistivity was measured using an LR700 resistance bridge (Linear Research). AC magnetic susceptibility was measured using home-made ac-susceptometer that fits inside the pressure cell. The signal was measured using an LR700 mutual-inductance bridge (Linear Research). A static field \(B_{\mathrm{dc}}\) of 0.01 T and a modulation field \(B_{\mathrm{ac}}\) of 1.3 mT at a frequency of 16 Hz were used for the measurements. AC calorimetry measurements were performed using a commercial heater and Cernox thermometer following the method described in Ref. [17].
Muon-spin relaxation measurements under pressure were performed at the \(\mu\)E1 beam-line using GPD spectrometer at the Paul Scherrer Institute (PSI), Switzerland (see Ref. [18; 19] for details of the high-pressure \(\mu\)SR technique at PSI). Synchrotron \({}^{57}\)Fe Mossbauer spectroscopy measurements under pressure were conducted at the Nuclear Resonance beamline (ID18) of the European Synchrotron Radiation Facility (ESRF), Grenoble [20] and at the beamline 3ID-B of the Advanced Photon Source, Argonne National Laboratory, USA [21; 22]. X-ray diffraction experiments using a diamond-anvil cell for generating pressure were performed at the XDS beamline at the Brazilian Synchrotron facility LNLS [23].
DFT calculations were performed using the plane-wave pseudopotential method implemented in the Vienna ab-initio simulation package (VASP) [24], applying the local density approximation (LDA) [25] and the general gradient approximation (GGA) [26] for the exchange-correlation functional. We use a plane-wave energy cutoff of 500 eV and a Regular Monkhorst-Pack grid of \(8\times 8\times 12\) to perform the ionic relaxation and \(10\times 10\times 14\) to achieve the self-consistent calculations. To obtain the density of states, the \(k\)-mesh was increased to \(22\times 22\times 24\) using the tetrahedron method. The optimization of the structures was carried out with a force convergence tolerance of 1 meV/A per atom. The tetragonal (T) and orthorhombic (O) structures have \(P4_{2}/mmm\) and \(Pnnm\) space group symmetry, respectively. In order to study the FM and various AFM magnetic states and respective structural distortion of the orthorhombic structure, we carried out collinear and non-collinear calculations. To perform our studies, we have considered the structural parameters given in this study at 10 K and ambient pressure and the structural data given in the previous work [14].
###### Acknowledgements.
This work was partly supported by Deutsche Forschungsgemeinschaft (DFG) through Research Training Group GRK 1621. U. Nitzsche is acknowledged for technical support for DFT calculations. RDdR acknowledges financial support from the Sao Paulo Research Foundation (FAPESP) (Grant 2018/00823-0) and from the Max Planck Society under the auspices of the Max Planck Partner Group R. D. dos Reis of the MPI for Chemical Physics of Solids, Dresden, Germany.
## Appendix: Field-induced metamagnetic transition
The transverse magnetoresistance MR(\(B\)) = \([\rho(B)-\rho(0)]/\rho(0)\) of LuFe\({}_{4}\)Ge\({}_{2}\) as a function of magnetic field measured at different pressures at \(T=2\) K is shown in Fig. 8a. MR(\(B\)) presents evidence for a field-induced metamagnetic transition under pressure; a tiny kink in MR(\(B\)) at about 5 T at pressures starting from 1.5 GPa (see inset of Fig. 8a). This feature becomes much pronounced upon increasing pressure and continuously shifts to lower fields. Eventually, a strong step-like decrease in the MR is observed at higher pressures with MR reaching \(-35\%\) at 7 T for \(p=2.35\) GPa. We note that high-field magnetization measurements on LuFe\({}_{4}\)Ge\({}_{2}\) at ambient pressure revealed a weak metamagnetic transition at an applied magnetic field of 47 T, yet without the tendency of saturation of the magnetization [16]. Such a metamagnetic transition could be related to a spin reorientation under the influence of applied magnetic field. This is also corroborated by the Mossbauer measurements at high pressure and a magnetic field of 6 T, where some part of the Fe moments seem to reorient in the direction of the applied field. It is also interesting to notice that the normalized resistivity \(\rho/\rho_{300\text{K}}\) at higher magnetic fields appears to have similar values for all applied pressures, as displayed in Fig. 8b. This suggests that the field-polarized phase could be the same in the entire pressure range, once again indicating the close similarity between the AFM1 and AFM2 phases.
Figure 8: (a) Magnetoresistance MR(\(B\)) of LuFe\({}_{4}\)Ge\({}_{2}\) measured at \(T=2\) K for different pressures. The inset shows an enlarged view of the low-pressure curves. (b) Normalized resistivity \(\rho/\rho_{300\text{K}}\) as a function of field at 2 K for several pressures. |
2306.01449 | SASMU: boost the performance of generalized recognition model using
synthetic face dataset | Nowadays, deploying a robust face recognition product becomes easy with the
development of face recognition techniques for decades. Not only profile image
verification but also the state-of-the-art method can handle the in-the-wild
image almost perfectly. However, the concern of privacy issues raise rapidly
since mainstream research results are powered by tons of web-crawled data,
which faces the privacy invasion issue. The community tries to escape this
predicament completely by training the face recognition model with synthetic
data but faces severe domain gap issues, which still need to access real images
and identity labels to fine-tune the model. In this paper, we propose SASMU, a
simple, novel, and effective method for face recognition using a synthetic
dataset. Our proposed method consists of spatial data augmentation (SA) and
spectrum mixup (SMU). We first analyze the existing synthetic datasets for
developing a face recognition system. Then, we reveal that heavy data
augmentation is helpful for boosting performance when using synthetic data. By
analyzing the previous frequency mixup studies, we proposed a novel method for
domain generalization. Extensive experimental results have demonstrated the
effectiveness of SASMU, achieving state-of-the-art performance on several
common benchmarks, such as LFW, AgeDB-30, CA-LFW, CFP-FP, and CP-LFW. | Chia-Chun Chung, Pei-Chun Chang, Yong-Sheng Chen, HaoYuan He, Chinson Yeh | 2023-06-02T11:11:00Z | http://arxiv.org/abs/2306.01449v1 | # SASMU: boost the performance of generalized recognition model using synthetic face dataset
###### Abstract
Nowadays, deploying a robust face recognition product becomes easy with the development of face recognition techniques for decades. Not only profile image verification but also the state-of-the-art method can handle the in-the-wild image almost perfectly. However, the concern of privacy issues raise rapidly since mainstream research results are powered by tons of web-crawled data, which faces the privacy invasion issue. The community tries to escape this predicament completely by training the face recognition model with synthetic data but faces severe domain gap issues, which still need to access real images and identity labels to fine-tune the model. In this paper, we propose SASMU, a simple, novel, and effective method for face recognition using a synthetic dataset. Our proposed method consists of spatial data augmentation (SA) and spectrum mixup (SMU). We first analyze the existing synthetic datasets for developing a face recognition system. Then, we reveal that heavy data augmentation is helpful for boosting performance when using synthetic data. By analyzing the previous frequency mixup studies, we proposed a novel method for domain generalization. Extensive experimental results have demonstrated the effectiveness of SASMU, achieving state-of-the-art performance on several common benchmarks, such as LFW, AgeDB-30, CA-LFW, CFP-FP, and CP-LFW.
## 1 Introduction
With the developments of deep learning techniques, state-of-the-art face recognition methods [33, 34, 6, 31, 7, 1] advance the performance to a great extent, such as over 99.5% validation accuracy on Labeled Faces in the Wild (LFW) [11] dataset and 97.70% TAR@FAR=1e-4 on IJB-C [23] dataset. Beyond these successes, researchers also extend the potentials of modern face recognition techniques to some special applications, such as the face with facial mask [50, 5] and under the near-infrared light scenario [16, 35, 25]. However, these methods are mostly trained with web-crawled datasets, such as MS1M [10], CASIA-Webface [46], and WebFace260M [51]. Several issues remain challenging, as listed below:
* Privacy issue: It is extremely difficult to collect consents from all enrolled participants, particularly for huge datasets, such as Webface260M [51] containing four million people and over 260 million face images.
* Long-tailed distribution: Large differences exist in datasets in terms of the numbers of images, poses, and expressions per person.
* Image quality: It is difficult to maintain the same quality level for each image in a large dataset.
* Noisy label: Web-crawled image dataset faces the issue of noisy labels when social networks automatically label face images among users and incorrect labeling may occur from time to time.
* Lack of attribute annotations: Detailed annotations for facial attributes, such as pose, age, expression, and lighting, are usually not available.
The most critical challenge is the privacy issue, which we define as whether to use recognizable information or not. To refrain from privacy invasion, unrecognizable noises or random-region masks can be added to face images [36], but the risk of leakage of real and distinguishable face images remains high. To solve the privacy issue once and for all, using synthetic data to train the face recognition model is a good practice. Thanks to the development of the generative model and computer graphics, we could generate realistic images by using computing resources [8, 2]. However, the domain gap is unavoidable, and the previous works [28, 2] access the real image and label to close the domain gap
which still violates privacy-preserving. In this work, our main contributions are summarized as followed:
* Analyzing the impacts and potentials of spatial data augmentation (SA), and showing the analytical results of possible options, such as grayscale, perspective operation, etc.
* Applying the spectrum mixup (**SMU**) to minimize the synthetic-to-real domain gap without using real face images for training.
* Achieving the state-of-the-art performance of face recognition without using any recognizable information.
## 2 Related works
Recent advancements in face recognition research have focused on improving the accuracy and efficiency of deep learning models. One important research area is to investigate various architectures, such as attention mechanisms [19, 41, 39, 38, 40, 20] and multi-task learning [47, 12, 27], to improve model capability in accommodation with variations in facial expressions, poses, and lighting conditions. Most importantly, the design of loss functions [34, 33, 18, 6, 7, 31, 24, 32, 17] and open source large dataset [46, 10, 37, 15, 51] have been drawing much attentions in this research field. Additionally, researchers have been investigating privacy-preserving face recognition techniques to protect individuals' privacy by not storing their raw face images in the system. Instead, they have developed methods to generate privacy-preserving representations for the faces that can be used for person identification.
Recently, researchers have explored the use of synthetic data and applied data augmentation techniques to improve the model generalization capability to compute unseen data. Promising results have been reported in reducing bias and improving the recognition accuracy on diverse datasets. However, there are still certain cases where noticeable gaps exist between real and synthetic images, especially in the frequency domain. For instance, the artifact on synthetic data can be revealed by using the frequency spectrum analysis and can be recognized by a simple classifier on spectrum [14, 9]. Therefore, reducing the domain gap between real and synthetic datasets can lead to more robust and effective recognition systems trained on synthetic data, which can have significant practical applications in areas such as security, surveillance, and biometrics.
To sum up, these recent advancements have the potential to improve the accuracy, efficiency, and fairness of face recognition systems, making them more suitable for real-world applications.
In this section, we first introduce deep face recognition using margin-based softmax loss functions. We then explore the performance gap between the models trained on synthetic and real datasets (SynFace and RealFace). Lastly, we introduce 1) identity mixup to enlarge the intra-class variations and 2) domain mixup to mitigate the domain gap between synthetic and real face images.
### Training of face recognition
Training of face recognition system is to teach a computer vision model to recognize and identify human faces in images or videos. The training process involves feeding the model with a large amount of labeled face data, that is, images or videos of people's faces along with corresponding identity labels. The model then learns to extract unique features from each face and uses those features to differentiate one's face from another.
During training, parameters of the model are determined through various techniques such as deep learning algorithms, convolutional neural networks (CNNs), and transfer learning to improve its accuracy while reducing errors in identifying faces. The accuracy of the face recognition model is evaluated with a separate dataset that is unseen to the model.
The ultimate goal of face recognition training is to develop a robust and reliable system that can accurately recognize and identify faces in real-world scenarios, such as security systems, access control, and biometric authentication. In the past few years, researchers mainly focus on the loss design for face recognition training.
#### 2.1.1 Losses for face recognition
In general, face recognition methods adopt unified (hybrid) loss functions, which combine three losses involving specific constraints on the angles between the weight vectors of different classes, and govern the distance metrics among them.
Let \(x_{i}\) be the input feature vector under class \(i\), \(y_{i}\) be the ground truth label of class \(i\), and \(W\) be the learnable weight matrix of the loss function with size of \(C\times D\), where \(C\) is the number of classes, and \(D\) is the feature dimension. The joint loss function can then be written as:
\[\begin{split} L&=-\frac{1}{N}\sum_{i=1}^{N}\log \frac{e^{s\cdot\delta}}{e^{s\cdot\delta}+\sum_{j\neq\gamma_{i}}e^{s\cdot\cos( \theta_{j})}}\\ \delta&=\cos(m_{1}\cdot\theta_{j}+m_{2})-m_{3}\:, \end{split} \tag{1}\]
where \(N\) is the batch size, \(m_{i\in{1,2,3}}\) is the margin parameter, \(s\) is the scale factor, and \(\theta_{j}\) indicates the angle between the weight \(W_{j}\) and the feature \(x_{i}\). In detail, sphereface [22], arcface [6], and cosface [34] have the parameters \((m_{1},0,0)\), \((1,m_{2},0)\), and \((1,0,m_{3})\), respectively. By using this joint loss function, we can train the network to learn discriminative features that are well-separated in the embedding space,
while also minimizing the computational overhead of multiple loss functions.
Recent research studies begin to investigate other designs of loss function, such as those considering the image quality effect [24, 32, 17]. In this paper, we conduct experiments on the SOTA method, Adaface [17], which proposed the adaptive margin function by approximating the image quality with feature norms. Their margins can be written as:
\[\begin{split} m_{1}^{Adaface}&=1\\ m_{2}^{Adaface}&=-m_{2}\cdot\widehat{\|z_{i}\|}\\ m_{3}^{Adaface}&=m_{3}\cdot\widehat{\|z_{i}\|}+m_ {3}\\ \widehat{\|z_{i}\|}&=\lfloor\frac{\|x_{i}\|-\mu_{z} }{\sigma_{z}/h}\rceil\,,\end{split} \tag{2}\]
where \(\|x_{i}\|\) means the norm of the feature vector \(x_{i}\), \(h\) is a constant, \(\mu_{z}\) and \(\sigma_{z}\) are the mean and standard deviation of all \(\|x_{i}\|\) within a batch.
### Privacy-preserving face recognition
Privacy-preserving face recognition is an emerging area of research. One approach is to use Masked Autoencoders in FaceMAE [36], where face privacy and recognition performance are considered simultaneously. Alternatively, learnable privacy budgets in the frequency domain [13] can be used. The other approach is to use differential privacy to convert the original image into a projection on eigenfaces and to add noises for better privacy. Specifically, differential privacy works by adding random noises to the data or query results in a way that guarantees whether the presence or absence of any individual in the dataset does not significantly affect the outcome. This means that an observer cannot determine whether any individual's data is included in the dataset or not. The amount of noise added is calibrated based on a privacy parameter called epsilon \(\epsilon\), which determines the level of privacy protection. A smaller value of \(\varepsilon\) provides stronger privacy protection but may result in less accurate query results. Differential privacy offers a theoretical guarantee of privacy [4]. However, the methods above still need to access real information such as RGB images and related identity information. To refrain from any privacy invasion, we need to use synthetic data and avoid any privacy information in the training pipeline. Synface [28] use the pre-trained GAN [8] to generate massive of synthetic images to decrease the needy of the real image, which only uses 1 over 10 real image [46] for domain mixup in the training pipeline.
## 3 Methods
To improve the accuracy of face recognition network, in this study, we propose a spatial augmentation and spectrum mixup (SASMU) module which operates in the spatial and frequency domains, respectively. We first describe how the synthetic faces are controlled, rendered, and aligned to prepare the dataset (Section 3.2). After providing the dataset statistics (Section 3.1), we introduce the proposed spectrum mixup method for minimizing the synthetic-to-real domain gap (Section 3.3).
### Dataset statistics
In this paper, we conduct our study upon the setting of SOTA face recognition with synthetic image data, Synface [3] and Digiface-1M [2]. These two works develop their face recognition model on two different types of datasets, in which the former one use the generated data from GAN and the later one create the images using the traditional simulation and rendering pipeline. Specifically, Synface [3] uses the pretrained DiscofaceGAN [8] to generate the facial images. DiscoFaceGAN [8] can generate realistic and diverse face images of virtual people with disentangled and controllable features. It uses 3D priors to learn latent representations for identity, expression, pose, and illumination, and then synthesizes face images by imitating a 3D face deformation and rendering process. DiscoFaceGAN [8] can produce high-quality face images with fine-grained control over each feature. Despite the fast synthetic images production, DiscofaceGAN [8] suffers from identity consistency, especially in large poses or severe environment conditions shown in Fig. 1. In contrast, Digiface-1M [2] leverage the well-developed 3D CG technique to render the image. They first create the synthetic 3D model of a person then follow a predefined instructions to obtain a set of synthesized face images. As shown in Fig. 1, Digiface-1M well produces a variety of face images for the same identity with given context settings. In the following experiment, we also show that the lack of identity consistency decreases the performance of face recognition.
### Data augmentation
Data augmentation is widely used in the vision tasks [30] to extend the amount and variety of training dataset. Starting from light augmentation, such as cropping, rescaling, and photometric jittering, to heavy augmentation including but not limited to random affine, random masking, and warping, performance improvements can be achieved by adopting data augmentation in various vision systems. In the face recognition research field, however, we surprisingly found that there is almost no additional data augmentation applied in the previous methods. One intuitive reason might be assocaited with the increasing amount of datasets [51], which could meet the requirement of scale issues for model training. In practice, we found that only slight data augmentation, for example, the applied probability of grayscale, \(p_{grayscale}=0.05\), is beneficial for using Adaface loss [17]
on real image dataset [46]. However, the performance decreases rapidly when \(p_{grayscale}\) becomes larger.
In contrast with the diversity of real images, synthetic images are usually monotonous. For example, the face in the real image could be occluded, over-exposed, and distorted. It is hard for GAN model [8] to mimic those artifacts without changing the identities. As for traditional rendering pipeline, the current technique [2] could generate various face images for the same identity, at the expense of high costs of hardware, software, and computational time.
In our study, the data augmentation is beneficial for using synthetic data. Taking Synface training method as an example, adding random erase (RE) with probability \(p_{RE}=1\) can increase the testing performance on LFW [11] by more than \(1\%\). It is worthwhile to investigate the data augmentation when using synthetic image to train the face recognition model.
We split the data augmentation into two groups, appearance-based and geometry-based methods. While keeping the structure of the face, the appearance-based method only changes the color tonality, such as grayscale, Gaussian noise, blur, salt and pepper, channel shuffle, equalization, and auto contrast. The geometry-based method changes the structure of face images, such as crop, flip, affine, and perspective.
### The proposed spectrum mixup (SMU)
The main objective of this study is to develop a privacy-preserving face recognition model by training it on a synthetic dataset. To this end, we propose a novel data augmentation technique called Spectrum Mixup (SMU) that reduces the domain gap between real and synthetic datasets by immigrating information from the amplitude spectrum of real images. In contrast to other mixup strategies in the frequency domain, as shown in Fig. 3, that use weighted sum operation or hard-assignment mask, we integrate the amplitude components of synthetic data and the amplitude components of real data using a Gaussian-based soft-assignment map, and enhance high-frequency information, as illustrated in Fig. 2. Our approach is based on the following hypotheses: 1) semantic content (identity information) is mainly encoded in the phase components; 2) incorporating amplitude information from real data into synthetic data results in a better fit to the distribution of the real dataset; and 3) enhancing high-frequency information is more effective than low-frequency information since deep neural networks prioritize fitting certain frequencies [43], usually from low to high, which indicates that synthetic data carry realistic low-frequency information but lack high-frequency details.
To obtain the frequency components of an image*\(\mathbf{x}\in\mathbb{R}^{M\times N}\), we use the 2D discrete Fourier transform, which can be expressed as:
Footnote *: For simplicity, the single-channel image is used to illustrate the procedure of discrete Fourier transform, while the extension to color images is straightforward by processing on each channel separately using the same way.
\[\mathcal{F}(\mathbf{x})(u,v)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}x(m,n)e^{-j2\pi(\frac {um}{M}+\frac{vm}{N})}, \tag{3}\]
where \((m,n)\) denotes the coordinate of an image pixel in the spatial domain; \(x(m,n)\) is the pixel value; \((u,v)\) represents the coordinate of a spatial frequency in frequency domain; \(\mathcal{F}(\mathbf{x})(u,v)\) is the complex frequency value of image \(\mathbf{x}\); \(e\) and \(j\) are Euler's number and the imaginary unit, respectively. Accordingly, \(\mathcal{F}^{-1}(\cdot)\) is the 2D inverse discrete Fourier transform which converts frequency spectrum
Figure 1: Comparison of DiscofaceGAN [8] and Digface1M [2] datasets. For each dataset, the images of the first and second rows are samples from two different identities. Obviously, DiscofaceGAN [8] fails to maintain identity consistency among generated face images, whereas Digface1M [2] performs great in generating various face images for the same identity.
Figure 2: The proposed spectrum mixup (SMU) module.
to spatial domain. Following Euler's formula:
\[e^{j\theta}=\cos(\theta)+j\sin(\theta), \tag{4}\]
the natural exponential function in Eq. (3) can be rewritten as:
\[e^{-j2\pi(\frac{um}{M}+\frac{vm}{N})}\!=\!\cos 2\pi\Big{(}\frac{um}{M}\!+\! \frac{vn}{N}\Big{)}\!-\!j\sin 2\pi\Big{(}\frac{um}{M}\!+\!\frac{vn}{N}\Big{)}\,. \tag{5}\]
According to Eq. (3) and Eq. (5), the image is decomposed into orthogonal sine and cosine functions which constitute the imaginary and real part of the frequency component \(\mathcal{F}(\textbf{{x}})\), respectively. Then, the amplitude and phase spectra of \(\mathcal{F}(\textbf{{x}})(u,v)\) are defined as:
\[\mathcal{A}(\textbf{{x}})(u,v)=\left(R^{2}(\textbf{{x}})(u,v)+I^{2}(\textbf{{ x}})(u,v)\right)^{1/2}, \tag{6}\]
\[\mathcal{P}(\textbf{{x}})(u,v)=\arctan\left(\frac{I(\textbf{{x}})(u,v)}{R( \textbf{{x}})(u,v)}\right), \tag{7}\]
where \(R(\textbf{{x}})\) and \(I(\textbf{{x}})\) represent the real part and imaginary part of \(\mathcal{F}(\textbf{{x}})\), respectively.
Furthermore, to implement our SMU method, we use a Gaussian kernel to create a soft-assignment map, denoted as \(G\). The soft-assignment map is defined as follows:
\[\textbf{{G}}(u,v)=e^{-D^{2}(u,v)/2D_{0}^{2}}, \tag{8}\]
where \(D_{0}\) is a positive constant that represents the cut-off frequency, and \(D_{0}^{2}\) is the distance between a point \((u,v)\) in the frequency domain and the center of the frequency rectangle, that is,
\[D(u,v)=\left((u-M/2)^{2}+(v-N/2)^{2}\right)^{1/2}, \tag{9}\]
where \(M\) and \(N\) represent the height and width of the frequency rectangle and image, respectively.
The SMU procedure for two randomly sampled images \(\textbf{{x}}_{syn}\) and \(\textbf{{x}}_{real}\), can be formalized as follows:
\[\textbf{{x}}_{syn}^{\prime}=\mathcal{F}^{-1}((\textbf{{1}}-\textbf{{G}}) \circ\mathcal{A}(\textbf{{x}}_{real})+\textbf{{G}}\circ\mathcal{A}(\textbf{{ x}}_{syn}),\mathcal{P}(\textbf{{x}}_{syn})), \tag{10}\]
where \(\circ\) denotes the element-wise multiplication operation. We maintain the low-frequency information of synthetic data and immigrate high-frequency details from the amplitude components of the real image. The resulting amplitude components are then combined with the phase components of \(\textbf{{x}}_{syn}\) to obtain the final augmented synthetic image \(\textbf{{x}}_{syn}^{\prime}\).
To summarize, the SMU procedure uses a soft-assignment map to combine the low-frequency components of the synthetic image with the high-frequency components of the real image, resulting in a more realistic augmented synthetic image. Note that, the SMU method only uses the amplitude spectra of the real images to obtain high-frequency components, while the labels or identity information of the real images is not used during the training process. It means that the method can be applied to any type of image dataset without the need for manual annotation or labeling, making it a useful tool for various applications in computer vision.
## 4 Experiments
### Experiment setup and evaluation protocol
In this paper, LResNet50E-IR model [6] is used as our backbone for face recognition. We conduct all experiments using the learning strategies and experimental hyperparameters of the state-of-the-art methods [2, 28] on 4 NVIDIA V100 GPUs. The batch size is 256, and the number of
Figure 3: Comparisons of different mixup strategies in frequency domain. Yang et al. [44] replaced the amplitude spectrum of the source image directly with the amplitude spectrum of the target image. Yang et al. [45] applied a frequency mask to swap the low-frequency components of the source amplitude spectrum. Xu et al. [42] integrated two amplitude spectra by using a weighted sum operation. Liu et al. [21] retained the high-frequency components of the source amplitude spectrum and combined the low-frequency components of the source amplitude spectrum with those of the target image.
epochs is 40. The initial learning rate is 0.1, and it is divided by 10 at 24-th, 30-th, and 36-th epoch. The loss function is Adaface loss [17], and SGD optimizer is used to train models. For a fair and steady comparison, our model is trained from scratch and no early stop strategy is used in our experiments. The trained model at the last epoch is used to conduct inference on the testing datasets.
### Datasets
In this study, the Digiface1M (synthetic) dataset [2] and the CASIA-WebFace (real) dataset [46] are used to train our face recognition model. For the Digiface1M dataset, there are 720k face images with a resolution of \(256\times 256\) in total, in which each identity consists of 72 images. For the CASIA-WebFace dataset, it consists of 528k images with a resolution of \(112\times 112\) preprocessed by [6] for 10575 identities. Note that only face images from the CASIA-WebFace dataset are used in the proposed SMU module; therefore, the identity information is not adopted in our training stage. For preprocessing, we randomly select 200 identities with 50 face images as our real image data on the CASIA-WebFace, and crop all of the images into a size of \(112\times 112\).
To evaluate the recognition performance, we test on five common face verification benchmarks, including LFW [11], AgeDB-30 [26], CALFW [49], CFP-FP [29], and CP-LFW [48] datasets, and compute verification accuracy for evaluation. The LFW is one of the most common benchmark datasets which contains 6000 pairs of images collected in-the-wild. The AgeDB-30 and CALFW aim to test the model performance under large age variation. The CFP-FP and CP-LFW datasets are used to evaluate recognition ability with large pose variation.
### Choice of synthetic dataset
Recently, a study [51] reports that a higher number of identities and image samples can improve recognition performance with diverse and robust embedding. Bae et al. [2] demonstrate that the robustness and efficiency of their pipeline can generate face images with the same identity in any conditions, and reveal a high correlation between image number and performance. Also, the experimental results of several studies [28, 8] suggest a similar conclusion to the above studies. However, there are significant biases between face images when considering the consistency of intra-identity images. These biases lead to difficulty recognizing whether they are the same identity even if generated by the same identity information, as shown in Fig. 1. We further conduct experiments to investigate the impact of sample numbers for each identity. As shown in Table 1, higher sample numbers with identity inconsistency affect the recognition performance, gradually. In this way, we choose the Digiface1M as our training dataset to develop a face recognition system.
### Ablation studies
#### 4.4.1 SA selection
In this study, we conduct several experiments to find the best combination of data augmentations for performance improvement. All details of the data augmentation setting are reported in our supplementary document. Finally, our combination of data augmentations consists of low-resolution operation, random cropping operation, photometric augmentation, grayscale operation, and perspective operations. As shown in Table 3 and Table 2, the experimental results suggest that better performance can be obtained when increasing the strength (probability) of data augmentation.
#### 4.4.2 Impact of the cut-off frequency in SASMU
In this experiment, the proposed SASMU method has been tested with different settings of the cut-off frequency \(D_{0}\) for performance evaluation. As shown in Table 4, the average accuracy is 83.96% without the SMU operation, and the performance can be improved by using proposed method except for \(D_{0}=15\). The best average accuracy (84.56%) is yielded by the proposed method with \(D_{0}=60\).
### Quantitative results
#### 4.5.1 Performance of frequency-based mixup methods
In this experiment, we compare the proposed method to other mixup methods in frequency domain. For fair comparison, we adopted SA method for all experiments. As shown in Table 5, we obtain the best accuracy on all datasets when using our SASMU method. It is noteworthy that the accuracy of other methods are lower than baseline which is without any mixup operation in frequency domain. The reason might be that they assumed that semantic information is mainly encoded in high-frequency space, and thus preserved them while combining the low-frequency components of the target image for domain adaptation. However, previous studies have claimed that there are serious domain gaps between real and synthetic data spaces, especially in
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \#Sample & LFW & AgeDB-30 & CA-LFW & CFP-FP & CP-LFW & Avg. \\ \hline
40 & **90.68** & **65.67** & **75.55** & **68.59** & 68.40 & **73.78** \\
50 & 88.90 & 65.45 & 74.82 & 68.57 & **68.63** & 73.27 \\
72 & 89.50 & 63.55 & 73.63 & 67.04 & 67.50 & 72.25 \\
100 & 88.95 & 63.17 & 73.52 & 66.01 & 66.48 & 71.63 \\
200 & 88.32 & 61.63 & 71.17 & 65.43 & 66.05 & 70.52 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The impact of sample number for each identity in DiscofaceGAN [8]. We have tested different settings with the default Adaface training processing [17] without any other data augmentation or modification methods.
frequency domain [14, 9], and learning high-frequency information is difficult than low-frequency [43]. This is the reason that we decide to immigrate the high-frequency information from the real space to synthetic data for minimizing the synthetic-to-real domain gap, instead of relying on the low-frequency components.
#### 4.5.2 Visualization of frequency-based mixup methods
To investigate the effect of mixup operations in frequency domain, we have visualized the augmented images using different methods, as shown in Fig. 4. The amplitude spectrum of synthetic images are combined with the amplitude spectrum of real image using the proposed SMU method and other methods. In these experiments, the optimal hyperparameters of other mixup methods are used. Yang et al. [44] directly replace the amplitude of the synthetic image with that of the real image, which causes the inconsistency between the phase and amplitude of the synthetic image. Yang et al. [45] swap low-frequency components using a square mask between two images, leading to a ringing effect on the augmented image as the square mask works as an ideal filter. Xu et al. [42] adopt weighted sum operation to combine amplitude spectra, without considering that different frequencies have different importance and information, which produces artifacts in the augmented images. Liu et al. [21] retain the high-frequency components of the synthetic amplitude spectrum and combined the low-frequency components of the source amplitude spectrum with those of the real image. However, their setting leads to only a few frequency points being adjusted on the synthetic image, resulting in only image intensities being changed in the spatial domain, merely. In other words, when enlarging the hyperparameter of their method, it will cause the ringing effect which is in line with the results of [45]. In addition, we compute the peak signal-to-noise ratio (PSNR) values of those augmented images and show them in Fig. 4. It indicates that our method can produce high-quality images which are similar to the original synthetic images.
## 5 Conclusion
In this study, we have investigated how to develop a robust face recognition system with consideration for the privacy-preserving issue using the synthetic dataset. The proposed SASMU method is used to increase data variation and minimize the synthetic-to-real domain gap, which consists of the best combination of spatial data augmentations (SA) and spectrum mixup (SMU). First, we have analyzed how common data augmentations improve the recog
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline Cut-off freq. & LFW & AgeDB-30 & CA-LFW & CFP-FP & CP-LFW & Avg. \\ \hline - & 95.32 & 78.37 & 81.37 & 85.21 & 79.53 & 83.96 \\ \(D_{0}=15\) & 95.03 & 78.53 & 81.23 & 84.14 & 79.27 & 83.64 \\ \(D_{0}=30\) & 95.72 & 79.75 & 81.72 & 85.33 & 79.78 & 84.46 \\ \(D_{0}=45\) & **95.77** & 78.90 & **82.32** & 85.60 & **80.18** & 84.55 \\ \(D_{0}=60\) & 95.75 & **79.72** & 81.97 & **85.63** & 79.75 & **84.56** \\ \(D_{0}\sim U(S)\) & 95.72 & 79.32 & 81.57 & 84.97 & 80.33 & 84.38 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ablation study of the proposed SASMU method for cut-off frequency \(D_{0}\) of Gaussian kernel (%). \(S\) denotes a sample space (\(S=\{15,30,45,60\}\) is used in this experiment) and \(U\) stands for uniform distribution. The best average accuracy is 84.56% with \(D_{0}=60\).
\begin{table}
\begin{tabular}{l|l|c c c c c c} \hline \hline Name & Description & \(p^{LR}\) & \(p^{Crop}\) & \(p^{Pho}\) & \(p^{Gray}\) & \(p^{Per}\) & \(p^{GB}\) & \(p^{GN}\) \\ \hline DA-S0 & Original in [17] & 0.2 & 0.2 & 0.2 & - & - & - & - \\ DA-S1 & Weakest SA & 0.2 & 0.2 & 0.2 & 0.01 & - & - & - \\ DA-S2 & Forth strongest SA & 0.5 & 0.5 & 0.5 & 0.2 & - & - & - \\ DA-S3 & Third strongest SA & 0.5 & 0.5 & 0.5 & 0.4 & - & - & - \\ DA-S4 & Second strongest SA & 0.5 & 0.5 & 0.5 & 0.4 & 0.4 & - & - \\ DA-S5 & Strongest SA & 0.5 & 0.5 & 0.5 & 0.4 & 0.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Notations and settings for SA. \(p^{*}\) denote the probability of adopting data augmentation.
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline Cut-off freq. & LFW & AgeDB-30 & CA-LFW & CFP-FP & CP-LFW & Avg. \\ \hline - & 95.32 & 78.37 & 81.37 & 85.21 & 79.53 & 83.96 \\ \(D_{0}=15\) & 95.03 & 78.53 & 81.23 & 84.14 & 79.27 & 83.64 \\ \(D_{0}=30\) & 95.72 & 79.75 & 81.72 & 85.33 & 79.78 & 84.46 \\ \(D_{0}=45\) & **95.77** & 78.90 & **82.32** & 85.60 & **80.18** & 84.55 \\ \(D_{0}=60\) & 95.75 & **79.72** & 81.97 & **85.63** & 79.75 & **84.56** \\ \(D_{0}\sim U(S)\) & 95.72 & 79.32 & 81.57 & 84.97 & 80.33 & 84.38 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of different mixup method in frequency domain (%). The SA method is used in all experiments. Our method achieves the best performance on all datasets than other methods.
nition model for considering different conditions in the real scene and various color spaces (e.g., RGB-/gray-space), and have found the best combination of data augmentations for face recognition when using synthetic dataset. Second, we have investigated the reason for causing the domain gap between real and synthetic datasets, and have proposed a novel mixup method on the frequency domain, SMU, to reduce the gap for improving recognition performance. Note that only synthetic data and real images (without labels) are used, and no data from the target dataset is used in the training stage. Extensive experimental results have demonstrated the effectiveness of SASMU, achieving state-of-the-art performance on several common face verification benchmarks, including LFW, AgeDB-30, CA-LFW, CFP-FP, and CP-LFW. Our experimental results suggest that 1) SASMU is a crucial and efficient training strategy for face recognition; 2) applying SA can improve the recognition performance, especially using synthetic data; and 3) using the SMU method to immigate high-frequency information from real data outperforms other methods with the opposite assumptions. For future work, we can further discuss and analyze heavy data augmentation for face recognition or other tasks. Furthermore, the SMU method may be applied to other computer vision tasks for domain generalization and may be used to develop a generalized foundation model for several studies.
|
2303.13693 | Convergence of a simple discretization of the finite Hilbert
transformation | For a singular integral equation on an interval of the real line, we study
the behavior of the error of a delta-delta discretization. We show that the
convergence is non-uniform, between order $O(h^{2})$ in the interior of the
interval and a boundary layer where the consistency error does not tend to
zero. | Martin Costabel | 2023-03-23T22:07:50Z | http://arxiv.org/abs/2303.13693v2 | # Convergence of a simple discretization of
###### Abstract.
For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order \(O(h^{2})\) in the interior of the interval and a boundary layer where the consistency error does not tend to zero.
Key words and phrases: singular integral equation, Hilbert transform, delta-delta discretization, method of discrete vortices, discrete dipole approximation 2020 Mathematics Subject Classification: 65R20, 45E05
## 1. Introduction
Let \(a,b\in\mathbb{R}\) with \(a<b\). On the interval \(\Omega=(a,b)\) we consider the singular integral equation, abbreviated as \((\lambda\mathbb{I}-A_{\Omega})u=f\),
\[\lambda u(x)-\mathrm{p.v.}\int_{\Omega}\frac{u(y)}{i\pi(x-y)}dy=f(x)\,,\quad x \in\Omega\,. \tag{1.1}\]
We discretize it in the simplest imaginable fashion. We choose \(N\in\mathbb{N}\) defining the mesh width \(h=\frac{1}{N}\) and fix some origin \(a^{N}\in\mathbb{R}\), thus defining the infinite regular grid
\[\Sigma^{N}=\{x_{m}^{N}=a^{N}+mh\mid m\in\mathbb{Z}\}\]
and its finite counterpart \(\Sigma^{N}\cap\Omega\) indexed by
\[\omega^{N}=\{m\in\mathbb{Z}\mid x_{m}^{N}\in\Omega\}\,,\]
and consider the system
\[\lambda u_{m}-\frac{1}{i\pi N}\sum_{n\in\omega^{N},m\neq n}\frac{u_{n}}{x_{m}^ {N}-x_{n}^{N}}=f(x_{m}^{N})\,,\quad(m\in\omega^{N})\,. \tag{1.2}\]
This system can easily be programmed and solved with a couple of lines of code, without any knowledge of analysis, and it produces surprisingly good results, see Figure 1 for some examples for which the exact solution of the integral equation is known.
This simple delta-delta scheme is similar to Lifanov's method of discrete vortices [1], except that we take the same grid for the quadrature points and the evaluation points, and we put zero on the diagonal.
There is also a similarity to the fully discrete Calderon calculus analyzed in [5], although the differences, namely that we consider an open interval and not a closed curve and that our integral operator is strongly singular, are too important to try a similar analysis.
Note that the approximation scheme (1.2) for the integral equation (1.1) is not a projection method (or Petrov-Galerkin scheme) in any meaningful sense, although it has the form (except for the diagonal term) of a Galerkin scheme with Dirac deltas as test and trial functions. This has the negative consequence that some tools are not available that one would like to use in order to generalize results proved for the model singular integral equation (1.1) to equations with more general strongly singular kernels, a variable multiplier \(\lambda\) instead of a constant, or equations with lower order terms. These tools use the persistence of stability under compact perturbations of the operator, and this is known for projection methods and also for more general discrete approximation schemes in the Stummel-Vainikko sense, see for example [10]. But it seems that our simple scheme does not fit into any of these frameworks.
The question of a convergence proof for the delta-delta approximation (1.2) came up in the context of our recent research into the error analysis of higher-dimensional simple discretization methods for strongly singular volume integral equations related to the Discrete Dipole Approximation (DDA). The latter has been a standard tool in computational electrodynamics for half a century [7, 11, 2], with a wide range of applications from interstellar dust clouds to nano-particles. Despite the popularity of DDA in computational physics, there is very little known about its mathematical properties, but we have now some results about its stability [4, 3]. In order to complete the convergence proof, one needs estimates for the consistency error, and for gaining insight on the behavior of this, the one-dimensional example (1.1), (1.2) presented itself as a "toy problem", where the error analysis should be more transparent, while still showing some essential peculiarities of the more complicated higher-dimensional situation. It turns out that this is indeed the case, but that the results are interesting by themselves, and we present this in the following.
## 2. Stability
Numerical stability of the system (1.2), which we abbreviate as \((\lambda\mathbb{I}-T^{N})U^{N}=F^{N}\), means that there exists a bound of \(U^{N}\) by \(F^{N}\) uniform in \(N\), that is, a uniform resolvent estimate in some operator norm \(\left\|(\lambda\mathbb{I}-T^{N})^{-1}\right\|\leq C_{S}\) for all \(N\).
Figure 1. 3 examples: Exact solutions and their computed p/w constant approximations
### Stability in \(\ell^{2}\)
Such a stability estimate has been proved in [4], and although most of it is based on quite well known arguments, for the sake of completeness we quote the result and the main points of its proof here.
**Proposition 2.1**.: _The matrix \(T^{N}\) of the system (1.2) is selfadjoint with its eigenvalues in \(\mathscr{C}=[-1,1]\). For \(\lambda\in\mathbb{C}\) the discretization method (1.2) is stable in the \(\ell^{2}\) norm if and only if the integral operator \(\lambda\mathbb{I}-A_{\Omega}\) is boundedly invertible in \(L^{2}(\Omega)\), and this is equivalent to \(\lambda\not\in\mathscr{C}\). For such \(\lambda\), there is an estimate for the operator norms_
\[\|(\lambda\mathbb{I}-T^{N})^{-1}\|_{\mathscr{C}(\ell^{2}(\omega^{N}))}\leq \operatorname{dist}(\lambda,\mathscr{C})^{-1}=\|(\lambda\mathbb{I}-A_{\Omega} )^{-1}\|_{\mathscr{C}(L^{2}(\Omega))}\,. \tag{2.1}\]
The arguments are based on Fourier analysis and on a Lax-Milgram or Galerkin style use of the notion of numerical range. We recall the definition of the numerical range \(W(B)\) of a bounded linear operator \(B\) in Hilbert space, namely the range of values on the unit sphere of the sesquilinear form associated with \(B\)
\[W(B)=\{(u,Bu)|\|u\|=1\}\,. \tag{2.2}\]
This is a bounded set contained in the disk with radius \(\|B\|\), convex by the Toeplitz-Hausdorff theorem, and its closure contains the spectrum of \(B\). Writing for any \(u\) of norm one and \(z=(u,Bu)\) the estimate
\[\operatorname{dist}(\lambda,W(B))\leq|\lambda-z|=|(u,\lambda u-Bu)|\leq\|( \lambda\mathbb{I}-B)u\|\,,\]
one immediately gets the resolvent estimate in the operator norm
\[\|(\lambda\mathbb{I}-B)^{-1}\|\leq\operatorname{dist}(\lambda,W(B))^{-1}\,. \tag{2.3}\]
Considering that the restriction to a subspace does not increase the numerical range, one gets the same resolvent estimate for operators defined from \(B\) by restricting the sesquilinear form to a subspace. This is the Lax-Milgram argument for stability of Galerkin methods, and it applies here both to the integral equation (1.1) and the discrete system (1.2).
The integral operator \(A_{\Omega}\) in (1.1) and the system matrix \(T^{N}\) in (1.2) have this feature in common: They are both projections to bounded sets of translation invariant (i.e. convolution) operators, and these can be diagonalized by Fourier analysis.
The finite Hilbert transformation \(A_{\Omega}\) is the restriction to \(\Omega\) of the Hilbert transformation \(A\) on \(\mathbb{R}\), that is the convolution with the kernel \(K\) defined as the distribution
\[K(x)=\frac{1}{i\pi}\,\mathrm{p.v.}\,\frac{1}{x}\]
that has the Fourier transform
\[\widehat{K}(\xi)=\operatorname{sign}\xi\quad(\xi\in\mathbb{R})\,.\]
From the Plancherel theorem follows that the convolution operator \(A\) acting in the Hilbert space \(L^{2}(\mathbb{R})\) is equivalent to the operator of multiplication with its symbol \(\widehat{K}\) in \(L^{2}(\mathbb{R})\). It follows in particular that \(A\) is a selfadjoint involution and its spectrum consists of two eigenvalues \(\pm 1\) of infinite multiplicity.
This implies \(W(A_{\Omega})\subset W(A)=\mathscr{C}\) and the resolvent estimate
\[\|(\lambda\mathbb{I}-A_{\Omega})^{-1}\|_{\mathscr{C}(L^{2}(\Omega))}\leq \operatorname{dist}(\lambda,\mathscr{C})^{-1}\,.\]
On the discrete side there is a parallel argument using Fourier series instead of the Fourier transform. The system matrix \(T^{N}\) is a finite section of a (bi-)infinite Toeplitz matrix \(T\) that, owing to
\[\frac{1}{i\pi N}\frac{1}{x_{m}^{N}-x_{n}^{N}}=\frac{1}{i\pi(m-n)}\,,\]
is independent of \(N\),
\[T=\big{(}\frac{1}{i\pi(m-n)}\big{)}_{m,n\in\mathbb{Z}}\quad\text{with zero on the diagonal.} \tag{2.4}\]
Let \(\sigma_{T}\) be its symbol (characteristic function) defined by the Fourier series
\[\sigma_{T}(\tau)=\sum_{m\in\mathbb{Z}}\frac{1}{i\pi m}e^{im\tau}=\sum_{m=1}^{ \infty}\frac{2\sin m\tau}{\pi m}=\operatorname{sign}\tau-\frac{\tau}{\pi}\quad (\tau\neq 0)\,,\quad\sigma_{T}(0)=0\,.\]
Then by the Plancherel theorem it follows that the discrete convolution operator defined by \(T\) acting on \(\ell^{2}(\mathbb{Z})\) is equivalent to the operator of multiplication with \(\sigma_{T}\) in \(L^{2}(-\pi,\pi)\). Therefore both its spectrum and the closure of its numerical range \(\overline{W(T)}\) are equal to \(\,\raise 0.645pt\hbox{/}\kern-1.0pt\mathcal{C}=[-1,1]\). Since \(T^{N}\) arises from \(T\) by restriction to \(\ell^{2}(\omega^{N})\), we get the inclusion \(W(T^{N})\subset\,\raise 0.645pt\hbox{/}\kern-1.0pt\mathcal{C}6\) and the uniform resolvent estimate in (2.1).
With these simple arguments, we have proved Proposition 2.1, except for two points: The assertion that stability implies that \(\lambda\not\in\,\raise 0.645pt\hbox{/}\kern-1.0pt\mathcal{C}\), and the last equality in (2.1), where we would only get an inequality.
For the first point we use the fact that whereas the system (1.2)
\[(\lambda\mathbb{I}-T^{N})U^{N}=F^{N}\,, \tag{2.5}\]
when considered as an approximation scheme for the integral equation (1.1) \((\lambda\mathbb{I}-A_{\Omega})u=f\), is not a projection method, it is indeed a standard Galerkin scheme when considered as an approximation scheme for the infinite system
\[(\lambda\mathbb{I}-T)U=F\quad\text{ in }\ell^{2}(\mathbb{Z})\,. \tag{2.6}\]
As such, it converges whenever it is stable; in detail: Denote by \(P_{N}\) the orthogonal projection operator in \(\ell^{2}(\mathbb{Z})\) onto the subspace of sequences vanishing outside \(\omega^{N}\), identified with \(\ell^{2}(\omega^{N})\), so that we can write \((\lambda\mathbb{I}-T^{N})P_{N}=P_{N}(\lambda\mathbb{I}-T)P_{N}\). If we take as the origin \(a^{N}\) of the grid a fixed point inside \(\Omega\), then
\[\lim_{N\to\infty}P_{N}U=U\quad\text{ in }\ell^{2}(\mathbb{Z})\text{ for any }U.\]
If \(U^{N}\) and \(U\) satisfy (2.5), (2.6) with \(F^{N}=P_{N}F\), then we have the identity
\[U-U^{N}=\big{(}\mathbb{I}-(\lambda\mathbb{I}-T^{N})^{-1}P_{N}(\lambda\mathbb{I }-T)\big{)}(U-P_{N}U)\,.\]
Now let \(\lambda\in\mathbb{C}\) be such that the scheme (2.5) is stable,
\[\|(\lambda\mathbb{I}-T^{N})^{-1}\|\leq C_{S}\,.\]
Then \(\|U-U^{N}\|\leq\big{(}1+C_{S}\|\lambda\mathbb{I}-T\|\big{)}\|U-P_{N}U\|\) (this is the quasi-optimality of the Galerkin scheme, Cea's lemma), hence \(U^{N}\to U\), and because of \(\|U^{N}\|\leq C_{S}\|F\|\), we find in the limit
\[\|U\|\leq C_{S}\|F\|\,.\]
This implies that \(\lambda\mathbb{I}-T\) is invertible with the norm of the inverse bounded by \(C_{S}^{-1}\), hence \(\lambda\not\in\mathscr{C}\), in fact \(\operatorname{dist}(\lambda,\mathscr{C})\geq C_{S}^{-1}\).
For the second point, we recall that the finite Hilbert transformation and its spectral theory is a well-studied classical object. In particular it can be diagonalized by a generalized Fourier transformation [6] implying that as soon as \(\Omega\) is a proper subinterval of \(\mathbb{R}\) (even a half line), its action in \(L^{2}(\Omega)\) is unitarily equivalent to the operator of multiplication by \(\sigma\) in \(L^{2}(-1,1)\) with \(\sigma(\xi)=\xi\). Both its spectrum and the closure of its numerical range \(\overline{W(A_{\Omega})}\) are therefore equal to \(\mathscr{C}\), which shows the last equality in (2.1).
### Solution behavior
The simplicity of the statement of Proposition 2.1 is somewhat deceptive and hides a more complicated situation: Whereas the solvability of the finite system (1.2) is, of course, independent of the norm we choose in the space of sequences, the latter only being used to describe the stability, this is quite different for the solvability of the integral equation (1.1). Here, owing to the singular behavior of the solution at the endpoints of \(\Omega\), the spectrum of the finite Hilbert transform depends on the function space. The generalized eigenfunctions
\[g_{\xi}(x)=\tfrac{1}{\pi}\sqrt{\tfrac{b-a}{2(x-a)(b-x)}}e^{\tfrac{i}{2\pi} \log\tfrac{\xi+1}{1-\xi}\log\tfrac{x-a}{b-x}}\]
for which it is shown in [6] that they diagonalize the operator via the Hilbert space isomorphism \(f\mapsto F:L^{2}(a,b)\to L^{2}(-1,1)\) with the transform pair
\[F(\xi)=\tfrac{1}{\sqrt{1-\xi^{2}}}\int_{a}^{b}\overline{g_{\xi}(x)}f(x)dx,\quad f (x)=\int_{-1}^{1}g_{\xi}(x)F(\xi)\tfrac{d\xi}{\sqrt{1-\xi^{2}}}\]
such that \(A_{\Omega}f\) is transformed to \(\xi F(\xi)\), are genuine eigenfunctions in \(L^{p}(\Omega)\) with \(p<2\). More precisely, one considers the arcs of circles
\[C_{\alpha_{0}}=\{\lambda\in\mathbb{C}\mid\tfrac{\lambda+1}{\lambda-1}=e^{2\pi i \alpha},\operatorname{Re}\alpha=\alpha_{0}\}\]
which connect the points \(-1\) and \(1\) inside the unit circle if \(\tfrac{1}{4}<\alpha_{0}<\tfrac{3}{4}\) and outside if \(\alpha_{0}\in(0,\tfrac{1}{4})\cup(\tfrac{3}{4},1)\). Then the spectrum of \(A_{\Omega}\) in \(L^{p}(a,b)\) is the domain between \(C_{1-\frac{1}{p}}\) and \(C_{\frac{1}{p}}\). Its interior consists of eigenvalues. For an eigenvalue \(\lambda\), define the exponent \(\alpha\) by the above relation
\[\alpha=\tfrac{1}{2\pi i}\log\tfrac{\lambda+1}{\lambda-1}\quad\text{ or equivalently }\quad\lambda=-i\cot\pi\alpha,\qquad 1-\tfrac{1}{p}<\operatorname{Re}\alpha< \tfrac{1}{p}\,.\]
Then the corresponding eigenfunction in \(L^{p}(a,b)\) is
\[u(x)=(x-a)^{-\alpha}(b-x)^{\alpha-1}.\]
The spectrum in the dual space \(L^{\frac{p}{p-1}}(\Omega)\) is the same set. Thus for \(p<2\) the solution of (1.1) exists, in general, but is not unique, and for \(p>2\) it is unique, but does not exist for all right hand sides. Only for \(p=2\), the two arcs of circle degenerate to the interval \(\mathscr{C}\), and for \(\lambda\in\mathscr{C}\) the operator \(\lambda\mathbb{I}-A_{\Omega}\) in \(L^{2}(\Omega)\) is injective with dense range.
Thus, the delta-delta discretization manages to mimick the solution behavior of the singular integral equation for the specific value \(p=2\).
For \(A_{\Omega}\), it is also known how to write the resolvent explicitly as a singular integral operator with weight, involving multiplication by powers of the distance to the points \(a\) and \(b\), see for example
[8] or [9]. For the discrete version \(T^{N}\), such explicit formulas for the diagonalization and resolvent do not seem to be available.
### Exact solutions
Examples where both \(u\) and \(f\) are explicitly known functions can be obtained from function-theoretic arguments. For \(u\in L^{1}(a,b)\) define the Cauchy integral for \(z\in\mathbb{C}\setminus[a,b]\)
\[w(z)=\frac{1}{2\pi i}\int_{a}^{b}\frac{u(y)}{y-z}dy\,.\]
Then \(w\) is holomorphic in \(\mathbb{C}\setminus[a,b]\), behaves as \(O(|z|^{-1})\) as \(|z|\to\infty\), and satisfies on \(\Omega=(a,b)\) the jump relations
\[w_{+}-w_{-}=u\,;\qquad w_{+}+w_{-}=A_{\Omega}u,\quad\text{ where }\quad w_{\pm}(x)=\lim_{ \varepsilon\to 0+}w(x\pm i\varepsilon)\,. \tag{2.7}\]
Therefore the integral equation (1.1) is equivalent to the jump condition
\[(\lambda-1)w_{+}-(\lambda+1)w_{-}=f\,.\]
Reciprocally, any function \(w\) that is holomorphic outside \([a,b]\), vanishes at infinity and has square integrable upper and lower traces on the branch cut \([a,b]\) provides a solution \(u\) to the integral equation (1.1) with right hand side \(f\), if \(u\) and \(f\) are defined from the traces \(w_{\pm}\) via formulas (2.7).
For example, the function \(w_{0}(z)=\log\frac{z-a}{z-b}\) is (or can be chosen to be) holomorphic in \(\mathbb{C}\setminus[a,b]\), is \(O(|z|^{-1})\) at infinity, and has the upper and lower traces
\[w_{0,+}(x)=\log\tfrac{x-a}{b-x}-i\pi\,,\quad w_{0,-}(x)=\log\tfrac{x-a}{b-x}+i\pi\]
Since these traces belong to \(L^{2}(a,b)\), the jump \(u_{0}(x)=-2\pi i\) solves the integral equation (1.1) if we choose the right hand side \(f_{0}(x)=-2\pi i\lambda-2\log\tfrac{x-a}{b-x}\).
Another example is \(w_{1}(x)=\sqrt{(z-a)(z-b)}-z+\tfrac{a+b}{2}\) that has the right branch cut and behavior at infinity. Traces on \((a,b)\) are
\[w_{1,+}(x)=i\sqrt{(x-a)(b-x)}-x+\tfrac{a+b}{2}\,,\quad w_{1,-}(x)=-i\sqrt{(x- a)(b-x)}-x+\tfrac{a+b}{2}\]
The jump \(u_{1}(x)=2i\sqrt{(x-a)(b-x)}\) therefore solves the integral equation (1.1) if we choose the right hand side \(f_{1}(x)=2i\lambda\sqrt{(x-a)(b-x)}+2x-a-b\).
Other examples can be obtained by applying an entire analytic function \(F\) to the preceding example functions \(w\), provided \(F(0)=0\). For a third example, we take the function \(F(w)=e^{\alpha w}-1\) and apply it to the first example above, that is, we choose an exponent \(\alpha\in\mathbb{C}\) and define \(w_{2}(z)=e^{\alpha\log\frac{z-a}{z-b}}-1=\big{(}\tfrac{z-a}{z-b}\big{)}^{ \alpha}-1\).
The traces on the branch cut \(w_{2,\pm}(x)=(x-a)^{\alpha}(b-x)^{-\alpha}e^{\mp i\pi\alpha}-1\) belong to \(L^{2}(a,b)\) if \(\alpha\in(-\tfrac{1}{2},\tfrac{1}{2})\). The corresponding exact solution \(u_{2}\) and right hand side \(f_{2}\) are given by
\[u_{2}(x)=-2i\sin\pi\alpha\Big{(}\frac{x-a}{b-x}\Big{)}^{\alpha}\quad\text{ with }A_{\Omega}u_{2}(x)=2\cos\pi\alpha\Big{(}\frac{x-a}{b-x}\Big{)}^{\alpha}-2\;\text{ and }f_{2}=\lambda u_{2}-A_{\Omega}u_{2}\,.\]
Note that if \(\lambda=i\cot\pi\alpha\), then \(f_{2}\) is a constant function.
## 3. Error estimates
### Approximate solutions
For sake of simplicity we assume from now on that the length of \(\Omega\) is a multiple of \(h\) and that the origin \(a^{N}\) is chosen such that the boundary points \(a\) and \(b\) are midpoints of the mesh.
\[b-a=Mh\,,\quad M\in\mathbb{N}\,,\quad\omega^{N}=\{1,\dots,M\}\,,\quad x_{1}^{N}= a+\tfrac{h}{2}\,,\quad x_{M}^{N}=b-\tfrac{h}{2}\,. \tag{3.1}\]
When no confusion is possible, we omit superscripts \(N\) and write the discrete system as
\[\lambda u_{m}-\frac{1}{i\pi N}\sum_{\begin{subarray}{c}n=1\\ n\neq m\end{subarray}}^{M}\frac{u_{n}}{x_{n}-x_{m}}=f_{m}\,,\quad m\in\omega^{ N}\,. \tag{3.2}\]
In the spirit of Lifanov's method of discrete vortices [1], the delta-delta scheme can be considered as a Nystrom-type quadrature method, where the integral in (1.1) is replaced by its approximation by the midpoint rule. The approximate solution \(u^{(N)}\) satisfies the modified equation
\[\lambda u^{(N)}(x)-\frac{1}{i\pi}\sum_{n\in\omega^{N}}\frac{1}{N}\frac{u^{(N)} (x_{n})}{x_{n}-x}=f(x)\,, \tag{3.3}\]
and its evaluation in the gridpoints \(x=x_{m}\) gives the system (3.2) for the nodal values \(u_{m}=u^{(N)}(x_{m})\).
Alternatively, one can also consider the delta-delta scheme as a one-point quadrature approximation of a collocation method with piecewise constant trial functions. This is the popular point of view for DDA, see for example the error analysis in [12].
Let \(\chi_{j}\) be the characteristic function of the interval \(I_{j}=[x_{j}-\tfrac{h}{2},x_{j}+\tfrac{h}{2}]\). We consider the space of piecewise constant functions \(S^{N,0}(\omega^{N})=\operatorname{span}\{\chi_{j}\mid j\in\omega^{N}\}\).
Then the collocation scheme is: Find \(u^{N}\in S^{N,0}(\omega^{N})\) such that
\[\lambda u^{N}(x_{m})-\frac{1}{i\pi}\int_{\Omega}\frac{u^{N}(y)}{y-x_{m}}dy=f(x _{m})\,,\quad m\in\omega^{N}\,. \tag{3.4}\]
From this we get our system (3.2) by taking the \(u_{n}\) as the coefficients of \(u^{N}\) in the basis of the \(\chi_{n}\),
\[u^{N}=\sum_{n\in\omega^{N}}u_{n}\chi_{n},\]
and for the off-diagonal matrix elements \(A_{\Omega}\chi_{n}(x_{m})\) approximating the integrals by a one-point quadrature rule:
\[\int\frac{\chi_{k}(y)}{y-x_{m}}dy\ \sim\ \begin{cases}0&\text{ if }m=n\\ \frac{1}{x_{n}-x_{m}}\,\frac{1}{N}&\text{ if }m\neq n\end{cases}\]
After this replacement of the integrals, the system (3.4) becomes (3.2). For the diagonal elements, note that the Cauchy principal value integral \(\int\frac{\chi_{n}(y)}{y-x_{n}}dy\) vanishes for symmetry reasons.
Notice that even if the coefficients \((u_{n})\) in (3.3) and (3.4) are the same, the approximate solutions \(u^{N}\) and \(u^{(N)}\) are not the same: \(u^{N}\) in (3.4) is a piecewise constant function, whereas \(u^{(N)}\), which
for \(\lambda\neq 0\) is defined by (3.3) once the \(u_{k}\) are known, is a sum of the possibly smooth function \(f\) and a rational function with simple poles in the grid points.
### Consistency error
We define the _consistency error_ as the vector \(c^{N}[u]\) with components
\[c^{N}_{m}[u]=\frac{1}{i\pi}\Big{(}\int_{a}^{b}\frac{u(y)}{y-x_{m}}dy-h\sum_{k \in\omega^{N}\setminus\{m\}}\frac{u(x_{n})}{x_{n}-x_{m}}\Big{)},\quad m\in \omega^{N}\,. \tag{3.5}\]
It can be written as \(c^{N}[u]=(R_{N}A_{\Omega}-T^{N}R_{N})u\), where \(R_{N}\) is the restriction operator that maps a function \(u\) to its nodal values \(U=(u(x_{m}))_{m\in\omega^{N}}\).
Thus it is the quadrature error for the midpoint rectangle rule applied to the Cauchy singular integral. This quadrature error has been estimated under various assumptions on the function \(u\) in [1, Section 1.3], from which we borrow some ideas. We study here only the case where \(u\) is Holder continuous up to the boundary, \(u\in C^{\alpha}([a,b])\) for some \(\alpha\in(0,1)\).
In the applications of the singular integral equation (1.1) studied in [1], mainly from fluid dynamics, more general functions \(u\) with singularities at the boundary points \(a\) and \(b\) need to be considered. But since we want to emphasize the analogy with the volume integral equation of the DDA method, and solutions of the latter tend to be smooth up to (smooth points of) the boundary, the simple situation of \(u\in C^{\alpha}([a,b])\) will be sufficient for our purpose.
An advantage of this simplification is that we can get convergence results in an easy way from combining the consistency estimates of this section with the stability estimates of the preceding section. Convergence results for the case of solutions with singularities seem to require a much more complicated analysis, see [1, Chapter 7].
We consider first the case where \(u\) is constant.
**Lemma 3.1**.: _Let_
\[s_{j}=\int_{a}^{b}\frac{dy}{y-x_{j}}-\sum_{k\in\omega^{N}\setminus\{j\}}\frac{ h}{x_{k}-x_{j}}\,.\]
_Then for all \(j\in\omega^{N}\)_
\[|s_{j}|\leq\frac{h^{2}}{8}\big{|}\frac{1}{(x_{j}-a)^{2}}-\frac{1}{(b-x_{j})^{2 }}\big{|}=|\tfrac{a+b}{2}-x_{j}|(b-a)\big{(}\frac{h}{2(x_{j}-a)(b-x_{j})}\big{)} ^{2}\,. \tag{3.6}\]
Proof.: We split
\[s_{j}=\sum_{k\in\omega^{N}\setminus\{j\}}s_{jk}\quad\text{ with }s_{jk}=\int_{I_{k}}\frac{dy}{y-x_{j}}-\frac{h}{x_{k}-x_{j}}\,.\]
First we observe that there are cancellations: \(s_{jk}+s_{jk^{\prime}}=0\) for \(k^{\prime}+k=2j\), because of the antisymmetry of the kernel. Therefore
\[s_{j}=\begin{cases}\sum_{k}\{s_{jk}\mid 2x_{j}-a+\frac{h}{2}<x_{k}<b\}& \text{ if }x_{j}\leq\tfrac{a+b}{2}\,,\\ \sum_{k}\{s_{jk}\mid a<x_{k}<2x_{j}-b-\frac{h}{2}\}&\text{ if }x_{j}\geq \tfrac{a+b}{2}\,.\end{cases}\]
Then we apply the classical error representation for the midpoint rule
\[\int_{-\frac{h}{2}}^{\frac{h}{2}}f(t)dt-hf(0)=\int_{-\frac{h}{2}}^{\frac{h}{2}} \tfrac{1}{2}(\tfrac{h}{2}-|t|)^{2}f^{\prime\prime}(t)dt\,,\]
valid for any \(C^{2}\) function, to the function \(f(t)=\frac{1}{t+x_{k}-x_{j}}\). Consider the case \(x_{j}\leq\frac{a+b}{2}\). Then for the relevant indices \(k\), we have \(x_{k}>x_{j}\) and therefore \(f^{\prime\prime}(t)=2(t+x_{k}-x_{j})^{-3}>0\). Hence
\[0<s_{jk}<\frac{h^{2}}{8}\int_{I_{k}}\frac{2dy}{(y-x_{j})^{3}}\quad\Longrightarrow \quad 0<s_{j}<\frac{h^{2}}{8}\int_{2x_{j}-a}^{b}\frac{2dy}{(y-x_{j})^{3}}= \frac{h^{2}}{8}\big{(}-\frac{1}{(b-x_{j})^{2}}+\frac{1}{(x_{j}-a)^{2}}\big{)}\,.\]
Likewise, for \(x_{j}\geq\frac{a+b}{2}\), the second derivative is always negative, and we obtain
\[0>s_{j}>\frac{h^{2}}{8}\int_{a}^{2x_{j}-b}\frac{2dy}{(y-x_{j})^{3}}=\frac{h^{ 2}}{8}\big{(}-\frac{1}{(b-x_{j})^{2}}+\frac{1}{(x_{j}-a)^{2}}\big{)}\,.\]
The lemma means that for \(x_{j}\) in any compact subinterval of \((a,b)\), the \(s_{j}\) converge to zero as \(O(h^{2})\). But when \(x_{j}\) tends to \(a\) or \(b\) as \(h\to 0\), then \(s_{j}\) will have non-zero limits. For example for \(x_{1}=a+\frac{h}{2}\), one finds \(s_{1}\to\log 2-\gamma\), where \(\gamma=0.5772..\) is Euler's constant. For this case, the estimate (3.6) gives \(s_{j}\leq\frac{1}{2}\) in the limit.
Another way to express this behavior is that for a given an error threshold \(\epsilon\), the points \(x_{j}\) where \(|s_{j}|\) exceeds \(\epsilon\) are confined to a boundary layer of thickness \(h/\sqrt{2\epsilon}\).
**Proposition 3.2**.: _For \(u\in C^{\alpha}([a,b])\) let \(c_{j}^{N}[u]\) be defined by (3.5). Then_
\[|c_{j}^{N}[u]|\leq C\left(h^{\alpha}|\log h|\|u\|_{C^{\alpha}([a,b])}+|u(x_{j} )|\frac{h^{2}}{(x_{j}-a)^{2}(b-x_{j})^{2}}\right). \tag{3.7}\]
_Here the constant \(C\) depends on \(a,b\) but not on \(h=\frac{1}{N}\) nor on \(u\). If, in addition, \(u(a)=u(b)=0\) holds, then the estimate simplifies to_
\[|c_{j}^{N}[u]|\leq C\,h^{\alpha}|\log h|\|u\|_{C^{\alpha}([a,b])}\,. \tag{3.8}\]
Proof.: Split
\[i\pi c_{j}^{N}[u] =c_{jj}+\sum_{k\neq j}c_{jk}+u(x_{j})s_{j}\quad\text{ with }\] \[c_{jj} =\int_{I_{j}}\frac{u(y)}{y-x_{j}}dy,\ c_{jk}=\int_{I_{k}}\big{(} \frac{u(y)-u(x_{j})}{y-x_{j}}-\frac{u(x_{k})-u(x_{j})}{x_{k}-x_{j}}\big{)}dy\]
and \(s_{j}\) as defined and estimated in Lemma 3.1.
For \(c_{jj}\) we find
\[|c_{jj}|=\big{|}\int_{I_{j}}\frac{u(y)-u(x_{j})}{y-x_{j}}dy\big{|}\leq\int_{I _{j}}|y-x_{j}|^{\alpha-1}dy\,\|u\|_{C^{\alpha}([a,b])}\leq h^{\alpha}\,\|u\|_ {C^{\alpha}([a,b])}\,.\]
For the term \(u(x_{j})s_{j}\) we use the estimate from Lemma 3.1. It remains to estimate \(\sum_{k\neq j}c_{jk}\). Write
\[c_{jk}=c_{jk}^{0}+c_{jk}^{1}\quad\text{ with }c_{jk}^{0}=\int_{I_{k}}\frac{u(y)-u(x_ {k})}{y-x_{j}}dy\,,\,\,c_{jk}^{1}=\int_{I_{k}}\big{(}u(x_{k})-u(x_{j})\big{)} \big{(}\tfrac{1}{y-x_{j}}-\tfrac{1}{x_{k}-x_{j}}\big{)}dy.\]
Then
\[\Big{|}\sum_{k\neq j}c_{jk}^{0}\Big{|}\leq\big{(}\frac{h}{2}\big{)}^{\alpha} \|u\|_{C^{\alpha}([a,b])}\Big{(}\int_{a}^{x_{j}-\frac{h}{2}}\frac{dy}{x_{j}-y} +\int_{x_{j}+\frac{h}{2}}^{b}\frac{dy}{y-x_{j}}\Big{)}\leq C\,h^{\alpha}|\log h |\,\|u\|_{C^{\alpha}([a,b])}\,.\]
Finally, \(c_{jk}^{1}=\int_{I_{k}}\big{(}u(x_{k})-u(x_{j})\big{)}\frac{x_{k}-y}{(y-x_{j}) (x_{k}-x_{j})}dy\) can be estimated as
\[|c_{jk}^{1}|\leq\|u\|_{C^{\alpha}}\int_{I_{k}}\frac{|x_{k}-y|}{|y-x_{j}|x_{k}- x_{j}|^{1-\alpha}}dy\leq C\,\|u\|_{C^{\alpha}}\,h^{2}\,|x_{k}-x_{j}|^{\alpha-2}=C\, \|u\|_{C^{\alpha}}\,h^{\alpha}\,|k-j|^{\alpha-2}\]
For \(\alpha<1\), the infinite series \(\sum_{k\in\mathbb{Z}\setminus\{j\}}|k-j|^{\alpha-2}\) converges, so that
\[\Big{|}\sum_{k\neq j}c_{jk}^{1}\Big{|}\leq C\,h^{\alpha}\,\|u\|_{C^{\alpha}}\,.\]
For \(\alpha=1\), we would pick up an extra factor of \(\log N=|\log h|\).
The estimate (3.7) is proved.
Finally, if \(u(a)=u(b)=0\), then \(|u(x_{j})|\leq C\,\|u\|_{C^{\alpha}}(x_{j}-a)^{\alpha}(b-x_{j})^{\alpha}\), hence
\[|u(x_{j})|\frac{h^{2}}{(x_{j}-a)^{2}(b-x_{j})^{2}}\leq C\,h^{\alpha}\|u\|_{C^{ \alpha}}\big{(}\frac{h}{(x_{j}-a)(b-x_{j})}\big{)}^{2-\alpha}\]
The last factor is uniformly bounded, and this completes the proof of (3.8)
### Discrete error
If \(u\) is the solution of the integral equation (1.1) with right hand side \(f\) and \(U^{N}=(u_{n})_{n\in\omega^{N}}\) is solution of the discrete system (1.2) with \(f_{m}=f(x_{m})\), then we define the _discrete error_\(E^{N}\) as vector with the components \(E_{m}=u(x_{m})-u_{m}\), \(m\in\omega^{N}\). It is easy to see that \(E^{N}\) satisfies the discrete system with the consistency error as right hand side,
\[(\lambda\mathbb{I}-T^{N})E^{N}=c^{N}[u]\,. \tag{3.9}\]
By a direct application of our stability and consistency estimates we obtain an estimate for the discrete error. We choose the situation where the simplified consistency estimate (3.8) holds.
**Theorem 3.3**.: _Assume \(\lambda\in\mathbb{C}\setminus[-1,1]\). Let \(u\) be solution of the integral equation (1.1) with right hand side \(f\) and \((u_{n})_{n\in\omega^{N}}\) be solution of the discrete system (1.2) with \(f_{m}=f(x_{m})\). If we assume that \(u\in C^{\alpha}([a,b])\) with \(\frac{1}{2}<\alpha\leq 1\) and \(u(a)=u(b)=0\), then there is a constant \(C\) independent of \(N\) such that the discrete error satisfies_
\[\|E^{N}\|_{\ell^{2}(\omega^{N})}\leq C\,N^{\frac{1}{2}-\alpha}\log N\,. \tag{3.10}\]
Proof.: From the error equation (3.9) and our stability estimate (2.1) we obtain
\[\|E^{N}\|_{\ell^{2}(\omega^{N})}\leq C\,\|c^{N}[u]\|_{\ell^{2}(\omega^{N})}\,.\]
Now our consistency estimate (3.8) is an \(\ell^{\infty}(\omega^{N})\) estimate, so we lose a factor of \(\sqrt{N}\):
\[\|c^{N}\|_{\ell^{2}(\omega^{N})}\leq|\omega^{N}|^{\frac{1}{2}}\|c^{N}\|_{\ell^{ \infty}(\omega^{N})}\leq C\,N^{\frac{1}{2}}\|c^{N}\|_{\ell^{\infty}(\omega^{N}) }\leq C\,N^{\frac{1}{2}-\alpha}\log N\|u\|_{C^{\alpha}([a,b])}\,.\]
### Convergence in \(L^{2}\)
When the solution \(u\) does not vanish on the boundary, we only have the consistency estimate (3.7), and the consistency error \(c^{N}[u]\) will not converge to zero in \(\ell^{\infty}\) and even less in \(\ell^{2}\), hence the discrete error will not converge to zero in \(\ell^{2}\), either. Instead, we can study the error of the piecewise constant approximation
\[e^{N}=u-u^{N}\quad\text{ with }\quad u^{N}=\sum_{k\in\omega^{N}}u_{k}\chi_{k}\,.\]
Its \(L^{2}(\Omega)\) norm can be bounded by
\[\|e^{N}\|_{L^{2}(a,b)}^{2} =\sum_{k\in\omega^{N}}\int_{I_{k}}|u(y)-u_{k}|^{2}dy\] \[\leq 2\sum_{k\in\omega^{N}}\Big{(}\int_{I_{k}}|u(y)-u(x_{k})|^{2 }dy+\int_{I_{k}}|u(x_{k})-u_{k}|^{2}dy\Big{)}\] \[\leq 2\sum_{k\in\omega^{N}}\Big{(}h^{2\alpha+1}\|u\|_{C^{\alpha} }^{2}+h\,|E_{k}|^{2}\Big{)}\] \[\leq C\left(h^{2\alpha}\|u\|_{C^{\alpha}([a,b])}^{2}+h\,\|E^{N} \|_{\ell^{2}(\omega^{n})}^{2}\right).\]
Thus we get convergence as soon as we can show that \(N^{-\frac{1}{2}}\|E^{N}\|_{\ell^{2}}\) tends to zero. By our \(\ell^{2}\) stability estimates, this is equivalent to the fact that \(N^{-\frac{1}{2}}\|c^{N}\|_{\ell^{2}}\) tends to zero. For this, as we have seen before, it would be _sufficient_ that the consistency error \(\|c^{N}\|_{\ell^{\infty}}\) tends to zero.
But this is not _necessary_: In fact, assume that the consistency estimate (3.7) is satisfied. It implies
\[|c_{j}^{N}|\leq C\,\|u\|_{C^{\alpha}([a,b])}\Big{(}h^{\alpha}|\log h|+\max \big{\{}\frac{h^{2}}{(x_{j}-a)^{2}},\;\frac{h^{2}}{(b-x_{j})^{2}}\big{\}} \Big{)}\]
Now \(x_{j}-a\) takes the values \(\frac{h}{2}\), \(\frac{3h}{2}\), \(\frac{5h}{2}\),..., and therefore the sum
\[\sum_{j\in\omega^{N}}\big{(}\frac{h^{2}}{(x_{j}-a)^{2}}\big{)}^{2}\]
is bounded independently of \(N\) by a constant
\[C^{\prime}=16\sum_{j=0}^{\infty}(2j+1)^{-4}<\infty\,.\]
This implies finally
\[h\|c^{N}\|_{\ell^{2}(\omega^{N})}^{2}\leq C\,\|u\|_{C^{\alpha}([a,b])}^{2} \Big{(}h^{2\alpha}|\log h|^{2}+h\,C^{\prime}\Big{)}\,.\]
We have proved the following error estimate.
**Theorem 3.4**.: _Assume \(\lambda\in\mathbb{C}\setminus[-1,1]\). Let \(u\) be solution of the integral equation (1.1) with right hand side \(f\), let \(U^{N}=(u_{k})_{k\in\omega^{N}}\) be solution of the discrete system (3.2) with \(f_{j}=f(x_{j})\), and let \(u^{N}=\sum_{k\in\omega^{N}}u_{k}\chi_{k}\) be the corresponding piecewise constant function. If we assume that \(u\in C^{\alpha}([a,b])\), then there is a constant \(C\) independent of \(N\) such that the error \(e^{N}=u-u^{N}\) satisfies_
\[\left\|e^{N}\right\|_{L^{2}(a,b)}\leq C\left(N^{-\alpha}\log N+N^{-\frac{1}{2} }\right).\]
The last term \(O(\sqrt{h})\) in this error estimate that prevents the estimate to improve with regularity above \(C^{\frac{1}{2}}\) is due to the boundary layer and the choice of the \(L^{2}\) norm for measuring the error. One could choose \(L^{p}\) for \(1\leq p\leq\infty\) instead. In the same way as above, one obtains
\[\left\|e^{N}\right\|_{L^{p}(a,b)}\leq C\big{(}h^{\alpha}\|u\|_{C^{\alpha}([a,b ])}^{2}+h^{\frac{1}{p}}\left\|E^{N}\right\|_{\ell^{p}(\omega^{n})}\big{)}\,.\]
This would be best for \(p=1\), but we are tied to \(p=2\) because we depend on our stability estimate.
## 4. Numerical experiments
In this section, we fix arbitrarily the interval \([a,b]=[-0.15,1.35]\) and choose \(N\) as an odd multiple of \(10\), so that \([a,b]\) is subdivided in \(3N/2\) subintervals of length \(h=\frac{1}{N}\). We use the \(3\) examples discussed in section 2.3, where we divide the first example by \(-2\pi i\), so that \(u0(x)=1\). In the second example, we normalize by dividing by \(2i\), so that \(u1(x)=\sqrt{(x-a)(b-x)}\). For the third example, we choose \(\alpha=0.25\), so that \(u2(x)=\left(\frac{x-a}{b-x}\right)^{\frac{1}{4}}/\sqrt{2}\). In Figure 1, we saw the \(3\) exact solutions with their computed piecewise constant approximation, computed for \(N=10\) with \(\lambda=2\).
### Behavior of consistency error and discrete error
Knowing the exact solution, we can define the consistency error \(c^{N}\) as in (3.5) and the discrete error \(E^{N}\) as in Section 3.3. In Figure 2, we plot for our 3 examples the absolute values of \(c^{N}\) and \(E^{N}\) as functions of \(x\in[a,b]\) for \(N=10,50,250,1250\). In the logarithmic scale, one can see the convergence in the interior of the interval with a power of \(N\), whereas near the boundary points there is no convergence, illustrating the analysis shown in Section 3. Note that we have not presented an analysis for the case of a singular solution such as the one in the third example, but the numerical results show a similar behavior as for the other examples.
### Convergence rates
We plot several error norms as functions of \(N=10\times 3^{j}\), \(j=0,...,6\) in a loglog scale: The \(\ell^{2}(\omega^{N})\) norms of the consistency error \(c^{N}\) and the discrete error \(E^{N}\) as well as the \(\ell^{2}\) norm of the restriction of \(E^{N}\) to the interior subinterval \([0,1.2]\) of \((a,b)\). Finally the \(L^{2}(a,b)\) norm of the error \(e^{N}\) of the piecewise constant approximation, which we compare with the normalized \(\ell^{2}\) norm of \(E^{N}\), \(N^{-\frac{1}{2}}\|E^{N}\|_{\ell^{2}(\omega^{N})}\). |
2301.05688 | CANE: A Cascade-Control Approach for Network-Assisted Video QoE
Management | Prior efforts have shown that network-assisted schemes can improve the
Quality-of-Experience (QoE) and QoE fairness when multiple video players
compete for bandwidth. However, realizing network-assisted schemes in practice
is challenging, as: i) the network has limited visibility into the client
players' internal state and actions; ii) players' actions may nullify or negate
the network's actions; and iii) the players' objectives might be conflicting.
To address these challenges, we formulate network-assisted QoE optimization
through a cascade control abstraction. This informs the design of CANE, a
practical network-assisted QoE framework. CANE uses machine learning techniques
to approximate each player's behavior as a black-box model and model predictive
control to achieve a near-optimal solution. We evaluate CANE through realistic
simulations and show that CANE improves multiplayer QoE fairness by ~50%
compared to pure client-side adaptive bitrate algorithms and by ~20% compared
to uniform traffic shaping. | Mehdi Hosseinzadeh, Karthick Shankar, Maria Apostolaki, Jay Ramachandran, Steven Adams, Vyas Sekar, Bruno Sinopoli | 2023-01-13T18:19:21Z | http://arxiv.org/abs/2301.05688v1 | # CANE: A Cascade-Control Approach for Network-Assisted Video QoE Management
###### Abstract
Prior efforts have shown that network-assisted schemes can improve the Quality-of-Experience (QoE) and QoE fairness when multiple video players compete for bandwidth. However, realizing network-assisted schemes in practice is challenging, as: i) the network has limited visibility into the client players' internal state and actions; ii) players' actions may nullify or negate the network's actions; and iii) the players' objectives might be conflicting. To address these challenges, we formulate network-assisted QoE optimization through a cascade control abstraction. This informs the design of CANE, a practical network-assisted QoE framework. CANE uses machine learning techniques to approximate each player's behavior as a black-box model and model predictive control to achieve a near-optimal solution. We evaluate CANE through realistic simulations and show that CANE improves multiplayer QoE fairness by \(\sim\)50% compared to pure client-side adaptive bitrate algorithms and by \(\sim\)20% compared to uniform traffic shaping.
Multi-Player Video Streaming, Fairness in Quality-of-Experience, Model Predictive Control, Cascade Control Framework, Resource Allocation, Network-Assisted scheme.
## I Introduction
In recent years, video streaming has become a considerable fraction of daily Internet traffic, in which the user-perceived Quality-of-Experience (QoE) is a critical factor [1]. The QoE impacts player engagement and the revenues of providers. Thus, most modern players use some form of Adaptive BitRate (ABR) algorithm to tailor the bitrate level to the dynamically changing network conditions [2].
As video traffic grows, more independently-developed video players compete for the bandwidth of bottleneck links. The lack of coordination across players naturally leads to unfairness or/and sub-optimal QoE [3, 4]. For instance, two users watching a video from different players and sharing the same network link might experience very different video quality. Similarly, two users on devices with different resolution requirements might end up using the same amount of bandwidth, resulting in a suboptimal overall experience. Notably, pure server- or client-side ABR schemes are fundamentally unable to control the achieved QoE across competing players. On the one hand, pure server-side schemes (e.g., [5]) only work when all players fetch data from the same video server, which is impractical. On the other hand, client-side schemes (e.g., MPC [6] and BOLA [7]) have limited visibility and knowledge about competing players, thus are often unable to achieve cross-player QoE optimality and fairness.
In this context, _network-assisted_ schemes, where an network device can allocate the bandwidth to each player, have the potential to control QoE across players [8, 9, 10, 11]. Indeed, prior research has shown that under ideal conditions, network-assisted solutions can be more effective in enforcing QoE fairness across players compared to server- or client-side ABR [12, 13]. Furthermore, network-assisted schemes are more flexible and can realize complex cross-player policy objectives, e.g., using different bandwidth allocation schemes for different player settings.
In practice, however, the benefits of network-assisted schemes are hard to realize [14, 15]. First, network-assisted schemes assume an _accurate_ global knowledge of the players' internal operation. This assumption is unrealistic in practice [16], due to the heterogeneity of video players and the complexity of the client-side ABR algorithms. Second, as players affect each others' operations in a complex manner, network-assisted schemes with no stability guarantees may lead to oscillations [17, 10] or bandwidth underutilization [18]. Finally, with the increasing use of end-to-end encryption in video streaming applications [14, 16, 19], the network layer has an inaccurate view into the player's state and actions. Thus, a network-assisted scheme can only be helpful if it is robust against errors. The lack of a practical network-assisted scheme leaves network operators with no mechanism for enforcing high-level cross-player policies, e.g., prioritizing higher-resolution screens or premium users.
In this work, we aim at providing a _mechanism_ that allows operators to implement various _policies_ based on their high-level objectives. To this end, we revisit the network-assisted QoE management problem through a control-theoretic lens. We observe that the interaction of the bandwidth allocation in the network layer coupled with the client-side ABR algorithms creates a hierarchy with two nested control loops [10]. We abstract this interaction as a _cascade control system_[20] in
which a primary controller determines the target QoE for each player in terms of allocated bandwidth, and a secondary controller controls the QoE of each player.
Building on these insights, we develop CANE, a CAscade control-based NEetwork-assisted framework that can improve QoE fairness across players while achieving a near-optimal network-assisted QoE. We envision CANE that runs at an edge network device (e.g., wireless access point or cell edge node); utilizes only the information it can infer from the traffic; and views each player as a black box. CANE uses Machine Learning (ML) techniques to model and predict each player's ABR adaptation behavior. CANE also benefits from the intrinsic ability of cascade control systems to mitigate the impact of disturbances (e.g., modeling errors), which can help alleviate the need for highly accurate models. Finally, CANE uses Model Predictive Control (MPC) [21, 22] for bandwidth allocation to provide flexible and robust control. CANE can help operators realize flexible policies to tradeoff between efficiency and fairness across players.
We investigate CANE's effectiveness in practice by simulating and testing it on realistic experiments over diverse combinations of players, and a wide range of real-world bandwidth traces [23, 24, 25]. We find that CANE improves QoE fairness by \(\sim\)50% on median in comparison with pure client-side ABR algorithms and by \(\sim\)20% on median in comparison with uniform traffic shaping.
The rest of the paper is organized as follows. We begin by reviewing the literature in Section II. We sketch the problem space in Section III and motivate this work. In Section IV, we provide insights from control theory and revisit the problem of QoE management as a cascade control problem. Section V introduces CANE. We then provide design details in Section VI. We evaluate the effectiveness of CANE via extensive experiments in Section VII. Finally, Section VIII concludes the paper with future work.
## II Related Work
Video streaming in today's world.With the increasing popularity of video streaming applications, sharing a bottleneck bandwidth is a common aspect of most of the emerging video streaming use cases. There are three key attributes in these use cases. First, video players may access video content through different platforms, e.g., Netflix and YouTube. Second, video content providers themselves provide a variety of subscription plans, where subscribers to an expensive plan expect better service (including higher video quality). Third, video players may access video content on different devices, including phones, smart TVs, and laptops. These attributes imply that the classic ABR algorithms that maximize single-player QoE are far from meeting QoE objectives (specifically, multiplayer QoE fairness) in today's complex use cases.
QoE modeling.QoE in multiplayer video streaming application includes two main aspects: i) application-level components; and ii) network-level components. Application-level components include any characteristics that have an influence on the QoE for the user, e.g., rebuffering, video quality, quality changes, and startup delay [26]. Network-level QoE components are efficiency [1] and fairness [3, 27]. In today's video streaming applications, a set of diverse players connect to the Internet by a single router; this makes the network-level QoE components to become more critical.
QoE management in emerging video applications.QoE management aims at addressing application- and network-level QoE components. Application-level components can be addressed by the client-side ABR algorithms, e.g., [28, 29, 30, 31, 32, 6]. Regarding the network-level QoE components, existing solutions for addressing network-level components of QoE can be classified as: i) server-side; and iii) network-assisted. Server-side schemes (e.g., [33, 34, 5]) can help video content providers to manage QoE only across their own players. Network-assisted schemes are more practical in addressing network-level QoE components. Many research studies have developed a network-assisted scheme to address network-level QoE components. In this respect, [11, 13, 9] aim at improving QoE fairness without taking into account cross-player differences, and [35, 8] only consider diversity of players with respect to streaming devices.
## III Motivation
### _Setting_
As video traffic becomes predominant, it is more likely that multiple video players with different preferences, client-side ABR algorithms, streaming devices, and importance (e.g., service agreement) will share bottleneck links and compete for bandwidth in the network. Such scenarios are common today in home networks and enterprise settings, in which multiple devices (e.g., HDTV, tablet, laptop, cell phone, etc.) connect to the Internet by a single WiFi router or a cellular edge network.
As multiple-player instances compete, multiplayer QoE fairness and single-player QoE become critical and often conflicting requirements. Achieving multi-player QoE fairness is particularly challenging as it requires accounting for player-, device-, and user-specific context (e.g., priority, conditions, screen size) of each distinct player instance. For instance, players using the same amount of bandwidth are not guaranteed to experience the same QoE. A natural question then is: _Are today's client-side ABR algorithms alone sufficient to achieve good QoE and fairness?_1
Footnote 1: We do not consider server-assisted schemes since typically, in a shared setting, the different clients/players are connecting to diverse servers.
Metrics.We use one metric to quantify single-player QoE and two for multi-player QoE fairness. For single-player QoE, we use a score that depends on the video bitrate and buffer level. We elaborate on the detailed formulation in Section VI-A. For multi-player fairness, we use the Jain's unfairness index [29] and the pairwise unfairness index [36] ; a lower value of the indices implies a more fair case.
### _A motivating example_
To illustrate the problems that arise when multiple players share the same bottleneck link, we run them in practice and observe the behavior. We consider two players from academic papers, namely MPC [6] and BOLA [7] implemented on
Puffer [32], and a commercial video player from YouTube. Observe that these are representative and realistic examples of players that are far from adversarial or aggressive players, which would further deteriorate fairness. We consider groups of four players with different compositions, meaning we run instances of the same player (homogeneous) or of different players (heterogeneous). We refer to each group composition as a scenario. In all scenarios, the four players connect to the Internet via a single router. The available bandwidth changes over time according to 50 randomly selected bandwidth traces from the publicly available ones by FCC [23], Norway [24], and OBOE [25].
**Observation #1: Client-side ABR algorithms may lead to QoE unfairness.** Figures 1(a) show two CDFs of our two fairness indices across \(50\) experiments of four distinct scenarios: two homogeneous in which all four players are Youtube or MPC, respectively, and two heterogeneous in which 3 and 2 are MPC and the rest Youtube, respectively. We observe that heterogeneous combination highly degrades multiplayer QoE fairness compared to homogeneous combinations. Of course, not all heterogeneous combinations degrade fairness. Some players co-exist harmoniously. Concretely, in Figures 1 (b) we plot the same indices as in Figures 1 (a) but for combinations of MPC and BOLA players. We observe that the heterogeneous and homogeneous combinations of MPC and BOLA players lead to comparable multiplayer QoE fairness.
**Observation #2: Client-side ABR algorithms may consistently and inadvertently prioritize a specific class of players.** To understand the reason for the fairness degradation, we take a deeper look at the QoE delivered to each player. Figure 2 shows CDFs of the QoE that each player experiences under the heterogeneous scenario. We add the QoE that each player would experience in a homogeneous scenario as a baseline. Fig. 2-left concerns the scenario of two MPC and two YouTube players, while Fig. 2-right presents the scenario of two MPC and two BOLA players. On the one hand, we observe that both YouTube players enjoy an overall higher QoE not only compared to MPC players but even compared to what they would experience had they been competing with other Youtube players (YouTube homogeneous line). On the other hand, the MPC players experience worse QoE in a heterogeneous scenario than if they were in a homogeneous scenario. Thus, we can conclude that the unfairness in the heterogeneous scenario resulted from the _aggressive_ prioritization of YouTube players. Observe that a more greedy player would have been prioritized even further, effectively starving other players further.
**Observation #3: Network-assisted schemes have the potential to address multiplayer QoE fairness.** We implement and test a simple network-assisted scheme that shares the bandwidth equally among all players. Fig. 3 plots the CDFs of the fairness metrics in the scenario containing two MPC and two YouTube players. We observe that even this simple scheme can improve multiplayer QoE fairness across players. While such a scheme is not general or robust, it highlights the potential of network-assisted schemes.
**Observation #4: Uniform traffic sharing is not enough.** Designing a network-assisted bandwidth management scheme is not trivial. Each player may use a different adaptation algorithm, which will react differently to the available bandwidth. Thus, simply splitting the bandwidth equally across players may lead to unfairness or wasted bandwidth. As an illustration, Fig. 4 shows the average QoE that an MPC and a YouTube player experience while having equal bandwidth available. The YouTube player enjoys a higher QoE than the MPC player, as the YouTube player fully uses the available bandwidth, while the MPC player does not.
Fig. 1: Unfairness indices for MPC and YouTube players (figure (a)) and for MPC and BOLA players (figure (b)) in various multiplayer video streaming scenarios.
Fig. 3: Effectiveness of a uniform traffic shaping in addressing multiplayer QoE fairness in a simple, yet representative scenario.
Fig. 2: Average QoE delivered when MPC and YouTube players compete (left) and when BOLA and MPC compete (right).
## IV A Cascade Control Formulation
### _A Control-Theoretic View_
Network-assisted QoE management schemes need to tackle a variety of complex challenges arising from the diversity of video-player algorithms, network scenarios, and user settings. Indeed, today's players use different ABR algorithms; users might watch videos via different devices; and/or network operators might offer different service-level agreements. A natural question that arises then is: _Is it possible for a network-assisted scheme to achieve the optimal QoE in practice despite these challenges?_ To answer this question, we revisit the problem of network-assisted QoE from first principles to shed light on the key requirements for a candidate design.
At a high level of a network-assisted scheme, there are two logical control systems that interact. Thus, this structure conceptually represents a _cascade control system_[37, 38, 39]. Cascade control is widely used in different domains, ranging from chemical processes [40] to motion control [41]. It involves two controllers: i) the primary controller that sets a target setpoint; and ii) the secondary controller that confronts the target setpoint of the primary controller.
In the case of a QoE management scheme, the primary controller is the network-assisted scheme and decides the bandwidth allocation for each player, and the secondary controller is the ABR algorithm, which runs at each player and decides the bitrate level for video chunks.
**Requirements for network-assisted QoE management scheme.** From viewing the network-assisted scheme as a cascade control system, we can infer from hierarchical control theory [42, 43, 44, 45] that the primary controller (i.e., the bandwidth allocation algorithm) should meet the following requirements to achieve the optimal network-assisted QoE:
* _Player-Algorithm-Awareness_: The control system should allocate bandwidth to each player while considering the player's expected reactions to it, as higher available bandwidth does not necessarily lead to higher QoE.
* _Player-State-Awareness_: The control system should share the available bandwidth among the players while taking into consideration the players' internal state, e.g., used bitrate, buffer, etc.
* _Robustness_: The control system must be robust to maintain stability and optimality in a diverse set of conditions and unexpected events.
* _Flexibility_: The system should be able to optimize different metrics according to the network operator's high-level objectives.
These requirements are hard to meet in real video streaming applications, and thus achieving optimal network-assisted QoE is challenging. Client-side ABR algorithms are usually sophisticated, and it is challenging to build an explicit mathematical structure to describe them and to predict their reactions to network changes. Even assuming a known client-side ABR algorithm, players use a set of different variables (player's state) to determine the bitrate level of video chunks, including [46] bandwidth, buffer level, previous bitrate level, download time, and chunk size. Only a subset of these variables can be measured in the network layer on mostly encrypted traffic. As the ABR algorithms and the player's state are so hard to infer, the robustness requirement becomes harder to meet. Finally, optimizing certain objectives on top of ensuring stability and robustness makes network-assisted schemes even harder.
Note that the above-mentioned control-theoretic modeling can serve as a principled framework to reason about the soundness of network-assisted schemes in today's complex video streaming applications. This is very useful, as a poorly-designed network-assisted scheme can further degrade fairness [35], harm stability [17], and hamper efficiency [18]. According to our principled framework, the network-assisted schemes developed in [35, 8] do not meet _Player-Algorithm-Awareness_ requirement, as they can only address the diversity of players in terms of streaming devices. Also, the server-side schemes presented in [33, 34] do not meet the requirements of _Player-Algorithm-Awareness_, _Player-State-Awareness_, and _Flexibility_, as they do have control over only their own players. We stress that extending these schemes to meet all requirements is not straightforward.
**Cascade control benefits for network-assisted QoE.** Cascade control has multiple properties that facilitate network-assisted QoE schemes [20, 47]. First, the primary controller views the secondary control system solely via its inputs and outputs, meaning it is unaware of its exact workings and internal state. Similarly, the bandwidth allocation views the client-side ABR algorithm solely through its available bandwidth (input) and its effective bitrate (output). Second, cascade control responds more effectively to disturbances and corrects upsets (i.e., unexpected events) more quickly. Indeed, a secondary loop reduces the severity of disturbances and limits the overall variability experienced by the system. Thus, the use of cascade control can alleviate the impact of modeling and estimation errors. Third, the secondary controller is faster than the primary controller. Similarly, the players quickly respond to their available bandwidth [31]; while the bandwidth allocation can only respond slowly, as it takes time to observe the impact of bandwidth changes on the multiplayer QoE fairness. Importantly, letting the bandwidth allocation react more conservatively is beneficial, as we show in Section VII-C.
Fig. 4: Average QoE delivered to MPC and YouTube players for an equal bandwidth.
## V CANE Approach
### _Overview_
We develop CANE (CAScade control-based NEWork-assisted scheme) to achieve a near-optimal and fair network-assisted QoE. CANE runs at the router (or a wireless access point) and sends no information/command to the video players, and requires no modification on the client side.
CANE addresses the _Player-Algorithm-Awareness_ requirement by leveraging ML techniques, the _Player-State-Awareness_ and _Flexibility_ requirements by employing Model Predictive Control (MPC), and the _Robustness_ requirement by leveraging the intrinsic ability of cascade control structures and robustness feature of MPC schemes.
### _Workflow_
The deployment of CANE has two stages: offline and online. In the offline stage, we build an input-output mathematical model for each client-side ABR algorithm and store the models on the router for future online use.
In the online stage, at any time, CANE manipulates the bandwidth using five modules: i) throughput predictor; ii) state estimator; iii) ML-based models; iv) bandwidth allocation algorithm (i.e., MPC); and v) traffic shaping. As our goal in this paper is not to design throughput prediction nor state estimation mechanisms, we rely on existing approaches.
The computational overhead of CANE depends on the number of players, the complexity of the ML-based models, and the complexity of the MPC problem. For typical home-network scales, the computation cost of CANE is very low, and it can thus run in real time, as we show in Section VII. We leave large-scale evaluations (e.g., cellular edge with hundreds of players) to future work (see Section VIII).
## VI CANE Detailed Design
### _Preliminaries--Modeling_
General framework.Consider a set of \(N\) video players sharing a single bottleneck link. We denote the \(i\)-th player by \(P_{i}\), where \(i\in\{1,\cdots,N\}\). We assume that the link is the only bottleneck along the Internet path from the video players to the servers. We use \(w_{i}(t)\in\mathbb{R}_{\geq 0}\) to denote the bandwidth allocated to player \(P_{i}\) at time \(t\). We denote the bitrate level chosen by player \(P_{i}\) at time \(t\) by \(r_{i}(t)\in\mathcal{R}_{i}\), where \(\mathcal{R}_{i}\subset\mathbb{Z}\) is the set of available bitrate levels from which the ABR algorithm selects for player \(P_{i}\). For simplicity and without loss of generality, we assume that the set of available bitrate levels for all video players is the same, i.e., \(\mathcal{R}_{1}=\cdots=\mathcal{R}_{N}\).
Buffer level.Each player has a buffer to store downloaded yet unplayed video, which typically holds few tens of seconds of video segments. Let \(b_{i}(t)\in[0,\bar{B}_{i}]\in\mathbb{R}_{\geq 0}\) be the buffer level of player \(P_{i}\) at time \(t\). Namely, \(b_{i}(t)\) represents the amount of playtime of the video in the buffer. The \(\bar{B}_{i}\in\mathbb{R}_{\geq 0}\) is the maximum buffer level of player \(P_{i}\), which depends on the storage limitations of the device, as well as the policy of the content provider.
The buffer accumulates as a new video is downloaded and drains as the video is played. Thus, the buffer dynamics of player \(P_{i}\) can be approximated [11] as:
\[b_{i}(t+1)=\min\left\{\max\left\{b_{i}(t)+\left(\frac{w_{i}(t)}{r_{i}(t)}-1 \right)\Delta T,0\right\},\bar{B}_{i}\right\}, \tag{1}\]
where the time interval \([t,t+1)\) is equivalent to \(\Delta T\) seconds and \(\Delta T\in\mathbb{Z}_{>0}\) is constant.
Video quality.The video quality represents the expected player's opinion regarding the visual quality of a video. In general, the video quality depends on the bitrate level of video chunks, viewing distance, screen size and resolution. Assuming that there is a decent distance between player \(P_{i}\) and the streaming device, as shown in [48, 49, 50, 51, 52], the bitrate-quality relationship can be expressed through a sigmoid-like function. In this paper, we use the following function to quantify the video quality:
\[v_{i}(\theta_{i},r_{i}(t))=1-e^{-\theta_{i}\cdot r_{i}(t)}, \tag{2}\]
where \(v_{i}:\mathcal{R}\rightarrow[0,1]\) is the video quality perceived by player \(P_{i}\) at time \(t\), and \(\theta_{i}\in\mathbb{R}_{>0}\) is a scalar depending on the size and resolution of player \(P_{i}\)'s screen. The function \(v_{i}(\cdot)\) should be positive and non-decreasing [6]. Among all possible functions, we selected the one mentioned in (2), since it is concave (which makes the final optimization problem convex; see Section VI-B) and models the quality-saturation characteristic in video streaming applications [35]. For instance, the video quality in 3 [Mbps] and 1 [Mbps] may be similar on a mobile device with a smaller screen.
QoE.While players may differ in their preferences, the key elements of perceived QoE for each player are [6]: i) video quality; ii) quality variations; and iii) rebuffering. We define the QoE for player \(P_{i}\) at time \(t\) as
\[\begin{split} u_{i}(t)=& v_{i}(\theta_{i},r_{i}(t ))-\alpha\left|v_{i}(\theta_{i},r_{i}(t))-v_{i}(\theta_{i},r_{i}(t-1))\right| \\ &-\beta\cdot f(b_{i}(t)),\end{split} \tag{3}\]
where \(v_{i}(\theta_{i},r_{i}(t))\) is as in (2), \(\alpha\in\mathbb{R}_{\geq 0}\) is a design parameter that defines the tradeoff between high QoE and less quality changes, and \(\beta\in\mathbb{R}_{\geq 0}\) is a design parameter that defines the tradeoff between high QoE and high buffer level. In (3), the function \(f(\cdot)\) is a penalty function on the buffer level, which penalizes likelihood of rebuffering. One possible choice is
\[f(b_{i}(t))=e^{-\lambda\cdot b_{i}(t)}, \tag{4}\]
where \(\lambda\in\mathbb{R}_{>0}\) is a design parameter. The penalty function as in (4) can prevent rebuffering, since its value increases exponentially as the buffer level decreases. Note that by a proper selection for \(\alpha\) and \(\beta\), it can be ensured that \(u_{i}(t)\in[0,1],\ i\in\{1,\cdots,N\}\).
Video source providers design client-side ABR algorithms to maximize the player-perceived QoE, i.e., \(u_{i}(t)\). Different providers may use different elements or may consider different values for \(\alpha\) and \(\beta\) in their optimization. CANE uses (3) for all players regardless of their providers, as it requires a single merit to address the network-assisted QoE management problem. As a result, the QoEs computed by CANE do not necessarily reflect the QoEs recognized by providers.
### _MPC-based Bandwidth Allocation_
**MPC-based algorithms are sufficient for bandwidth allocation.** To meet _Player-Algorithm-Awareness_, _Player-State-Awareness_, and _Robustness_ requirements the control system should rely on feedback from the process. In control-theory literature, there are at least four such control algorithms: i) Proportional-Integral-Derivative (PID) control [53]; ii) Optimal control [54]; iii) Markov Decision Process (MDP) based control [55]; and iv) Model Predictive Control (MPC) [21]. Among them, an MPC-based algorithm is clearly the most suitable as it meets all the requirements we describe in SectionIV-A. While MPC is an adequate solution to our problem, we do not claim that it is also unique.
**Our contribution.** Despite the literature on MPC-based bandwidth allocation schemes (e.g., [11, 14]), to the best of our knowledge, they do not fully consider the cross-player differences, which is an important factor in today's video streaming applications. Our main originality is to use the systematic insights mentioned in Section IV to support the idea of using MPC for bandwidth allocation and to develop an MPC-based scheme that optimizes network-assisted QoE by taking into account the cross-player differences (i.e., players' importance/specifics).
**Control objectives.** Consider the prediction horizon \([t,t+T_{p}]\), where \(T_{p}\in\mathbb{Z}_{\geq 0}\) is the window size. It is assumed that the available bandwidth during this horizon is predicted _almost_ correctly. Concretely, at any \(t\), we know \(W(k),\ k=t,\cdots,t+T_{p}\), where \(W(k)\in\mathbb{R}_{\geq 0}\) is the total bandwidth at time instant \(k\). Since network conditions are stable and usually do not change drastically over a short horizon (i.e., tens of seconds) [30, 56], it is plausible to assume that the available bandwidth can be predicted for a short future horizon.
The average QoE of player \(P_{i}\) over the horizon \([t,t+T_{p}]\) can be predicted at time \(t\) as
\[U_{i}(t)=\frac{1}{T_{p}}\sum_{k=t}^{t+T_{p}}u_{i}(k), \tag{5}\]
where \(U_{i}\in[0,1]\). Based on this averaged QoE prediction, we can mathematically formulate our objectives in video streaming applications as control objectives as follows.
In multiplayer video streaming, two natural objectives are [11]: i) efficiency; and ii) multiplayer QoE fairness. The efficiency objective (a.k.a. social welfare) is the sum of utilities (i.e., QoE) of all players. It is noteworthy that the efficiency objective corresponds to \(\alpha\)-fairness [57], when \(\alpha=0\). Regarding the QoE fairness objective, we consider the pairwise fairness index, as it allows us to easily specify the importance of players through a set of scalars.
_Efficiency._ The efficiency objective can be formulated as
\[J_{e}\big{(}w_{i}(t:t+T_{p}),\ \forall i\big{)}=\frac{1}{N}\sum_{i=1}^{N}U_{ i}(t), \tag{6}\]
where \(J_{e}\in[0,1]\), and \(w_{i}(t:t+T_{p})=[w_{i}(t)\ \cdots\ w_{i}(t+T_{p})]^{\top}\in\mathbb{R}^{T_{p}+1}, \ i\in\{1,\cdots,N\}\).
_Multiplayer QoE Fairness._ Let \(\eta_{i}\in(0,1]\) be a weighting parameter defining the _importance_ of player \(P_{i}\), where \(\eta_{i}<\eta_{j}\) if player \(P_{i}\) is more important than player \(P_{j}\), and \(\eta_{i}=\eta_{j}\) if players \(P_{i}\) and \(P_{j}\) are equally important. The multiplayer QoE fairness objective can be formulated as
\[J_{f}\big{(}w_{i}(t:t+T_{p}),\ \forall i\big{)}=\sum_{i=1}^{N-1}\sum_{j>i }^{N}\frac{1}{N}\left|\eta_{i}U_{i}(t)-\eta_{j}U_{j}(t)\right|. \tag{7}\]
The objective function (7) addresses the players' importance. For instance, if \(\eta_{i}<\eta_{j}\) (i.e., player \(P_{i}\) in more important than player \(P_{j}\)), it implies that \(U_{i}(t)>U_{j}(t)\) when \(|\eta_{i}U_{i}(t)-\eta_{j}U_{j}(t)|=0\), which means that the important player receives a higher average QoE.
**Final optimization problem.** At this stage, we formulate the final optimization problem of MPC. The control goal is to maximize efficiency (i.e., \(\max J_{e}\)), and to minimize the pairwise difference of QoE (i.e., \(\min J_{f}\)). We can address this max-min problem via a convex combination of objective functions. That is, the final objective function is
\[J\big{(}w_{i}(t:t+T_{p}),\ \forall i\big{)}= -(1-\gamma)\cdot J_{e}\big{(}w_{i}(t:t+T_{p}),\ \forall i\big{)}\] \[+\gamma\cdot J_{f}\big{(}w_{i}(t:t+T_{p}),\ \forall i\big{)} \tag{8}\]
where \(\gamma\in[0,1]\) is the tradeoff coefficient which defines the attitude of the designer toward the above-mentioned control objectives. Finally, the optimization problem is given in (9), where (10) indicates the identified black-box models which will be discussed later.
According to (8), when \(\gamma\) has a high value, CANE optimizes the convex combination of efficiency and fairness with a high weight on the multiplayer QoE fairness. Our experimental results (see Section VII-C) suggest that, by setting \(\gamma\) to a large value, CANE can improve multiplayer QoE fairness, while achieving the same efficiency as the uniform traffic shaping described in Section III. This indicates that CANE performs well in the tradeoff space that involves efficiency and fairness that was observed in prior work [11, 27].
### _Black-Box Modeling of Video Players_
**Problem statement and strawman solutions.**_Player-Algorithm-Awareness_ requirement means that CANE requires
an approximation model that mimics the input-output behavior of each ABR algorithm, and mirrors actions of each ABR algorithm in response to changes in the inputs. To build such approximation models, there are some possible approaches. One may analyze Javascript code [58] or use manual experimentation [59] to understand the behavior of client-side ABR algorithms. A different approach is to model the external behavior of ABR algorithms without inspecting their internal workings. More specifically, one can encapsulate each client-side ABR algorithm into a black-box model, and then use ML techniques to build a model to approximate the input-output behavior of the algorithm [60]. There are two well-known candidate structures for this purpose: i) Neural Network (NN) [61]; and ii) Decision Tree (DT) [46].
The approaches that rely on code analysis are not suitable for CANE. First, decision-making may reside on the server side, which makes it impossible to infer in the network layer. Second, they increase the computational overhead. For instance, YouTube uses more than 80k lines of Javascript code [60], which cannot be analyzed by CANE in real-time applications. The NN-based or DT-based black-box models meet accuracy requirements and can mimic ABR algorithms accurately. In general, these models are too complex and increase the computational overhead of network-assisted schemes that typically run on routers.
**Our approach.** Maintaining real-time applicability of CANE requires limiting the complexity of black-box models. We use polynomial regressions to approximate video players' behavior, as: i) our analysis shows that polynomial regressions yield simple, yet sufficiently accurate approximations for our purpose; ii) polynomial regressions are easy to implement, which reduces the computational overhead of CANE; and iii) polynomial regressions make the feedback mechanism in CANE (i.e., the optimization problem (9)) easily computable.
Now, we describe the modeling details and discuss the obtained results. To understand a client-side ABR algorithm, the first step is to collect data about its behavior across controlled network conditions and different videos. For this purpose, we randomly select traces from FCC [23], Norway [24], and OBOE [25] datasets, and observe the input-output behavior of each client-side ABR algorithm. Each trace has a duration of 320 seconds; the minimum bandwidth is 0.1 [Mbps], and the maximum bandwidth is 12 [Mbps].
It should be noted that we consider only information that is accessible/estimable in the network layer. More precisely, we collect only the bandwidth, buffer, and previous bitrate level of the players. Note that, since manipulating the bandwidth occurs at the router, the bandwidth allocated to each player is available at the router. Also, there are some methods to estimate the players' buffers (e.g., [62] that reconstruct the player's buffer conditions by extracting HTTP information from traces, including request initiation times, range requests, their encoding rates, etc.) and to estimate bitrate levels (e.g., [63] that estimates the average bitrate by extracting the chunk size and chunk duration from HTTP logs) at the router.
We approximate the ABR algorithm of player \(P_{i}\) as
\[r_{i}(t)=h_{i}(b_{i}(t-T_{b}:t),w_{i}(t-T_{w}:t),r_{i}(t-1)), \tag{10}\]
where \(b_{i}(t-T_{b}:t)=[b_{i}(t-T_{b})\ \cdots\ b_{i}(t)]^{\top}\) and \(w_{i}(t-T_{w}:t)=[w_{i}(t-T_{w})\ \cdots\ w_{i}(t)]^{\top}\) with \(T_{b},T_{w}\in\mathbb{Z}_{\geq 0}\), and \(h_{i}:\mathbb{R}^{T_{b}+1}\times\mathbb{R}^{T_{w}+1}\times\mathcal{R}\to \mathcal{R}\) is a polynomial.
We use Polynomial Regression (ML technique) to compute the degree and coefficients of the polynomial \(h_{i}(\cdot)\) given in (10). More precisely, we use curve fitting techniques to identify the polynomial associated with each client-side ABR algorithm. We use 80% of collected data for training and the rest 20% for testing. Setting \(T_{b}=T_{w}=3\) and the degree of the polynomial \(h_{i}\) to 5, we build an approximation model for MPC players with 64.6% accuracy, for BOLA players with 67.5% accuracy, for BBA players with 54.4% accuracy, and for YouTube players with 51.5% accuracy. We define accuracy as the percentage of approximated bitrates that fall in the same resolution category as the original bitrate. Although it may be possible to improve accuracy, for instance, by increasing history depths and/or polynomial degree, this will increase the computational complexity of CANE. Note that we make no claim of our approximation models being highly accurate; rather, our goal is to build simple, yet sufficient approximation models that are applicable to CANE. In the next section, we will show that CANE is robust and can improve network-assisted QoE even with these mildly accurate approximation models.
### _Implementation Roadmap and Feasibility_
The focus of this paper is on the algorithmic and analytical aspects of the CANE framework. Hence, we do not have a fully working implementation (e.g., on a wireless access point or a cellular edge router). Still, for completeness, we discuss practical implementation challenges and how we can address them, leveraging available tools from previous work.
The main challenge in realizing CANE in practice is inferring details about the video sessions from encrypted traffic. Fortunately, this task has been the goal of previous research. We briefly mention a few, which can serve as a starting point for a full-fledged implementation and leave it as future work to realize an end-to-end system. This problem has been studied in depth in [64], where the authors use DNS requests to infer the start of video sessions and use machine learning models to predict application-level features such as video resolution. There are several methods to estimate the players' buffers (e.g., [62, 65]) and to estimate bitrate levels (e.g., [63, 64, 66, 67]) at the router. Also, some prior work (e.g., [65]) presents different ways to infer more data about video streaming sessions from encrypted video traffic. Thus, in an end-to-end setting, CANE can be combined with these methods at the router.
A secondary challenge in realizing CANE in practice is the compute and storage requirements [68, 69] of running the CANE algorithms and models. Note that we envision CANE running at the network edge, e.g., a wireless access point or a cellular edge router, not a "core" router. As such, the compute and storage requirements are pretty minimal, and we can build proof-of-concept implementations even on low-end hardware, e.g., Raspberry Pi.
## VII Evaluation
### _Setup_
**Players.** We consider players from MPC [6], BOLA [7], BBA [28], and YouTube connecting to Internet by a single router (i.e., \(N=4\)). In this paper, we consider a fully heterogeneous combination of players, i.e., one MPC, one BOLA, one BBA, and one YouTube player.
**Compared schemes.** We compare the following schemes:
* _Pure Client-Side ABR._ There is no control over the bandwidth allocated to the players.
* _Uniform Traffic Shaping._ The naive network-assisted scheme discussed in Section III, which shares the bandwidth equally among all players.
**Metrics.** We use the following performance metrics:
* _Social welfare (efficiency)._ Normalized sum of QoE of all players, as in (6).
* _Pairwise unfairness._ The sum of the absolute differences between the QoE of the players normalized by the number of the players, as in (7).
* _Weighted sum index._ The weighted sum of social welfare and pairwise unfairness, as in (8).
**Parameters.** The time horizon is discretized with \(\Delta T=1\) [s]. We use the weights \(\alpha=0.1\), \(\beta=0.1\), and \(\lambda=0.5\). We let \(\theta=2.1\times 10^{-3}\), \(T_{p}=4\), \(\gamma=0.75\), and \(\eta=1\).
**Throughput traces.** We use randomly selected traces from the FCC [23], Norway [24], and OBOE [25] datasets, which have a duration of 320 seconds and range from 0.1 to 12 [Mbps].
### _Experimental Set-up_
We have implemented the academic ABR algorithms (i.e., MPC, BBA, and BOLA) using Puffer [32]. For YouTube, we use the actual player. We use the tc (traffic control) tool [70] in Linux to change the bandwidth according to the throughput trace. We use the Selenium framework version 3.141.59 [71] in Python to automate the running of the players on different bandwidth traces.
**CANE implementation.** Regarding the offline stage, we use the same setup as in [60], and utilize mitmproxy [72] as the proxy. We use scikit-learn toolbox version 0.24.2 [73] for black-box modeling. Regarding the online stage of CANE, we implement it on Python 3.6, and use the SQLS algorithm from scipy.optimize toolbox of SciPy version 1.6.3 [74] for solving the optimization problem.
### _Results_
#### Vii-C1 Effectiveness investigation
We conduct three realistic video streaming scenarios to assess CANE's performance. In this section, we let \(\gamma=0.75\); we will confirm this selection later. While we evaluate CANE without implementing the estimation techniques to collect network-level and application-level information at the network device, our results are representative of a complete end-to-end implementation. Indeed, by incorporating a 4% uniform error for bitrate (worst case reported in [64]) and an 18% uniform error for buffer level (worst case reported in [62]), we observe a minor degradation in CANE's evaluation metrics. In particular, (in the median case) in all three scenarios, the social welfare metric decreases by 0.47%, the pairwise unfairness metric increases by 11.8%, and the weighted sum index increases by 8.16%. This observation demonstrates the ability of CANE to mitigate the impact of estimation errors.
**Scenario#1--Diverse players.** We first evaluate CANE when players are diverse but of the same importance, and use a similar device for streaming. Figure 5 shows the results obtained. With respect to social welfare, CANE shows 10.18% (up to 45.03%) gain over the pure client-side ABR, and has 2.88% median (up to 43.41%) gain over the uniform traffic shaping. With respect to pairwise unfairness, CANE has 52.54% median (up to 92.47%) gain over the pure client-side ABR, and has 19.4% median (up to 85.31%) gain over the uniform traffic shaping. With respect to the weighted sum index, CANE has 31.13% median (up to 62.82%) gain over the pure client-side ABR, and has 7.81% median (up to 42.64%) gain over the uniform traffic shaping.
**Scenario#2--Diverse players with different level of importance.** In this scenario, we assume that the MPC player is very important and needs to be prioritized. To address these cross-player differences, we use \(\eta=0.7\) for MPC player and \(\eta=1\) for others in the multiplayer QoE fairness objective given in (7). Fig. 6 shows the results obtained for this scenario. This figure shows that CANE has 1.68% median (up to 46.37%) gain over the pure client-side ABR with respect to social welfare. CANE achieves on-par social welfare, in most cases, comparable to that of the uniform traffic shaping. Still, CANE has up to 37.08% gain over the uniform traffic shaping. With respect to pairwise unfairness, CANE has 43.67% median (up to 70.36%) gain over the pure client-side ABR, and has 17.31% median (up to 64.51%) gain over the uniform traffic shaping. With respect to the weighted sum index, CANE has 27.13% median (up to 45.99%) gain over the pure client-side ABR, and has 8.87% median (up to 35.93%) gain over the uniform traffic shaping.
**Scenario#3--Diverse players with different streaming devices.** In this scenario, we assume that the BOLA player is streaming on a smaller screen. To address this characteristic, we use \(\theta=3.1\times 10^{-3}\) in (2) for BOLA player. Fig. 7 shows the obtained results for this scenario. Similarly to scenario #2, CANE yields median social welfare that is on par with that of uniform traffic shaping. CANE has up to 25.49% over uniform traffic shaping. Fig. 7 shows that CANE has 9.44% median (up to 44.57%) gain over the pure client-side ABR with respect to social welfare. With respect to pairwise unfairness, CANE has 47.97% median (up to 72.27%) gain over the pure client-side ABR, and has 19.92% median (up to 54.81%) gain over the uniform traffic shaping. With respect to the weighted sum index, CANE has 27.29% median (up to 46.67%) gain over the pure client-side ABR, and has 7.44% median (up to 25.65%) gain over the uniform traffic shaping.
#### Vii-C2 Sensitivity Analysis
We perform a sensitivity analysis of the performance of CANE
with respect to key design parameters, i.e., \(\gamma\) as in (8) and look-ahead horizon \(T_{p}\).
**Impact of the trade-off parameter \(\gamma\).** We consider two MPC and two YouTube players. Fig. 8 shows the impact of \(\gamma\) on the weighted sum index obtained. We see that as \(\gamma\) decreases (i.e., as we increase the weight on social welfare) the value of CANE above pure client-side ABR and uniform traffic shaping decreases. This observation verifies that CANE is more efficient at reducing unfairness than at increasing social welfare. More precisely, as we increase \(\gamma\), we put more weight on the metric that CANE is less efficient. As a result, the gain of CANE over the strawman schemes (i.e., pure client-side ABR and uniform traffic shaping) decreases. With this in mind, we selected \(\gamma=0.75\) in our simulations. Yet, a network operator can make an informed decision based on her high-level objectives.
**Impact of the look-ahead horizon \(T_{p}\).** Fig. 9 shows how the prediction window size impacts the performance and computation time of CANE. From Fig. 9-left, we see that as the look-ahead horizon increases, the performance of CANE improves as it takes into account more information about future conditions. However, as we look further into the future, the performance of CANE drops as the prediction accuracy reduces. Fig. 9-right shows that as the look-ahead horizon increases, the computation time of CANE climbs since it increases the size and complexity of the optimization problem 9. As a result, we selected \(T_{p}=4\), as it yields the best performance with an affordable computing time. A network operator can again select a parameter based on its computational power, the complexity of the optimization problem, and how effective she needs the system to be.
## VIII Conclusion
Prior work shows the potential benefits of network-assisted schemes in improving multiplayer QoE fairness. However, they do not provide a concrete roadmap for addressing the challenges of the practical realization of these schemes. Our work bridges this gap by developing a systematic and principled understanding of network-assisted schemes through the lens of a cascade control theory framework and builds
Fig. 5: Simulation results for diverse players of the same importance and similar device (Senario#1).
Fig. 6: Simulation results for diverse players with different level of importance (Senario#2).
Fig. 7: Simulation results for diverse players with different streaming devices (Senario#3).
Fig. 8: Impact of the tradeoff parameter \(\gamma\) on the weighted sum index of CANE.
based on these control-theoretic insights to develop CANE. We demonstrated a practical implementation of CANE. Our experimental results confirmed the effectiveness of CANE in achieving a near-optimal network-assisted QoE. In particular, CANE improves multiplayer QoE fairness by \(\sim\)50% on the median compared to the pure client-side ABR, and by \(\sim\)20% on median in comparison with the uniform traffic shaping. Our work prepares the ground for handling multiple mobile video players competing at the cellular edge, or other applications competing for bandwidth.
|
2303.02946 | Study on the weak decay between two heavy baryons $
\mathcal{B}_i(\frac{1}{2}^+)\to \mathcal{B}_f(\frac{3}{2}^+)$ in the
light-front quark model | In this work, we study the weak decay between two heavy baryons $
\mathcal{B}_i(\frac{1}{2}^+)\to \mathcal{B}_f(\frac{3}{2}^+)$ in the
light-front quark model where three-quark picture is employed for baryon. We
derive general form of transition amplitude of $
\mathcal{B}_i(\frac{1}{2}^+)\to \mathcal{B}_f(\frac{3}{2}^+)$, and analyze two
specific cases of transitions: the weak decays of single heavy baryon
$\Sigma_{b} \to \Sigma_{c}^*$ and the decays of double-charmed baryon
$\Xi_{cc}\to \Sigma_{c}^*(\Xi_{c}^*)$. We compute the hadronic form factors for
the transitions and apply them to study the decay widths of the semi-leptonic
$\mathcal B_i(\frac{1}{2}^+)\to\mathcal B_f(\frac{3}{2}^+) l\bar{\nu}_l$ and
non-leptonic $\mathcal B_i(\frac{1}{2}^+)\to\mathcal B_f(\frac{3}{2}^+)M$.
Previously we studied the transition $\Sigma_{b} \to \Sigma_{c}^*$ with the
quark-diquark picture of baryon in the light-front quark model. Here we revisit
this transition with three-quark picture of baryon. At the quark level, the
transition $\Sigma_{b} \to \Sigma_{c}^*$ is induced by the $b\rightarrow c$
transition.The subsystem of the two unchanged light quarks which possesses
definite and same spin in initial and final state can be viewed as a spectator,
so the spectator approximation can be applied directly. For the weak decay of
doubly charmed baryon $\Xi_{cc}$, a $c$ quark decays to a light quark $q_1$, so
both the initial state $cc$ and final state $q_1q_2$ ($q_1$ and the original
$q_2$ in initial state may be the same flavor quarks) which possess definite
spin are no longer spectators. A rearrangement of quarks for initial and final
states is adopted to isolate the unchanged subsystem $cq_2$ which can be viewed
as the spectator approximately. Future measurements on these channels will
constrain the nonperturbative parameter in the wavefunctions and test the model
predictions. | Fang Lu, Hong-Wei Ke, Xiao-Hai Liu, Yan-Liang Shi | 2023-03-06T07:32:16Z | http://arxiv.org/abs/2303.02946v2 | # Study on the weak decay between two heavy baryons
###### Abstract
In this work, we study the weak decay between two heavy baryons \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})\) in the light-front quark model where three-quark picture is employed for baryon. We derive general form of transition amplitude of \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})\), and analyze two specific cases of transitions: the weak decays of single heavy baryon \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) and the decays of double-charmed baryon \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})\). We compute the hadronic form factors for the transitions and apply them to study the decay widths of the semi-leptonic \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})l\bar{ \nu}_{l}\) and non-leptonic \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})M\). Previously we studied the transition \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) with the quark-diquark picture of baryon in the light-front quark model. Here we revisit this transition with three-quark picture of baryon. At the quark level, the transition \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) is induced by the \(b\to c\) transition.The subsystem of the two unchanged light quarks which possesses definite and same spin in initial and final state can be viewed as a spectator, so the spectator approximation can be applied directly. For the weak decay of doubly charmed baryon \(\Xi_{cc}\), a \(c\) quark decays to a light quark \(q_{1}\), so both the initial state \(cc\) and final state \(q_{1}q_{2}\) (\(q_{1}\) and the original \(q_{2}\) in initial state may be the same flavor quarks) which possess definite spin are no longer spectators. A rearrangement of quarks for initial and final states is adopted to isolate the unchanged subsystem \(cq_{2}\) which can be viewed as the spectator approximately. Future measurements on these channels will constrain the nonperturbative parameter in the wavefunctions and test the model predictions.
pacs: 13.30.-a, 12.39.Ki, 14.20.Lq
Introduction
Over the last decade, a great interest has been aroused in the field of hadron physics, especially heavy baryons. Significant progresses have been achieved in both experiment and theory. For example, the LHCb collaboration observed the doubly charmed baryon \(\Xi_{cc}^{++}\) in the final state \(\Lambda_{c}K^{-}\pi^{+}\pi^{+}\)[1] and it has been confirmed in the decays \(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}\)and \(\Xi_{cc}^{++}\to\Xi_{c}^{{}^{\prime}+}\pi^{+}\)[2; 3]. On the theory side, multiple approaches have been developed to study the physical properties of heavy baryons, such as decay width. For example, in Refs. [4; 5; 6] the authors employed light-front quark model (LFQM) to explore the weak decays of doubly heavy baryons. In Ref. [7] the weak decays of doubly heavy baryons were studyed within light-cone sum rules. In retrospect, weak decays of the heavy baryons were also studied under the heavy quark limit [8], the relativistic quark model with quark-diquark picture [9], the relativistic three-quark model [10] and the Bethe-Salpeter approach [11].
In Refs. [4; 5; 6; 12; 13; 14; 15] the LFQM was extended to study the weak decays of the heavy baryons with the quark-diquark picture. However, although the quark-diquark picture is an effective approximation when the diquark is a spectator in the transition, it is not very suitable for studying the decay where the diquark will be broken. Instead, three-quark picture was employed to study the weak decays between two heavy baryons with \(J^{P}=\frac{1}{2}^{+}\) in Refs. [16; 17; 18; 19]. In Ref. [12] the transition between \(J^{P}=\frac{1}{2}^{+}\) heavy baryon (\(\mathcal{B}_{i}(\frac{1}{2}^{+})\)) and \(J^{P}=\frac{3}{2}^{+}\) one (\(\mathcal{B}_{f}(\frac{3}{2}^{+})\)) was studied with the quark-diquark picture. In this work we employ the three-quark picture to study the transition \(\mathcal{B}_{i}(\frac{1}{2}^{+})\to\mathcal{B}_{f}(\frac{3}{2}^{+})\) within the LFQM framework. The LFQM is a relativistic quark model which has been applied to study transitions among mesons [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33] and has been extended to the case of baryon decay [4; 5; 6; 12; 13; 14; 15]. We derive general forms of transition amplitude \(\mathcal{B}_{i}(\frac{1}{2}^{+})\to\mathcal{B}_{f}(\frac{3}{2}^{+})\). Then we revisit the transition \(\Sigma_{b}\to\Sigma_{c}^{*}\) and study the relative weak decays of the double charmed baryon \(\Xi_{cc}\).
For the weak decay of single heavy baryon, the \(b\) quark in the initial state would transit into the \(c\) quark in the final state by emitting \(W\) bosons. The two light quarks do not take part in the process of the weak decay and the subsystem of the two light quarks can be regarded as a spectators. Under the three-quark picture, the three quarks are regarded as independent individuals. Although the two light quarks are no longer point-like diquark, the subsystem where they reside still has a definite spin, color, isospin and all the quantum numbers of the subsystem keep unchanged so it is treated as the spectator. For the weak decay of doubly charmed baryon (\(ccq_{2}\)), one \(c\) quark in the initial state decays to a lighter quark \(q_{1}\) by emitting \(W\) bosons via the weak interaction. The resulting new quark forms a subsystem (\(q_{1}q_{2}\)) with the light quark \(q_{2}\) which comes from the initial state. As a result, both \(cc\) pair in the initial state and \(q_{1}q_{2}\) in the final state which have definite spin cannot be treated as the spectators in the decay. A rearrangement of quarks for the initial and final baryons is necessary to isolate the unchanged subsystem \(cq_{2}\), which can be viewed as the spectator approximately.
In LFQM, we need to specify the vertex functions of the initial state and the final state according to the total spin and parity of the baryon. Here the quantum numbers of the initial state and the final state are \(\frac{1}{2}^{+}\) and \(\frac{3}{2}^{+}\), respectively. We adopt the vertex function of
\(\frac{1}{2}^{+}\) baryon in Ref. [16] and construct the three-body vertex function of \(\frac{3}{2}^{+}\) in analog to Ref. [16]. Then we derive the transition matrix element, extract the form factors and compute them numerically.
Using these form factors we calculate the associated non-leptonic decays and semileptonic decays. The leptons are not involved in the strong interaction which means the semileptonic decay is less contaminated by the non-perturbative QCD effect, hence the study on semileptonic decay can be very helpful to constrain the model parameters and test model predictions. For the non-leptonic two-body decays \(\Sigma_{b}\to\Sigma_{c}^{*}M\), \(\Xi_{cc}\to\Sigma_{c}^{*}M\) and \(\Xi_{cc}\to\Xi_{c}^{*}M\), by neglecting the interaction between the final states, the transition element is factorized to a product of the meson decay constant and the transition amplitude \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) under the factorization assumption. Future measurement on these channels will be necessary to further test model predictions and elucidate the underlying mechanism of heavy hadron decay.
This paper is organized as follows: in section II we present the vertex functions of the heavy flavor baryons, and derive the form factors of the transition \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) in the LFQM. In section III we present numerical results for the transition \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) along with all necessary input parameters. We calculate the form factors and the decay widths of related semi-leptonic and non-leptonic decays. Section IV is devoted to our conclusions and discussions.
## II \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) in the light-front quark model
In this paper we study the transition \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) with the three-quark picture of baryon in LFQM. We focus on the weak decay of single heavy baryon \(\Sigma_{b}\) to \(\Sigma_{c}^{*}\) and the doubly charmed baryon \(\Xi_{cc}\) to single charmed baryon \(\Sigma_{c}^{*}\) or \(\Xi_{c}^{*}\).
### the vertex functions
In Ref. [16], the vertex function of a baryon \({\cal B}\) with the total spin \(S=1/2\) and total momentum \(P\) was discussed1. For a single heavy baryon it could be expressed as
Footnote 1: Since the orbital angle momentum we study is 0, the total angle momentum \(J\) is equal to the total spin \(S\) of the baryon.
\[|{\cal B}(P,S,S_{z})\rangle=\int\!\{d^{3}\tilde{p}_{1}\}\{d^{3} \tilde{p}_{2}\}\{d^{3}\tilde{p}_{3}\}\,2(2\pi)^{3}\delta^{3}(\tilde{P}-\tilde {p}_{1}-\tilde{p}_{2}-\tilde{p}_{3})\] \[\times\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\Psi^{SS_{z}}_{S _{q_{1}q_{2}}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_ {2},\lambda_{3}){\cal C}^{\alpha\beta\gamma}{\cal F}_{Q,q_{1},q_{2}}\,|\,Q_{ \alpha}(p_{1},\lambda_{1})q_{1\beta}(p_{2},\lambda_{2})q_{2\gamma}(p_{3}, \lambda_{3})\rangle, \tag{1}\]
where \(Q\) denotes heavy quark (\(b\) or \(c\)), \(q_{1}(q_{2})\) denotes the flavor of the light quark, \(\lambda_{i}\) and \(p_{i}\,(i=1,2,3)\) are helicities and light-front momenta of the quarks, \({\cal C}^{\alpha\beta\gamma}\) and \({\cal F}_{Q,q_{1},q_{2}}\) are the color and flavor factors. The total spin of \(q_{1}\) and \(q_{2}\) is denoted to \(S_{q_{1}q_{2}}\) which is 0 for
\(\Lambda_{b}\left(\Lambda_{c}\right)\) and \(1\) for \(\Sigma_{b}\left(\Sigma_{c}\right)\), respectively. In light-front approach, the on-mass-shell light-front momentum \(p\) is defined as
\[\tilde{p}=(p^{+},p_{\perp}),\quad,p_{\perp}=(p^{1},p^{2}),\quad p^{-}=\frac{m^{ 2}+p_{\perp}^{2}}{p^{+}}\,\quad\{d^{3}p\}=\frac{dp^{+}d^{2}p_{\perp}}{2(2\pi)^{3}}. \tag{2}\]
The momentum-space wave function \(\Psi_{0}^{SS_{z}}\) and \(\Psi_{1}^{SS_{z}}\)[16, 34] are
\[\Psi_{0}^{SS_{z}}(\tilde{p}_{i},\lambda_{i})= A_{0}\bar{u}(p_{3},\lambda_{3})[(\bar{\not{P}}+M_{0})\gamma_{5}]v(p_{2}, \lambda_{2})\bar{u}(p_{1},\lambda_{1})u(\bar{P},S)\phi_{\cal B}(x_{i},k_{i \perp}),\] \[\Psi_{1}^{SS_{z}}(\tilde{p}_{i},\lambda_{i})= A_{1}\bar{u}(p_{3},\lambda_{3})[(\bar{\not{P}}+M_{0})\gamma_{ \perp}^{\beta}]v(p_{2},\lambda_{2})\bar{u}(p_{1},\lambda_{1})\gamma_{\perp \beta}\gamma_{5}u(\bar{P},S)\phi_{\cal B}(x_{i},k_{i\perp}). \tag{3}\]
For a single heavy baryon \({\cal B}\) with total spin \(S=3/2\) the vertex function is the same as that in Eq. (1) except \(\Psi_{1}^{SS_{z}}\) is replaced by \(\Psi_{1}^{{}^{\prime}SS_{z}}\)[35], which is given by
\[\Psi_{1}^{{}^{\prime}SS_{z}}(\tilde{p}_{i},\lambda_{i})= A_{1}^{\prime}\bar{u}(p_{3},\lambda_{3})[(\bar{\not{P}}+M_{0}) \gamma_{\perp}^{\alpha}]v(p_{2},\lambda_{2})\bar{u}(p_{1},\lambda_{1})u_{\alpha }(\bar{P},S)\phi_{\cal B}(x_{i},k_{i\perp}). \tag{4}\]
With the normalization of the state \({\cal B}\)
\[\langle{\cal B}(P^{\prime},S^{\prime},S^{\prime}_{z})|{\cal B}(P,S,S_{z}) \rangle=2(2\pi)^{3}P^{+}\delta^{3}(\tilde{P}^{\prime}-\tilde{P})\delta_{S^{ \prime}S}\delta_{S^{\prime}_{z}S_{z}}, \tag{5}\]
and
\[\int(\prod_{i=1}^{3}\frac{dx_{i}d^{2}k_{i\perp}}{2(2\pi)^{3}})2(2\pi)^{3} \delta(1-\sum x_{i})\delta^{2}(\sum k_{i\perp})\phi_{\cal B}^{*}(x_{i},k_{i \perp})\phi_{\cal B}(x_{i},k_{i\perp})=1, \tag{6}\]
we can compute the factors \(A_{0}\), \(A_{1}\) and \(A_{1}^{\prime}\):
\[A_{0} =\frac{1}{4\sqrt{P^{+}M_{0}^{3}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e _{3})}},\] \[A_{1} =\frac{1}{4\sqrt{3P^{+}M_{0}^{3}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+ e_{3})}},\] \[A_{1}^{\prime} =\frac{1}{4\sqrt{2P^{+}M_{0}^{3}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+ e_{3})}}, \tag{7}\]
where \(p_{i}\cdot\bar{P}=e_{i}M_{0}\,(i=1,2,3)\) is used, \(\bar{P}=p_{1}+p_{2}+p_{3}\) and \(e_{i}\) is defined in Eq. (12).
For the weak decay of doubly charmed baryon, the vertex functions with total spin \(S=1/2\) and momentum \(P\) is
\[|{\cal B}(P,S,S_{z})\rangle=\int\{d^{3}\tilde{p}_{1}\}\{d^{3} \tilde{p}_{2}\}\{d^{3}\tilde{p}_{3}\}\,2(2\pi)^{3}\delta^{3}(\tilde{P}-\tilde {p}_{1}-\tilde{p}_{2}-\tilde{p}_{3})\] \[\times\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\Psi_{S_{QQ}}^{SS _{z}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2}, \lambda_{3}){\cal C}^{\alpha\beta\gamma}{\cal F}_{Q,Q,q_{2}}\,|\,Q_{\alpha}(p_ {1},\lambda_{1})Q_{\beta}(p_{2},\lambda_{2})q_{2\gamma}(p_{3},\lambda_{3})\rangle. \tag{8}\]
In order to describe the momenta of the constituent quarks, the intrinsic variables \((x_{i},k_{i\perp})\) ( \(i=1,2,3\)) are introduced through
\[p_{i}^{+}=x_{i}P^{+},\qquad p_{i\perp}=x_{i}P_{\perp}+k_{i\perp}\qquad x_{1}+x _{2}+x_{3}=1,\qquad k_{1\perp}+k_{2\perp}+k_{3\perp}=0, \tag{9}\]
where \(x_{i}\) is the momentum fraction with the constraint \(0<x_{1},x_{2},x_{3}<1\). The variables \((x_{i},k_{i\perp})\) are Lorentz-invariant since they are independent of the total momentum of the hadron. The invariant mass square \(M_{0}^{2}\) is defined as a function of the internal variables \(x_{i}\) and \(k_{i\perp}\):
\[M_{0}^{2}=\frac{k_{1\perp}^{2}+m_{1}^{2}}{x_{1}}+\frac{k_{2\perp}^{2}+m_{2}^{2} }{x_{2}}+\frac{k_{3\perp}^{2}+m_{3}^{2}}{x_{3}}. \tag{10}\]
The internal momenta are defined as
\[k_{i}=(k_{i}^{-},k_{i}^{+},k_{i\perp})=(e_{i}-k_{iz},e_{i}+k_{iz},k_{i\perp})=( \frac{m_{i}^{2}+k_{i\perp}^{2}}{x_{i}M_{0}},x_{i}M_{0},k_{i\perp}). \tag{11}\]
It is easy to obtain
\[e_{i} = \frac{x_{i}M_{0}}{2}+\frac{m_{i}^{2}+k_{i\perp}^{2}}{2x_{i}M_{0}},\] \[k_{iz} = \frac{x_{i}M_{0}}{2}-\frac{m_{i}^{2}+k_{i\perp}^{2}}{2x_{i}M_{0}}, \tag{12}\]
where \(e_{i}\) is the energy of the \(i\)-th constituent, and they obey the condition \(e_{1}+e_{2}+e_{3}=M_{0}\). The transverse \(k_{i\perp}\) and \(z\) direction \(k_{iz}\) components constitute a momentum vector \(\vec{k}_{i}=(k_{i\perp},k_{iz})\).
The spatial wave function \(\phi_{\cal B}(x_{i},k_{i\perp})\)[36; 37] is defined as
\[\phi_{\cal B}(x_{1},x_{2},x_{3},k_{1\perp},k_{2\perp},k_{3\perp})=\frac{e_{1} e_{2}e_{3}}{x_{1}x_{2}x_{3}M_{0}}\varphi(\overrightarrow{k}_{1},\beta_{1}) \varphi(\frac{\overrightarrow{k}_{2}-\overrightarrow{k}_{3}}{2},\beta_{23}) \tag{13}\]
with \(\varphi(\overrightarrow{k},\beta)=4(\frac{\pi}{\beta^{2}})^{3/4}{\rm exp}( \frac{-k_{z}^{2}-k_{z}^{2}}{2\beta^{2}})\), where \(\beta\) is a non-perturbative parameter that characterizes the shape of wave function. We will discuss the parameter selection in section III.
### the form factors of \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) in LFQM
The form factors for the transition from initial heavy baryon \({\cal B}_{i}(\frac{1}{2}^{+})\) (i.e. \(|{\cal B}(P,1/2,S_{z})\)) to final heavy baryon \({\cal B}_{f}(\frac{3}{2}^{+})\) (\(|{\cal B}(P^{\prime},3/2,S^{\prime}_{z})\)) are defined as
\[\langle{\cal B}_{f}(\frac{3}{2}^{+})\mid\bar{Q}_{2}\gamma^{\mu}(1 -\gamma_{5})Q_{1}\mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle=\] \[\bar{u}_{\alpha}(P^{\prime},S^{\prime}_{z})\left[\gamma^{\mu}P^{ \alpha}\frac{f_{1}(q^{2})}{M_{{\cal B}_{i}}}+\frac{f_{2}(q^{2})}{M_{{\cal B}_{ i}}^{2}}P^{\alpha}P^{\mu}+\frac{f_{3}(q^{2})}{M_{{\cal B}_{i}}M_{{\cal B}_{f}}}P^{ \alpha}P^{\prime\mu}+f_{4}(q^{2})g^{\alpha\mu}\right]u(P,S_{z})-\] \[\bar{u}_{\alpha}(P^{\prime},S^{\prime}_{z})\left[\gamma^{\mu}P^{ \alpha}\frac{g_{1}(q^{2})}{M_{{\cal B}_{i}}}+\frac{g_{2}(q^{2})}{M_{{\cal B}_{ i}}^{2}}P^{\alpha}P^{\mu}+\frac{g_{3}(q^{2})}{M_{{\cal B}_{i}}M_{{\cal B}_{f}}}P^{ \alpha}P^{\prime\mu}+g_{4}(q^{2})g^{\alpha\mu}\right]\gamma_{5}u(P,S_{z}), \tag{14}\]
where momentum \(q\equiv P-P^{\prime}\). \(Q_{1}\) and \(Q_{2}\) denote heavy quark operators (See Fig. 1). \(M_{{\cal B}_{i}}\) and \(M_{{\cal B}_{f}}\) represent the masses of heavy baryon \({\cal B}_{i}\) and \({\cal B}_{f}\), respectively. For the deday of doubly charmed baryon \(Q_{2}\) should be replaced by \(q_{1}\) (See Fig. 2).
Here the momenta of initial and final baryons \(P\) and \(P^{\prime}\) obey the on-shell relations \(E^{(^{\prime})2}=P^{(^{\prime})2}+M^{(^{\prime})2}\). However, in Eq. (3) and (4) the spinors are function of \(\bar{P}\) and \(\bar{P}^{\prime}\), which are the sums of the momenta of the involved constituent quarks and do not obey physical on-shell relations. To reconcile the conflict, here we follow the approach in the previous study [12] and assume form factors ( \(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{4}\), \(g_{1}\), \(g_{2}\), \(g_{3}\), \(g_{4}\)) are the same in both physical and unphysical form. Then Eq. (14) is re-written as the following equation where the spinors are off-shell:
\[\langle{\cal B}_{f}(\frac{3}{2}^{+})\mid\bar{Q_{2}}\gamma^{\mu}(1 -\gamma_{5})Q_{1}\mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle=\] \[\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})\left[\gamma^{ \mu}\bar{P}^{\alpha}\frac{f_{1}(q^{2})}{M_{{\cal B}_{i}}}+\frac{f_{2}(q^{2})} {M_{{\cal B}_{i}}^{2}}\bar{P}^{\alpha}\bar{P}^{\mu}+\frac{f_{3}(q^{2})}{M_{{ \cal B}_{i}}M_{{\cal B}_{f}}}\bar{P}^{\alpha}\bar{P}^{\prime\mu}+f_{4}(q^{2})g ^{\alpha\mu}\right]u(\bar{P},S_{z})-\] \[\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})\left[\gamma^{ \mu}\bar{P}^{\alpha}\frac{g_{1}(q^{2})}{M_{{\cal B}_{i}}}+\frac{g_{2}(q^{2})} {M_{{\cal B}_{i}}^{2}}\bar{P}^{\alpha}\bar{P}^{\mu}+\frac{g_{3}(q^{2})}{M_{{ \cal B}_{i}}M_{{\cal B}_{f}}}\bar{P}^{\alpha}\bar{P}^{\prime\mu}+g_{4}(q^{2})g ^{\alpha\mu}\right]\gamma_{5}u(\bar{P},S_{z}). \tag{15}\]
#### ii.2.1 the transition \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\)
The lowest order Feynman diagram responsible for the \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) ( \(\Sigma_{b}\) and \(\Sigma_{c}^{*}\) are \({\cal B}_{i}(\frac{1}{2}^{+})\) and \({\cal B}_{f}(\frac{3}{2}^{+})\), respectively) weak decay is shown in Fig. 1. Following the approach given in Refs. [13; 14; 36; 37] the transition matrix element can be calculated with the vertex functions of \(\mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle\) and \(\mid{\cal B}_{f}(\frac{3}{2}^{+})\rangle\),
\[\langle{\cal B}_{f}(\frac{3}{2}^{+})\mid\bar{Q_{2}}\gamma^{\mu}(1 -\gamma_{5})Q_{1}\mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle=\] \[\int\frac{\{d^{3}\tilde{p}_{2}\}\{d^{3}\tilde{p}_{3}\}\phi^{*}_{B_ {f}}(x^{\prime},k^{\prime}_{\perp})\phi_{B_{i}}(x,k_{\perp}){\rm Tr}[\gamma_{ \perp}^{\alpha}(\not{\bar{P}^{\prime}}\!\!+M_{0}^{\prime})(\not{p}_{3}\!\!+m_{ 3})(\not{\bar{P}}\!\!+M_{0})\gamma_{\perp}^{\beta}(\not{p}_{2}\!\!-m_{2})]}{16 \sqrt{6p_{1}^{+}p_{1}^{\prime+}p_{1}^{\prime+}P^{\prime}}\!\!+\!M_{0}^{3}M_{0 }^{\prime 3}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e_{3})(m_{1}^{\prime}+e_{1}^{\prime} )(m_{2}^{\prime}+e_{2}^{\prime})(m_{3}^{\prime}+e_{3}^{\prime})}\] \[\times\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})(\not{p}_{ 1}^{\prime}+m_{1}^{\prime})\gamma^{\mu}(1-\gamma_{5})(\not{p}_{1}\!\!+m_{1}) \gamma_{\perp\beta}\gamma_{5}u(\bar{P},S_{z}), \tag{16}\]
where for the weak decay of single heavy baryons,
\[m_{1}=m_{b},\qquad m_{1}^{\prime}=m_{c},\qquad m_{2}=m_{q_{1}},\qquad m_{3}=m_ {q_{2}}.\]
Figure 1: The Feynman diagram for single heavy baryon weak decay, where \(\bullet\) denotes \(V-A\) current vertex.
\(p_{1}\) denotes the four-momentum of the heavy quark \(b\), \(p_{1}^{\prime}\) denotes the four-momentum of the quark \(c\), \(P\) (\(P^{\prime}\)) stands as the four-momentum of \({\cal B}_{i}\) (\({\cal B}_{f}\)). Setting \(\tilde{p}_{2}=\tilde{p}_{2}^{\prime}\), \(\tilde{p}_{3}=\tilde{p}_{3}^{\prime}\) we have
\[x_{i}^{\prime}=\frac{P^{+}}{P^{\prime+}}x_{i},\qquad k_{1\perp}^{\prime}=k_{1 \perp}-(1-x_{1})q_{\perp},\qquad k_{2\perp}^{\prime}=k_{2\perp}+x_{2}q_{\perp}, \qquad k_{3\perp}^{\prime}=k_{3\perp}+x_{3}q_{\perp}. \tag{17}\]
Multiplying the following expressions \(\bar{u}(\bar{P},S_{z})\gamma_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime},S_{z} ^{\prime})\), \(\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}^{\prime}\bar{P}^{\xi}u_{\xi}(\bar{P}^{ \prime},S_{z}^{\prime})\), \(\bar{u}(\bar{P},S_{z})g_{\mu}^{\xi}u_{\xi}(\bar{P}^{\prime},S_{z}^{\prime})\) to the right sides of both Eq. (15) and Eq. (16) and then summing over the polarizations of all states, we can obtain four algebraic equations and each equation contains the form factors \(f_{1}\), \(f_{2}\), \(f_{3}\) and \(f_{4}\). Solving these equations, we can get the explicit expressions of these form factors \(f_{i}(i=1,2,3,4)\) (See Appendix \(A\) for detail).
Similarly, multiplying these expressions \(\bar{u}(\bar{P},S_{z})\gamma_{\mu}\bar{P}^{\xi}\gamma_{5}u_{\xi}(\bar{P}^{ \prime},S_{z}^{\prime})\), \(\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}^{\prime}\bar{P}^{\xi}\gamma_{5}u_{\xi}( \bar{P}^{\prime},S_{z}^{\prime})\), \(\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}\bar{P}^{\xi}\gamma_{5}u_{\xi}(\bar{P}^{ \prime},S_{z}^{\prime})\), and \(\bar{u}(\bar{P},S_{z})g_{\mu}^{\xi}\gamma_{5}u_{\xi}(\bar{P}^{\prime},S_{z}^{ \prime})\) to the right sides of both Eq. (15) and Eq. (16) and solving for four algebraic equations, we can obtain analytical expressions of \(g_{i}(i=1,2,3,4)\).
#### iii.1.2 the transition \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})\)
For the weak decay of doubly charmed baryon \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})\), one \(c\) quark in the initial state (\([cc]q_{2}\)) decays to a light quark \(q_{1}\) in the final state (\(c[q_{1}q_{2}]\)) by weak interaction (Fig. 2). This light quark \(q_{1}\) combines with the light quark \(q_{2}\) from the initial state to form a physical subsystem \(q_{1}q_{2}\) which possesses definite spin i.e. the quark composition in the initial state is \([cc]q_{2}\), and that in the final state is \(c[q_{1}q_{2}]\) (\(q_{1}\) and \(q_{2}\) may be same or not), which means neither the original \([cc]\) nor the final \([q_{1}q_{2}]\) are spectators. The physical subsystem (diquark) is not spectator in the transition, so the quarks need to be rearranged. Therefore, the \([cc]\) diquark in the initial state needs to be destroyed and then rearranged with another light quark using a Racah transformation. The related transformations are [5]
\[[c^{1}c^{2}]_{1}[q_{2}]=\frac{\sqrt{2}}{2}(-\frac{\sqrt{3}}{2}[c^{2}][c^{1}q_{ 2}]_{0}+\frac{1}{2}[c^{2}][c^{1}q_{2}]_{1}-\frac{\sqrt{3}}{2}[c^{1}][c^{2}q_{ 2}]_{0}+\frac{1}{2}[c^{1}][c^{2}q_{2}]_{1}), \tag{18}\]
Figure 2: The Feynman diagram for doubly charmed baryon weak decay, where \(\bullet\) denotes \(V-A\) current vertex.
where the subscript after brackets 0 or 1 denotes the total spin of two quarks in the brackets. The superscript for every \(c\) quark is added to distinguish each other. After the rearrangement \(\Psi^{SS_{z}}_{Sc_{c}}(\tilde{p}_{i},\lambda_{i})\) can be expressed to \(-\frac{\sqrt{6}}{2}\Psi^{SS_{z}}_{0}(\tilde{p}_{i},\lambda_{i})+\frac{\sqrt{2}} {2}\Psi^{SS_{z}}_{1}(\tilde{p}_{i},\lambda_{i})\) where the subscript 0 and 1 are the total spin of \(cq_{2}\) subsystem.
Since the final state baryon \({\cal B}_{f}\) (\(\Sigma_{c}^{*}\) or \(\Xi_{c}^{*}\)) has spin of 3/2, the Racah transformation leads to the following rearrangement:
\[[c][q_{1}q_{2}]_{1}=[q_{1}][cq_{2}]_{1}=[q_{2}][cq_{1}]_{1}, \tag{19}\]
and the expression \(\Psi^{{}^{\prime}SS_{z}}_{cq_{1}}(\tilde{p}^{\prime}_{i},\lambda^{\prime}_{i})\) or \(\Psi^{{}^{\prime}SS_{z}}_{cq_{2}}(\tilde{p}^{\prime}_{i},\lambda^{\prime}_{i})\) is just \(\Psi^{{}^{\prime}SS_{z}}_{1}(\tilde{p^{\prime}}_{i},\lambda^{\prime}_{i})\).
After the rearrangement the spectators in the process are isolated so the spectator approximation can be used in the calculation. In terms of the rearrangement the expressions of the form factors for the transition \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})\) have an additional factor \(\frac{\sqrt{2}}{2}\) relative to those for \(\Sigma_{b}\to\Sigma_{c}^{*}\).
## III Numerical results
### The \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\) form factors
In order to compute the relative transition rates of semi-leptonic decays and non-leptonic decays of \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})\), we need to calculate the aforementioned form factors numerically. First of all, we need to fix the free parameters of the model, including masses of quarks and baryons, and wavefunction \(\beta\) parameters. The masses of quarks given in Ref. [23] and the masses of baryons taken from Refs. [1; 38] are listed in table 1.
There is no precise measure of the parameters \(\beta_{1}\), \(\beta_{23}\), \(\beta^{\prime}_{1}\) and \(\beta^{\prime}_{23}\) in the wave functions of the initial and the final baryons. Generally the reciprocal of \(\beta\) is related to the electrical radium of the baryon. Since the strong coupling strength between \(q_{1}\) and \(q_{2}\) is half of that between \(q_{1}\bar{q}_{2}\). Assuming a Coulomb-like potential, one can expect the radius of \(q_{1}q_{2}\) to be \(1/\sqrt{2}\) times that of \(q_{1}\bar{q}_{2}\) i.e. \(\beta_{q_{1}q_{2}}\approx\sqrt{2}\beta_{q_{1}\bar{q}_{2}}\). In Ref. [39] in terms of the binding energy the authors also obtained the same results. Therefore in our work we use the \(\beta\) values in the mesons case (\(\beta_{q_{1}\bar{q}_{2}}\)) [23] to estimate \(\beta_{q_{1}q_{2}}\). As for the value of \(\beta_{1}\) we refer to the \(\beta\) values of the heavy m
\begin{table}
\begin{tabular}{c c c c c c c c c c} \(m_{c}\) & \(m_{b}\) & \(m_{d(u)}\) & \(m_{s}\) & \(\Sigma_{b}\) & \(\Sigma_{c}^{*}\) & \(\Omega_{b}\) & \(\Omega_{c}^{*}\) & \(\Xi_{b}^{{}^{\prime}}\) & \(\Xi_{cc}\) & \(\Xi_{c}^{*}\) \\ \hline
1.3 & 4.4 & 0.25 & 0.5 & 5.811 & 2.517 & 6.065 & 2.768 & 5.937 & 3.621 & 2.646 \\ \end{tabular}
\end{table}
Table 1: The masses of the involved quarks and baryons (in units of GeV).
Since the form factors \(f_{i}(i=1,2,3,4)\) and \(g_{i}(i=1,2,3,4)\) is calculated in the frame \(q^{+}=0\) i.e. \(q^{2}=-q_{\perp}^{2}\leq 0\) (the space-like region) one needs to extend them into the time-like region to evaluate the transition rates. In Refs. [36; 37] the form factors were parameterized using the three-parameter form
\[F(q^{2})=\frac{F(0)}{\left(1-\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right) \left[1-a\left(\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}\right)+b\left(\frac{q^{2} }{M_{\mathcal{B}_{i}}^{2}}\right)^{2}\right]}. \tag{20}\]
However for the decay of doubly charmed baryon this parametric form doesn't work well, instead a polynomial form was employed [17]. So here we also use the three-parameter polynomial form
\[F(q^{2})=F(0)+a\frac{q^{2}}{M_{\mathcal{B}_{i}}^{2}}+b\left(\frac{q^{2}}{M_{ \mathcal{B}_{i}}^{2}}\right)^{2}, \tag{21}\]
where \(F(q^{2})\) represents the form factors \(f_{i}\) and \(g_{i}\).
Using the form factors calculated numerically in the space-like region we fit the parameters \(a,b\) and \(F(0)\) in the un-physical region and then extrapolate to the physical region with \(q^{2}\geq 0\) through Eq. (21).
#### ii.1.1 the transition \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\)
Based on the forementioned discussion, we set \(\beta_{1}=\sqrt{2}\beta_{b\bar{s}}\) and \(\beta_{1}^{\prime}=\sqrt{2}\beta_{c\bar{s}}\). However \(u\) and \(d\) quarks can be regarded as a \(1^{+}\) diquark which means the distance between \(u\) and \(d\) quarks should be smaller than normal case so \(\beta_{23}\) will bigger than \(\sqrt{2}\beta_{u\bar{d}}\). In Ref. [16] we fixed \(\beta_{23}=\beta_{23}^{\prime}=2.9\beta_{u\bar{d}}\) and \(\beta_{u\bar{d}}=0.263\) GeV [23], since \(u\) and \(d\) quarks are in the \({}^{3}S_{1}\) state. The relevant values of \(\beta\) are presented in table 2. With these parameters we calculate the form factors and make theoretical predictions on the transition rates. The fitted values of \(a,\ b\) and \(F(0)\) in the form factors \(f_{i}\) and \(g_{i}\) are presented in Table 3. The dependence of the form factors on \(q^{2}\) is depicted in Fig. 3.
From Fig. 3, one can find that the absolute values of the form factors \(f_{1}(q^{2})\) and \(g_{2}(q^{2})\) are close to 0. The absolute values of \(f_{2}(q^{2})\) and \(f_{3}(q^{2})\) are almost close to each other at the small value of \(q^{2}\) (\(q^{2}<6\) Gev). Compared with the results in Ref. [12], the values of the \(f_{i}\) and \(g_{i}\) here are similar to those obtained in Scheme II of Ref. [12] where the polarization of the \([ud]\) diquark depends on the momentum of the diquark itself. Considering \(1^{+}\) diquark is a so called bad diquark [41], which means the distance of \(ud\) here is bigger than that for a \(0^{+}\) diquark case, we estimate \(\beta_{23}=\beta^{\prime}_{23}<2.9\beta_{u\bar{d}}\) so we also set \(\beta_{23}=\beta^{\prime}_{23}=2.0\beta_{u\bar{d}}\) and \(2.5\beta_{u\bar{d}}\) to do the same calculation and compare the results on the semileptonic and nonleptonic decays later.
#### iii.2.2 the transition \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})\)
For the doubly baryon \(\Xi_{cc}\), two \(c\) quarks consist of a \(1^{+}\) diquark so we set \(\beta_{1}=\beta_{c[cq_{2}]}=2.9\beta_{c\bar{c}}\) and \(\beta_{23}=\beta^{\prime}_{23}=\sqrt{2}\beta_{c\bar{q}_{2}}\). Since the rearrangement for \(\Sigma_{c}^{*}(\Xi_{c}^{*})\) we choose \(\beta^{\prime}_{1}=\beta_{d[cq_{2}]}=\sqrt{2}\beta_{c\bar{d}}\,(\beta^{\prime} _{1}=\beta_{s[cq_{2}]}=\sqrt{2}\beta_{c\bar{s}})\). The relevant parameters are collected in table 2.
The fitted values of \(a,\ b\) and \(F(0)\) in the form factors \(f_{i}\), \(g_{i}\) are presented in Table 4. The dependence of the form factors on \(q^{2}\) is depicted in Fig. 4. Compared with the curves of the form factors in Fig. 3, those in Fig. 4 change more significantly with \(q^{2}\), especially when \(q^{2}>2\) GeV. Similarly we also set \(\beta_{1}=\beta_{c[cq_{2}]}=2.0\beta_{c\bar{c}}\) and \(\beta_{1}=\beta_{c[cq_{2}]}=2.5\beta_{c\bar{c}}\) in the calculation.
Semi-leptonic decay of \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})+l\bar{\nu}_{l}\)
#### iii.2.1 Semi-leptonic decay of single heavy baryon: \(\Sigma_{b}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\)
Employing these form factors obtained in Sec. III.1.1, we evaluate the rates of \(\Sigma_{b}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\). At the same time we also depict the differential decay rates of the \(\Sigma_{b}\to\Sigma_{c}^{*}+l\bar{\nu}_{l}\) which depend on \(\omega\) ( definition can be found in Appendix B) in Fig. 5. We calculate the total decay widths, longitudinal decay widths, transverse decay widths and the ratio of the longitudinal to transverse decay rates \(R\) with \(\beta_{23}=\beta_{23}^{\prime}=2.0\beta_{u\bar{d}}\), \(2.5\beta_{u\bar{d}}\), and \(2.9\beta_{u\bar{d}}\), respectively. The results are listed in table 5.
As we can see in table 5, the width increases slowly when the value of \(\beta_{23}\) decreases and the decay width is a little smaller than that in Ref. [12] where quark-diquark picture for baryons within the LFQM was employed. The longitudinal decay rate, transverse decay rate and their ratio \(R\) are also a little difference under the two types of models of baryon. In Ref. [10] the authors employed relativistic three-quark model to calculate theses form factor and their predictions on the decay width of \(\Sigma_{b}\to\Sigma_{c}^{*}+l\bar{\nu}_{l}\) are almost twice larger than this work. In summary, our results on the decay width of \(\Sigma_{b}\to\Sigma_{c}^{*}+l\bar{\nu}_{l}\) are less than the results in Refs. [9; 10; 11; 12], which means a small \(\beta_{23}\) (\(\beta_{23}^{\prime}\)) is more reasonable if we want to make the prediction close to those in references.
We also use the same method to calculate the decay width of \(\Omega_{b}\to\Omega_{c}^{*}l\bar{\nu}_{l}\), \(\Xi_{b}^{{}^{\prime}}\to\Xi_{c}^{*}l\bar{\nu}_{l}\) with \(\beta_{23}=\beta_{23}^{\prime}=2.0\beta_{q_{1}\bar{q}_{2}}\) and the results are shown in table 6. From table 6, we can see that the widths of \(\Sigma_{b}\to\Sigma_{c}^{*}l\bar{\nu}_{l},\Omega_{b}\to\Omega_{c}^{*}l\bar{ \nu}_{l}\) and \(\Xi_{b}^{{}^{\prime}}\to\Xi_{c}^{*}l\bar{\nu}_{l}\) are close to each other. Similarly, they are lower than those in the Ref. [9].
iii.2.2 Semi-leptonic decay of doubly charmed baryon: \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) and \(\Xi_{cc}\to\Xi_{c}^{*}l\bar{\nu}_{l}\)
For the weak decay of doubly charmed baryon, we calculate the decay rates of \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) and \(\Xi_{cc}\to\Xi_{c}^{*}l\bar{\nu}_{l}\). The curves of the differential decay widths depending on \(\omega\) for \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) are depicted in Fig. 6 which is very similar to that for \(\Sigma_{b}\to\Sigma_{c}^{*}+l\bar{\nu}_{l}\). The curves for \(\Xi_{cc}\to\Xi_{c}^{*}l\bar{\nu}_{l}\) are similar to those in Fig. 6 except their peak values are 20 times bigger than those for \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) so we omit the figure. The numerical results with \(\beta_{1}=2.0\beta_{c\bar{c}}\), \(2.5\beta_{c\bar{c}}\), \(2.9\beta_{c\bar{c}}\) are presented in VII and VIII, respectively. One can find that the differential decay rate of \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})l\bar{\nu}_{l}\) increases with the decrease of the \(\beta_{1}\) value and the differential decay rate with \(\beta_{1}=2.0\beta_{c\bar{c}}\) is close to that in Refs. [5; 42] where the decay was explored with the quark-diquark picture in the LFQM. When \(\beta_{1}=2.9\beta_{c\bar{c}}\), the result of \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) is about half of the Refs. [5; 42]. Our results here indicate \(\beta_{1}\) prefers a small number, such as \(2.0\beta_{c\bar{c}}\) if the predictions in references are accurate.
\begin{table}
\begin{tabular}{l|c c c c} & \(\Gamma\) (\(10^{10}\)s\({}^{-1}\)) & \(\Gamma_{L}\) & \(\Gamma_{T}\) & \(R\) \\ \hline this work(\(\beta_{23}=2.0\beta_{u\bar{d}}\)) & 2.12 & 1.17 & 0.953 & 1.23 \\ \hline this work(\(\beta_{23}=2.5\beta_{u\bar{d}}\)) & 2.01 & 1.11 & 0.898 & 1.24 \\ \hline this work(\(\beta_{23}=2.9\beta_{u\bar{d}}\)) & 1.95 & 1.09 & 0.860 & 1.27 \\ \hline the results in Ref. [9] & 3.23 & 1.61 & 1.62 & 0.99 \\ \hline the results in Ref. [10] & 4.56 & 2.49 & 2.07 & 1.20 \\ \hline the results in Ref. [11] & 3.75 & - & - & - \\ \hline the results in Ref. [12] & 3.17\(\pm\)0.30 & 1.58\(\pm\)0.16 & 1.59\(\pm\)0.13 & 0.994\(\pm\)0.024 \\ \end{tabular}
\end{table}
Table 5: The widths and polarization asymmetries of \(\Sigma_{b}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\).
### Non-leptonic decays of \({\cal B}_{i}(\frac{1}{2}^{+})\to{\cal B}_{f}(\frac{3}{2}^{+})+M\)
Because of the strong interaction, the non-leptonic decays are more complicated than the semi-leptonic processes. Here we adopt the theoretical framework of factorization assumption, where the hadronic transition matrix element can be factorized into a product of two independent matrix elements of currents.
#### iii.2.1 Non-leptonic decays of single heavy baryon: \(\Sigma_{b}\to\Sigma_{c}^{*}+M\)
For \(b\to c\) transition, the hadronic transition matrix element is
\[\langle{\cal B}_{f}(\frac{3}{2}^{+})M\mid{\cal H}\mid{\cal B}_{i}( \frac{1}{2}^{+})\rangle \tag{22}\] \[= \frac{G_{F}V_{bc}V_{qa_{a}q_{b}}^{*}}{\sqrt{2}}\langle M\mid\bar{ q}_{b}\gamma^{\mu}(1-\gamma_{5})q_{a}\mid 0\rangle\langle{\cal B}_{f}(\frac{3}{2}^{+}) \mid\bar{c}\gamma^{\mu}(1-\gamma_{5})b\mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle,\]
where the term \(\langle M\mid\bar{q}_{b}\gamma^{\mu}(1-\gamma_{5})q_{a}\mid 0\rangle\) can be written as the decay constant of meson \(M\) (where \(q_{a}\) and \(q_{b}\) denote heavy or light quark flavors) and the second one \(\langle{\cal B}_{f}(\frac{3}{2}^{+})\mid\bar{c}\gamma^{\mu}(1-\gamma_{5})b \mid{\cal B}_{i}(\frac{1}{2}^{+})\rangle\) is determined by the form factors we obtained. The Fermi constant and CKM matrix elements are selected from Ref.[38]
\[G_{F}=1.1664\times 10^{-5}{\rm GeV}^{-2},\] \[V_{cb}=0.0416,\qquad V_{ud}=0.9738,\qquad V_{us}=0.2257,\qquad V _{cd}=0.230,\qquad V_{cs}=0.957.\]
\begin{table}
\begin{tabular}{c|c c c|c c c} & \(\Gamma\) (\(10^{10}{\rm s}^{-1}\)) & \(\Gamma_{L}\) & \(\Gamma_{T}\) & \(R\) & \(\Gamma\) (\(10^{10}{\rm s}^{-1}\)) [9] & \(\Gamma_{L}\) & \(\Gamma_{T}\) & \(R\) \\ \hline \(\Sigma_{b}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\) & 2.12 & 1.17 & 0.953 & 1.23 & 3.23 & 1.61 & 1.62 & 0.99 \\ \hline \(\Omega_{b}\to\Omega_{c}^{*}l\bar{\nu}_{l}\) & 1.99 & 1.08 & 0.913 & 1.18 & 3.09 & 1.52 & 1.57 & 0.97 \\ \hline \(\Xi_{b}^{*}\to\Xi_{c}^{*}l\bar{\nu}_{l}\) & 2.08 & 1.13 & 0.947 & 1.19 & 3.03 & 1.48 & 1.55 & 0.95 \\ \end{tabular}
\end{table}
Table 6: The widths and polarization asymmetries of the semileptonic decay between two single heavy baryons (\(\beta_{23}=2.0\beta_{u\bar{d}}\)).
\begin{table}
\begin{tabular}{c|c c c c} & \(\Gamma\) (\(10^{10}{\rm s}^{-1}\)) & \(\Gamma_{L}\) & \(\Gamma_{T}\) & \(R\) \\ \hline this work(\(\beta_{1}=2.0\beta_{c\bar{c}}\)) & 2.02 & 0.991 & 1.03 & 0.962 \\ \hline this work(\(\beta_{1}=2.9\beta_{c\bar{c}}\)) & 1.67 & 0.831 & 0.841 & 0.988 \\ \hline the results in [5] & 2.39 & - & - & - \\ \hline the results in [42] & 2.64 & - & - & - \\ \end{tabular}
\end{table}
Table 7: The widths and polarization asymmetries of the transition \(\Xi_{cc}\to\Sigma_{c}^{*}l\bar{\nu}_{l}\).
In table 9, we present the results of the main two-body decay channels for \(\Sigma_{b}\) with the different value of \(\beta_{23}(\beta_{23}^{\prime})\). The decay constants are derived from the Ref. [23]. From the table 9 one also can notice that the widths increase with the decrease of the value of \(\beta_{23}\left(\beta_{23}^{\prime}\right)\) and the results with \(\beta_{23}=\beta_{23}^{\prime}=2.0\beta_{u\bar{d}}\) are close to those with the heavy quark limit in Ref.[12].
#### iv.2.2 Non-leptonic decays of doubly charmed baryon: \(\Xi_{cc}\to\Sigma_{c}^{*}(\Xi_{c}^{*})+M\)
The non-leptonic decays results of the doubly charmed baryon are listed in table 10. As we can see from the table that the non-leptonic decay widths increase significantly with the decrease of the value of \(\beta_{1}\). A smaller value of \(\beta_{1}\) is more consistent with results in previous studies, such as Ref.[5]. Future experiments are needed to constrain the \(\beta\) parameters and test our model predictions.
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & \(\Gamma(\beta_{1}=2.0\beta_{c\bar{c}})\) & \(\Gamma(\beta_{1}=2.5\beta_{c\bar{c}})\) & \(\Gamma(\beta_{1}=2.9\beta_{c\bar{c}})\) & \(\Gamma\)[5] \\ \hline \(\Xi_{cc}\to\Sigma_{c}^{*}\pi\) & 0.116 & 0.0924 & 0.0758 & 0.176 \\ \hline \(\Xi_{cc}\to\Sigma_{c}^{*}\rho\) & 0.459 & 0.369 & 0.304 & 0.574 \\ \hline \(\Xi_{cc}\to\Sigma_{c}^{*}K\) & \(7.34\times 10^{-3}\) & \(5.77\times 10^{-3}\) & \(4.70\times 10^{-3}\) & \(7.55\times 10^{-3}\) \\ \hline \(\Xi_{cc}\to\Sigma_{c}^{*}K^{*}\) & \(2.22\times 10^{-2}\) & \(1.79\times 10^{-2}\) & \(1.47\times 10^{-2}\) & \(2.43\times 10^{-2}\) \\ \hline \(\Xi_{cc}\to\Xi_{c}^{*}\pi\) & 2.39 & 1.91 & 1.55 & 3.39 \\ \hline \(\Xi_{cc}\to\Xi_{c}^{*}\rho\) & 9.03 & 7.33 & 5.98 & 7.07 \\ \hline \(\Xi_{cc}\to\Xi_{c}^{*}K\) & 0.135 & 0.106 & 0.0857 & 0.106 \\ \hline \(\Xi_{cc}\to\Xi_{c}^{*}K^{*}\) & 0.355 & 0.288 & 0.235 & 0.190 \\ \hline \hline \end{tabular}
\end{table}
Table 9: The non-leptonic decay widths (in unit \(10^{10}\)s\({}^{-1}\)) of the doubly charmed baryon to the single charmed baryon.
Conclusions and Discussions
In this paper, we study the weak decays between two heavy baryons \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})\) with the three-quark picture of baryon in the LFQM. We derive the general form of transition amplitude, and obtain analytical expression of form factors for specific transition processes: \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) and \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})\). For weak decay of \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\), the \(b\) quark decays to \(c\) quark and \(ud\) subsystem with definite spin can be regarded as spectator in initial and final states. For the transition \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})\), the \(cc\) system in initial state and the \(ud\) (\(su\) or \(sd\)) system in final state possess definite spins, but they are not spectators. In that case, quark rearrangement is adopted in our calculation of form factors. We then compute numerical values of these form factors with reasonable assumptions of model parameters. Last, we calculate rate of semi-leptonic and non-leptonic decays of \(\Sigma_{b}\) (\(\Sigma_{b}\rightarrow\Sigma_{c}^{*}l\bar{\nu}_{l}\), \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}+M\)) and \(\Xi_{cc}\) ( \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})l\bar{\nu}_{l}\), \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})+M\)) based on numerical results of these form factors.
The weak decays of \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\) were studied in Ref. [12] where the quark-diquark picture was employed in the LFQM. Instead of quark-diquark picture, here we use the three-quark picture to revisit the transition of \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\). During the transition, the two light quarks serve as the spectators and maintain their all quantum numbers (spin, color). The associated momentum is also unchanged. The \(b\) quark in initial state transits to \(c\) quark by emitting two leptons mediated by gauge bosons \(W^{\pm}\).
We calculate the form factors for the transition \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}\). The values of the \(f_{i}\) and \(g_{i}\) are similar to those obtained in Scheme II of Ref. [12] where the polarization of the \([ud]\) diquark depends on the momentum of the diquark itself. Considering \(1^{+}\) diquark is a so-called bad diquark, i.e. the distance between the two quarks is larger than a good diquark case, we estimate \(\beta_{23}=\beta_{23}^{\prime}<2.9\beta_{u\bar{d}}\) so we also set \(\beta_{23}=\beta_{23}^{\prime}=2.0\beta_{u\bar{d}}\) and \(2.5\beta_{u\bar{d}}\) to perform the same calculation. With computed form factors we evaluate the decay widths of the semi-leptonic \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}l\bar{\nu}_{l}\) and non-leptonic \(\Sigma_{b}\rightarrow\Sigma_{c}^{*}+M\) with different values of \(\beta_{23}\). Comparing the results in other references, we find \(\beta_{23}\) and \(\beta_{23}^{\prime}\) prefer a small value if we want our predictions close to those in the references. We also compute the semi-leptonic decay width of \(\Omega_{b}\rightarrow\Omega_{c}^{*}l\bar{\nu}_{l}\) and \(\Xi_{b}^{{}^{\prime}}\rightarrow\Xi_{c}^{*}l\bar{\nu}_{l}\) with \(\beta_{23}=\beta_{23}^{\prime}=2.0\beta_{q_{1}\bar{q}_{2}}\).
For the weak decay of the doubly charmed baryon \(\Xi_{cc}\), a heavy quark \(c\) in the initial state decays to a light quark (\(s\) or \(d\)) in the final state through the weak interaction. However the \(cc\) pair in initial state and \(ud\) (\(us\) or \(ds\)) in the final state possess definite spin which can be regarded as diquarks. In this way, neither the initial physical diquark \(cc\) nor the final physical diquark \(ud\) (\(us\) or \(ds\)) are spectators. Therefore, the three-quark picture is more suitable here. We have rearranged the quarks in the initial and final states using the Racah transformation so the effective spectator can be isolated from the baryon. We calculate the form factors in the space-like region and then extend them to the time-like region (the physical region) by using the three-parameter form. Using these form factors we calculate the widths of semi-leptonic decay \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})l\bar{\nu}_{l}\) and non-leptonic decay \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})+M\), respectively. For the weak decay of \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*})l\bar{\nu}_{l}\), the decay width increases when the \(\beta_{1}\) value decreases. Our results on the semi-leptonic decay \(\Xi_{cc}\rightarrow\Sigma_{c}^{*}(\Xi_{c}^{*}l\bar{\nu}_{l})l\bar{\nu}_{l}\) with \(\beta_{1}=2.0\beta_{c\bar{c}}\)
are close to the Ref. [5]. Future experiments will be necessary to precisely determine the parameters and test model predictions.
## Acknowledgement
This work is supported by the National Natural Science Foundation of China (NNSFC) under the Contracts No. 12075167, 11975165 and 12235018.
Appendix A the form factor of \(\mathcal{B}_{i}{(\frac{1}{2}^{+})}\rightarrow\mathcal{B}_{f}{(\frac{3}{2}^{+})}\)
\(\bar{u}(\bar{P},S_{z})\gamma_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime},S^{ \prime}_{z})\), \(\bar{u}(\bar{P},S_{z})\bar{P}^{\prime}_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime },S^{\prime}_{z})\), \(\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime},S^{ \prime}_{z})\),
\(\bar{u}(\bar{P},S_{z})g^{\xi}_{\mu}u_{\xi}(\bar{P}^{\prime},S^{\prime}_{z})\) are multiplied to the right side of Eq.(16), and then we have
\[F_{1} = \int\frac{dx_{2}d^{2}k_{2\perp}^{2}}{2(2\pi)^{3}}\frac{dx_{3}d^{ 2}k_{3\perp}^{2}}{2(2\pi)^{3}}\frac{\phi^{*}_{\mathcal{B}_{f}}(x^{\prime},k^ {\prime}_{\perp})\phi_{\mathcal{B}_{i}}(x,k_{\perp})Tr[\gamma^{\alpha}_{\perp} (\not{\bar{P}^{\prime}}\!\!+M^{\prime}_{0})(\not{p}_{3\!\!\!+}\,M_{0})\gamma^{ \beta}_{\perp}(\not{p}_{2\!\!\!-}\,m_{2})]}{16\sqrt{6x_{1}x^{\prime}_{1}M^{3}_ {0}M^{\prime 3}_{0}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e_{3})(m^{\prime}_{1}+e^{ \prime}_{1})(m^{\prime}_{2}+e^{\prime}_{2})(m^{\prime}_{3}+e^{\prime}_{3})}} \tag{15}\] \[\times\sum_{S_{z},S^{\prime}_{z}}\mathrm{Tr}[u_{\xi}(\bar{P}^{ \prime},S^{\prime}_{z})\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})(\not {p}^{\prime}_{1}\!\!+m^{\prime}_{1})\gamma^{\mu}\gamma_{5}(\not{p}_{1\!\!\!+} \,m_{1})\gamma_{\perp\beta}\gamma_{5}u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z}) \gamma_{\mu}\bar{P}^{\xi}],\] \[F_{2} = \int\frac{dx_{2}d^{2}k_{2\perp}^{2}}{2(2\pi)^{3}}\frac{dx_{3}d^{ 2}k_{3\perp}^{2}}{2(2\pi)^{3}}\frac{\phi^{*}_{\mathcal{B}_{f}}(x^{\prime},k^ {\prime}_{\perp})\phi_{\mathcal{B}_{i}}(x,k_{\perp})Tr[\gamma^{\alpha}_{\perp }(\not{\bar{P}^{\prime}}\!\!+M^{\prime}_{0})(\not{p}_{3\!\!\!+}\,m_{3})(\not{ \bar{P}}\!\!+M_{0})\gamma^{\beta}_{\perp}(\not{p}_{2\!\!\!-}\,m_{2})]}{16\sqrt {6x_{1}x^{\prime}_{1}M^{3}_{0}M^{\prime 3}_{0}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e_{3})(m^{ \prime}_{1}+e^{\prime}_{1})(m^{\prime}_{2}+e^{\prime}_{2})(m^{\prime}_{3}+e^{ \prime}_{3})}}\] (16) \[\times\sum_{S_{z},S^{\prime}_{z}}\mathrm{Tr}[u_{\xi}(\bar{P}^{ \prime},S^{\prime}_{z})\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})(\not {p}^{\prime}_{1}\!\!+m^{\prime}_{1})\gamma^{\mu}\gamma_{5}(\not{p}_{1\!\!\!+} \,m_{1})\gamma_{\perp\beta}\gamma_{5}u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z}) \bar{P}^{\prime}_{\mu}\bar{P}^{\xi}],\] \[F_{3} = \int\frac{dx_{2}d^{2}k_{2\perp}^{2}}{2(2\pi)^{3}}\frac{dx_{3}d^{ 2}k_{3\perp}^{2}}{2(2\pi)^{3}}\frac{\phi^{*}_{\mathcal{B}_{f}}(x^{\prime},k^ {\prime}_{\perp})\phi_{\mathcal{B}_{i}}(x,k_{\perp})Tr[\gamma^{\alpha}_{\perp }(\not{\bar{P}^{\prime}}\!\!+M^{\prime}_{0})(\not{p}_{3\!\!\!+}\,m_{3})(\not{ \bar{P}}\!\!+M_{0})\gamma^{\beta}_{\perp}(\not{p}_{2\!\!\!-}\,m_{2})]}{16\sqrt {6x_{1}x^{\prime}_{1}M^{3}_{0}M^{\prime 3}_{0}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e_{3})(m^{ \prime}_{1}+e^{\prime}_{1})(m^{\prime}_{2}+e^{\prime}_{2})(m^{\prime}_{3}+e^{ \prime}_{3})}}\] (17) \[\times\sum_{S_{z},S^{\prime}_{z}}\mathrm{Tr}[u_{\xi}(\bar{P}^{ \prime},S^{\prime}_{z})\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})(\not {p}^{\prime}_{1}\!\!+m^{\prime}_{1})\gamma^{\mu}\gamma_{5}(\not{p}_{1\!\!\!+} \,m_{1})\gamma_{\perp\beta}\gamma_{5}u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z}) \bar{P}_{\mu}\bar{P}^{\xi}],\] \[F_{4} = \int\frac{dx_{2}d^{2}k_{2\perp}^{2}}{2(2\pi)^{3}}\frac{dx_{3}d^{ 2}k_{3\perp}^{2}}{2(2\pi)^{3}}\frac{\phi^{*}_{\mathcal{B}_{f}}(x^{\prime},k^ {\prime}_{\perp})\phi_{\mathcal{B}_{i}}(x,k_{\perp})Tr[\gamma^{\alpha}_{\perp }(\not{\bar{P}^{\prime}}\!\!+M^{\prime}_{0})(\not{p}_{3\!\!\!+}\,m_{3})(\not{ \bar{P}}\!\!+M_{0})\gamma^{\beta}_{\perp}(\not{p}_{2\!\!\!-}\,m_{2})]}{16 \sqrt{6x_{1}x^{\prime}_{1}M^{3}_{0}M^{\prime 3}_{0}(m_{1}+e_{1})(m_{2}+e_{2})(m_{3}+e_{3})(m^{ \prime}_{1}+e^{\prime}_{1})(m^{\prime}_{2}+e^{\prime}_{2})(m^{\prime}_{3}+e^{ \prime}_{3})}}\] (18) \[\times\sum_{S_{z},S^{\prime}_{z}}\mathrm{Tr}[u_{\xi}(\bar{P}^{ \prime},S^{\prime}_{z})\bar{u}_{\alpha}(\bar{P}^{\prime},S^{\prime}_{z})(\not {p}^{\prime}_{1}\!\!+m^{\prime}_{1})\gamma^{\mu}\gamma_{5}(\not{p}_{1\!\!\!+} \,m_{1})\gamma_{\perp\beta}\gamma_{5}u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z}) \delta^{\xi}_{\mu}].\]
Simultaneously, \(\bar{u}(\bar{P},S_{z})\gamma_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime},S^{ \prime}_{z})\), \(\bar{u}(\bar{P},S_{z})\bar{P}^{\prime}_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{ \prime},S^{\prime}_{z})\), \(\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}\bar{P}^{\xi}u_{\xi}(\bar{P}^{\prime},S^{ \prime}_{z})\),
\(\bar{u}(\bar{P},S_{z})g^{\xi}_{\mu}u_{\xi}(\bar{P}^{\prime},S^{\prime}_{z})\) are multiplied to the right side of Eq.(15), one can obtain
\[F_{1} = \mathrm{Tr}\{u_{\xi}(\bar{P}^{\prime},S^{\prime}_{z})\bar{u}_{ \alpha}(\bar{P}^{\prime},S^{\prime}_{z})\left[\gamma^{\mu
\[F_{3} = \mbox{Tr}\{u_{\xi}(\bar{P}^{\prime},S^{\prime}_{z})\bar{u}_{\alpha}( \bar{P}^{\prime},S^{\prime}_{z})\left[\gamma^{\mu}\bar{P}^{\alpha}\frac{f_{1}(q^ {2})}{M_{{\cal B}_{i}}}+\frac{f_{2}(q^{2})}{M^{2}_{{\cal B}_{i}}}\bar{P}^{ \alpha}\bar{P}^{\mu}+\frac{f_{3}(q^{2})}{M_{{\cal B}_{i}}M_{{\cal B}_{f}}}\bar {P}^{\alpha}\bar{P}^{\prime\mu}+f_{4}g^{\alpha\mu}\right] \tag{10}\] \[u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z})\bar{P}_{\mu}\bar{P}^{\xi}\},\] \[F_{4} = \mbox{Tr}\{u_{\xi}(\bar{P}^{\prime},S^{\prime}_{z})\bar{u}_{ \alpha}(\bar{P}^{\prime},S^{\prime}_{z})\left[\gamma^{\mu}\bar{P}^{\alpha}\frac {f_{1}(q^{2})}{M_{{\cal B}_{i}}}+\frac{f_{2}(q^{2})}{M^{2}_{{\cal B}_{i}}}\bar {P}^{\alpha}\bar{P}^{\mu}+\frac{f_{3}(q^{2})}{M_{{\cal B}_{i}}M_{{\cal B}_{f} }}\bar{P}^{\alpha}\bar{P}^{\prime\mu}+f_{4}g^{\alpha\mu}\right]\] (11) \[u(\bar{P},S_{z})\bar{u}(\bar{P},S_{z})g^{\xi}_{\mu}\}.\]
After solving the Eqs. (11), (12), (12) and (11), \(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{4}\) can be expressed by \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}\) which can be numerically evaluated through Eqs. (10), (11), (11) and A(4). The polarization sum formula for a particle with \(S=3/2\) is
\[\sum_{S_{z}}u_{\xi}(\bar{P},S_{z})\bar{u}_{\alpha}(\bar{P},S_{z})=-(\bar{P}+M_ {0})[T_{\xi\alpha}(\bar{P})-\frac{1}{3}\gamma^{\rho}T_{\rho\xi}(\bar{P})T_{ \alpha\sigma}(\bar{P})\gamma^{\sigma}], \tag{12}\]
with
\[T_{\xi\alpha}(\bar{P})=g_{\xi\alpha}-\frac{\bar{P}_{\xi}\bar{P}_{\alpha}}{M^{ 2}_{0}}. \tag{13}\]
Appendix B Semi-leptonic decay of \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})l\bar{\nu}\)
The helicity amplitudes are expressed in terms of the form factors for \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})\)[43; 44]
\[H^{V,A}_{1/2,\,0} = \mp\frac{1}{\sqrt{q^{2}}}\frac{2}{\sqrt{3}}\sqrt{M_{{\cal B}_{i} }M_{{\cal B}_{f}}(w\mp 1)}[(M_{{\cal B}_{i}}w-M_{{\cal B}_{f}}){\cal N}^{V,A}_{4}(w)\] \[\mp(M_{{\cal B}_{i}}\mp M_{{\cal B}_{f}})(w\pm 1){\cal N}^{V,A}_{1 }(w)+M_{{\cal B}_{f}}(w^{2}-1){\cal N}^{V,A}_{2}(w)\] \[+M_{{\cal B}_{i}}(w^{2}-1){\cal N}^{V,A}_{3}(w)],\] \[H^{V,A}_{1/2,\,1} = \sqrt{\frac{2}{3}}\sqrt{M_{{\cal B}_{i}}M_{{\cal B}_{f}}(w\mp 1)}[{ \cal N}^{V,A}_{4}(w)-2(w\pm 1){\cal N}^{V,A}_{1}(w)],\] \[H^{V,A}_{3/2,\,1} = \mp\sqrt{2M_{{\cal B}_{i}}M_{{\cal B}_{f}}(w\mp 1)}{\cal N}^{V,A}_{4 }(w), \tag{14}\]
where again the upper (lower) sign corresponds to \(V(A)\), \({\cal N}^{V}_{i}\equiv g_{i}\), \({\cal N}^{A}_{i}\equiv f_{i}\) (\(i=1,2,3,4\)) and the \(q^{2}\) is the lepton pair invariant mass. The remaining helicity amplitudes can be obtained using the relation
\[H^{V,A}_{-{\cal N}^{\prime},\,-\,\lambda_{W}}=\mp H^{V,A}_{{\cal N}^{\prime},\, \lambda_{W}}.\]
Partial differential decay rates can be represented in the following form
\[\frac{d\Gamma_{T}}{dw} = \frac{G^{2}_{F}}{(2\pi)^{3}}|V_{Q_{1}Q_{2}}|^{2}\frac{q^{2}M^{2}_ {{\cal B}_{f}}\sqrt{w^{2}-1}}{12M_{{\cal B}_{i}}}[|H_{1/2,\,1}|^{2}+|H_{-1/2,\, -1}|^{2}+|H_{3/2,\,1}|^{2}+|H_{-3/2,\,-1}|^{2}],\] \[\frac{d\Gamma_{L}}{dw} = \frac{G^{2}_{F}}{(2\pi)^{3}}|V_{Q_{1}Q_{2}}|^{2}\frac{q^{2}M^{2}_ {{\cal B}_{f}}\sqrt{w^{2}-1}}{12M_{{\cal B}_{i}}}[|H_{1/2,\,0}|^{2}+|H_{-1/2,\, 0}|^{2}], \tag{15}\]
where \(p_{c}=M_{{\cal B}_{f}}\sqrt{w^{2}-1}\) is the momentum of \({\cal B}_{f}\) in the reset frame of \({\cal B}_{i}\).
The differential decay width of the \({\cal B}_{i}(\frac{1}{2}^{+})\rightarrow{\cal B}_{f}(\frac{3}{2}^{+})l\bar{\nu}_ {l}\) can be written as
\[\frac{d\Gamma}{dw}\;=\;\frac{d\Gamma_{T}}{dw}+\frac{d\Gamma_{L}}{dw}. \tag{40}\]
Integrating over the parameter \(\omega\), we can obtain the total decay width
\[\Gamma=\int_{1}^{\omega_{\rm max}}d\omega\frac{d\Gamma}{dw}, \tag{41}\]
where \(\omega=v\cdot v^{\prime}\) and the upper bound of the integration \(\omega_{\rm max}=\frac{1}{2}(\frac{M_{{\cal B}_{i}}}{M_{{\cal B}_{f}}}+\frac{ M_{{\cal B}_{f}}}{M_{{\cal B}_{i}}})\) is the maximal recoil.
The ratio of the longitudinal to transverse decay rates \(R\) is defined by
\[R=\frac{\Gamma_{L}}{\Gamma_{T}}=\frac{\int_{1}^{\omega_{\rm max}}d\omega\ q^{2} \ p_{c}\left[|H_{\frac{1}{2},0}|^{2}+|H_{-\frac{1}{2},0}|^{2}\right]}{\int_{1 }^{\omega_{\rm max}}d\omega\ q^{2}\ p_{c}\left[|H_{1/2,\,1}|^{2}+|H_{-1/2,\,-1 }|^{2}+|H_{3/2,\,1}|^{2}+|H_{-3/2,\,-1}|^{2}\right]}. \tag{42}\]
|
2307.11573 | Control- & Task-Aware Optimal Design of Actuation System for Legged
Robots using Binary Integer Linear Programming | Athletic robots demand a whole-body actuation system design that utilizes
motors up to the boundaries of their performance. However, creating such robots
poses challenges of integrating design principles and reasoning of practical
design choices. This paper presents a design framework that guides designers to
find optimal design choices to create an actuation system that can rapidly
generate torques and velocities required to achieve a given set of tasks, by
minimizing inertia and leveraging cooperation between actuators. The framework
serves as an interactive tool for designers who are in charge of providing
design rules and candidate components such as motors, reduction mechanism, and
coupling mechanisms between actuators and joints. A binary integer linear
optimization explores design combinations to find optimal components that can
achieve a set of tasks. The framework is demonstrated with 200 optimal design
studies of a biped with 5-degree-of-freedom (DoF) legs, focusing on the effect
of achieving multiple tasks (walking, lifting), constraining the mass budget of
all motors in the system and the use of coupling mechanisms. The result
provides a comprehensive view of how design choices and rules affect reflected
inertia, copper loss of motors, and force capability of optimal actuation
systems. | Youngwoo Sim, Guillermo Colin, Joao Ramos | 2023-07-21T13:23:49Z | http://arxiv.org/abs/2307.11573v1 | # Control- & Task-Aware Optimal Design of Actuation System
###### Abstract
Athletic robots demand a whole-body actuation system design that utilizes motors up to the boundaries of their performance. However, creating such robots poses challenges of integrating design principles and reasoning of practical design choices. This paper presents a design framework that guides designers to find optimal design choices to create an actuation system that can rapidly generate torques and velocities required to achieve a given set of tasks, by minimizing inertia and leveraging cooperation between actuators. The framework serves as an interactive tool for designers who are in charge of providing design rules and candidate components such as motors, reduction mechanism, and coupling mechanisms between actuators and joints. A binary integer linear optimization explores design combinations to find optimal components that can achieve a set of tasks. The framework is demonstrated with 200 optimal design studies of a biped with 5-degree-of-freedom (DoF) legs, focusing on the effect of achieving multiple tasks (walking, lifting), constraining the mass budget of all motors in the system and the use of coupling mechanisms. The result provides a comprehensive view of how design choices and rules affect reflected inertia, copper loss of motors, and force capability of optimal actuation systems.
## I Introduction
Athletic robots such as the MIT cheetah and Cassie have emerged to show superior performance to their semi-static counterparts. These machines possess the ability to execute agile movements, such as utilizing momentum to jump or push heavy objects. Achieving such athletic motions requires a comprehensive approach to whole-body control and systematic actuation system design [1, 2, 3], wherein multiple actuators are built to operate cooperatively across the robot's body. While the literature has made significant strides in whole-body control of humanoids, the design problem pertaining to high-DoF systems still demands further investigation.
The ability of athletic robots to generate large forces and acceleration is harnessed by incorporating design principles at two distinct layers of the actuation system. At the actuator level, the quasi-direct drive (QDD) paradigm [4] suggests minimizing the inertia of the limbs. Using smaller gear ratios minimizes the reflected inertia of individual actuators and reduces inertial torques required to swing the limbs. In cases where gear ratios are not large enough to amplify torque sufficiently, this can be addressed at the system level by employing cooperative (coupled or parallel) actuation [5], where actuators are coupled together to combine their torque output and meet the torque requirements of specific joints.
However, as the number of actuators of a system grows, designers face multi-faceted issues in integrating the aforementioned principles. First, the design considerations for multi-actuator systems, such as humanoid legs, are not trivial. Design principles often provide general intuition but leave specific design choices to the discretion of designers. Consequently, designers need to analyze the trade-offs between component choices and their impact on performance metrics in a multi-dimensional sense. A popular choice of performance metrics are, namely, inertia matrix which measures kinetic energy of the entire system [6] and force capability which measures available force that end-effectors can exert on the environment [7]. In legged robot design, coupled actuation has shown potential benefits of amplifying joint torques without increasing reflected inertia extensively [5]. However, under coupling, modifying even a single gear ratio out of many actuators affects torque and velocity capabilities of other actuators, which complicates design analysis. Second, it is difficult to smoothly parameterize characteristics of available design candidates. Designers typically source motors or reduction mechanisms from the market, where there can be significant jumps in properties between available options. This demands designers reflect the discontinuities in design studies, even complicating the design process.
Fortunately, a broad spectrum of methods exists to address the challenges associated with designing multi-DoF systems. One straightforward approach involves defining high-level tasks and identifying optimal component choices, such as gear ratios, motors, and limb lengths, with the goal of optimizing system performance metrics such as power consumption [8, 9, 10] or force capability [11]. Another family of methods is called the co-design paradigm, wherein the design and control aspects are jointly optimized through the
Fig. 1: Conceptual overview of the proposed design optimization of multi-actuator system. Video: [https://youtu.be/DEXrp0TsqRI](https://youtu.be/DEXrp0TsqRI)
addition of a controller and simulation of design cases [12]. This co-design approach emphasizes control-related aspects such as sensitivity [13] and robustness [14, 15].
However, existing optimal design tools lack certain crucial aspects of the design process, often yielding theoretical designs that lack practicality in real-world applications. First, the expert knowledge of designers, which includes non-parametric design rules, proves essential in humanoid design. For example, the actuation system design in humanoids involves intricate space optimization to accommodate multiple actuators within the robot's body. These considerations are better defined as logical constraints rather than parametric ones. Secondly, designers typically rely on optimal design software to gain a comprehensive view of multiple design scenarios, rather than seeking a singular theoretically optimal solution. Moreover, designers engage in an iterative process of obtaining optimal designs from software and making necessary modifications. In this process, design tools assist in reasoning and justifying these modifications. Hence, the software employed must be relatively fast and capable of incorporating human intuitions. Regrettably, current state-of-the-art frameworks heavily rely on computationally intensive algorithms, such as genetic algorithms, or the number of optimization variables is limited.
This paper proposes a _design tool_ that assists designers in creating high-DoF actuation systems, addressing the aforementioned issues. First, we want a design tool that guarantees a system design that is capable of various humanoid tasks such as carrying objects, weight-lifting, and/or running. However, it is a chicken-and-egg problem; without a robot design, one cannot reason about tasks and vice versa. Thus, tasks or motions are planned based on a reduced-order model (ROM) of a robot consisting of high-level design parameters such as overall mass, inertia, and center of mass (CoM) height with fixed morphology and ordering of joints. These assumptions are consistent with existing motion planners based on single rigid body dynamics (SRBD) [16] or centroidal dynamics [1], allowing us to use similar ROM-based motion planners. Second, in this framework, designers are encouraged to use their expert knowledge to populate and curate libraries of design candidates such as motors, gear ratios, and coupling mechanisms between actuators and joints. Moreover, the high-level intuition of designers is translated into design rules, which designers can experiment with to observe how optimal solutions behave under these rules. Third, our objective is to create dynamic humanoids capable of rapid limb acceleration and impact mitigation. To achieve these goals, [4] suggests that minimizing the reflected inertia of actuation systems is the key, which sets the objective of the proposed design optimization to choosing components from design libraries that minimize reflected inertia while ensuring task completion. Lastly, to solve the optimization more efficiently, the optimal design problem is translated into a binary integer linear program which yields optimal solutions faster than existing methods.
This framework is demonstrated by design studies of a biped with 5-DoF legs focusing on the effects of 1) achieving a set of tasks, 2) employing actuator-joint couplings (parallel mechanism) and 3) limiting the mass budget of all motors in a system. The optimization study was given two tasks that involve different joint usage patterns: walking and weight-lifting in a snatch style. Next, design libraries of motors, gear ratio, and coupling mechanisms were compiled from existing humanoids and unprecedented-but-practical designs for comparison. From the results of 200 optimal design studies with varied motor mass budgets and usage of couplings, several trends in reflected inertia, force/torque capability, and copper loss of motors were observed.
The contribution of the proposed framework is as follows. First, this tool allows designers to quickly navigate vast design spaces and reveals a comprehensive overview of optimal multi-actuator system designs for legged robots. Second, the framework serves as a reasoning tool for high-DoF system design. Designers can quickly run a large number of optimal studies to observe how optimal solutions respond to modifying tasks, design libraries, or design rules. For example, providing a larger set of tasks would result in general-purpose humanoid design whereas providing a single task would yield a more dedicated system design. Moreover, novice engineers can update components available in the market to create better designs, while experts can explore combinations of novel mechanisms for comparison.
## II Background: Representation of Design Library & Combinations
The binary variables are efficient tools for handling two essential aspects of design: 1) determining whether to select or exclude component candidates from libraries and 2) expressing combinations of components.
**Definition 1**: _(Binary Combination)_ Let \(\mathbf{p}\!=\!\{p_{1},\cdots,p_{n}\}\) be a library of \(n\) objects, \(\mathbf{x}\!=\!(x_{1},\cdots,x_{n})\!\in\!\{0,1\}^{n}\) be a vector of binary variables. An arbitrary singular choice of an object \(p(\mathbf{x})_{B}\) out of the library \(\mathbf{p}\) is obtained by a _binary combination of_\(\mathbf{p}\)_and_\(\mathbf{x}\) as follows,
\[p(\mathbf{x})_{B}\coloneqq\sum_{i=1}^{n}p_{i}x_{i} \tag{1}\] \[\implies \sum_{i=1}^{n}x_{i}=1\quad\text{(exclusivity)}. \tag{2}\]
For instance, a selection of \(p_{j}\) is achieved by assigning 1 to the corresponding variable \((x_{j}\!=\!1)\) and 0 to the rest \((x_{k}\!=\!0,k\!\neq\!j,k\!\in\![n])\). The definition itself implies an exclusivity constraint that enforces a single choice. It should be noted that the objects \(p_{i}\) can be scalars such as maximum torque or matrices like the Jacobian of coupling mechanism that maps actuator velocities to joint velocities.
**Remark 1**: _(Linearization of Multilinear Polynomials)_ Products of binary variables commonly arise when combining components; e.g., an actuator is a combination of a motor and a reduction mechanism. A useful property of the product of binary variables is that it can be perfectly linearized (as opposed to linearly approximated). The product of binary variables \(x\) and \(y\), denoted as \(xy\), is linearized by introducing a new binary variable \(z\) in place of \(xy\) and adding the following linear constraints:
\[z\leq x,\quad z\leq y,\quad z\geq x+y-1. \tag{3}\]
For linearization of general multilinear forms, refer to [17].
## III Optimal Design of Transmission System for Humanoid Robots
### _Overview_
We introduce a design optimization tool that finds mechanical components for the actuation system of athletic humanoids. Most importantly, this framework is composed as an interactive tool for designers and addresses difficulties in designing a high-DoF actuation system. Hence, the framework expects designers to provide their knowledge and intuition and run several iterations to gain an understanding of optimal designs of humanoids.
The workflow of the proposed tool is divided into three steps. (Fig. 2). The first step is to prepare key ingredients for design optimization; 1) ROMs for the generation of task trajectories, 2) a set of task trajectories generated by motion planning software using ROMs, 3) libraries of candidate components to create the robot, 4) relevant design rules, and 5) design goals. As to the preparation of task trajectories, first, ROMs are created from known kinematics and parameters of high-level design envelopes and behavior targets set by designers (e.g., overall mass, inertia, CoM height or desired velocity) depending on the tasks. Then, using ROM-based motion planning approaches (e.g., SRBD-based trajectory optimization [18] or kino-dynamic planning [19]), task-space trajectories of desired tasks are obtained. Finally, the task-space trajectories are converted to joint-space trajectories using inverse kinematics, inverse dynamics, or whole-body controllers. It is important to note that the ROM assumes fixed limb lengths and joint ordering. This is because humanoids are typically designed to have similar proportions and kinematics to humans. Also, as there exist only a few practical joint orderings that mimic human limb kinematics, design studies for every joint ordering can be carried out. It is also assumed that the actuators are placed very closely to the center of the robot (proximal actuation) and actuator dynamics is neglected. Hence, the actuator placements and their inertial torques do not affect the dynamics of the robot. Lastly, the framework does not take into account the structural design of limbs because we assume that the apparent inertia and mass could be significantly minimized compared to the reflected inertia of the actuation system.
The second step of the framework is to integrate tasks, design library, and design rules and translate them to linear binary integer programming. The overarching concept is expressed as follows with \(\mathbf{x}\) as design choices of mechanical components,
\[\min_{\mathbf{x}} \text{ReflectedInertia}(\mathbf{x}), \tag{4a}\] \[\mathrm{s.t.} \mathbf{x}\in\text{DesignLibrary},\] (4b) \[\text{TaskTrajectory}\subseteq\text{OperationDomain}(\mathbf{x}),\] (4c) \[\text{DesignRules}(\mathbf{x}). \tag{4d}\]
The contents of (4) are briefed as follows.
**Control-Awareness** (4a): The objective is to minimize the reflected inertia of the entire actuation system perceived at the joint-level (joint-space reflected inertia). This objective aims to improve force control performance, enhance the system's ability to regulate external impacts and enable agile limb movements [4]. The objective aligns with the QDD paradigm, although designers have the flexibility to modify it to prioritize energy efficiency or minimum torque expenditure, depending on their interest.
**Discrete Design Space** (4b): In this framework, designers are responsible for providing and modifying libraries of candidate motors, reduction mechanisms, and actuator-joint couplings. An optimization solver then searches for the optimal design by exploring all possible combinations of provided components. Designers are encouraged to use actuator-joint coupling mechanisms from existing robots, utilize their expert knowledge to curate practical component candidates, or include novel components whose utility is in question.
**Task-Awareness** (4c): Robots are expected to achieve multiple tasks. This framework ensures the feasibility of these tasks within the capability of motors. In other words, the given tasks are mapped to trajectories of motor torques and velocities. Then, these trajectories are constrained to remain within the motor limits such as voltage and thermal limit (operation domain) as in Fig. 3(b). The safety of a controller can be tuned by parameters that define allowable closeness between extremes of task trajectory and the boundary of the operating domain.
**Design Rules** (4d): Additional constraints can be incorporated into the optimization process, such as limiting the sum of motor mass for a subset of joints or limiting the
Fig. 2: Overview of design optimization of high-DoF actuation system for humanoid robots
copper loss on motors for a certain duration. As long as the constraints are multilinear polynomial, they can be perfectly linearized (see Sec. II). Although not presented in this paper, logical constraints can be imposed for compact arrangement of large components such as motors.
As the last step of the workflow, designers need to analyze the resulting optimal solutions and modify the aforementioned ingredients for design optimization. For instance, if the optimization is infeasible due to high torque requirements, one can expand the design library by adding more torque-dense motors, relax the design rules, or relax the task requirements (e.g., smaller total mass, smaller desired CoM velocity of the ROM).
### _Maximizing Force Control Bandwidth_
Smaller reflected inertia of actuators contributes to faster limb swing or acceleration and improves the ability to mitigate impacts from contact by quickly retracting from contacts (e.g., ground contact). Both capabilities are critical requirements for dynamic humanoids.
Hereafter, the goal of minimizing reflected inertia of the actuation system is reconnected with the earlier optimization sketch (4) to formulate objective functions using binary variables. Although [4] suggests minimizing reflected inertia in task-space, obtaining such expression using binary variables in linearizable form is prohibitive due to rational forms arising from matrix inversions [6, Eqn. (17)]. As a proxy to task-space inertia, a joint-space reflected inertia matrix is used. The downside of using this proxy is that it neglects the lever effect of limb lengths to joints which is beyond the scope of this paper. Nevertheless, some relevant observations are stated in Sec. V-C.
Let us build the joint-space reflected inertia matrix of an actuation system consisting of \(n_{d}\) joints (or actuators) from libraries of motors, gearboxes, and actuator-joint couplings with associated binary variables \(\mathbf{x}_{i}\!\in\!\mathbb{B}^{n_{m_{i}}}\), \(\mathbf{y}_{i}\!\in\!\mathbb{B}^{n_{g_{i}}}\), \(i\!\in\![n_{d}]\), and \(\mathbf{z}\!\in\!\mathbb{B}^{n_{c}}\), where \(n_{(\cdot)}\) is the number of candidates in the respective libraries. The reflected inertia of an \(i\)'th actuator is the product of binary combinations of rotor inertia \(I^{r}_{i}(\mathbf{x}_{i})_{B}\) and gear ratio squared \(N^{2}_{i}(\mathbf{y}_{i})_{B}\),
\[I^{a}_{i}(\mathbf{x}_{i},\mathbf{y}_{i})=I^{r}_{i}(\mathbf{x}_{i})_{B}N^{2}_{ i}(\mathbf{y}_{i})_{B}. \tag{5}\]
Next, the reflected inertia of each actuator forms a diagonal matrix of actuator-space reflected inertia \(\mathbf{H}^{a}_{\mathrm{r}}(\mathbf{x},\mathbf{y})=\mathrm{diag}(I^{a}_{1}, \ldots,I^{a}_{n_{d}})\). This inertia matrix is projected on to joint-space by a binary combination of actuator-joint coupling Jacobian \(\mathbf{C}(\mathbf{z})_{B}\) which represents a mechanical connection between actuators and joints. The coupling Jacobian maps actuator velocities \(\dot{\boldsymbol{\psi}}\in\mathbb{R}^{n_{a}}\) to joint velocities \(\dot{\mathbf{q}}\in\mathbb{R}^{n_{a}}\) as \(\mathbf{C}:\dot{\boldsymbol{\psi}}\rightarrow\dot{\mathbf{q}}\). The joint-space reflected inertia is as follows,
\[\overline{\mathbf{H}}^{i}_{\mathrm{r}}(\mathbf{x},\mathbf{y},\mathbf{z},k)= \mathbf{C}^{-\top}\mathbf{H}^{a}_{\mathrm{r}}\mathbf{C}^{-1}, \tag{6}\]
where \(k\) denotes the time instance at which the inertia is evaluated along the given tasks and \(\mathbf{x}\), \(\mathbf{y}\) are lifted binary variables of \(\mathbf{x}_{i}\), \(\mathbf{y}_{i}\).
Lastly, the _size of the inertia matrix_ is measured using the \(\mathrm{trace}()\) operator. The trace of an inertia matrix is the sum of its eigenvalues which relate to the lengths of semi-axes of the inertia ellipsoid (Fig. 3(a)). Hence, the trace operator allows the optimizer to minimize reflected inertia on every dimension until motor mass becomes too small and, subsequently, a motor does not have sufficient torque to achieve given tasks.
### _Task Feasibility_
Task feasibility assures that the desired task trajectories are accommodated within the capabilities of the motors. The torque and velocity capability or an operation domain of a motor is approximated as a convex polygon \(\mathrm{convhull}(\mathbf{v}_{1},\ldots,\mathbf{v}_{n})\) (colored polygon in Fig. 3(b)). This polygon comprises vertices \(\mathbf{v}_{i}=(\alpha_{i}\omega_{\diamond},\beta_{i}\tau_{\diamond})\), whose coordinates are defined by motor parameters \(\omega_{\diamond},\tau_{\diamond}\) such as rated velocity and torque with coefficients \(\alpha_{i},\beta_{i}\). Next, the vertices are associated with the motor library through a binary combination, \(\mathbf{v}_{i}(\mathbf{x})=(\alpha_{i}\omega_{\diamond}(\mathbf{x})_{B}, \beta_{i}\tau_{\diamond}(\mathbf{x})_{B})\). Consequently, the operation domain is expressed as follows,
\[P(\mathbf{x},\boldsymbol{\alpha},\boldsymbol{\beta})=\mathrm{convhull}\left( \mathbf{v}_{1}(\mathbf{x})_{B},\ldots,\mathbf{v}_{k}(\mathbf{x})_{B}\right), \tag{7}\]
where \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\) are vectors of coefficients \(\alpha_{(\cdot)}\) and \(\beta_{(\cdot)}\).
Next, the joint-space trajectories of given tasks are concatenated and projected to motor-space. The joint-space trajectory of length \(T\), \(U^{i}=\{(\boldsymbol{\omega},\boldsymbol{\tau})_{k}\,|\,\boldsymbol{\omega}, \boldsymbol{\tau}\in\mathbb{R}^{m},k\in[T]\}\), is projected to actuator-space using actuator-joint coupling Jacobian \(\mathbf{C}(\mathbf{z})_{B}\), then projected to motor-space using gear ratio Jacobian \(\mathbf{N}(\mathbf{y})=\mathrm{diag}(N_{1}(\mathbf{y})_{B},\ldots,N_{m}( \mathbf{y})_{B})\). The tasks trajectory on motor-space \(U^{\mathsf{m}}(\mathbf{y},\mathbf{z})\) is obtained as a function of choices of gearboxes and couplings,
\[\begin{split} U^{\mathsf{m}}(\mathbf{y},\mathbf{z})=\{( \boldsymbol{\omega},\boldsymbol{\tau})\,|\,\boldsymbol{\omega}=\mathbf{N} \mathbf{C}^{-1}\boldsymbol{\omega}_{\mathrm{j}},\\ \boldsymbol{\tau}=\mathbf{N}^{-\top}\mathbf{C}^{\top}\boldsymbol{ \tau}_{\mathrm{j}},(\boldsymbol{\omega}_{\mathrm{j}},\boldsymbol{\tau}_{ \mathrm{j}})\in U^{\mathrm{j}}\}.\end{split} \tag{8}\]
Finally, the following task feasibility constraint ensures the task trajectory to be contained within the operation domain
\[U^{m}(\mathbf{y},\mathbf{z})\subseteq P(\mathbf{x},\boldsymbol{\alpha}, \boldsymbol{\beta}), \tag{9}\]
which easily converts to a set of linear constraints.
### _Design Rules_
In this study, we bound the motor mass budget. If the motor mass budget is left unbounded, the optimizer will always prefer heavier, larger, and more torque-dense motors.
Fig. 3: (a) The size of reflected inertia matrix is measured as the sum of semi-axes lengths which is obtained by trace operation (b) Task trajectory projected on to motor-level is constrained to remain within the operation domain of a motor defined by vertices \(\mathbf{v}_{i}\).
However, there is a limit to the diameter of the motor that fits inside the robot's body. Moreover, the sum of motor mass may take up to \(40\%\) of the entire mass of a robot [20]. Hence, a smaller motor mass budget may leave significant room for a larger payload. The motor mass budget constraint is enforced as follows,
\[m_{t}(\mathbf{x})_{B}=\sum_{i=1}^{m}m_{i}x_{i}\leq m_{t}^{\text{UB}}, \tag{10}\]
where \(m_{t}\) is total mass of a design case, \(m_{i}\) is mass of candidate motors and \(m_{t}^{\text{UB}}\) is the total mass budget.
Designers can further employ various design rules such as limits on torque, power, etc. at various levels such as motor, actuator, and joint-levels. It is because system capability and task trajectories can be explicitly expressed using binary combinations. Second, logical constraints are also allowed. A statement, 'If a component \(A_{1}\) is used, then \(A_{2}\) also is used', is translated as \(x_{1}-x_{2}\geq 0\) where \(x_{1}\), \(x_{2}\) are binary variables that associate components \(A_{1}\), \(A_{2}\). For more logical constraints, refer to [21].
## IV Humanoid Design Study
### _Robot Model and Tasks_
A humanoid robot is modeled as a single rigid body with massless limbs and weight \(20\,\)kg that closely resembles Tello in [5]. The joint ordering is hip yaw, hip roll, hip pitch, and knee pitch followed by ankle pitch.
Next, based on the ROM, two task trajectories were generated (Fig. 4). The first task corresponds to moderately fast walking, with a peak speed of 1.3 m/s, a step frequency of 0.3 s. Joint torques and velocities of single support phase (0.15 s) are used. The walking trajectories are generated by simulating the SRBD of Tello. A step placement feedback controller that tracks a predefined walking speed is implemented [22]. An optimization-based balance controller, similar to in [23], computes the ground reaction forces required to realize the desired center-of-mass walking dynamics. The SRBD task-space trajectories are converted to joint-space trajectories using inverse kinematics. The resulting joint velocities, which contain high-frequency signals, are low-pass filtered at a cutoff frequency of 30 Hz to obtain meaningful solutions. The second task trajectory is to lift a point mass of 15 kg in a snatch style over 3.6 s. This trajectory is generated by extracting the joint trajectory from a video of an athlete performing a snatch. The trajectory is scaled down to match the size of the robot, and joint torques are obtained using inverse dynamics. The lifting task only recruits the pitch axes of the hip, knee, and ankle joints.
It is important to note that designers may choose other tasks and this study is merely showcasing an example. Depending on the choice of task, optimal designs may vary significantly.
### _Design Library and Rules_
The design space is formulated by combinations of the following component libraries. The motor library contains \(n_{m}=10\) motors from CubeMars (AK, R, RI series [24]). The reduction library consists of ideal transmission without frictional loss and gear ratio ranges from 1 to 12 at whole number interval \((n_{r}=12)\). The vertices of the operation domains of the motors are defined by peak torque and rated velocity from datasheets and vectors of tuning coefficients \(\boldsymbol{\alpha}=(1.5,-2.5,-2.5,1.5)\), \(\boldsymbol{\beta}=(1.2,1.2,-1.2,-1.2)\).
There are 8 couplings studied in this paper including 1 serial and 7 parallel couplings. The parallel couplings are based on modules of 2-DoF differential coupling whose Jacobian is
\[\mathbf{C}_{d}=\frac{1}{2}\begin{bmatrix}1&1\\ -1&1\end{bmatrix}. \tag{11}\]
The couplings are labeled as'serial', 'par-(\(n_{1}m_{1}\))' and 'par-(\(n_{1}m_{1}\))-(\(n_{2}m_{2}\))' where \(n_{i},m_{i}\) denotes the index of joints involved in \(i\)'th differential coupling (see Table. I). Since a mechanical coupling between faraway joints is typically impractical, the couplings are limited to consecutive joints. The hardware realization of Tello [5] utilizes 'par-23-45'.
The total mass of 5 motors is subject to a constraint (motor mass budget), with 25 values ranging from 2.2 kg to 4.0 kg at intervals of 0.075 kg. After a few runs of modifying the values of the motor mass budget, it was found that the optimization was infeasible below 2.2 kg and solutions did not vary beyond 4.0 kg.
### _Design Optimization_
The optimization problem is detailed as follows. Inclusion of library data is implied by the use of binary optimization
Fig. 4: Sketches of tasks used in humanoid design study. (a) walking forward at moderately fast speed and (b) lifting an object in snatch style.
variables \(\mathbf{x},\mathbf{y},\mathbf{z}\).
\[\begin{array}{ll}\underset{\mathbf{x},\mathbf{y},\mathbf{z}}{\text{min}}&\sum_ {k=1}^{T}\text{trace}(\overline{\mathbf{H}}_{r}^{i}(\mathbf{x},\mathbf{y}, \mathbf{z},k))\\ \mathrm{s.t.}&U^{m}(\mathbf{y},\mathbf{z})\subseteq P(\mathbf{x},\boldsymbol{ \alpha},\boldsymbol{\beta}),\\ &m_{t}(\mathbf{x})_{B}\leq m_{t}^{\text{tUB}},\\ &\mathbf{x}\in\mathbb{B}^{n_{a}\times n_{m}},\mathbf{y}\in\mathbb{B}^{n_{a} \times n_{r}},\mathbf{z}\in\mathbb{B}^{n_{c}},\\ &\text{exclusivity}(\mathbf{x},\mathbf{y},\mathbf{z})\end{array} \tag{12}\]
To explicitly study the effect of constraining total mass budget for all 5 motors and employing different coupling Jacobian, the optimization was run for all combination of mass budget constraints and coupling Jacobians (total \(25\times 8=200\) runs). After optimal solutions are obtained, system characteristics such as inertia matrix and force/torque capability polytopes (FCP/TCP) in joint and task-space and copper loss on actuators are retrieved. The FCP/TCP represents the maximum force/torque that can be applied for a given configuration, based on the limits of actuators [7].
The optimization problem (12) is linearized using techniques introduced in Sec. II. The design optimization is written in MATLAB and calls an integer linear programming solver (intlinprog, Gurobi). The solver was simultaneously run on CPU (20 cores, Intel i7-12700H) with default solver parameters. Computation of total 200 runs took less than 100 s.
## V Results and Discussion
### _Motor Mass Budget vs. Reflected Inertia vs. Copper Loss_
The first row of Fig. 5(a) gives an overview of optimal designs obtained with varied motor mass budgets and use of different couplings. First, as the motor mass budget decreases, availability of larger and more torque-dense motors decreases. Consequently, the optimal solutions leans towards smaller and weaker motors, necessitating the usage of elevated gear ratios. In the end, the use of higher gear ratios inflates reflected inertia or contributes to higher cost (optimal objective function value).
As to the effect of couplings, a few coupling cases were found infeasible (par-34, par-12-34) across all motor mass budgets, whereas other couplings become infeasible as the motor mass budget decreases. When the motor mass budget was limited to extremely small values, only coupled mechanisms provided sufficient torques and velocities (feasible). The serial actuation (no coupling) demonstrated the least reflected inertia at the cost of high copper loss for walking task across all cases of motor mass budget. In conclusion, this result suggests that 1) the framework can filter out couplings that cannot generate sufficient torque or speed, 2) at extremely low motor mass budget, actuators need to cooperate so that they can amplify joint torques to achieve desired tasks, and 3) use of coupling increases the reflected inertia for designs with a large motor mass budget.
The second and third row of Fig. 5(a) shows the combined copper loss of all motors of each task. As the motor mass budget decreases, the copper loss also decreases for most couplings. This tendency is opposite to that of reflected inertia because larger reflected inertia implies higher gear ratios, which again implies less motor torque and motor current draw. Two optimal designs following this trend are compared in Table. II. However, these implications are not strict: in the walking task, a few coupling cases did not follow this tendency (Fig. 4(a), second row, serial, par-45, par-23-45, around 3.2\(\sim\)3.7 kg). This is because different choices of gear ratio and motor can emerge as long as they are optimal and satisfy design rules.
If this study is posed as a multi-objective optimization with minimization of reflected inertia and the copper loss, the optimal solutions can be plotted in 3D space as Fig 5(b). Pareto optimal solutions are emphasized as larger dots over other optimal solutions. To aid visual understanding, optimal solutions are projected onto 2D slices.
### _Effect of Actuator-Joint Coupling_
The influence of the usage of couplings on task-space FCP and motor-level trajectory is visualized in Fig. 6 which presents a comparative analysis between two design cases, specifically, 'par-12' and 'par-23-45'. The latter utilizes more a higher degree of coupling than the former while both are designed with an identical motor mass budget.
Fig. 5: (a) Optimal costs (trace of joint-space reflected inertia) vs. total mass budget for 5 motors for every coupling case. Total copper losses per walking and snatch lifting task (b) Optimal designs’ motor mass budget, optimal cost and copper loss of snatch task. The bigger dots represent Pareto front.
A notable point in this comparison involves the velocity-dependent nature of FCP, attributed to the contour of the operational domain of motors. When motor velocities are high, the available torque (T1) near the boundary of the operation domain is reduced due to high motor velocity (Fig. 6(d1), (d2)).
The latter case, with additional degrees of coupling, has more chances for cooperative actuation which means that the load on a joint can be shared among the coupled actuators. Thanks to the load sharing, the individual actuators experience lower torque requirements. This allows employing smaller gear ratios which renders smaller motor velocities. Hence, employing more degrees of coupling has a benefit of keeping the motor-level trajectory (velocity) at a safe distance from the boundary of the operational domain. The degradation of motor torque capability in design case 'par-12' manifests as a significantly smaller FCP at 129 ms (Fig. 6(a1), (b1)), in contrast to a larger FCP in (Fig. 6(a2), (b2)).
The state-dependent nature of the FCP could be related to the margin of admissible error of feedback controllers. For instance, if joint velocity deviates from a (pre)planned trajectory towards the boundary of the operation domain of motors, the torque capability diminishes. Simultaneously, a feedback controller will attempt to bring the state back to reference. At that moment, if the torque margin is insufficient, the torque commands from the feedback controller could suffer from saturation which may lead to failure. Designers can prevent this by modifying the approximated operation domain to be smaller than the actual operation domain of motors by tuning the coefficients \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\).
### _Characterization of Inertia and Capability Polytopes_
The relation between reflected inertia and FCP/TCP in both joint- and task-space is analyzed to address two possible claims; 1) minimization of reflected inertia should take place at task-space level, not joint-space and 2) maximization of TCP/FCP could be also taken into account as well as minimization of the reflected inertia. Unfortunately, the authors could not obtain direct quantification of reflected inertia in task-space and the volume of TCP/FCP using binary integer linear programming. Still, a few trends were observed by plotting the relation between the size of reflected inertia versus volume of TCP/FCP in joint-space and task-space as in Fig. 7. Optimal designs with smaller motor mass budgets are plotted with more opacity. The TCP/FCP were calculated without considering the degradation in torque margin due to high velocity. Since FCP varies as configuration (joint angles) changes, its distribution is illustrated as an 0.7-sigma ellipse. In each space, two similar trends were observed; 1) serial actuation shows more cohesiveness in the relation between the size of reflected inertia and FCP/TCP compared to coupled ones, 2) a combination of coupled actuation and smaller mass budget leads to larger size and variation of
Fig. 6: Comparison of two optimal designs under identical motor mass budget and with different usage of coupling mechanism. (a1), (a2) TCP of contact leg in contact along walking. (b1), (b2) Frontal views show significant difference in size the FCP. (c1), (c2) FCPs of both designs are large enough to contain ground reaction forces to achieve moderately fast walking. (d1), (d2) Couplings affect how task trajectories are projected onto motor-level and motor torque capability margins (T1 vs. T2).
reflected inertia, and 3) larger motor mass budget allows larger FCP/TCP.
## VI Conclusion
The existing design principles and optimal design tools for the design of multi-DoF actuation systems often fall short in providing practical designs or understanding of optimal designs. To resolve this issue, an interactive optimal design tool for actuator system for humanoids is introduced. This tool is designed to incorporate expert designers' knowledge and rapidly solve large numbers of design studies to expand understanding of optimal designs.
This paper closes with several limitations of the present study and suggestions for future research. First, ROM employed in this study assumes massless limbs and actuator dynamics is neglected. Hence, the inertial forces exerted on the limbs and rotors within the actuation system are not accounted for. Future work may incorporate inertial forces associated with the limbs' apparent inertia and the reflected inertia of the actuators into the optimization process. Second, the applicability of optimal design solutions produced by the proposed tool remains unclear. An essential next step would be to integrate this tool with simulation software for the validation of proposed designs.
## Acknowledgment
To research advisors Patrick Wensing and Donghyun Kim, Johannes Englsberger, and my colleagues Donghoon Baek and Seunghyun Bang for guidance and fruitful discussion.
|
2303.03480 | Can an Embodied Agent Find Your "Cat-shaped Mug"? LLM-Guided Exploration
for Zero-Shot Object Navigation | We present LGX (Language-guided Exploration), a novel algorithm for
Language-Driven Zero-Shot Object Goal Navigation (L-ZSON), where an embodied
agent navigates to a uniquely described target object in a previously unseen
environment. Our approach makes use of Large Language Models (LLMs) for this
task by leveraging the LLM's commonsense reasoning capabilities for making
sequential navigational decisions. Simultaneously, we perform generalized
target object detection using a pre-trained Vision-Language grounding model. We
achieve state-of-the-art zero-shot object navigation results on RoboTHOR with a
success rate (SR) improvement of over 27% over the current baseline of the
OWL-ViT CLIP on Wheels (OWL CoW). Furthermore, we study the usage of LLMs for
robot navigation and present an analysis of various prompting strategies
affecting the model output. Finally, we showcase the benefits of our approach
via \textit{real-world} experiments that indicate the superior performance of
LGX in detecting and navigating to visually unique objects. | Vishnu Sashank Dorbala, James F. Mullen Jr., Dinesh Manocha | 2023-03-06T20:19:19Z | http://arxiv.org/abs/2303.03480v2 | # Can an Embodied Agent Find Your "Cat-shaped Mug"?
###### Abstract
We present LGX, a novel algorithm for Object Goal Navigation in a "_language-driven, zero-shot manner_", where an embodied agent navigates to an _arbitrarily described_ target object in a _previously unexplored_ environment. Our approach leverages the capabilities of Large Language Models (LLMs) for making navigational decisions by mapping the LLMs implicit knowledge about the semantic context of the environment into sequential inputs for robot motion planning. Simultaneously, we also conduct generalized target object detection using a pre-trained Vision-Language grounding model. We achieve state-of-the-art zero-shot object navigation results on RoboTHOR with a success rate (SR) improvement of over 27% over the current baseline of the OWL-VIT CLIP on Wheels (OWL CoW). Furthermore, we study the usage of LLMs for robot navigation and present an analysis of the various semantic factors affecting model output. Finally, we showcase the benefits of our approach via _real-world_ experiments that indicate the superior performance of LGX when navigating to and detecting visually unique objects.
## I Introduction
Humans do not conform to preset class labels when referring to objects, instead describing them with free-flowing natural language. Robot agents performing _object goal navigation_ in household environments must be able to comprehend and efficiently navigate to this seemingly infinite, arbitrary set of objects defined using natural language. For instance, a human may ask the robot agent to find its "cat-shaped mug." An agent trained on rigid class labels may interpret this as the human asking for a "cat" or a "mug" when the human is really referring to a mug in the shape of a cat. These types of unique objects typically lie outside the domain of the object categories commonly found in large image datasets such as ImageNet 21k [1] and OpenImages V4 [2] preventing an agent from detecting them prior to the humans command. Additionally, agents deployed in household environments may be required to navigate to these target objects without explicitly having a map or layout of the house available.
In our work, we aim to address these issues by tackling the _L-ZSON_ task [3]. _L-ZSON_ or Language-Driven Zero-Shot Object Navigation involves the agent taking a natural language description of an object, and tasking an agent with finding it in a "zero-shot" manner, without ever having seen the environment _nor_ the target object beforehand. This is in contrast to the conventional tasks in the literature of Object Goal Navigation [4, 5], where the agent's goal is to find an object in the environment from a specified set of object categories, and Zero-Shot Object Navigation (ZSON) [6, 7], where the target object is similarly from pre-set object categories, but the environment is previously unseen.
Simulation environments for object goal navigation tasks, including RoboTHOR [8] and AI Habitat [9], only contain common day-to-day household objects described using simple language (eg. Mug, Table, Bed). However, humans tend to use unconstrained, natural language when talking to agents [10], leading to ambiguity in the agents' interpretation [11]. This problem becomes more apparent in the sim2real transfer of common object navigation models [12]. In this work, we seek to address this issue by carrying out real-world experiments with unique object references (eg. olive-colored jacket).
Common approaches to solving Object Goal Navigation are based on fully supervised learning [4, 13], which is not practical for an agent that is expected to detect arbitrarily
Fig. 1: **LLM-Based Navigation**: Our method, LGX approaches the problem of Language-driven Zero-Shot Object Navigation or L-ZSON. To navigate to and detect an unseen, arbitrarily described object class in an unknown environment, we first extract visual semantic information about the environment. This information is utilized to develop a prompt for the Large Language Model (LLM), whose output provides us with either object sub-goals or cartesian directions to guide the embodied agent towards the target. Meanwhile, GLIP searches for the environment for the target object, which in this case is a “_cat-shaped mug_”.
described objects and perform consistently in dynamic real-world environments. While some recent works address generalizability to new locations via the ZSON task [14], even fewer address the issue of generalizing to novel objects [15] with the L-ZSON task, and none study real-world test cases that contain an abundance of unconstrained language. These works utilize large-scale pre-trained models including CLIP [16] and GLIP [17] to perform zero-shot open-vocabulary object detection in the wild. The downstream transfer of such 'foundation models' [18] has shown great improvement in various vision and language tasks such as image captioning [19] and question answering [20]. This transfer to robotics is non-trivial however, as unlike vision and language, robot tasks usually involve some form of experiential decision-making as the agent continuously interacts with the environment. As such, exploiting the implicit knowledge contained by these models to compose robot actions presents a unique challenge.
**Main Contributions:** Motivated by the challenges presented above, we present, LGX, a novel approach to leverage the implicit knowledge capabilities of large pre-trained vision and language models to solve the issues of efficient exploration and detection of arbitrarily described objects in the L-ZSON task.
The success of an object navigation task, including the specific case of L-ZSON, significantly relies on the performance of two major components involved -- _Sequential Decision Making_ and _Target Object Grounding_. The former refers to making exploratory decisions at each timestep, while the latter refers to locating a target object specified by its natural language description in an image. In this work, we seek to effectively utilize large-scale open vocabulary models including Large Language Models (LLMs) and Vision-Language (VL) Models to address generalizability issues that hinder the performance of both these components.
The performance of LLMs is highly dependent on the quality of the prompts used [21] as input. As the success of our object navigation task is directly influenced by this factor, we study various visual and semantic factors affecting the formulation of these prompts and present a case-by-case analysis of the effect of various prompt types. Additionally, we also study the usage of VL models in Target Object Grounding, aiming to show improved performance with unique object references. We make the following novel contributions:
1. We present LGX, a novel approach to tackle L-ZSON, a language-guided zero-shot object goal navigation task. Our approach localizes objects described by unconstrained language by making use of large-scale Vision-Language (VL) models and leverages semantic connections between objects built into Large Language Models (LLMs). Specifically, we study the implicit capabilities of LLMs in assisting the sequential navigational decisions necessary to perform zero-shot object navigation.
2. Our approach utilizes visual scene descriptions of the environment to _formulate prompts_ for LLM's, the output of which drives our navigation scheme. We study various types of prompts and provide insights into successfully using these prompts for robot navigation.
3. Our approach shows a 27% improvement on the state-of-the-art zero-shot success rate (SR) and success weighted by path length (SPL) on RoboTHOR.
4. Finally, we also present a transfer of our method onto a real-world robotics platform and study the various complexities involved in this setting. To the best of our knowledge, ours is the first approach to evaluate the performance of L-ZSON methods in the real world.
## II Related Work
### _Language-Guided Robotics_
Using language to guide robots is a popular task in literature, with work ranging from using generalized grounding graphs [22] for robot manipulation [23, 24] to performing language-guided navigation [25, 26]. Tellex et. al in [27] have recently presented a useful survey on using language from a robotics perspective.
Thomas et. al in [28] presents an approach to parse unconstrained natural language via a systematic probabilistic graph-based approach. More recent work tackling this problem by Jesse et. al. [29, 30] and Gao et. al. [31] has explored the use of human-robot dialogue to gather relevant information for completing tasks. Parsing unconstrained natural language is very relevant in our work, and we are motivated by the techniques developed by these papers.
### _Object Goal Navigation_
In Embodied AI, using language to provide navigational guidance towards a target object presents many popular tasks including Vision-and-Language Navigation (VLN) [32, 33] and Object Goal Navigation [4, 5]. Fully-supervised learning approaches for object navigation in the past have often combined both these components utilizing either Reinforcement or Imitation Learning [6], with some form of embedded memory [34]. These approaches however work only with a specific set of objects, and the performance is constrained to the domain of the dataset they're trained on. In our work, we explicitly seek to address the generalizability issues involved in both these components by effectively capturing the implicit commonsense knowledge captured by large pre-trained vision and language models.
### _Language-Driven Zero-Shot Navigation_
Recent works have attempted to use CLIP [16] for performing zero-shot embodied navigation. CLIP is a large pre-trained Vision-Language model that is capable of zero-shot object detection. Dorbala et. al. in [35] use CLIP to perform Vision-and-Language navigation in a zero-shot manner, while Gadre et.al in [15] have used it to perform object goal navigation. Both these works work under the assumption of unseen environments.
L-ZSON introduced by Gadre et. al in [3] approaches the problem of zero-shot object navigation, using uncommon target objects. They obtain a baseline for this task using OWL-ViT, a finetuned vision transformer for object grounding, and frontier-based exploration (FBE) [36]. In contrast, our approach uses GLIP [17], a pre-trained VL model for zero-shot object grounding. To explore the environment, we incorporate GPT-3 [37], an LLM, to make navigational decisions.
### _LLMs for Language-Guided Navigation_
The adaptation of Large Language Models in robotics has not been widespread, given the various real-world challenges involved with implementation. A few recent works have looked at using generative models for navigation, specifically, LM-Nav [38] and VLMaps [39]. Both these works look at solving the Vision-and-Language Navigation (VLN) problem, where the input is the unconstrained language describing a _path to the goal_. The latter uses GPT-3 [37] to obtain "landmarks" or subgoals, while the former focuses on using an LLM for "code-writing" [40]. In contrast, our focus is on translating visual scene semantics into input prompts for the LLM to obtain navigational guidance in the form of actions. We directly incorporate the LLM output into a sequential decision-making pipeline such that it drives our agent's navigation scheme.
## III Solving L-ZSON: LLM + GLIP For Exploring The Environment
### _Method Overview_
We present an overview of our method in Figure 2. The core component of our approach involves using a Large Language Model (LLM) to predict where the agent needs to navigate. To do this, we first extract the context from the scene in the form of objects or captions. Either of these contextual cues are then used to devise a prompt asking the LLM about which how the agent should proceed to explore the environment. The LLM output is then either a cartesian direction or a sub-goal object in the agent's vicinity that it needs to move towards.
Simultaneously, we use a Vision-Language model, GLIP [17] to obtain a threshold score for target object grounding. Once the GLIP threshold is met, the agent assumes the target object to be in its egocentric view and raises a STOP condition. An episode is rendered successful if the target object is in the agent's view.
### _Scene Understanding_
Our agent first observes the environment, gathering RGB as well as Depth images for inspection. To gather as much information as possible, we have the robot rotate in place 360 degrees, taking images \(im_{i}\) at a set resolution \(r\). This leaves us with a set of \(360/r\) RGB images, \(IM\), while the depth images are used to construct a 2D costmap of the environment. Every image in \(IM\) is then fed into either an object detection or an image captioning model. Both these models give us different types of results, which we discuss in the experimentation section. For object detection, we use YOLO [41], while BLIP [42] gives us image captions. BLIP produces descriptive captions \(C\) of the environment, which requires a higher resolution. On the other hand, YOLO gives us a list of objects \(O\) around the agent that it can potentially navigate towards. Either \(C\) or \(O\) form the basis of our prompt to the large language model.
The full circle rotation by the robot is significant in that it increases the observational capacity, enabling it to make a fully informed navigation decision from its current position
Fig. 2: An overview of our approach. We first gather observational data from the environment by performing a 360 degree rotation to obtain depth and RGB images around the agent. The RGB images give us semantic information about the objects in the agent’s view, while the depth image allows us to create a costmap. We then synthesize prompts for the LLM by utilizing the extracted object labels. Finally, the LLM drives the navigational scheme by producing an output from the object list, which tells the agent which direction to head towards. Simultaneously, we attempt to ground the target object in the scene with GLIP. When the target is found, we exit decision making loop and navigate directly to it.
in the environment. Without it, the robot would proceed toward seen objects over unknown space, even if none of the seen objects were related to the goal object, \(o_{g}\). For example, if the robot goal object \(o_{g}\) is a "blue pillow," but it is initialized facing a kitchen and we see objects such as "microwave," "mug," and "table," the robot will proceed to explore near those objects because it does not know that directly behind it is a "bed" or a "couch."
Simultaneously, while performing the full circle rotation, the agent uses its depth images to construct a costmap of the environment. Planning on this costmap helps the agent easily avoid obstacles that it may have encountered without it.
We also run GLIP on each of the RGB images with the target object as an input. When the grounding accuracy of GLIP is beyond a threshold \(G_{t}h\), we assume that the target object is in view of the agent, which triggers a STOP signal. The episode is rendered successful if the ground truth target object lies in the view.
### _Intelligent Exploration with Large Language Models_
We utilize the extracted semantics to devise a prompt for our LLM, GPT-3. There are two scenarios we explore,
1. YOLO + LLM: In this case, we utilize the list of objects \(O\) that YOLO detects around the agent to synthesize a prompt. For improving object detection, we set the rotate resolution \(r\) to a lower value here.
2. BLIP + LLM: Here, we utilize image captions \(C\) generated from the previous step to create a prompt. The rotate resolution \(r\) is set to 90 here, referring to cartesian directions that the agent can take while exploring.
The LLM output upon using the YOLO + LLM approach gives us an object from \(O\) to navigate towards. The agent then reorients itself towards this object. While using BLIP + LLM, the output gives us a cardinal direction towards which the agent reorients itself. When the LLM output does not follow these expected outcomes, the agent chooses a random direction.
### _Goal Detection and Motion Planning_
While completing the 360\({}^{\circ}\) rotation-in-place, we also run GLIP [17] with the goal object \(o_{g}\) as its target label. Should GLIP find the target object \(o_{g}\) with a high enough confidence, we terminate the exploration loop.
Once the agent has reoriented itself toward either a target object or a direction, we use the 2D costmap to perform navigation. The costmap allows us to avoid obstacles that might be obstructing a clear path towards the selected sub goal.
When using YOLO, the sub-goal is the 3D position of the target object, while using BLIP, we select a random point a fixed distance in front of the robot.
## IV Analysis of our Approach
### _Using GLIP for Zero-Shot Detection_
Open-vocabulary grounding models have demonstrated strong zero-shot performance to object-level recognition tasks and proved to generalize well across numerous data sources. A key distinguishing factor in these models is the need to input text, or prompt, describing objects of interest to the model alongside an image. For our method we chose to use GLIP for its inclusion of a bounding box around the detected objects of interest. The bounding box provides us key information for localizing the object necessary to drive the navigation towards it. GLIP outputs can be defined as
\[\{o_{t,i},b_{t,i}\}=GLIP(I_{t},P_{o}) \tag{1}\]
where \(o_{t,i}\) and \(b_{t,i}\) are the object detections and bounding boxes respectively. \(I_{t}\) and \(P_{o}\) represent the input image and the input prompt, defining the objects of interest, respectively. For our L-ZSON task, GLIP enables us to detect and localize objects described with natural language. An example of this can be found in figure 3 showcasing how GLIP can not only identify the object defined using natural language, the "Cat-shaped mug," but differentiate it from related objects. Because of these behaviors, running GLIP during our rotate-in-place procedure allows us to confidently detect our goal objects \(o_{g}\) irrespective of how they are described.
### _Examining LLM Prompts for Scene Exploration_
The outcome of an LLM is greatly influenced by the prompts that they are given. With too little context the LLM is unlikely to make informed decisions and too much context potentially confuses the model. The structure of the prompt to the LLM can also greatly affect the responses the model provides, with some structures producing output significantly easier to parse. Should the LLM not provide us with a valid response the agent moves in a randomized direction. This situation is not ideal so it is also important that the prompt is phrased in a way that minimizes invalid responses as much as possible.
We compare seven different LLM prompts along three different axis with a base prompt of "You are controlling a home robot. The robot wants to find a \(o_{g}\) in my house. Which object from \(\{O\}\) should the robot go towards? Reply with ONE object from the list of objects." We first explore how the perspective of the prompt alters the LLM feedback. These prompts are below:
* **Robot-Prompt**: "You are controlling a home robot. The robot wants to find a \(o_{g}\) in my house. Which object from \(\{O\}\) should the robot go towards? Reply with ONE object from the list of objects."
* **I-Prompt**: "I want to find a \(o_{g}\) in my house. Which object from \(\{O\}\) should I go towards? Reply in ONE word."
Fig. 3: An example of GLIP output when fed with the input string “Cat-shaped mug. Cat. Mug” on the image given. GLIP can successfully locate a unique object, like a “cat-shaped mug” and differentiate between it and related objects like a cat or a mug.
* **Third-Person-Prompt**: "A \(o_{g}\) is in a house. Which object from \(\{O\}\) is likely closest to \(o_{g}\)? Reply with ONE object from the list of objects."
The Second axis we explore is the structure of the prompt. We vary what information is in the prompt first versus last.
* \(\{O\}\)**-First-Prompt**: "You are controlling a home robot. You must select one object from \(\{O\}\) that the robot should go towards to try to find \(o_{g}\) in my house. Reply with ONE object from the list of objects."
* **Get-Closest-Prompt**: "You are controlling a home robot. The robot wants to find a \(o_{g}\) in my house. Which object from \(\{O\}\) is probably the closest to \(o_{g}\)? Reply with ONE object from the list of objects."
* **"ONE word"-First-Prompt**: "Reply with ONE word. You are controlling a home robot. The robot wants to find a \(o_{g}\) in my house. Which object from \(\{O\}\) should the robot go towards?"
Last, we test creating prompts with natural language captions of the scene.
* **BLIP-Prompt**: "I want to find a \(o_{g}\) in my house. In FRONT of you there is.... To your RIGHT, there is.... BEHIND you there is.... To your LEFT there is.... Which direction from Front, Left, Right, Behind should I go towards? Reply in ONE word."
## V Experiments and Results
### _Experiment Setup_
We use the RoboTHOR [8] validation set as a simulation environment for our experiments. It contains 1800 validation episodes with 15 validation environments. 12 different goal object categories are present. A distinguishing factor of RoboTHOR relative to [9] based environments is the goal objects consisting of mainly small objects. This makes RoboTHOR more reminiscent of real-world tasks and more challenging relative to the alternatives.
**Prompt Selection Setup.** We ran each prompt on a subset of the RoboTHOR validation set that consisted of the episodes we regularly performed the worst and the best on for a sampling of performance.
**Metrics.** We report and compare Success Rate (SR) and Success Rate weighted by inverse path length (SPL) [43]. SPL is the primary metric used in both the Habitat and RoboTHOR challenges. For our prompt ablations, we define a new metric, **Prompt Success Rate (PSR)** as:
\[PSR=\frac{p_{suc}}{p_{total}} \tag{2}\]
where \(p_{suc}\) denotes the number of instances where the LLM chooses a valid response, and \(p_{total}\) denotes the total number of times the agent prompts the LLM. A valid LLM response is when it chooses either an object detected by the agent or a direction for navigation, depending on the semantic extraction scheme used.
**Real World Setup.** Motivated by the lack of unique target classes in RoboTHOR, we conducted a two-phase experiment with a TurtleBot 2 robot agent to further validate our method, LGX. The environment is set up with four rooms modeled after common household rooms with each room containing two large common objects, and two 'target objects' defined with natural language. The specific objects utilized and their room assignment can be found in Table I. The task for the robot is to navigate from a room that does not contain the target object, through a 'hallway,' and then into the room that contains the target object, before localizing the target object. To replicate a household layout where rooms can commonly be seen from the hallway, we assume that the robot agent can perceive the large, common objects in the adjacent rooms. A success case is defined by a successful GLIP detection of the target object.
### _Baselines and Ablations_
We compare our method, LGX with two state-of-the-art methods and an ablative method:
**CLIP-on-Wheels (CoW).**[3] use Grad-CAM, a gradient-based visualization technique with CLIP [16] to localize a goal object in the egocentric view. CoW employs a frontier-based exploration technique for zero-shot object navigation.
**OWL CoW.**[3] utilizes the OWL-ViT transformer, in place of a CLIP model. OWL-ViT is a model created through that turns CLIP-like models into object detectors by fine-tuning on a set prediction task. This detector then replaces CLIP in the CoW method.
**Random with GLIP.** As a baseline for our real-world analysis of LGX we also choose a random direction selector as an exploration module. The agent here does not rely on the LLM output and entirely takes random decisions.
### _Comparison with Baselines_
We compare the performance of our method with other models set up for the L-ZSON task in Table II. Our method significantly outperforms the OWL CoW and the original CoW with an improvement in the success rate of 7.5% and an improvement in the SPL metric. The improvement over both CoWs makes sense as its frontier-based exploration is relatively naive and likely prone to failures in the complex RoboTHOR environment. While GLIP extends CLIP for
\begin{table}
\begin{tabular}{l c c} Room & Target Objects & Common Objects \\ \hline \hline Kitchen & Red Bull can, Stevia sugar packets & sink, fridge \\ \hline Living Room & remote control, coffee table & couch, tv \\ \hline Bedroom & bust, olive-colored jacket & bed, blanket \\ \hline Office & silver pen, whiteboard & desk, computer \\ \hline \hline \end{tabular}
\end{table} TABLE I: Setup for our real-world experiments. We use four common household rooms and populate them with common and uncommon target objects that are likely found in them. The common objects are used to help navigate from room to room as they are large and easily perceived. The target objects are what the robot is trying to find to complete the task.
object detection similar to the OWL model, the original CoW utilized Grad-CAM on CLIP likely resulting in more failures to directly localize the target object.
In Figure 4 we compare directly with CoW and OWL across the different target objects in RoboTHOR. Our method outperformed than both baselines across smaller objects like 'bowl' and 'vase.' Our performance was similar to OWL for larger objects like 'television.' These results showcase the performance deficit of CoW that is likely due to the inability of CLIP to localize the target object in the image effectively.
### _Comparison of Prompt Tuning Strategies_
As seen in Table III, the natural language-based prompts from BLIP produced worse performance relative to the object-based prompts despite a perfect PSR. We believe this is due to the limited action space when under the BLIP-based prompting scheme. The object-based prompts gave the LLM many different pathing options while the BLIP-based prompts were by definition associated with the four cardinal directions and potentially leading the agent towards a dead end. For example, we noted episodes where the LLM continuously picked a cardinal direction where there was no valid path.
No significant difference in task SR was captured over our second axis of prompt tuning denoting the perspective of the LLM relative to the robot. This is despite a wide array of PSRs for the different perspectives. The robot-perspective prompt exhibited the highest SR, but also the lowest PSR of the perspectives explored. Notably, when using the robot-perspective prompt, the LLM responded with 'no' or 'nothing' more frequently over the empty responses more commonly seen in the other prompts.
Across our changes to the structure of the prompt, there was no significant difference in SR for the object-set-first prompt or the get-closest-object prompt. However, the "ONE word" first prompt, denoting the placement of the "reply with ONE word" phrase before the rest of the prompt exhibited significantly worse SR and PSR. We believe this is due to the LLM no longer heading this instruction when placed _before_ the remainder of the prompt. The high PSR of the get-closest-prompt indicates that picking the likely closest object may be a simpler problem for the LLM to approach. Similarly, the high PSR of the object-set-first prompt indicates that the LLM could better reference the object-set when it was placed at the beginning of the prompt.
We believe that the insignificant performance differences in task SR, despite large changes in PSR, is another indicator of RoboTHOR providing a skewed basis for this type of context dependent, intelligent exploration of the scene.
### _Comparison Against Baselines in the Real World_
In our real world experimentation, we found that our method significantly outperformed the available baselines, improving upon the SR of GoW by 26.4% and the SR of Random with GLIP by 47.3%. This resulted in GoW navigating to the correct room 33% of the time while Random with GLIP explored the objects in the starting scene with the same frequency as exploring the hallway. All of
Fig. 4: The class breakdown of LGX versus the OWL CoW and original CoW on RoboTHOR. LGX provides a strong improvement in localizing the baseball bat, bowl, laptop, spray bottle, and vase classes. Similar performance is noted on larger classes like television and garbage can.
Fig. 5: An sampling of GLIP success and failure cases in our real-world experimentation. When the goal object was present in the scene, GLIP accurately detected it 87.5% of the time. Conversely, when the goal object was not present in the scene, GLIP falsely detected it 8.3% of the time.
\begin{table}
\begin{tabular}{l c c} \multicolumn{3}{c}{RoboTHOR} \\ \cline{2-3} Model & Success-Rate (\%) \(\uparrow\) & PSR (\%) \(\uparrow\) \\ \hline \hline BLIP-Prompt & 29.2 & 100 \\ \hline I-Prompt & 33.3 & 87.7 \\ Robot-Prompt & 33.8 & 71.1 \\ Third-Person & 31.3 & 99.4 \\ \hline \{_O_\}first & 33.8 & 95.3 \\ Get-Closest-Object & 32.1 & 95.7 \\ “ONE word” first & 28.1 & 52.3 \\ \end{tabular}
\end{table} TABLE III: Comparison of seven different prompts across three axis of change on RoboTHOR. The object-based prompts (middle and bottom) outperform than natural language-based prompts.
the success rates were also effected by the failure cases of GLIP, specifically false negatives when attempting to detect the'stevia sugar packets' and false positives for the 'Red Bull can' and the'stevia sugar packets' (see Figure 5).
The LLM behavior in our method during our real world experimentation is characterized by three potential cases as shown in Figure 7. The success case occurs 54.2% of the time and is a result of the robot agent successfully navigating from the starting room into the hallway, then into the room that contains the target, before detecting the target with GLIP.
In a phase 1 failure case, the agent does not enter the hallway as the LLM believes one of the objects in the starting scene likely will lead to the target. One example of this we noted was when the target object is 'Red Bull can,' the LLM would output 'desk' when the starting scene was the office. While we placed the 'Red Bull can' in the kitchen for this experiment, it is plausible that you would find it on a 'desk,' explaining this output from the LLM. In the phase 2 failure case, the agent enters a room that does not include the target object. This occurred 20.8% of the time with our method. This case is associated with the LLM poorly relating the target object other objects in the target room. One example of this case is the 'olive-colored jacket' which the LLM typically believed would be found near the 'desk.' A breakdown of the system performance for each target object is found in Figure 6. Our method failed to localize the 'bust' believing it to be associated with the 'desk.' However, the relative success of the baselines indicates that GLIP succeeded in detecting the 'bust' once inside the correct room.
## VI Limitations, Conclusions, and Future Work
In this work we present a novel algorithm for language-based zero-shot object goal navigation. Our method leverages the capabilities of Large Language Models (LLMs) for making navigational decisions and open-vocabulary grounding models for detecting objects described using natural language. We showcase state of the art results on the RoboTHOR baseline, study the structure and phrasing of the LLM prompts that power our exploration, and validate our approach with real-world experiments.
**Limitations and Future Work.** Note that our method still includes a number of failure cases, especially when the LLM incorrectly localizes the target object. Future work should explore how varying the context fed to the LLM by filtering the list of objects detected or providing a history of visited objects.
|
2304.08572 | Strong-randomness renormalization groups | This is a very brief review article, written for a book (in preparation) in
memory of Michael E. Fisher and to celebrate 50+ years since the Wilson-Fisher
renormalization group. Strong-randomness renormalization groups were first
developed to treat various quantum critical ground states, especially in
one-dimensional systems. After briefly reviewing some of the earlier work with
these methods, the recent application of this approach to the many-body
localization (MBL) phase transition is reviewed. | David A. Huse | 2023-04-17T19:26:34Z | http://arxiv.org/abs/2304.08572v1 | # Strong-randomness renormalization groups
###### Abstract
This is a very brief review article, written for a book (in preparation) in memory of Michael E. Fisher and to celebrate 50+ years since the Wilson-Fisher renormalization group. Strong-randomness renormalization groups were first developed to treat various quantum critical ground states, especially in one-dimensional systems. After briefly reviewing some of the earlier work with these methods, the recent application of this approach to the many-body localization (MBL) phase transition is reviewed.
## I Introduction
Renormalization group approaches to systems in critical states have been developed for a wide variety of different types of systems. As a consequence of this variety, there are many very different types of renormalization group (RG) calculations. What do they all have in common? Almost all RGs coarse-grain a system of many degrees of freedom by some type of "integrating out" of the highest-energy or in some sense "stiffest" degrees of freedom. Good analytical control of the RG usually requires some level of simplicity of the couplings between the degrees of freedom that are integrated out and the remaining degrees of freedom.
For many translationally-invariant systems the RG is implemented in momentum space, as in the seminal Wilson-Fisher paper Wilson and Fisher (1964); Fisher (1965). In many cases, the highest-energy (or stiffest) degrees of freedom are the remaining modes with the highest momenta, and these are integrated out, thus decreasing the ultraviolet momentum cutoff. For systems of many degenerate fermions "highest momenta" may be replaced with momenta farthest in energy from the Fermi energy (see, e.g., Ref. Wilson and Fisher (1964)).
Some other renormalization groups work instead in real space, one notable example being the Kosterlitz-Thouless RG for two-dimensional superfluids, which can be formulated with a real-space cutoff on the distance between a vortex and an antivortex Kosterlitz and Thouless (1971); Kosterlitz and Thouless (1972). Vortex-antivortex pairs at this cutoff distance are integrated out, thus increasing the cutoff distance. The good analytic control of the Kosterlitz-Thouless RG is due to the fixed line governing the superfluid phase and its critical point being at zero density of vortices and antivortices, so the pairs that are being integrated out are dilute when we are near the fixed line. A similar real-space RG for some one-dimensional spin systems, where one integrates out pairs of domain walls at the cutoff distance, was developed earlier, in Ref. Huse (2010).
For systems with strong quenched spatial randomness, the highest-energy (or stiffest) degrees of freedom are typically at some particular location in real space where the couplings are the strongest. This naturally leads to a real-space RG, as is discussed below, and as has been reviewed thoroughly by Igloi and Monthus Igloi and Monthus (1971); Monthus (1972), and more briefly by Refael and Altman Refael and Altman (1997). Such strong-randomness renormalization groups are the focus of this very brief review, which begins with some discussion of their use for quantum critical ground states, and finishes with more recent generalizations that address the many-body localization (MBL) phase transition that can occur in highly excited quantum states.
## II Random-Singlet ground state
The first strong-randomness RG was developed by Ma, Dasgupta and Hu (1979) for the low temperature properties of random quantum antiferromagnets Ma (1979); Ma and Fisher (1989); Ma (1990); Ma and Fisher (1989). The system they considered is an antiferromagnetic Heisenberg spin-1/2 chain, with quenched random nearest-neighbor spin interactions. The Hamiltonian is:
\[H=\sum_{n}J_{n}\mathbf{\sigma}_{n}\cdot\mathbf{\sigma}_{n+1}\, \tag{1}\]
where the couplings \(J_{n}>0\) are drawn from a probability distribution \(P(\log J)\) that is broad, and \(\mathbf{\sigma}_{n}\) is the vector of Pauli operators for the spin-1/2 at site \(n\). The ground state of this spin chain is a so-called random-singlet state Ma (1979); Ma and Fisher (1989); Ma (1990); Ma and Fisher (1989). This random-singlet ground state consists, to first approximation, of pairs of spins that are in their total spin zero singlet state, with the two spins in each such pair being at various distances along the chain determined by local details of that sample's Hamiltonian.
The strong-randomness renormalization group treatment of this random-singlet ground state proceeds as follows: Each coarse-graining step of the RG "integrates out" the two spins, \(n\) and \(\left(n+1\right)\), that are coupled by the strongest remaining renormalized coupling \(J_{n}\). The order in which the spins are integrated out is dictated by energy, so proceeds in a sequence that is specific to each particular sample. Since these pairs of strongly-coupled spins are each located at some position in real space, this RG is often called a real-space RG. But really it more fundamentally works in "energy space". The two spins to be integrated out are also coupled to their other neighbors via the couplings \(J_{n-1}\) and \(J_{n+1}\), which are both weaker than \(J_{n}\). The small parameters that allow good analytic control of this RG are the ratios \(J_{n-1}/J_{n}\) and \(J_{n+1}/J_{n}\). When the distribution \(P(\log J)\) is broad, this makes these ratios typically small, so this RG approach
is a controlled approximation. Assuming that these two ratios are small, those weaker couplings can be treated in low-order perturbation theory.
Thus we have the three terms in the local Hamiltonian:
\[H=...+J_{n-1}\boldsymbol{\sigma}_{n-1}\cdot\boldsymbol{\sigma}_{n}+J_{n} \boldsymbol{\sigma}_{n}\cdot\boldsymbol{\sigma}_{n+1}+J_{n+1}\boldsymbol{\sigma }_{n+1}+\ldots, \tag{2}\]
with the middle term the strongest. The lowest order approximation to the ground state is: spins \(n\) and \((n+1)\) are in their total spin zero singlet state. If we stop at that very lowest perturbative order, the chain is cut, so this does not address what are the ground-state correlations between, for example, spins \((n-1)\) and \((n+2)\). The leading term that produces an interaction across the two spins that we are "integrating out" is produced at second order in perturbation theory in the weaker couplings, resulting in the renormalized Hamiltonian:
\[H^{\prime}=...+\delta E_{0}+J^{\prime}\boldsymbol{\sigma}_{n-1}\cdot \boldsymbol{\sigma}_{n+2}+..., \tag{3}\]
with renormalized coupling \(J^{\prime}=J_{n-1}J_{n+1}/(2J_{n})+...\) as well as a contribution \(\delta E_{0}=-3J_{n}+...\) to the ground state energy. The...'s refer to higher-order perturbative effects, which are asymptotically RG-irrelevant. So this RG step puts spins \(n\) and \((n+1)\) in the local ground state, thus removing those degrees of freedom, and introduces a new renormalized coupling between spins \((n-1)\) and \((n+2)\), producing a renormalized spin chain that is two spins shorter in length.
As the initial authors understood [10; 11; 12], and as was analysed in much more detail by Daniel Fisher [13], this approach results in a _functional_ renormalization group for the probability distribution \(P(\log J)\) of the nearest-neighbor couplings. Under the RG flow, this distribution broadens without limit due to the very weak new renormalized couplings that are produced. This produces a RG flow to an _infinite-randomness fixed point_, where the approximations \(J_{n\pm 1}\ll J_{n}\) used in the RG become asymptotically exact, so this RG gives the correct low-energy description of this system. Because the three couplings that entered in setting the renormalized coupling \(J^{\prime}\) are all removed from the renormalized Hamiltonian \(H^{\prime}\), no correlations between different couplings \(J_{m}\) are generated by the leading-order RG. Thus although one could have a joint probability distribution \(P(\{J_{m}\})\) with short-range correlations between the couplings on different bonds, such correlations are RG-irrelevant: the RG flow reduces these correlations and the fixed point distribution is asymptotically uncorrelated, which facilitates the analysis [13].
This random-singlet ground state is a quantum critical state. Any spin-1/2 chain of this form, with random nearest-neighbor-only antiferromagnetic couplings drawn from a continuous joint probability distribution with only short-range correlations in the couplings, has a ground state governed by this same infinite-randomness fixed point. Asymptotically, when the remaining spins are at typical distance \(r\) bare lattice spacings, the typical renormalized couplings \(J\) scale as: \(-\log J\sim r^{1/2}\) for large \(r\). Thus there is an exponential dynamical scaling (dynamical critical exponent \(z\to\infty\)), with the energy scales decreasing as the exponential of a power of the length scale. Such exponential dynamical scaling is typical of infinite-randomness quantum critical points. The equal-time spin-spin correlations in this ground state are very broadly distributed, with the mean correlation falling off with distance as \(\sim r^{-2}\) due to rare strongly-correlated spins, while the typical correlations fall off with distance exponentially in \(r^{1/2}\), as do the renormalized couplings [13].
## III Other infinite-randomness fixed points
There are (infinitely) many other one-dimensional models, both quantum and classical, whose asymptotic ground state and/or low-frequency properties can be systematically and correctly obtained with a strong-randomness RG approach [7; 8; 9]. One notable example is the quantum critical point of the transverse-field Ising spin chain, whose infinite-randomness fixed point is very closely related to that of the random-singlet state [14; 15]. An infinite series of discretely different infinite-randomness fixed points was also found. These govern the quantum critical ground states of certain spin chains with larger spin \(S\), as well as chains of interacting non-Abelian anyons [7; 8; 16; 17].
For many of these one-dimensional models, the strong-randomness RG can be solved analytically. When the method is generalized to systems in more than one dimension (\(d>1\)), parts of the calculations must instead be implemented numerically. It is found that, unlike in one dimension, for \(d>1\) and most models the RG flow near infinite randomness is instead towards weaker randomness, so such systems are not governed by infinite-randomness fixed points. The principal exception to this is the quantum critical point of the random transverse-field Ising model, which appears to remain governed by infinite-randomness fixed points for all dimensions \(d\)[7; 8].
## IV Many-body localization
One recent development in the use of strong-randomness renormalization groups is in the study of many-body localization (MBL). Some of this RG work was reviewed in Refs. [8; 9], so here I will focus on the more recent work.
Many-body localization (MBL) is Anderson localization for many interacting particles or spins in highly-excited states at thermodynamic conditions that correspond to nonzero entropy density. The system is assumed to be isolated from and not interacting with any other system or environment. The key question that is asked is whether or not this isolated many-body system "thermalizes": does it successfully serve as a thermal "bath"
for itself and, under its own unitary quantum dynamics, bring all of its subsystems to thermal equilibrium with each other. The MBL (localized) phase is the part of this system's phase diagram where the system remains (Anderson) localized near any nonthermal initial state, so it fails to bring itself to thermal equilibrium: it fails to thermalize. Some recent reviews about MBL include [18; 19; 20; 21].
Most uses of the strong-randomness RG for many-body quantum systems are for the study of the system's ground state and low-lying excited states. In the study of MBL, on the other hand, the focus is on highly-excited states, often typical states that at thermal equilibrium would correspond to infinite temperature. One main question that we want to address for MBL systems is whether or not the eigenstates of the system's dynamics, as well as its long-time dynamical states, are at thermal equilibrium. The MBL phase is the regime where the system does not go to thermal equilibrium nor are the eigenstates of its dynamics at thermal equilibrium, even in the limits of large systems and long times. It appears that a true MBL phase that remains localized in the standard thermodynamic limit and in the infinite-time limit is a possibility only for one-dimensional systems with short enough range interactions [22; 23], so those are the systems the remainder of this review considers.
The dynamics of a MBL system may be due to a time-independent Hamiltonian, or due to a Floquet unitary operator that produces the dynamics for one period of a Hamiltonian that is periodic in time. Many-body localization does not require randomness, for example it may occur due to nonrandom quasiperiodicity of the system [24], but here we only consider MBL due to quenched randomness. An example Hamiltonian is a Heisenberg spin-\(1/2\) chain with a random field:
\[H=\sum_{n}[h_{n}S_{n,z}+\mathbf{S}_{n}\cdot\mathbf{S}_{n+1}]\, \tag{4}\]
with the quenched random fields \(\{h_{n}\}\) at each site \(n\) drawn independently and uniformly from \([-W,+W]\). This is one of the most studied MBL models [25]; it is sometimes called the "standard model" for MBL, but only because it is highly studied and not because it is the best choice for a model in which to study MBL. The MBL phase is at large \(W\) (strong random fields) for this model. The precise location of the phase boundary is still not clear, but it appears to be at \(W_{c}>15\). [26; 27]
The MBL phase is a gapless quantum-critical dynamic phase, and the strong-randomness RG has been used within the MBL phase by various authors; much of this work is reviewed in [8]. Here I instead review some of the RG studies of the dynamic quantum phase transition between the MBL phase, where the system does not thermalize, and the thermal phase, where the system does thermalize in the long time and large system limits. Early numerical work on this MBL phase transition in one-dimensional systems found behavior consistent with exponential dynamical scaling (dynamic critical exponent \(z\to\infty\)), suggesting that it might be governed by an infinite-randomness fixed point [25]. Since then, there has been a series of publications steadily exploring and developing strong-randomness RG treatments of this MBL phase transition in one-dimensional systems with short-range interactions and/or hoppings [28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38].
When one uses a strong-randomness RG to study a many-body quantum ground state, this is usually done working directly with the microscopic degrees of freedom of the system, and the highest-energy remaining local excitations are usually integrated out, leaving a renormalized system with fewer degrees of freedom, as in most renormalization groups. For the MBL phase transition, on the other hand, a direct fully controlled RG calculation has not been found, so the RG is instead phenomenological and/or approximate, and in most cases both. The number of degrees of freedom sets the density of states (many-body states per unit energy) and thus the thermodynamic entropy, which is a key ingredient in the physics of the MBL phase transition, so no true "integrating out" that removes degrees of freedom is done. Thus the RG renormalizes (coarse-grains) the dynamics, scaling to longer times and larger length scales, but without the usual reduction of the number of degrees of freedom. As a result, the RG in most cases does not work directly with the microscopic degrees of freedom, but instead with more coarse-grained measures of the local dynamics.
In strong-randomness RG treatments of the MBL phase transition, the one-dimensional system is to some extent coarse-grained in to "blocks", which are segments of the chain with certain coarse-grained properties [28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38]. In many of these works, the blocks are assumed to be of only two types: locally thermalizing blocks that are locally less random and/or more interacting, so these thermal blocks do become strongly entangled; and locally MBL blocks that are locally more random and/or less interacting, so these locally MBL blocks have localized states that remain area-law entangled unless thermalized by a nearby thermal block. Such a "binary" classification of the local properties is only approximate when applied at small length scales. However, since the resulting RG does exhibit a flow to infinite randomness in the MBL phase and at the phase transition, this binary description then becomes self-consistent and thus potentially asymptotically correct. Multiple assumptions are used in developing these RGs, so there certainly remains the possibility that some important and relevant physics has been left out.
Near the fixed point governing the MBL phase transition, the locally thermalizing blocks are rare, occupying an asymptotically vanishing fraction of the length of the one-dimensional system as one renormalizes to the limit of low energy scales. These widely-spaced thermal blocks are randomly placed, and their lengths have a probability distribution. This strong-randomness RG is, as usual, a _functional_ RG, with this distribution of thermal block lengths the function that is being renormalized.
Each thermal block is locally thermalizing the nearby
locally MBL regions. As we renormalize to lower energy scales in this RG the locally thermal blocks thermalize the spins in the adjacent MBL blocks out to a distance that grows logarithmically with the (inverse) energy scale. There are two "events" that happen as this RG flow runs: (i) A single thermal block may reach the limit of how many spins in the nearby MBL regions it can thermalize. Once we renormalize below that energy scale, that thermal block is found to instead be localized (no longer able to spread more thermalization), so it is removed and that region is instead localized at this lower energy scale. (ii) If two nearby thermal blocks manage to thermalize the entire MBL region that lies in between them, then they merge with each other and with this intermediate region that they have thermalized to make a new, much longer renormalized thermal block [35; 36]. The latter process broadens the distribution of the lengths of the thermal blocks. The energy scale of a thermal block is set by its many-body energy-level spacing which is exponentially small in its length, so this length distribution becoming broader means a flow to infinite randomness.
A nice analogy with the Kosterlitz-Thouless (KT) RG and transition has been found and developed [32; 33; 34; 35; 36]: In Kosterlitz-Thouless, if vortices and antivortices are forbidden, then the two-dimensional superfluid remains stable. This critical KT superfluid phase is governed by a RG fixed line, with the superfluid stiffness divided by the temperature being the dimensionless parameter that varies along this fixed line. The KT phase transition occurs where the RG flow near this fixed line changes from being stable to being unstable to allowing vortices and antivortices [4; 5].
For the MBL transition in one-dimensional systems with quenched randomness, the role of vortices and antivortices is instead played by locally thermalizing rare regions (thermal blocks) [18; 35; 22; 36]. In this approach, the MBL phase is governed by a fixed line of the RG, with the dimensionless parameter that varies along the fixed line being the ratio of two lengths: one being the length per bit of thermodynamic entropy, and the other length being a decay length for the exponential dependence of the relaxation rate of spins on their distance from the nearest thermal block [35; 22; 36]. The MBL phase is governed by the part of this fixed line that is stable to allowing locally thermalizing rare regions (thermal blocks); in the MBL phase these rare regions are RG-irrelevant, so the RG flow goes to this fixed line. In the MBL phase these thermal blocks only manage to locally thermalize a limited number of spins in the adjacent MBL blocks before the energy scale is reached where this thermalization stops and the thermal block becomes instead localized. The MBL-to-thermal phase transition is then governed by the point on this fixed line where these locally thermalizing rare regions (added thermal blocks) become RG-relevant, so the fixed line becomes unstable in the thermal phase and the RG flow heads off towards thermalization. This occurs when, under the RG flow, enough of the thermal blocks can merge and produce much longer thermal blocks, so the fraction of the system occupied by the thermal blocks instead increases under the RG flow.
Although there is this strong _qualitative_ analogy to the Kosterlitz-Thouless (KT) RG, the strong-randomness RG flow for the MBL transition is mathematically distinct [36]. For KT, the RG is a two-parameter flow, with the vortex fugacity and the reduced superfluid stiffness being the two parameters. For MBL, on the other hand, it is instead a _functional_ RG, with the function being the probability distribution of the lengths of the locally thermalizing rare regions. When two of these rare regions (thermal blocks) are close enough together to thermalize the locally MBL typical region in between them, then, under the RG flow, they merge in to a much longer rare region, extending the distribution function to those longer lengths [35; 36]. This produces a rather direct connection within the functional RG flow between two very different length scales, a feature that is not present in the simpler Kosterlitz-Thouless RG flow.
Recent numerical work to quantitatively estimate where this asymptotic MBL-to-thermal phase transition occurs in the phase diagram of microscopically-defined spin-chain models has shown that it is actually very deep in the regime where the behavior of those models appears to be strongly localized for sample sizes and times accessible to standard numerical and experimental methods [26; 27]. Thus it currently appears that the physics of the asymptotic MBL phase transition as found in the strong-randomness RG may only apply on rather large length scales and thus only for extremely low energy scales (equivalently, extremely long times). One way to put a "positive spin" on our present understanding of this situation is to note that the strong-randomness RG method allows one to develop what appears to be a controlled asymptotic low-energy theory of this phase transition, even though other approaches to studying this novel and theoretically challenging phase transition have not been able to reach those very low energy scales.
## V Conclusion
The strong-randomness renormalization group methods are versions of the renormalization group (RG) that are useful for treating certain systems with quenched randomness, particularly those whose low-energy behavior is governed by an infinite-randomness fixed point. These methods have wide applicability in certain one-dimensional quantum systems, as well as in various other systems, as reviewed in Refs. [7; 8; 9]. More recently, such strong-randomness RG methods have been generalized to study the dynamic phase transition between many-body localization (MBL) and thermalization, as I have briefly reviewed here.
Acknowledgement
I thank Michael Fisher for being a wonderful teacher, graduate adviser, and collaborator, and for all that he taught me about critical phenomena, the renormalization group, and many other things. I am deeply indebted to him for his contributions to my education and development as a scientist.
|
2306.09773 | Unraveling cradle-to-grave disease trajectories from multilayer
comorbidity networks | We aim to comprehensively identify typical life-spanning trajectories and
critical events that impact patients' hospital utilization and mortality. We
use a unique dataset containing 44 million records of almost all inpatient
stays from 2003 to 2014 in Austria to investigate disease trajectories. We
develop a new, multilayer disease network approach to quantitatively analyse
how cooccurrences of two or more diagnoses form and evolve over the life course
of patients. Nodes represent diagnoses in age groups of ten years; each age
group makes up a layer of the comorbidity multilayer network. Inter-layer links
encode a significant correlation between diagnoses (p $<$ 0.001, relative risk
$>$ 1.5), while intra-layers links encode correlations between diagnoses across
different age groups. We use an unsupervised clustering algorithm for detecting
typical disease trajectories as overlapping clusters in the multilayer
comorbidity network.
We identify critical events in a patient's career as points where initially
overlapping trajectories start to diverge towards different states. We
identified 1,260 distinct disease trajectories (618 for females, 642 for males)
that on average contain 9 (IQR 2-6) different diagnoses that cover over up to
70 years (mean 23 years). We found 70 pairs of diverging trajectories that
share some diagnoses at younger ages but develop into markedly different groups
of diagnoses at older ages.
The disease trajectory framework can help us to identify critical events as
specific combinations of risk factors that put patients at high risk for
different diagnoses decades later. Our findings enable a data-driven
integration of personalized life-course perspectives into clinical
decision-making. | Elma Dervić, Johannes Sorger, Liuhuaying Yang, Michael Leutner, Alexander Kautzky, Stefan Thurner, Alexandra Kautzky-Willer, Peter Klimek | 2023-06-16T11:11:18Z | http://arxiv.org/abs/2306.09773v1 | # Unraveling cradle-to-grave disease trajectories from multilayer comorbidity networks
###### Abstract
We aim to comprehensively identify typical life-spanning trajectories and critical events that impact patients' hospital utilization and mortality. We use a unique dataset containing 44 million records of almost all inpatient stays from 2003 to 2014 in Austria to investigate disease trajectories. We develop a new, multilayer disease network approach to quantitatively analyse how cooccurrences of two or more diagnoses form and evolve over the life course of patients. Nodes represent diagnoses in age groups of ten years; each age group makes up a layer of the comorbidity multilayer network. Inter-layer links encode a significant correlation between diagnoses (p < 0.001, relative risk > 1.5), while intra-layers links encode correlations between diagnoses across different age groups. We use an unsupervised clustering algorithm for detecting typical disease trajectories as overlapping clusters in the multilayer comorbidity network.
Complexity Science Hub Vienna, Josefstadter Strasse 39, 1080 Vienna, Austria;
Medical University of Vienna, Section for Science of Complex Systems, CeMSIIS, Spidasse 23, 1090 Vienna, Austria;
Medical University of Vienna, Department of Internal Medicine III, Clinical Division of Endocrinology and Metabolism, Wahringer Gurtel 18-20, A-1090 Vienna, Austria;
Medical University of Vienna, Department of Psychiatry and Psychotherapy, Wahringer Gurtel 18-20, A-1090 Vienna, Austria;
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.
Gender Institute, A-3571 Gars am Kamp, Austria
## Introduction
Multimorbidity, the occurrence of two or more diseases in one patient, is a frequent phenomenon [1, 2]. Today's reality of a 100-year lifespan brings a shifting multimorbidity burden and increased healthcare and long-term care costs [3, 4]. It was estimated that mmore than 50 million people in Europe show more than one chronic condition [5]. In [6], authors estimated that 16-57% of adults in developed countries are diagnosed with more than one
chronic disease and predicted a dramatic rise of multimorbidity rates in the next years. The WHO World Report on Ageing and Health emphasizes the importance of research to better understand the dynamics and consequences of ageing [7]. Studies on multimorbidity patterns may contribute to successful ageing by the prevention of disease progression by identifying critical events which lead to a rapid deterioration of health [8, 9].
As diseases tend to co-occur and interact with each other (in a way that can worsen the course of both), they cannot be studied separately from each other [10]. The analysis of multimorbidity has recently been catalyzed by the massive collection of patient health information on diagnoses, medication, results of laboratory tests in electronic health records (EHR), and other clinical registries. Comorbidity networks have been established as tools to analyse multimorbidity in such datasets [11, 12]. Age and sex-specific analyses can further be conducted to address age- and sex-dependent associations between diagnoses [13, 14]. These works confirm that patients mainly develop diseases in close network proximity to disorders they already suffer.
The concept of disease trajectories has been proposed to formally describe the progression of multimorbidity over time. Disease trajectories are frequently occurring patterns or sequences of diagnoses at particular times and are typically extracted from the medical history of millions of patients. Thus, apart from the pairwise disease associations, uncovering complex disease patterns and assessing their temporal directionality is crucial for estimating disease progression, developing prediction models [15, 16], analysing trajectories [17, 18] and their temporal patterns using clustering algorithms [19, 20]. Many studies used data on in-hospital stays to construct such trajectories. A summary of applications of machine learning tools to understand the structure and formation of multimorbidity in the population was given in [21]. However, studies of multimorbidity patterns over the full life span of patients, from cradle to grave, remain scarce, as studies frequently take cross-sectional approaches [2, 22].
Longitudinal analysis of multimorbidity requires large population-wide disease registries which span over multiple years, if not decades. Such analyses are challenging as they require custom-made methods and that are often computationally challenging [17]. Taken together, a life span perspective on multimorbidities addressing the need for more comprehensive knowledge on disease trajectories and their critical events is largely missing to date [23].
Here, we propose a novel approach to dynamical comorbidity networks from longitudinal population-wide healthcare data to comprehensively identify disease trajectories in an entire population. A multilayer comorbidity network is constructed where nodes correspond to diagnoses, layers to age groups, intralayer links to disease co-occurrence relations, and interlayer links encode the directionality of disease pairs (which diagnosis tends to occur first). We identify temporal disease trajectories as communities in this multilayer network. In some cases, these tightly connected communities share some nodes and be referred to as overlapping communities.
The central assumption of our approach is that communities of nodes in the comorbidity network represent patients' diseases trajectories. We identify overlapping communities rather than exclusive clusters as the same diseases (nodes) can naturally be part of different disease trajectories, i.e. sleep disorders in patients with and without obesity and diabetes mellitus type 2. We further try to identify critical events as points along trajectories, where two initially identical trajectories start to diverge and will lead to different outcomes in terms of disease burden (hospital utilization) and mortality.
Figure 1 illustrates the suggested methodology of this large-scale disease trajectory study. We analysed data from an electronic health registry covering almost all of 8.9 million Austrians with more than 44 million in-hospital stays over 17 years, from 1997-2014. To ensure the comparability of the health status of our study population, we restricted the
Figure 1: Workflow of the research presented in this article.
analysis to patients who were "healthy" at the beginning of the observed period between 2003 and 2014. Therefore, in the first step of the analysis, we identified as the study population all patients with at least one hospital stay between 1997 and 2002 with a diagnosis from the range A00-N99 (in total 1,081 diagnoses). Moreover, in the early 2000s, Austria transitioned from the previous ICD coding system to ICD-10 2001. It was crucial to avoid combining various classification systems as it would have compromised the reliability of the analysis, Figure 1 (blue box).
In a next step we then constructed a multilayer comorbidity network to explore how different disease conditions co-occur and develop over time. We separated our data into 10-year age groups. For every age group we introduced a layer in the multilayer comorbidity network. In this network, two types of link can be found, links that connect nodes in the same layer (intralayer links) and links that connect nodes from different layers (interlayer links). All identified significant correlations of diagnoses in the same age group are defined as intralayer links, while interlayer links represent the correlation between diagnoses in different age groups, Figure 1 (green box). Nodes without any intralayer links were removed, Figure 1 (red box).
We used an algorithm based on the local optimization of a fitness function presented in [24] to identify overlapping communities in the multilayer network, Figure 1 (orange box). Note that the detected communities typically encompass more than one age layer. We analysed the age structure of the detected overlapping communities and the number of chapters of diagnoses inside the communities. More concrete, we conceptualize disease trajectories as groups of diagnoses that occur at different age groups (layers in the network) and that are more closely connected to other diagnoses in the same community compared to diagnoses outside of the community.
As disease trajectories can overlap, this enables us to comprehensively study relationships between disease trajectories across more than one age group. We defined pairs of trajectories as converging if they do not overlap (no shared diagnoses) in younger age groups while they have a nonzero overlap in older age groups. Additionally, diverging pairs of trajectories overlap at the beginning, in younger age groups, but have different pathways in older age groups.
From this we can identify critical events in patient careers. Critical events are defined as combinations of diagnoses in a specific age group, mainly chronic conditions, that signal that the disease trajectories are about to diverge towards paths that lead to different levels of mortality or lengths of hospital stays in the following age groups. Critical events can be thought of as bifurcation points of disease trajectories that can lead to trajectories associated with strongly varying outcomes. These events can support the identification of patients at risk for more severe multimorbidity trajectories and associated adverse outcomes in the next decade and thereby provide leverage points for targeted preventive actions.
## Data and Methods
### Data
The analysed dataset spans 17 years of nationwide in-hospital data from all hospitals in Austria. Each hospital stay is recorded with primary and secondary diagnoses, age in the resolution of 5 years, sex, admission and release date, release type (i.e., release, transfer, death...). This dataset covers the period from 1997 until 2014 and the vast majority of Austria's population with 8.9 unique patients. Diagnoses are coded with the three-digit International Classification of Diseases, 10th Revision (ICD-10) codes. We restricted our analysis to 1,081 codes from A00 to N99, excluding codes describing health encounters that can not be directly related to diseases (i.e., O00-O9A - Pregnancy, childbirth, and the puerperim, S00-T88 - Injury, poisoning and certain other consequences of external causes...). The data always reports a primary diagnosis as the main reason for hospitalization, along with a variable number of secondary diagnoses.
In this study, we assigned equal importance to both primary and secondary diagnoses [19, 25]. To ensure that our study population's health state was comparable at the beginning of the observation period and not in the middle of connected hospitalization episodes, we introduced a wash-out period and limit the analysis to patients who had no hospital visits between 1997 and 2002. Consequently, excluding these patients also ensured that analysed data has only one ICD coding system, as in the early 2000s, Austria updated its ICD coding system to ICD-10 2001 [19, 25, 26].
### Multilayer Comorbidity Network
Formally, we construct the multilayer comorbidity network given by the tensor \(M_{i,j}^{\alpha,\beta}\) where \(i\) and \(j\) refer to nodes (diagnoses) on layers (age groups) \(\alpha\) and \(\beta\), respectively. We refer to entries in \(M\) with \(\alpha=\beta\) as intralayer links and with \(\alpha\neq\beta\) as interlayer links. The analysis was performed separately for male and female patients.
#### Intralayer links
Intralayer links give the correlation between diagnoses within the same age group. The analysed dataset was stratified by six time windows of two years each, from 2003 to 2014. A contingency table is created for each pair of diagnoses in each stratum (for each sex and age group, the intralayer analysis includes six strata, each covering two calendar years). We used all contingency tables with more than four patients in each subgroup to compute relative risks (RR) and the p-value for rejecting the null hypothesis that the co-occurrence of two analysed diagnoses is statistically independent [26]. A weighted average of the estimates of the risk ratios and odds ratios across the stratified data were calculated by using the Cochran-Mantel-Haenszel method [27].
Subsequently, all correlations with RR higher than 1.5 and p-value smaller than 0.05 were extracted and presented as intralayer links [14]. These links are bidirectional, and we use a normalized RR as the link weight. The normalization of RR was done such that the sum of all total weights of all intralayer links with the same target was one.
#### Interlayer links
To estimate directionality or time order in pairs of diagnoses, we split the observation period in two time frames \(T1=[2003,2008]\) and \(T2=[2009,2014]\). We investigate if a patient diagnosed with \(i\) in \(T1\) elevates the risk of being diagnosed with
in \(T2\) and compute the interlayer link weight as
\[M_{i,j}^{\alpha\neq\beta}=\frac{P(j_{T2}^{\beta}|i_{T1}^{\alpha})}{P(j_{T2}^{ \beta})}\,. \tag{1}\]
Overlapping community detection in multilayer networkWe deleted all nodes without at least one inbound and one outbound link. Further, we normalized all link weights to range from 0 to 1 by dividing each link's weight by the sum of all links of the same type of a target node
\[M_{ij}^{\alpha\beta}=\frac{M_{ij}^{\alpha\beta}}{\sum_{j}M_{ij}^{\alpha\beta}}\,. \tag{2}\]
The algorithm for detecting the overlapping and hierarchical community structure in complex networks proposed in [24] was applied. This unsupervised clustering algorithm does not have a predefined number of communities. The detection procedure is initiated starting with a random node, which represents one community by itself.
Community's fitness is \(f_{G}=\frac{k_{G}^{G}}{(k_{in}^{G}+k_{out}^{G})^{a}}\),
where \(k_{in}^{G}\) are the total internal degrees of the nodes in the community \(G\) and \(k_{out}^{G}\) are the total external degrees of the nodes in the community \(G\).
As long as the \(f_{G}\) improves, neighboring nodes are added, or nodes already community members are removed.
The entailed resolution parameter \(a\) enables us to uncover different hierarchical levels of a system, natural choice is \(a=1\). Fitness is calculated at each step. Once the fitness cannot be increased anymore by a node removal or addition step, that community is "completed" and "closed." The community detection process ends when all nodes have been assigned to at least one community. To parallelize and optimize this computationally costly process, we identify the community of every node and delete duplicates among the discovered communities.
Identified communities usually consist of diseases in different age groups that tend to co-occur more frequently among themselves than diseases that are not part of the community. Hence, these communities represent typical disease trajectories; we denote a trajectory \(X\) as a set of diagnosis-age tuples, \(X=\{(i_{1},\alpha_{1}),(i_{2},\alpha_{2}),(i_{3},\alpha_{3})...\}\), where \(i\) is an ICD10 code ranging from [A00, N99] and \(\alpha\) is the age group from \([1,8]\).
We measure the similarity of trajectories by the Jaccard coefficient between two trajectories consisting of tuples with diagnoses \(i\) and age groups \(\alpha\), \((i,\alpha)\). That is, two trajectories have a non-zero overlap if they share diagnoses within the same age groups.
Identifying converging and diverging trajectoriesWe performed a comprehensive classification with respect to all pairwise relations between every pair of trajectories. Provided
that two trajectories share at least one diagnosis, they can be related in one of four different ways, namely (i) diverging, (ii) converging, (iii) nested, or (iv) persistent, Figure 2.
Diverging trajectories have some overlapping elements at younger ages, but they develop into markedly different sets of diagnoses at older ages.
More formally, trajectories \(X=\{(i_{11},\alpha_{11}),(i_{12},\alpha_{12}),(i_{13},\alpha_{13})...\}\) and
\(Y=\{(i_{21},\alpha_{21}),(i_{22},\alpha_{22}),(i_{23},\alpha_{23})...\}\) are diverging if it holds that
\[\begin{array}{c}\left\{\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}=\alpha_{\min} ^{X})\}\cap\{(i_{2i},\alpha_{2i})\in Y|\alpha_{2i}=\alpha_{\min}^{X})\}\right\} \cup\\ \left\{\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}=\alpha_{\min}^{Y})\}\cap\{(i_{ 2i},\alpha_{2i})\in Y|\alpha_{2i}=\alpha_{\min}^{Y})\}\right\}\neq\emptyset \mbox{ and }\\ \qquad\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}>\alpha_{\min}^{X})\}\;\neq\;\{ (i_{2i},\alpha_{2i})\in Y|\alpha_{2i}>\alpha_{\min}^{X})\}\mbox{ and }\\ \qquad\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}>\alpha_{\min}^{Y})\}\;\neq\;\{ (i_{2i},\alpha_{2i})\in Y|\alpha_{2i}>\alpha_{\min}^{Y})\},\end{array} \tag{3}\]
where \(\alpha_{\min}^{X}=\min_{(i,\alpha)\in X}\;\alpha\;,\;\alpha_{\min}^{Y}=\min_{ (i,\alpha)\in Y}\;\alpha\).
Converging trajectories overlap at older ages but are clearly different at younger ages. Trajectories \(X\) and \(Y\) are converging if it holds that
\[\begin{array}{c}\left\{\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}=\alpha_{\max }^{X})\}\cap\{(i_{2i},\alpha_{2i})\in Y|\alpha_{2i}=\alpha_{\max}^{X})\} \right\}\cup\\ \left\{\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}=\alpha_{\max}^{Y})\}\cap\{(i_{ 2i},\alpha_{2i})\in Y|\alpha_{2i}=\alpha_{\max}^{Y})\}\right\}\neq\emptyset \mbox{ and }\\ \qquad\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}<\alpha_{\max}^{X})\}\;\neq\;\{ (i_{2i},\alpha_{2i})\in Y|\alpha_{2i}<\alpha_{\max}^{X})\}\mbox{ and }\\ \qquad\{(i_{1i},\alpha_{1i})\in X|\alpha_{1i}<\alpha_{\max}^{Y})\}\;\neq\;\{ (i_{2i},\alpha_{2i})\in Y|\alpha_{2i}<\alpha_{\max}^{Y})\},\end{array} \tag{4}\]
where \(\alpha_{\max}^{X}=\max_{(i,\alpha)\in X}\;\alpha\;,\;\alpha_{\max}^{Y}=\max_ {(i,\alpha)\in Y}\;\alpha\).
Two trajectories are nested if one of them is a subset of another one, \(X\subset Y\) or \(Y\subset X\). Persistent trajectories \(X\) and \(Y\) can overlap in the highest age group of \(X\) and lowest age group of \(Y\), or vice versa.
**Identifying critical events** We define critical events by one or a combination of diagnoses and age groups where two trajectories begin to diverge and where one of the diverging trajectories has patients with considerably higher number of diagnoses, higher mortality or more extended hospital stays in the subsequent age group(s) compared to the other diverging trajectory. Mortality of a trajectory for a certain age group is calculated as \(M=\sum_{i}m_{i}*\prod_{j\neq i}(1-m_{j})\), where \(m\) is the in-hospital mortality of a diagnosis (defined as the percentage of patient diagnosed with the diagnose in a specific age group who die
Figure 2: Visual representation of pairwise relations between pair of trajectories
in-hospital) which is a member of a trajectory. Length of hospital stay of a trajectory in a certain age group is defined as the average number of days spent in hospital for patients who are diagnosed with at least half of all diagnoses from a trajectory.
## Results
### Multilayer Comorbidity Network
We constructed the multilayer comorbidity network based on hospital data, basic characteristics of the database are shown in Figure S1. We used all 3-digits ICD10 codes from the range A00-N99 and one more newly introduced code for patients without any diagnosis, in total 1,082 codes. Nodes in the constructed network are ICD10 codes appearing in one of eight different age groups, i.e. E66-0-9, E66-10-19, etc. Hence, we used 8,648 nodes to construct a multilayer comorbidity network with eight layers (one for each ten years age group, 0-9, 10-19,... 70-79 years old). We filtered the network by removing nodes without any intralayer links. This reduced the network from 8,648 nodes to 4,923 nodes for males and 4,764 nodes for females. The average degree in the filtered male network is 11.6 SD 39.7, for the female network the average degree is 15.8 SD 46. The number of hospital stays, Figure 3(A), and nodes \(N\) Figure 3(B) increases with age, reaches a peak at ages 60 to 69 and decreases for older ages. We see similar age trends in Figure 3(C) the total number of links and Figure 3(D) the average degree for intralayer as well as in- or outbound interlayer links for males and females.
### Trajectories
The unsupervised community detection algorithm discovered 642 distinct disease trajectories in the male and 618 in the females network; they are listed in SI, Tables S1-S2, and shown in Figure 4. These trajectories contain on average 9 (IQR
Figure 3: (A) Total number of hospital stays per age group, Network properties: (B) number of nodes, (C) number of all inter- and intralayer links and, (D), the average degrees \(<\)\(k\)\(>\).
that range over up to 7 age groups (mean: 2.3 age groups), meaning that these trajectories range on average over 20-30 years and in some cases over up to 70 years of life. Besides trivial examples like a trajectory with the only diagnosis being K51 (ulcerative colitis) in each age group in males, we also found more complex trajectories spanning 70 years. For instance, for female patients there is a trajectory that starts with personality disorder (F61) at the age of 20-29y. Over the following decades there is an accumulation of mental disorders including depression (F33), post-traumatic stress disorder (F43) and eating disorders (F50) in 50-59y, followed by anxiety disorders (F40) and a few more non-chronic diagnoses in 60-69y.
The distribution of the size of the trajectories (number of diagnoses-age tuples) is presented in Figure 5 (A). Most trajectories contain between 3 and 5 diagnoses-age combinations; while a few trajectories contain more than hundred e
Figure 4: (A) Multilayer comorbidity network of female patients. Nodes represent diagnoses in age groups of ten years; each age group makes up a layer of the comorbidity multilayer network. The node color indicates the diagnose chapter; the diagnosis prevalence of the node in the network scales node size. All identified trajectories of female (B) and male (C) patients. The members of the trajectories are nodes of a multilayer comorbidity network (diagnose + 10 years age group). The Y-axis approximates the age groups of nodes; the node color indicates the diagnose chapter, and each grey area around nodes is one trajectory. More detailed visualization of these plots can be found in the interactive web application: (A) [https://vis.csh.ac.at/netwiewer/](https://vis.csh.ac.at/netwiewer/) and (B) & (C) [https://vis.csh.ac.at/comorbidity_network_graphics/](https://vis.csh.ac.at/comorbidity_network_graphics/)
ries into seven groups based on the number of age groups in the trajectory and analysed the number of different disease chapters in one trajectory Figure 5 (B). This shows that trajectories typically span heterogeneous chapters of ICD codes, meaning that they often span diagnoses affecting quite different organ systems. We calculated the Jaccard index to inspect the pairwise similarity and dissimilarity of trajectories; see the distribution of this index in Figure 5 (C). Jaccard indices range between zero and one, indicating varying degrees of similarity between two trajectories. The most common relationship among pairs of trajectories is nested, which explains the peak at one in the Jaccard index. Figure 5 (D) shows frequency statistics of different types of trajectory pairs.
In Figure 6 we show a more detailed view of some of the trajectories from Figure 4. We show two examples for trajectories (grey areas) departing from (A) hypertension (I10) at an age of 10-19y in females and (B) sleep disorders (G47) at an age of 20-29y in males. In both cases, different combinations of other diagnoses appear in subsequent decades. The hypertension trajectory diverged into chronic kidney diseases (2,289 patients) or (1,027 patients) a combination of metabolic (obesity, disorders of lipoprotein metabolism) and digestive disorders (liver diseases, cholleithiasis) with nicotine abuse. The sleep disorder trajectory diverged either toward the metabolic syndrome (including obesity and type 2 diabetes) in 115 patients or towards a combination of movement disorders, hernia, obesity and diseases of the middle ear (316 patients).
In total, we identified 35 pairs of such diverging trajectories in females and 35 in males; see Figure 5 D). On average, diverging trajectories have 2.9 SD 0.8 age groups, 3.5 SD 1.8 different diagnoses chapters, and 8.1 SD 4.7 different diseases for females, and for males 3.0 SD 1 age groups, 3.5 SD 2.9 different diagnoses chapters, and 11 SD 11 different diseases. While there are 64 pairs of converging trajectories in females and 95 in males, converging trajectories in females have: 2.8 SD 0.9 age groups, 4.2 SD 3.2 different diagnoses chapters, and 26 SD 79 different diseases, in males: 3 SD 1 age groups, 3.8 SD 3.5 different diagnoses chapters, and 22 SD 68 different diseases. Some of the trajectories are persistent (16 pairs of trajectories in females, 14 in males). These can be combined as they overlap at the end of X trajectory and the beginning of the Y trajectory.
The most frequent relationship between trajectories was the complete overlap of shorter and longer trajectories, which we defined as nested. We found 314 pairs of nested trajectories among female trajectories and 266 in males trajectories.
We designed and implemented an online visualization tool that allows a user to interactively explore the comorbidity network structure and the underlying diagnose data, [https://vis.csh.ac.at/netviewer/](https://vis.csh.ac.at/netviewer/).
### Outcomes of trajectories
For every trajectory, we calculated (in-hospital) mortality and the number of days spent in the hospital for each age group, Figure 7. In-hospital mortality for each trajectory is shown
in the yellow outer circle. The analysis reveals notable variations in mortality rates across trajectories, with younger age groups generally exhibiting lower mortality. Moreover, it is evident that certain trajectories undergo significant shifts in mortality as they progress into older age groups. The green circle represents the average duration of hospitalization for trajectories, while the blue circle denotes the number of diagnoses, and the purple inner circle signifies the count of patients who followed at least 50% of a given trajectory. Notably, the green circle highlights discernible differences in the number of hospital days
Figure 5: Properties of trajectories. We show for males and females (A) the distribution of the sizes of the trajectories, (B) the distributions of the number of different ICD chapters per each group of trajectories, i.e. trajectories which span over one age group - first blue plot, C) the histogram of the pairwise Jaccard index and (D) frequency statistics of different types of trajectory pairs.
among different trajectories. Some trajectories have a clearly higher number of hospital days compared to other trajectories; these trajectories mainly consist of mental and behavioral disorders (F chapter) and infectious and parasitic diseases (B chapter) in males, while in females, besides these we see diseases of musculoskeletal and connective tissue (M chapter) and diseases of the nervous system (G chapter).
We also compared outcomes of diverging trajectories; some examples are shown in Table 1 (extended tables in SI, Table S8 and S9). We calculated an average number of hospital diagnoses, hospital days, and hospital stays for each age group in each trajectory over all patients following these trajectories. We calculated the ratio of each outcome of trajectories in each diverging pair to check if these trajectories develop into different outcomes in terms of disease burden and mortality. For example, both trajectories from the pair starting with N81 in 50s are characterised with a similar average number of hospital diagnoses in the 20s, while in the 30s patients of the second trajectory have, on average, 24% more hospital diagnoses. In the same example, we see that patients of the first trajectory, on average have more days spent in hospital and hospital stays in the 20s (Ratio of average number of / number of days spent in hospital = 1.547 / hospital stays = 1.548), but in 30s patients of the second trajectory, have remarkably more days spent in hospital and hospital stays (Ratio of average number of / number of days spent in hospital = 0.331 / hospital stays = 0.551), Table 1.
Figure 6: Two examples of diverging trajectories, (A) departing from hypertension (I10) at an age of 10-19y in females diverges to the ”kidney-trajectory” and the ”metabolic trajectory” (B) departing from sleep disorders (G47) at an age of 20-29y in males diverges to metabolic trajectory with diabetes mellitus type 2 (E11), obesity (E66), lipid disorders (E78) and hyperuricemia (E79) and path with movement disorders or otitis media (G25), obesity (E66) and abdominal hernia (K46).
Figure 7: Outcomes of trajectories that span more than one age group (A) Females (B) Males, the outer yellow circle shows mortality of each trajectory in each age group, the green circle presents an estimation of the average number of hospital days of patients of the trajectory, while the blue circle presents an estimation of the average number of diagnoses, the inner purple circle shows a number of patients who are following each trajectory in each age group.Each trajectory is represented by a single line within each circle, which is further divided based on the age groups the trajectory encompasses.
## Discussion and conclusion
In this work we introduced a novel method to identify life-course disease trajectories, in some cases spanning up to 70 years of life, in terms of sequences and combinations of hospital diagnoses that form and change over time. Our comprehensive analysis identified 642 disease trajectories in males and 618 in females ranging over the entire diagnostic spectrum (41% of males and 42% of female trajectories contained diagnoses from more than one ICD chapter). While the most common length of these trajectories was two diagnoses for both sexes, on average they contained 5.3 SD 5.1 and 5.4 SD 5.5 diagnoses for males and females, respectively, emphasizing the heterogeneous and widespread nature of multimorbidity in the general population.
There is a substantial variation in the number of patients that follow a trajectory. We count patients for each trajectory for each age group if they have at least 50% of diagnoses from a trajectory. In general, shorter trajectories tend to be followed by more patients (more than 10,000 patients per trajectory per age group) than longer, more specific ones that typically contain approximately a hundred patients. The number of patients in a trajectory typically increases with age.
The trajectories foster the rapid identification of critical events. These can take the form of bifurcation points where a trajectory "splits up" into multiple diverging trajectories at a specific age group. More concrete, we found 35 pairs of diverging trajectories for females and 35 pairs for males. For example, in females diagnosed with arterial hypertension (I10) between 10 and 19 years, two major trajectories were identified by the model. The first trajectory lead to the additional diagnosis of chronic kidney disease (N18) at an age of 20-29 years. This is clinically relevant as the number of pediatric arterial hypertension is increasing worldwide [28] and it is well known that aHTN is closely related to chronic kidney disease. So our results point out that from a clinical point of view, a strict monitoring for arterial hypertension should be established especially in children at high risk, such as obese children or children with the metabolic syndrome. Arterial hypertension does not only mean increased risk for chronic kidney disease, but also other complications such as cardiovascular disease. The second trajectory was characterized by patients with the metabolic syndrome; these patients were disproportionally diagnosed with obesity (E66), lipid disorders (E78), steatosis hepatitis (K76), cholleithiasis (K80) and nicotine abuse (N17) in their further life. In general, we therefore have two trajectories in females initially diagnosed with arterial hypertension, which are in principal dangerous conditions - the "kidney-trajectory" and the "metabolic trajectory". We found that approximately 2,289 patients follow the "kidney-" and 1,027 patients follow the metabolic trajectory. These trajectories are mostly important as metabolic diseases belong to the most common diseases worldwide and also chronic kidney disease is a disease which is related to multi morbidity and increased mortality rate.
In a different example we found that sleeping disorders (G47) in males diagnosed in the age groups between 20-39 years were also followed by a metabolic trajectory which was defined by an over-representation of later diagnoses of diabetes mellitus type 2 (E11), obesity (E66), lipid disorders (E78) and hyperuricemia (E79). The other trajectory, diverging from sleeping disorders, is characterized by a higher chance of being diagnosed with movement disorders or otitis media (G25), obesity (E66) and abdominal hernia (K46). We found substantial differences in the average number of diagnoses and hospital days between patients of different branches of these diverging trajectories. While patients who followed these two trajectories showed similar average numbers of diagnoses at age 20-29 (3.3 diagnoses in both cases), patients who followed a metabolic trajectory had, on average, 3.9 diagnoses ten years later while patients who followed the other trajectory had, on average, 5.1 diagnoses. The number of sleeping disorders is on the rise and these results show that patients with sleeping disorders have to be monitored for several diseases in different trajectories. Our analysis also identified several instances were diverging trajectories differed substantially in their mortality, in some cases of up to 18 times.
In terms of mortality we identify trajectories that develop into a combination of diagnoses with high mortality in older age groups. For instance, a trajectory consisting of chronic bronchitis and COPD at an age of 40-49y, bronchiectasis and intraoperative and postprocedural complications at 50-59y and finally in sequelae of tuberculosis, inflammatory polyneuropathy, conjunctivitis, bronchitis, bronchiectasis, eosinophil and again intraoperative and postprocedural complications in 60-69y in males had eight times higher mortality in the age group 60-69y compared to its' mortality ten years earlier (mortality increased from \(0.089\) in 40-49y to \(0.013\) before jumping to \(0.11\) in 60-69y). Trajectories with the highest mortality usually contain cancer diagnoses, but cardiovascular or respiratory
diseases also feature in the trajectories with high mortality.
## Strengths and Limitations
Strengths of this study include its comprehensive population-wide in-hospital database, containing information on about 9 million individuals. Non-systematic errors, such as randomly missing diagnoses, have little impact on our research because of the volume of the data set. However, this study has some limitations caused by data quality and limitations in data availability, in particular, the lack of information on outpatient visits, medication and lifestyle. Consequently, we cannot evaluate the outcomes of outpatient visits, blood tests, examinations, or imaging because primary care diagnoses are not recorded in this dataset; only hospital diagnoses coded with ICD10 codes were available for analysis.
Another drawback is that the database was designed for billing purposes, so diagnoses that did not result in financial compensation were frequently not reported. Therefore, we have to point out that some diseases, such as alcohol related disorders or nicotine dependence, are often not recorded correctly in our data. Further, socio-economic indicators for individual patients were also not available in the dataset, leaving it yet to be explored how socio-economic status impacts on these trajectories. An additional constraint associated with the dataset is the exclusive availability of in-hospital mortality data. On a methodological level, it is also important to bear in mind that a constructed multilayer comorbidity network has two types of links (with normalized links weights); but these types are not distinguishable by the used community detection algorithms.
In summary, we presented a novel and statistically grounded way of studying disease progression over time based on a population-wide and decade-spanning data set of hospital diagnoses. We proposed an age multilayer comorbidity network as a base for our modelling approach. We showed that this kind of network is a promising approach for better understanding disease trajectories and their dynamics as patients age. While some of the identified trajectories in this study have been described in previously published studies, many novel disease trajectories and their decades-long time dynamics have been revealed. A better understanding of diseases, their correlations and the sequences in they occur has the potential to improve the prevention of focal diseases. Early detection and identification of a patient's projected disease trajectory might enable prompt and timely treatments next to targeted preventive action. Consequently, that will help transition health systems from single-disease models to more effective life-spanning and individualized multimorbidity models [29].
|
2304.07127 | OPI at SemEval 2023 Task 1: Image-Text Embeddings and Multimodal
Information Retrieval for Visual Word Sense Disambiguation | The goal of visual word sense disambiguation is to find the image that best
matches the provided description of the word's meaning. It is a challenging
problem, requiring approaches that combine language and image understanding. In
this paper, we present our submission to SemEval 2023 visual word sense
disambiguation shared task. The proposed system integrates multimodal
embeddings, learning to rank methods, and knowledge-based approaches. We build
a classifier based on the CLIP model, whose results are enriched with
additional information retrieved from Wikipedia and lexical databases. Our
solution was ranked third in the multilingual task and won in the Persian
track, one of the three language subtasks. | Sławomir Dadas | 2023-04-14T13:45:59Z | http://arxiv.org/abs/2304.07127v1 | OPI at SemEval 2023 Task 1: Image-Text Embeddings and Multimodal Information Retrieval for Visual Word Sense Disambiguation
###### Abstract
The goal of visual word sense disambiguation is to find the image that best matches the provided description of the word's meaning. It is a challenging problem, requiring approaches that combine language and image understanding. In this paper, we present our submission to SemEval 2023 visual word sense disambiguation shared task. The proposed system integrates multimodal embeddings, learning to rank methods, and knowledge-based approaches. We build a classifier based on the CLIP model, whose results are enriched with additional information retrieved from Wikipedia and lexical databases. Our solution was ranked third in the multilingual task and won in the Persian track, one of the three language subtasks.
## 1 Introduction
Visual word sense disambiguation (VWSD) is a task in the field of multimodal natural language processing, in which the goal is to identify the intended meaning of a target word in a given context by selecting the most appropriate image from a set of candidate images. Finding images corresponding to the correct meaning of the word might improve the performance of methods combining text and visual information such as image search engines, visual question answering, or image generation models.
SemEval 2023 workshop hosted a task on visual word sense disambiguation. The task involved selecting the best matching image out of ten candidates given a short textual description. The descriptions usually consisted of two words: the target word and the context word [1]. For example, the phrase _andromeda tree_ contains the ambiguous target word _andromeda_ and the context word _tree_, which indicates a specific meaning of the target word. The task organizers provided three datasets, of which the trial and training datasets contained phrases in English, while the test dataset was multilingual and consisted of English, Italian, and Persian subsets. Participants were allowed to submit their solutions for a particular language or for all three languages. The systems were ranked according to the average accuracy score from three language-specific subtasks.
In this paper, we describe our system for the VWSD shared task. The backbone of our solution is a classifier using multimodal CLIP embeddings [1, 1], which has been enriched with features extracted from Wikipedia and dictionaries. These knowledge sources are used to retrieve textual and image data, providing additional information useful for determining the correct meaning of the target word. Our system was ranked third in the multilingual task and took first place in the Persian subtask. The source code of our system is publicly available, as well as the fine-tuned models and other resources required to reproduce our experiments.1
Footnote 1: [https://github.com/sdadas/wssd](https://github.com/sdadas/wssd)
## 2 System description
The proposed approach consists of several modules which together constitute a visual word sense disambiguation system. The three core components of our approach are: 1) a classifier based on CLIP image-text embeddings, 2) a Wikipedia retrieval module, 3) a learning to rank (LTR) model whose role is to generate the final ranking of images based on the features provided by the other modules. A high-level overview of the system is shown in Figure 1.
### CLIP-based classifier 1
CLIP (Contrastive Language-Image Pretraining) is a method for learning multimodal embeddings for language and vision by aligning the representations of images and their captions. The original CLIP models published by OpenAI [1]
were trained on a dataset of 400 million image-text pairs. Recently, CLIP architectures trained on even larger datasets and with more parameters have been released by the LAION group, achieving state-of-the-art results on several image retrieval and zero-shot image classification tasks (Cherti et al., 2022). Visual word sense disambiguation can be viewed as a text-to-image matching task, the type of problems for which CLIP is particularly effective. Therefore, we chose this model as the basis for our classifier.
We utilize CLIP in zero-shot mode, using a pre-trained checkpoint, to assign a score for each context-image pair from the data sample. Specifically, we compute vector representations of textual context \(\mathbf{c}\) and image \(\mathbf{x}\), and then calculate the similarity between these vectors using the following formula:
\[score(\mathbf{c},\mathbf{x})=sim(\mathbf{c},\mathbf{x})-p(\mathbf{x}) \tag{1}\]
in which \(sim(\mathbf{c},\mathbf{x})\) denotes a standard cosine similarity and \(p(\mathbf{x})\) is a score penalty for the image \(\mathbf{x}\). The penalty is calculated for each image as the mean similarity between that image and all the contexts in the dataset, normalized by the frequency of image occurrence. The rationale for using penalties is the observation that some images have high cosine similarity to many contexts, leading the model to incorrectly prefer them for the majority of samples in which they appear. The penalty lowers the similarity for these cases without affecting the results for the other images. We calculate it using the following formula:
\[p(\mathbf{x})=\left(\frac{1}{|C|}\sum_{\mathbf{c_{i}}\in C}sim(\mathbf{c_{i}},\mathbf{x}) \right)\cdot\frac{card(\mathbf{x})}{\max\limits_{\mathbf{x_{j}}\in X}card(\mathbf{x_{j}})} \tag{2}\]
in which \(C\) is the set of all contexts, \(X\) is the set of all images, and \(card(\mathbf{x})\) denotes the number of samples in which image \(\mathbf{x}\) appears.
#### 2.1.1 Multilingual classification
Publicly available CLIP models were trained on a set of English image captions, and are therefore not adapted for generating vector representations for texts in other languages. Consequently, the described method cannot be applied directly to Italian and Persian. However, it is possible to use transfer learning methods to train a multilingual or language-specific text encoder aligned with the representations generated by the original model. Such methods have been used in the past to create multilingual versions of CLIP (Reimers and Gurevych, 2020; Carlsson et al., 2022). The basic idea is to fine-tune a language model using bilin
Figure 1: A diagram showing our visual word sense disambiguation system. Given the target word and its context, our method outputs a relevance ranking of candidate images. The ranking is produced by a fine-tuned learning to rank (LTR) model, which utilizes features extracted from the CLIP-based classifier, Wikipedia retrieval module, and global statistics calculated from the dataset.
gual or multilingual corpora. The original CLIP model generates a vector representation of the English text, while the language model produces a representation for the translation of that text. The difference between these vectors is then used to compute mean squared error (MSE) loss and the language model is optimized to approximate the representations generated by CLIP.
We employed this technique to train Italian and Persian text encoders, using OpenCLIP H/14 (Cherti et al., 2022)2 as the teacher model and XLM-R large (Conneau et al., 2020)3 as the student model. To train the encoders, we collected 10.5 million English images captions, which we then translated to Italian and Persian using publicly available neural machine translation models (Tiedemann and Thottingal, 2020; Khashabi et al., 2021). The dataset for training was obtained from the following three sources:
Footnote 2: [https://huggingface.co/laion/CLIP-ViT-H-14-laion2B-s32B-B79K](https://huggingface.co/laion/CLIP-ViT-H-14-laion2B-s32B-B79K)
Footnote 3: [https://huggingface.co/xlm-roberta-large](https://huggingface.co/xlm-roberta-large)
* English subset of Wikipedia-based Image Text (WIT) dataset (Srinivasan et al., 2021).
* SBU Captions dataset (Ordonez et al., 2011).
* A subset of 7 million English captions from Conceptual 12M (Changpinyo et al., 2021).
For multilingual classification, we use the same scoring procedure as described in the previous section. The only difference is that we replace the original CLIP text encoder with our fine-tuned models.
#### 2.1.2 Context augmentation
One way to improve the performance of the described classifier is to expand the textual context with additional phrases associated with the actual meaning of the target word, which is expected to increase the similarity between the context and the correct image. We can do this with lexical databases by finding the sense of the target word and then extracting additional information from the definition of that sense. In our solution, we use multilingual resources available in Extended Open Multilingual WordNet (Bond and Paik, 2012; Bond and Foster, 2013). Specifically, we utilize the following lexical resources:
* For English, we use Princeton WordNet database (Miller, 1995).
* For Italian, we use two lexical databases: one included in MultiWordNet (Pianta et al., 2002) and another one from EuroWordNet (Toral et al., 2010).
* For all three languages, we employ additional multilingual resources: Wiktionary and the Common Locale Data Repository (CLDR).
Our context expansion procedure works by appending alternative names extracted from a specific word sense, as well as from senses that are linked to it through hypermym, instance hypernym, member meronym, or substance meronym relations. For example, the context _andromeda tree_ is expanded to: _andromeda tree, andromeda, japanese andromeda, lily of the valley tree, pieris japonica, shrub, bush_.
In order to find the correct sense, we retrieve a list of available senses of the target word from all lexical databases for a specific language and then compare the descriptions of these senses with the context word. The description is constructed from definitions, alternative names, and examples of use of a given sense, as well as senses linked to it by hypernym or instance hypernym relations. In our solution, we implemented two algorithms for matching sense and context:
* **Exact matching**, which involves finding exact occurrences of the context word in the sense description. The similarity between the context and the description is computed as the number of matched words divided by the total number of words in the description.
* **Similarity matching**, involving the comparison of word vectors extracted from the word embedding model. In this method, we convert the context word and words from the sense description into their vector representations using multilingual FastText models (Grave et al., 2018). The similarity between context and sense is calculated as the maximum cosine similarity between the representation of the context word and the representations of all the words from the description.
We select the sense with the highest similarity to the context. For English, only exact matching is used. This method, however, has a low recall for languages other than English. For Italian and Persian we use exact matching first, and if no sense is found, we use similarity matching as a fallback method.
#### 2.1.3 Drawbacks of CLIP-based methods
Although CLIP offers high zero-shot performance for the visual word sense disambiguation task, we also noticed certain problems in using this model that we could not fully eliminate. We share our observations below, which may provide suggestions for future research:
\(\bullet\) The model is sensitive to images containing text. It also tends to assign high scores to images, which contain the target or context word. For example, for the context _blue mood_, it assigned the highest similarity to an image showing just the word _blue_ on a blue background.
\(\bullet\) The model performs best with images, which directly show the object being described. However, it has trouble modeling more abstract relationships between textual context and image, especially when the context describes non-physical concepts such as emotions, actions, or events.
\(\bullet\) The model has a bias toward more commonly used word senses. As a result, in some cases even expanding the context with additional phrases directing the model to the correct prediction does not help, it still chooses the image relating to the more popular meaning of the target word.
### Wikipedia retrieval module 2
Apart from the classifier described in the previous sections, our solution also includes a Wikipedia-based retrieval module, which returns an independent set of scores for each context-image pair. To apply this method, we first download publicly available Wikipedia dumps for the languages of interest, and then create BM25 (Robertson et al., 2009) indexes with texts extracted from Wikipedia articles. For each document, we include a set of URLs to images attached to the article. We utilize WIT (Srinivasan et al., 2021) dataset to obtain a mapping between articles and images.
During inference, we use the following procedure to process the data sample, consisting of textual context and a set of images:
1. Full context is used to query the index. In response, we retrieve the top 10 articles sorted by their relevance to the query. If no relevant documents are found, we retry the search using only the target word as a query.
2. We download all the images attached to the retrieved articles. Next, we transform both the downloaded images and the images from the sample to their vector representations using the CLIP model.
3. We compare the sets of sample and article vectors using cosine similarity. The final score for each sample image is equal to the maximum of all similarities to the retrieved images.
### Learning to rank 5
The last element of our solution is the learning to rank model (LTR), which leverages the results returned by the other modules of the system to generate the final ranking of images. At the same time, it is the only component, which requires fine-tuning, as the modules described previously operate in zero-shot mode. Our approach is based on the LambdaMART algorithm (Burges, 2010), which transforms the ranking problem into a pairwise classification task. It uses a loss function that compares the relative ordering of two items, rather than absolute scores. This allows the model to better capture the relative importance of different items in the ranking process.
In our case, each data sample is represented by ten vectors consisting of numerical features, with each vector describing a comparison between the context and one of the sample images. We use the training set provided by the task organizers to optimize the model. The numerical features are computed from the outputs of the CLIP classifier and the Wikipedia retrieval module, as well as calculated from the dataset statistics. In the case of scoring modules, the following features are extracted from each: the score assigned to the image, the average and maximum of the scores assigned to the other images from the same sample, the difference between the current score and the average, the difference between the current score and the maximum. We also include the penalty value \(p(\mathbf{x})\) for the image, extracted from the CLIP-based classifier. As for other features, we include the following values in the model 3:
\(\bullet\) Similarity values computed by CLIP between the image vector and the individual word vectors from the sample - separately for the target word and the context word.
\(\bullet\) Two frequency-related features, calculated as the logarithm of the number of occurrences of the image and the context word in the entire dataset.
Input features and hyperparameters of the LTR model are detailed in the Appendix.
## 3 Experiments and results
This section contains a discussion of the official results of the visual word sense disambiguation task. We have also included a description of other variants of our system which were not used in the submitted solution. We conducted post-evaluation experiments using the gold labels provided by the organizers to analyze the results obtained by alternative versions of our approach.
### Official results
The shared task consisted of three language subtasks, and the final ranking of the submitted solutions was based on the average of the results obtained in these subtasks. The primary metric used to evaluate the systems was accuracy, but the organizers also reported mean reciprocal rank (MRR) as an additional metric. 54 teams participated in the shared task. Our solution was ranked third in the main classification and won the Persian language subtask. The results of the top three ranked solutions, the official baseline, and the best results for each language subtask are shown in Table 1.
The team which won the task achieved consistently high accuracy in all three languages, despite not winning on any of the subtasks. The other teams, including us, scored lower in one or more languages. The weak point of our solution was Italian, on which we ranked only 9th, with a difference of 14% accuracy to the winning system. The performance of the best solutions on the Italian subtask turned out to be as high as on English, which was a surprise considering that no training data was available for this language.
### Post-evaluation results
One of the most important components of our approach is the CLIP model, used for both text-to-image and image-to-image comparisons. In our solution, we employed _OpenCLIP H/14_ model published by LAION, which until recently was the largest CLIP variant available. In 2023, an even larger _G/14_ model was released. To study the impact of the selected CLIP version on the accuracy of the whole system, we tested our solution on available OpenAI and LAION models. The results of this experiment are shown in Table 2. Since we do not have multilingual versions of these models, the results presented are for the English subtask only. The conclusions of the experiment are consistent with results on other datasets found in the literature. LAION models achieve significantly higher accuracy than OpenAI models, and larger models outperform smaller ones. The only surprising finding is the weaker performance of the largest _G/14_ model. It is possible that other hyperparameters of
\begin{table}
\begin{tabular}{l|c c c|c c c|c c c|c c c} \hline \hline \multirow{2}{*}{**System**} & \multicolumn{3}{c|}{**Average**} & \multicolumn{3}{c|}{**English**} & \multicolumn{3}{c|}{**Italian**} & \multicolumn{3}{c}{**Persian**} \\ & **ACC** & **MRR** & **R** & **ACC** & **MRR** & **R** & **ACC** & **MRR** & **R** & **ACC** & **MRR** & **R** \\ \hline Organizers’ baseline & 37.20 & 54.39 & - & 60.48 & 73.87 & - & 22.62 & 42.61 & - & 28.50 & 46.70 & - \\ Best result for each language & & & & **84.02** & **89.55** & - & **84.26** & **89.05** & - & **64.00** & **74.39** & - \\ \hline South China Normal University & **72.56** & **82.22** & **1** & 80.13 & 87.42 & 4 & 77.05 & 86.05 & 3 & 60.50 & 73.19 & 2 \\ Samsung Research China (Beijing) & 71.82 & 80.72 & 2 & **84.02** & **89.55** & **1** & 72.46 & 82.08 & 5 & 59.00 & 70.51 & 3 \\ Our system & 70.49 & 79.80 & 3 & 77.97 & 85.88 & 6 & 69.50 & 79.15 & 9 & **64.00** & **74.39** & **1** \\ \hline \hline \end{tabular}
\end{table}
Table 1: The performance of three top-rated teams in the visual word sense disambiguation task compared to the baseline solution provided by the organizers, as well as the highest scores achieved for each language, according to the official results. We show the average scores across all subtasks, as well as the results obtained on each language subtask. The table includes accuracy (ACC), mean reciprocal rank score (MRR), and the rank of each team (R). Bold values indicate the best result in a category.
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **ACC** & **MRR** \\ \hline
**CLIP models** & & \\ \hline _OpenAI CLIP models_ & & \\ \hline clip-vit-base-patch16 & 70.63 & 79.70 \\ clip-vit-base-patch32 & 71.92 & 80.56 \\ clip-vit-large-patch14 & 73.00 & 82.38 \\ \hline _LAION CLIP models_ & & \\ \hline CLIP-ViT-B-32-laion2B-s34B-b79K & 73.00 & 82.63 \\ CLIP-ViT-L-14-laion2B-s32B-b82K & 74.30 & 83.89 \\ CLIP-ViT-H-14-laion2B-s32B-b79K & **77.97** & **85.88** \\ CLIP-ViT-bigG-14-laion2B-39B-b160k & 76.89 & 85.48 \\ \hline
**Context expansion methods** & & \\ \hline WordNet only & 77.97 & 85.88 \\ T5 only & 75.16 & 84.26 \\ WordNet + T5 & **78.83** & **86.26** \\ \hline \hline \end{tabular}
\end{table}
Table 2: The performance of our system on the English subtask using alternative CLIP models or context expansion methods. Text in blue indicates the methods which were used in the submitted solution.
our system would need to be readjusted in order to achieve the optimal performance for the largest model.
Another aspect of the system we examined is the context expansion method. In the submitted solution, we used a method based on WordNet and other lexical resources. While developing the system, we also explored an alternative technique using a sequence-to-sequence model. We employed the recently released Flan-T5 Chung et al. (2022) for this task. Our approach was to send the following prompt to the model: _What is the meaning of [context]?_ In response, the model would generate a definition of the given word sense, which we added to the context. The advantage of this approach is that it allows the context to be expanded for every sample, unlike dictionary-based methods which only expand the context with known definitions. The main disadvantage is the quality of the generated answers. In some cases, they were incorrect, which had a negative impact on the accuracy of the system. Therefore, we decided not to include this method in our solution.
Table 2 shows the results of the three context expansion methods. We tested the performance of the approach based on the T5 model and lexical resources. We also tested a hybrid approach, in which we first try to expand the context using WordNet and other databases, and if that fails, we use the T5 model. As we expected, the standalone T5 model turned out to be worse than the lexical method. However, the hybrid approach managed to improve the accuracy of the English subtask over our original solution.
### Ablation study
As part of the experiments, we performed an ablation study to better understand the impact of the various elements on the accuracy of our solution. The results are shown in Table 3.
The experiment involved performing an evaluation on a system in which a specific component was disabled. We disabled the following functionalities: penalties \(p(x)\) used in the CLIP-based classifier (_no penalties_), learning to rank model (_no LTR_), context expansion (_no expansion_), and Wikipedia retrieval module (_no Wikipedia_). We also tested a version of the system stripped of all the above components, based only on the CLIP model (_CLIP only_). In cases where LTR module is disabled, we instead use a simple heuristic to select the best matching image. We choose the image found by the Wikipedia retrieval module if the value assigned to that image is higher than 0.9, and the value assigned to the all other images is lower than 0.8. Otherwise, we select the highest rated image by the CLIP-based classifier.
As we can see, each of the components we proposed contributed to the performance of the final solution. The simplest version of the system, based only on the CLIP model, performs at least 10% worse than the submitted solution. However, we can also observe that the effect of each functionality varies for different languages. For example, without Wikipedia, the English and Italian subtasks only lose approximately 2% accuracy, while the Persian subtask scores 12% lower. Context expansion is a feature, which has a significant performance impact on each of the three languages.
## 4 Conclusion
In this paper, we described our solution for the visual word sense disambiguation shared task. Our system was ranked third in the multilingual track and won in the Persian track, one of the three language-specific subtasks. In the publication, we demonstrated how to build a system, which incorporates different approaches to the problem: image-text embeddings, lexical resources, image and text retrieval. We showed that each of these components can improve the performance of the overall solution. We have also pointed out some drawbacks of our approach, which can be a starting point for creating better methods in the future.
|
2308.01683 | Growth of Torsion Groups of Elliptic Curves Over Number Fields without
Rationally Defined CM | For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove
that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$
such that if $d$ is a positive integer whose minimal prime divisor is greater
than $p_{\mathcal{K}}$, then for any extension $L/\mathcal{K}$ of degree d and
any elliptic curve $E/\mathcal{K}$, we have
$E\left(L\right)_{\operatorname{tors}} =
E\left(\mathcal{K}\right)_{\operatorname{tors}}$. By not assuming the GRH, this
is a generalization of the results by Genao, and Gon\'alez-Jim\'enez and
Najman. | Bo-Hae Im, Hansol Kim | 2023-08-03T10:50:27Z | http://arxiv.org/abs/2308.01683v2 | # Growth of torsion groups of elliptic curves over number fields without rationally defined CM
###### Abstract.
For a quadratic field \(\mathcal{K}\) without rationally defined complex multiplication, we prove that there exists of a prime \(p_{\mathcal{K}}\) depending only on \(\mathcal{K}\) such that if \(d\) is a positive integer whose minimal prime divisor is greater than \(p_{\mathcal{K}}\), then for any extension \(L/\mathcal{K}\) of degree \(d\) and any elliptic curve \(E/\mathcal{K}\), we have \(E\left(L\right)_{\text{tors}}=E\left(\mathcal{K}\right)_{\text{tors}}\). By not assuming the GRH, this is a generalization of the results by Genao, and Gonzalez-Jimenez and Najman.
Key words and phrases:elliptic curve, torsion subgroup, prime degree isogeny 2010 Mathematics Subject Classification: Primary: 11G05, Secondary: 14H52, 14K02 Bo-Hae Im was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by grant funded by the Korea government (MSIT) (NRF-2023R1A2C1002385).
Another approach to study the torsion parts of elliptic curves over \(K\) is via isogeny. For any elliptic curve \(E/K\) and any finite subgroup \(V\) of \(E\left(\overline{K}\right)\), there is a unique isogeny \(\alpha:E\to E^{\prime}\) defined over \(\overline{K}\) with kernel \(V\). Moreover, \(V\) is \(\operatorname{Gal}\left(\overline{K}/K\right)\)-invariant if and only if \(\alpha\) is \(K\)-rational ([18, Exercise 3.13e]). Therefore, we may identify \(V\) with \(\alpha\). Moreover, a \(\operatorname{Gal}\left(\overline{K}/K\right)\)-invariant finite subgroup of \(E\left(\overline{K}\right)\) corresponds to a \(K\)-rational isogeny from \(E\). Hence, for positive integers \(m\) and \(n\) such that \(m\mid n\), the existence of a \(K\)-rational isogenies whose kernel is isomorphic to \(\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\) is weaker than the existence of an elliptic curve over \(K\) whose torsion part containing \(\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\). Any \(K\)-rational isogeny \(\alpha:E\to E^{\prime}\) whose kernel is isomorphic to \(\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\) factors as \(\beta\circ[m]\) for a unique \(K\)-rational isogeny \(\beta:E\to E^{\prime}\) and \(\ker\beta\) is isomorphic to \(\mathbb{Z}/\frac{n}{m}\mathbb{Z}\). An isogeny with the cyclic kernel is called _a cyclic isogeny_. Obviously, any isogeny of prime degree is cyclic. If \(V\) is \(\operatorname{Gal}\left(\overline{K}/K\right)\)-invariant and cyclic, then so are all subgroups of \(V\) since they are of form \(\left\{P\in V:[n]\,P=O\right\}\). Hence, any \(K\)-rational cyclic isogeny is a composition of \(K\)-rational cyclic isogenies of prime degree. So the torsion subgroups of elliptic curves \(E/K\) can be studied by investigating whether there is a \(K\)-rational isogeny of prime degree. Over \(\mathbb{Q}\), Mazur proves [13, Theorem 3] which proves that there are \(12\) prime degrees of \(\mathbb{Q}\)-rational isogenies.
For any finite extension \(L\) of \(K\), it is obvious that \(E\left(L\right)_{\text{tors}}\supseteq E\left(K\right)_{\text{tors}}\) and \(r_{E\left(L\right)}\geq r_{E\left(K\right)}\). So we might expect that the torsion parts and the ranks of elliptic curves might grow upon base change. It is natural to ask which finite extensions \(L/K\) preserve the torsion parts of elliptic curves \(E/K\) without any growth upon base change from \(K\) to \(L\). Regarding this question, Gonalez-Jimenez and Najman ([5]) gave a partial answer for elliptic curves over \(\mathbb{Q}\) as the follows:
**Theorem** ([5, Theorem 7.2.i]).: _Let \(d\) be a positive integer whose minimal prime divisor is greater than \(7\). For any extension \(L/\mathbb{Q}\) of degree \(d\) and any elliptic curve \(E/\mathbb{Q}\), \(E\left(L\right)_{\text{tors}}=E\left(\mathbb{Q}\right)_{\text{tors}}\)._
Motivated by [5, Theorem 7.2.i], the following question captures our primary attention and focus.
**Question 1**.: _Let \(K\) be a number field. Does \(K\) satisfy the following_ **Property \(\mathcal{P}\left(K\right)\)**_?_
**Property \(\mathcal{P}\left(K\right):\)** There exists a prime \(p_{K}\) depending only on \(K\) such that any elliptic curve
\[E/K\text{ satisfies }E\left(L\right)_{\text{tors}}=E\left(K \right)_{\text{tors}}\text{ for any extension }L/K\text{ of degree whose minimal prime divisor is greater than }p_{K}.\]
A partial answer to Question 1 is given by Genao in [3]. To describe the results in [3], we recall the definition of RCM which stands for rationally defined complex multiplication.
**Definition 1.1** ([3, p.2]).: _We say that a number field \(K\) has RCM (rationally defined complex multiplication) if there exists a CM elliptic curve defined over \(K\) whose endomorphism ring is \(K\)-rational._
It is pointed out in [3, Theorem 3] that the answer to Question 1 is affirmative only for \(K\) without RCM. Moreover, assuming GRH (generalized Riemann hypothesis), [3, Theorem 1] states that the answer to Question 1 is affirmative if and only if \(K\) has no RCM.
Not assuming GRH, we prove that the answer to Question 1 is affirmative for quadratic fields without RCM as a generalization of the results by Genao([3]) and Gonalez-Jimenez and Najman ([5]). We call a quadratic extension of \(\mathbb{Q}\)_a quadratic field_ simply.
**Theorem 1.2**.: _Every quadratic field \(\mathcal{K}\) without RCM satisfies_ **Property \(\mathcal{P}\left(\mathcal{K}\right)\)**_._
The main ingredient of the proofs of our main results is Proposition 1.8 below which gives the structure of the Galois group \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) for a prime \(\ell\) under certain assumptions without RCM, whose proof is given in Section 2. We might expect from [5, Theorem 7.2.i] that if **Property \(\mathcal{P}\left(K\right)\)** is satisfied, then such a prime \(p_{K}\) is at least \(7\). Therefore, we are directed to study the Galois groups \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) for primes \(\ell\) and elliptic curves \(E\) over \(K\) with a point \(R\in E\left[\ell\right]-\left\{O\right\}\) such that the degree \(\nmid\left[K\left(R\right):K\right]\) is not divisible by either the small primes \(2\) or \(3\). First, we analyze it independently of assuming the RCM, and we give a less robust result than Proposition 1.8 for the purpose.
**Proposition 1.3**.: _Let \(K\) be a number field. There exists a positive integer \(N_{K}^{\prime}\) depending only on \(K\) such that for every prime \(\ell>N_{K}^{\prime}\) and elliptic curve \(E/K\) with a point \(R\in E\left[\ell\right]-\left\{O\right\}\) such that \(2,3\nmid\left[K\left(R\right):K\right]\), the Galois group \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) is isomorphic to a subgroup of the Borel subgroup of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\)._
Proposition 1.3 can be put in another way as follows: Under its assumption, \(E\left[\ell\right]\) has a Galois invariant \(1\)-dimensional \(\mathbb{F}_{\ell}\)-subspace \(V\) and the pair \(\left(E,V\right)\) is a non-cuspidal \(K\)-raional point of the modular curve \(X_{0}\left(\ell\right)\). We may identify \(\left(E,V\right)\) to the \(K\)-rational isogeny \(\alpha\) from \(E\) with kernel \(V\) in the sense as in Section 1. Hence, if we let \(Y_{0}\left(\ell\right)\) be the set of all non-cuspidal points in \(X_{0}\left(\ell\right)\), then \(\ell\) is a **p**rime **degree** of an **isogeny** if and only if \(Y_{0}\left(\ell\right)\left(K\right)\neq\emptyset\). Let
\[\mathcal{PDI}\left(K\right):=\left\{\text{a prime }\ell:Y_{0}\left(\ell \right)\left(K\right)\neq\emptyset\right\}. \tag{1}\]
Then, Question 1 is related to the following question.
**Question 2**.: _For each number field \(K\), is the set \(\mathcal{PDI}\left(K\right)\) finite?_
Momose gave a sufficient condition to the finiteness of the set \(\mathcal{PDI}\left(K\right)\) of this question as follows.
**Theorem** ([15, Theorem B]).: _Let \(\mathcal{K}\) be a quadratic field. If \(\mathcal{K}\) is not an imaginary quadratic field of class number \(1\), then \(\mathcal{PDI}\left(\mathcal{K}\right)\) is finite._
**Remark 1.4**.: Let \(K\) be a number field. It is well known that \(\mathcal{PDI}\left(K\right)\) is infinite if \(K\) contains the Hilbert class field of an imaginary quadratic field. Obviously, an imaginary quadratic field has class number \(1\) if and only if it is the Hilbert class field of itself. This fact points out why the condition that the base field has no RCM is crucial, again. Conversely, under GRH, \(\mathcal{PDI}\left(K\right)\) is infinite if and only if \(K\) contains the Hilbert class field of an imaginary quadratic field (see [15, Remark 8]).
**Remark 1.5**.: The proofs of [3, Theorem 1] and [3, Theorem 3] depend on the finiteness of \(\mathcal{PDI}\left(K\right)\). For a number field \(K\) without RCM, under the GRH, if \(\mathcal{PDI}\left(K\right)\) is finite, then **Property \(\mathcal{P}\left(K\right)\)** holds. However, **Property \(\mathcal{P}\left(K\right)\)** may not hold for \(K\) with RCM in which case \(\mathcal{PDI}\left(K\right)\) is infinite.
Moreover, for a number field \(K\), Momose classified in [15] all \(K\)-rational isogenies of sufficiently large prime degrees in \(\mathcal{PDI}\left(K\right)\) into three types. If \(K\) does not have RCM, only two types can occur as follows.
**Theorem 1.6** ([15, Theorem A]).: _Let \(K\) be a number field without RCM. For a prime \(\ell\) and \(\left(E,V\right)\in Y_{0}\left(\ell\right)\left(K\right)\), the natural representation \(\lambda:\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right) \rightarrow\operatorname{GL}\left(V\right)\cong\mathbb{F}_{\ell}^{\times}\) is called the isogeny character of \(\left(E,V\right)\). There is a constant prime number \(C_{K}\) depending only on \(K\) such that any \(\ell>C_{K}\) is one of the following types:_
_Type 1._ \(\lambda^{12}\) _or_ \(\left(\operatorname{cyc}_{\ell}^{-1}\lambda\right)^{12}\) _is unramified._
_Type 2._ \(\lambda^{12}=\operatorname{cyc}_{\ell}^{6}\) _and_ \(\ell\equiv 3\pmod{4}\) _where_ \(\operatorname{cyc}_{\ell}\) _is the_ \(\ell\)_-cyclotomic character._
**Definition 1.7**.: _For a number field \(K\) without RCM, we denote by \(\mathcal{PDI}_{2}\left(K\right)\) the set of all primes \(\ell\equiv 3\pmod{4}\) such that there is a \(K\)-rational isogeny of degree \(\ell\) whose isogeny character \(\lambda\) satisfies that \(\lambda^{12}=\operatorname{cyc}_{\ell}^{6}\)._
For a number field \(K\) without RCM, we can elaborate on Proposition 1.3 to provide more precise properties.
**Proposition 1.8**.: _Let \(K\) be a number field without RCM, there exists a positive integer \(N_{K}\) depending only on \(K\) such that the following holds: for every prime \(\ell>N_{K}\) and an elliptic curve \(E/K\), if there exists \(R\in E\left[\ell\right]-\left\{O\right\}\) such that \(2,3\nmid\left[K\left(R\right):K\right]\), then we have the following;_
1. _the Galois group_ \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) _is isomorphic to a subgroup of the Borel subgroup of_ \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) _containing an unipotent element,_
2. _there exists a prime factor of_ \(\ell-1\) _which divides_ \(\left[K\left(S\right):K\right]\)_, for all_ \(S\in E\left[\ell\right]-\left\{O\right\}\) _and_
3. \(\ell\equiv 7,11,23,31,35\pmod{36}\) _and_ \(\ell\in\mathcal{PDI}_{2}\left(K\right)\)_._
Proof.: It follows from Proposition 2.12 since except for finitely many primes \(\ell\) satisfy all three conditions (i)-(iii) in Proposition 2.12.
Then, by applying Proposition 1.8, we prove the following theorem whose proof is given in Section 3.
**Theorem 1.9**.: _Let \(K\) be a number field without RCM and_
\[\mathcal{R}\left(K\right):=\left\{\begin{aligned} &\text{prime divisors of }\ell-1:\begin{aligned} &\ell\in\mathcal{PDI}_{2}\left(K\right)\text{ in }\left(1\right),\\ &\ell\equiv 7,11,23,31,35\pmod{36}\end{aligned}\right\}. \tag{2}\]
_If \(\mathcal{R}\left(K\right)\) is finite, then \(K\) satisfies_ **Property**_\(\mathcal{P}\left(K\right)\)._
Finally, our main result, Theorem 1.2 is proved by applying Theorem 1.9 and the following result [15, Theorem 4] by Momose and we give its proof in Section 3.
**Theorem** ([15, Theorem 4]).: _For any quadratic field \(\mathcal{K}\), \(\mathcal{PDI}_{2}\left(\mathcal{K}\right)\) is finite._
**Remark 1.10**.: The proofs of Theorem 1.2 and Theorem 1.9 rely on the finiteness of \(\mathcal{PDI}_{2}\left(K\right)\), while [3, Theorem 1] is proved under the assumption on finiteness of \(\mathcal{PDI}\left(K\right)\).
The \(\ell^{\infty}\)-torsion subgroups for sufficiently large primes \(\ell\) : the proofs of Proposition 1.3 and Proposition 1.8
In this section, we investigate the possible structures of subgroups of \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) and we prove Proposition 1.3 and Proposition 1.8 by applying them.
Subgroups of \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) and their corresponding subgroups of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\)
Let \(K\) be a number field and \(E/K\) be an elliptic curve over \(K\). For a positive integer \(N\), recall that \(E\left[N\right]\cong\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}\). So we consider the Galois group \(\operatorname{Gal}\left(K\left(E\left[N\right]\right)/K\right)\) as a subgroup of \(\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})\) depending on a basis \(\mathcal{B}=\left\{P,Q\right\}\) for \(E\left[N\right]\) via the following map.
**Definition 2.1**.: _Let \(K\) be a number field, \(E/K\) an elliptic curve over \(K\), and \(N\) a positive integer. For a given (ordered) basis \(\mathcal{B}=\left\{P,Q\right\}\) for \(E\left[N\right]\), we have the injective group homomorphism,_
\[\rho_{\mathcal{B}}:\operatorname{Gal}\left(K\left(E\left[N\right] \right)/K\right) \to\operatorname{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right)\text{ defined by}\] \[\sigma \mapsto\rho_{\mathcal{B}}\left(\sigma\right):\begin{pmatrix}P\\ Q\end{pmatrix}\mapsto\begin{pmatrix}P^{\sigma}\\ Q^{\sigma}\end{pmatrix}.\]
_Let \(G\left(\mathcal{B}\right):=\rho_{\mathcal{B}}\left(\operatorname{Gal}\left(K \left(E\left[N\right]\right)\right)/K\right)\), the image of \(\rho_{\mathcal{B}}\)._
For an odd prime \(\ell\) and a given basis \(\mathcal{B}=\left\{P,Q\right\}\) for \(E\left[\ell\right]\), we consider the right action of the image \(G\left(\mathcal{B}\right)\) on the row vectors in \(\mathbb{F}_{\ell}^{2}\).
**Definition 2.2**.: _Let \(G\) be a subgroup of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\). For each non-zero row vector \(\left(c\,d\right)\in\mathbb{F}_{\ell}^{2}\), we define the following subgroups of \(G\):_
* \(G_{c\,d}:=\left\{A\in G:\left(c\,d\right)A=\left(c\,d\right)\right\}\)_._
* \(G^{\circ}:=G\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\)_._
_We note that \(G_{c\,d}^{\circ}=G_{c\,d}\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell} \right)=G_{c\,d}\cap G^{\circ}\)._
**Lemma 2.3**.: _Let \(K\) be a number field and \(E/K\) be an elliptic curve over \(K\). For an odd prime \(\ell\), let \(\zeta_{\ell}\) be a primitive \(\ell\)th root of unity, and for a given basis \(\mathcal{B}=\left\{P,Q\right\}\) for \(E\left[\ell\right]\), we let \(G=G\left(\mathcal{B}\right)\). Then, for any non-zero row vector \(\left(c\,d\right)\in\mathbb{F}_{\ell}^{2}\), we have the following:_
* \(G_{c\,d}=\rho_{\mathcal{B}}\left(\operatorname{Gal}\left(K\left(E\left[\ell \right]\right)/K\left(\left[c\right]P+\left[d\right]Q\right)\right)\right)\)_, and_ \(\left[G:G_{c\,d}\right]=\left[K\left(\left[c\right]P+\left[d\right]Q\right):K\right]\)_._
* \(G_{c\,d}^{\circ}=\rho_{\mathcal{B}}\left(\operatorname{Gal}\left(K\left(E \left[\ell\right]\right)/K\left(\zeta_{\ell}\right)\right)\right)\)_,_ \(G/G^{\circ}\cong\operatorname{Gal}\left(K\left(\zeta_{\ell}\right)/K\right)\)_, and_ \(\left[G:G^{\circ}\right]=\left[K\left(\zeta_{\ell}\right):K\right]\)_._
* \(G_{c\,d}^{\circ}=\rho_{\mathcal{B}}\left(\operatorname{Gal}\left(K\left(E \left[\ell\right]\right)/K\left(\zeta_{\ell},\left[c\right]P+\left[d\right]Q \right)\right)\right)\)_, and_ \(\left[G:G_{c\,d}^{\circ}]=\left[K\left(\zeta_{\ell},\left[c\right]P+ \left[d\right]Q\right):K\right]\)_._
* \(G_{c\,d}^{\circ}\) _is trivial, or it is conjugate to_ \(\left\langle U\right\rangle\)_, where_ \(U=\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)\)_. In particular,_ \(\left|G_{c\,d}^{\circ}\right|\bigm{|}\ell\)_._
* \(G_{c\,d}=\begin{cases}G_{1\,d/c},&\text{ if }c\neq 0,\\ G_{01},&\text{ if }c=0.\end{cases}\)__
Proof.: (a), (b), and (c) follow by the definitions of \(\rho_{\mathcal{B}}\) and the Weil pairing directly. In fact, the Weil pairing \(e_{\ell}:E\left[\ell\right]\times E\left[\ell\right]\to\left\{z\in\mathbb{A} ^{1}:z^{\ell}-1=0\right\}\) is a non-degenerate alternative multi-linear form as a \(K\)-morphism of varieties, so \(e_{\ell}\left(P,Q\right)\) is a primitive \(\ell\)th root of unity \(\zeta_{\ell}\), and \(\zeta_{\ell}\in K\left(E\left[\ell\right]\right)\).
For (d), \(G_{c\,d}^{\circ}\) consists of matrices with only one eigenvalue \(1\) and such non-trivial subgroups of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) are conjugates of \(\left\langle U\right\rangle\). Hence, \(G_{c\,d}^{\circ}\) is trivial, or of order \(\ell\).
(e) is obvious.
Now, we study the structures of certain subgroups of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) which are corresponding to the subgroups of \(G\left(\mathcal{B}\right)\) of our interest.
We fix a generator \(\alpha\) of \(\mathbb{F}_{\ell}^{\times}\). Then, \(\mathbb{F}_{\ell^{2}}=\mathbb{F}_{\ell}\left(\sqrt{\alpha}\right)\), so we denote the non-trivial Galois action of \(\operatorname{Gal}\left(\mathbb{F}_{\ell^{2}}/\mathbb{F}_{\ell}\right)\) by \(\overline{a+b\sqrt{\alpha}}:=a-b\sqrt{\alpha}\) for \(a,b\in\mathbb{F}_{\ell}\). We call \(\beta\in\mathbb{F}_{\ell^{2}}\)_rational (resp. irrational)_ if \(\beta\in\mathbb{F}_{\ell}\) (resp. \(\beta\notin\mathbb{F}_{\ell}\)). We let
\[U:=\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right),\text{ and }\operatorname{diag}\left(a,d\right):= \left(\begin{smallmatrix}a&0\\ 0&d\end{smallmatrix}\right)\text{ for }a,d\in\mathbb{F}_{\ell}^{\times}.\]
We recall the Borel subgroup \(\mathscr{B}(\ell)\), the split Cartan subgroup \(\mathscr{C}_{s}(\ell)\), and the non-split Cartan subgroup \(\mathscr{C}_{ns}(\ell)\) of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\). If there is no confusion, we denote them by \(\mathscr{B},\mathscr{C}_{s},\mathscr{C}_{ns}\) respectively omitting \(\ell\).
The Borel subgroup is defined by \(\mathscr{B}:=\left\{\left(\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}\left(\mathbb{F}_{ \ell}\right)\right\}\) which is the normalizer of each subgroup of \(\mathscr{B}\) containing \(U\).
The split Cartan subgroup \(\mathscr{C}_{s}\subseteq\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) is defined by the group of all invertible diagonal matrices. We denote an invertible diagonal matrix \(\operatorname{diag}\left(\alpha^{a},\alpha^{b}\right)\) by \(\operatorname{diagexp}\left(a,b\right)\). Then, we see that \(\mathscr{C}_{s}\cong\left(\mathbb{Z}/\left(\ell-1\right)\mathbb{Z}\right)^{2}\) via the group isomorphism, \(\operatorname{diagexp}:\left(\mathbb{Z}/\left(\ell-1\right)\mathbb{Z}\right) ^{2}\to\mathscr{C}_{s}\) defined by \(\left(a,b\right)\mapsto\operatorname{diagexp}\left(a,b\right)\) in the above.
The non-split Cartan subgroup \(\mathscr{C}_{ns}\subseteq\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) is defined by
\[\left\{\begin{pmatrix}a&b\alpha\\ b&a\end{pmatrix}:\left(0,0\right)\neq\left(a,b\right)\in\mathbb{F}_{\ell}^{2 }\right\}.\]
Then, we see that \(\mathscr{C}_{ns}\cong\left(\mathbb{F}_{\ell}\left(\sqrt{\alpha}\right)\right) ^{\times}\) via a group isomorphism \(\left(\mathbb{F}_{\ell}\left(\sqrt{\alpha}\right)\right)^{\times}\to\mathscr{C }_{ns}\) defined by \(a+b\sqrt{\alpha}\mapsto\left(\begin{smallmatrix}a&b\alpha\\ b&a\end{smallmatrix}\right)\).
We note that for each \(\left(a,b\right)\in\mathbb{F}_{\ell}^{2}-\left\{\left(0,0\right)\right\}\), the matrix \(\left(\begin{smallmatrix}a&b\alpha\\ b&a\end{smallmatrix}\right)\in\mathscr{C}_{ns}\) has two conjugate eigenvalues \(\delta\) and \(\bar{\delta}\) in \(\mathbb{F}_{\ell^{2}}\).
It is clear that
\[\left|\mathscr{B}\right|=\ell(\ell-1)^{2},\quad\left|\mathscr{C}_{s}\right|=( \ell-1)^{2},\quad\text{ and }\quad\left|\mathscr{C}_{ns}\right|=\ell^{2}-1.\]
**Lemma 2.4**.: _Let \(H\) be an abelian subgroup of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(\ell\nmid\left|H\right|\). Then, there exists \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(T^{-1}HT\subseteq\mathscr{C}_{s}\) or \(T^{-1}HT\subseteq\mathscr{C}_{ns}\)._
Proof.: Since \(H\) is abelian and \(\ell\nmid\left|H\right|\), all matrices in \(H\) are simultaneously diagonalizable over \(\mathbb{F}_{\ell^{2}}\).
If for each \(A\in H\), all eigenvalues of \(A\) are rational, then \(T^{-1}HT\subseteq\mathscr{C}_{s}\) for some \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\).
Suppose that there exists \(A\in H\) with two irrational eigenvalues. There exists \(T_{1}\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell^{2}}\right)\) such that \(T_{1}^{-1}HT_{1}\) consists of diagonal matrices over \(\mathbb{F}_{\ell^{2}}\). So \(D:=T_{1}^{-1}AT_{1}=\operatorname{diag}\left(\lambda,\overline{\lambda}\right)\), for some irrational element \(\lambda\in\mathbb{F}_{\ell^{2}}\), and for any \(A^{\prime}\in H\), \(D^{\prime}:=T_{1}^{-1}A^{\prime}T_{1}\) is diagonal. Then, the diagonal entries of either \(D^{\prime}\) or \(DD^{\prime}\) are irrational. In other words, the diagonal entries of one of \(D^{\prime}\) or \(DD^{\prime}\) are conjugate to each other. Thus, \(D^{\prime}=\operatorname{diag}\left(\lambda^{\prime},\overline{\lambda^{\prime}}\right)\) for an eigenvalue \(\lambda^{\prime}\) of \(A^{\prime}\). Hence, \(T_{1}^{-1}HT_{1}\) is contained in \(\left\{\operatorname{diag}\left(\delta,\overline{\delta}\right):\delta\in \mathbb{F}_{\ell^{2}}^{\times}\right\}\). Since \(\mathbb{F}_{\ell^{2}}^{\times}\) is cyclic, there is an irrational element \(\mu\in\mathbb{F}_{\ell^{2}}^{\times}\) such that \(T_{1}^{-1}HT_{1}=\left\langle\operatorname{diag}\left(\mu,\overline{\mu}\right)\right\rangle\). We let \(\mu=a+b\sqrt{\alpha}\) for some \(a,b\in\mathbb{F}_{\ell}\). There are \(t,u,v\in\mathbb{F}_{\ell}\) such that \(\left(\begin{smallmatrix}t&u\\ v&2a-t\end{smallmatrix}\right)=T_{1}\operatorname{diag}\left(\mu,\overline{\mu} \right)T_{1}^{-1}\in H\). We let \(P:=\left(\begin{smallmatrix}u&u\\ a+b\sqrt{\alpha}-t&a-b\sqrt{\alpha}-t\end{smallmatrix}\right)\), \(Q:=\left(\begin{smallmatrix}b&\alpha\\ b\sqrt{\alpha}&-b\sqrt{\alpha}\end{smallmatrix}\right)\), \(R:=\left(\begin{smallmatrix}t&u\\ v&2a-t\end{smallmatrix}\right)\), and \(S:=\left(\begin{smallmatrix}a&b\alpha\\ b&a\end{smallmatrix}\right)\). Then, we can see that \(P^{-1}RP=\operatorname{diag}\left(\mu,\overline{\mu}\right)=Q^{-1}SQ\) and that \(T:=PQ^{-1}\) is \(\operatorname{Gal}(\mathbb{F}_{\ell^{2}}/\mathbb{F}_{\ell})\)-invariant. Hence, \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) and \(T^{-1}HT=\left\langle S\right\rangle\subseteq\mathscr{C}_{ns}\).
We denote the normalizers of \(\mathscr{C}_{s}\) and of \(\mathscr{C}_{ns}\) in \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) by \(\mathscr{N}_{s}\) and \(\mathscr{N}_{ns}\), respectively. Then,
\[\mathscr{N}_{s}=\left\langle\mathscr{C}_{s},\left(\begin{smallmatrix}0&1\\ 0&1\end{smallmatrix}\right)\right\rangle\text{ and }\mathscr{N}_{ns}=\left\langle\mathscr{C}_{ns},\left( \begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix}\right)\right\rangle.\]
**Lemma 2.5**.: _Let \(H\) be a subgroup of \(\mathscr{C}_{s}\) or \(\mathscr{C}_{ns}\) such that \(H\not\subseteq\left\{kI_{2}:k\in\mathbb{F}_{\ell}^{\times}\right\}\) and \(\mathscr{N}\) be the normalizer of \(H\) in \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\). Then,_
\[\mathscr{N}\subseteq\begin{cases}\mathscr{N}_{s},&\text{ if }H\subseteq \mathscr{C}_{s},\\ \mathscr{N}_{ns},&\text{ if }H\subseteq\mathscr{C}_{ns}.\end{cases}\]
Proof.: Suppose \(H\subseteq\mathscr{C}_{s}\). Then since \(H\not\subseteq\left\{kI_{2}:k\in\mathbb{F}_{\ell}^{\times}\right\}\), there exists \(B=\operatorname{diag}\left(a,d\right)\in H\) where \(a,d\in\mathbb{F}_{\ell}^{\times}\) such that \(a\neq d\). For any \(X\in\mathscr{N}\), since \(XBX^{-1}\in H\subseteq\mathscr{C}_{s}\), we conclude that \(XBX^{-1}\) is \(B\) or \(\operatorname{diag}\left(d,a\right)\) considering the eigenvalues of \(B\). Then, in either case, we can show that \(X\in\mathscr{N}_{s}\) by direct calculation, which implies that \(\mathscr{N}\subseteq\mathscr{N}_{s}\).
Suppose \(H\subseteq\mathscr{C}_{ns}\). Then, there exists \(B^{\prime}=\left(\begin{smallmatrix}a&b\alpha\\ b&a\end{smallmatrix}\right)\in H\) where \(a,b\in\mathbb{F}_{\ell}\) such that \(b\neq 0\). For any \(X\in\mathscr{N}\), since \(XB^{\prime}X^{-1}\in H\subseteq\mathscr{C}_{ns}\), we conclude that \(XB^{\prime}X^{-1}\) is \(B^{\prime}\) or \(\left(\begin{smallmatrix}a&-b\alpha\\ -b&a\end{smallmatrix}\right)\) considering the eigenvalues of \(B^{\prime}\). Then, in either case, we can show that \(X\in\mathscr{N}_{ns}\) by direct calculation, which implies that \(\mathscr{N}\subseteq\mathscr{N}_{ns}\).
Next, we investigate the structures of subgroups of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) that satisfy certain conditions when its intersection with \(\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) is considered in the following lemmas.
**Lemma 2.6**.: _Let \(H\) be a subgroup of \(\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(2,\ell\nmid\left|H\right|\). Then, \(H\) is cyclic._
Proof.: If \(H=\left\{I_{2}\right\}\), it is trivially true. We assume that \(H\neq\left\{I_{2}\right\}\). Then, \(H\) is solvable by the Feit-Thompson theorem([19]). So we let \(n_{0}\) be the smallest positive integer such that \(H^{\left(n_{0}\right)}=\left\{I_{2}\right\}\) where \(H^{\left(n\right)}\) denotes the \(n\)th derived subgroup of \(H\). Since \(H\) is not trivial, \(n_{0}>0\) and the non-trivial normal subgroup \(N:=H^{\left(n_{0}-1\right)}\subseteq H\) is abelian. Since \(\ell\nmid\left|N\right|\), all matrices in \(N\) are simultaneously diagonalizable over \(\mathbb{F}_{\ell^{2}}\). Thus, there exists \(P\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell^{2}}\right)\) such that \(P^{-1}NP\) consists of diagonal matrices over \(\mathbb{F}_{\ell^{2}}\). Since any subgroup of \(\mathbb{F}_{\ell^{2}}^{\times}\) is cyclic and \(\det\left(P^{-1}NP\right)=\left\{1\right\}\), we have that \(P^{-1}NP\) is cyclic. Let \(D\) be a generator of \(P^{-1}NP\). Considering the eigenvalues of \(D\) for each \(X\in P^{-1}HP\), there exists \(\epsilon\left(X\right)\in\left\{\pm 1\right\}\) such that \(XDX^{-1}=D^{\epsilon\left(X\right)}\) since \(P^{-1}NP\unlhd P^{-1}HP\), so this defines a group homomorphism \(\epsilon:P^{-1}HP\rightarrow\left\{\pm 1\right\}\). Since \(\left|H\right|\) is odd, \(\epsilon\) is trivial. So any \(X\in P^{-1}HP\) commutes with elements of \(P^{-1}DP\). Since \(N\) is a non-trivial subgroup of \(\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) of odd order, \(D\) has two distinct diagonal entries (otherwise, \(D=\pm I_{2}\) and \(-I_{2}\) has order \(2\)). Hence, any \(X\in P^{-1}HP\) is diagonal and \(H\) is abelian. By Lemma 2.4, we conclude that \(H\) is contained in \(\mathscr{C}_{s}\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) or \(\mathscr{C}_{ns}\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) up to conjugacy and this completes the proof since \(\mathscr{C}_{s}\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) and \(\mathscr{C}_{ns}\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) are cyclic.
**Lemma 2.7**.: _Let \(H\) be a subgroup of \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(2,\ell\nmid\left|H^{\circ}\right|\) where \(H^{\circ}=H\cap\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\). Then, there exists \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(T^{-1}HT\subseteq\mathscr{N}_{s}\) or \(T^{-1}HT\subseteq\mathscr{N}_{ns}\)._
Proof.: If \(H^{\circ}\subseteq\left\{\pm I_{2}\right\}\), then \(H\) is abelian since \(H/H^{\circ}\) is isomorphic to a subgroup of \(\mathbb{F}_{\ell}^{\times}\) which is cyclic. By Lemma 2.4, there exists \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(T^{-1}HT\) is contained in \(\mathscr{C}_{s}\) or \(\mathscr{C}_{ns}\).
If \(H^{\circ}\not\subseteq\left\{\pm I_{2}\right\}\), then \(H^{\circ}\not\subseteq\left\{kI_{2}:k\in\mathbb{F}_{\ell}^{\times}\right\}\). So by Lemma 2.6, \(H^{\circ}\) is cyclic and by Lemma 2.4, there exists \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(T^{-1}H^{\circ}T\) is contained in \(\mathscr{C}_{s}\) or \(\mathscr{C}_{ns}\). Let \(\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)}\left(T^{-1}H^{ \circ}T\right)\) be the normalizer of \(T^{-1}H^{\circ}T\) in \(\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\). Then, since \(T^{-1}H^{\circ}T\not\subseteq\left\{kI_{2}:k\in\mathbb{F}_{\ell}^{\times}\right\}\), Lemma 2.5 implies that \(\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)}\left(T^{-1}H^{ \circ}T\right)\subseteq\mathscr{N}_{ns}\) or \(\mathscr{N}_{s}\). Since \(H^{\circ}\unlhd H\), we have that \(T^{-1}HT\subseteq\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell} \right)}\left(T^{-1}H^{\circ}T\right)\), which completes the proof.
At last, the following elementary lemma will be useful.
**Lemma 2.8**.: \(\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) _is generated by \(U=\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)\) and \(U^{t}=\left(\begin{smallmatrix}1&0\\ 1&1\end{smallmatrix}\right)\)._
Proof.: First, we note that for each \(s\in\mathbb{F}_{\ell}\),
\[\begin{pmatrix}1&s\\ 0&1\end{pmatrix}=U^{s},\text{ and }\begin{pmatrix}1&0\\ s&1\end{pmatrix}=(U^{t})^{s},\]
and for any non-zero \(s\in\mathbb{F}_{\ell}\),
\[\begin{pmatrix}0&-s^{-1}\\ s&0\end{pmatrix} =\begin{pmatrix}1&-s^{-1}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ s&1\end{pmatrix}\begin{pmatrix}1&-s^{-1}\\ 0&1\end{pmatrix}\text{ and }\] \[\begin{pmatrix}s&0\\ 0&s^{-1}\end{pmatrix} =\begin{pmatrix}1&0\\ -1&1\end{pmatrix}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -1&1\end{pmatrix}\begin{pmatrix}0&-s^{-1}\\ s&0\end{pmatrix}.\]
For any \(\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\), at least one of \(a\) and \(c\) is non-zero. If \(a\neq 0\), then
\[\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}1&0\\ ca^{-1}&1\end{pmatrix}\begin{pmatrix}a&b\\ 0&d-bca^{-1}\end{pmatrix}=\begin{pmatrix}1&0\\ ca^{-1}&1\end{pmatrix}\begin{pmatrix}a&b\\ 0&a^{-1}\end{pmatrix}=\begin{pmatrix}1&0\\ ca^{-1}&1\end{pmatrix}\begin{pmatrix}1&ab\\ 0&1\end{pmatrix}\begin{pmatrix}a&0\\ 0&a^{-1}\end{pmatrix}.\]
If \(c\neq 0\), then
\[\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}a&\left(ad-1\right)c^{-1}\\ c&d\end{pmatrix}=\begin{pmatrix}1&ac^{-1}\\ 0&1\end{pmatrix}\begin{pmatrix}0&-c^{-1}\\ c&d\end{pmatrix}=\begin{pmatrix}1&ac^{-1}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -cd&1\end{pmatrix}\begin{pmatrix}0&-c^{-1}\\ c&0\end{pmatrix}.\]
### The proof of Proposition 1.8
First, we consider the case when \(\ell\geq 5\) is a prime and an elliptic curve \(E/K\) contains a torsion point \(R\) of order \(\ell\) such that \(2\nmid\left[K\left(R\right):K\right]\).
Recall that \(K(E[\ell])\) contains a primitive \(\ell\)th root of unity \(\zeta_{\ell}\).
**Proposition 2.9**.: _Let \(K\) be a number field, \(E/K\) be an elliptic curve over \(K\), and \(\ell\geq 5\) be a prime. If there exists \(R\in E\left[\ell\right]-\left\{O\right\}\) of \(E\) such that \(2\nmid\left[K\left(R\right):K\right]\). Then, there is a basis \(\mathcal{B}\) of \(E\left[\ell\right]\) such that \(G\left(\mathcal{B}\right)\) is contained in \(\mathcal{B}\), \(\mathscr{N}_{s}\), or \(\mathscr{N}_{ns}\)._
Proof.: We let \(K_{0}=K\left(\zeta_{\ell}\right)\). First, we show that the extension \(K_{0}\left(R\right)\) is normal over \(K_{0}\). Suppose not. First, note that for any \(P\in E\left[\ell\right]-\left\langle R\right\rangle\), the set \(\left\{P,R\right\}\) is a basis for \(E\left[\ell\right]\). We fix a basis \(\mathcal{B}^{\prime}:=\left\{P,R\right\}\) and let \(H:=G\left(\mathcal{B}^{\prime}\right)\). Then since \(K_{0}\left(R\right)\) is not normal over \(K_{0}\), \(H_{01}^{\circ}\subseteq H^{\circ}\) is not a normal subgroup. So \(H_{01}^{\circ}\) is non-trivial and there exists \(A\in H^{\circ}\) - \(\mathscr{N}_{H^{\circ}}\left(H_{01}^{\circ}\right)\), where \(\mathscr{N}_{H^{\circ}}\left(H_{01}^{\circ}\right)\) is the normalizer of \(H_{01}^{\circ}\) in \(H^{\circ}\). Moreover, since \(H_{01}^{\circ}\) is a non-trivial subgroup of \(\left\langle U\right\rangle\) of order \(\ell\) by Lemma 2.3(d), we see that \(H_{01}^{\circ}=\left\langle U\right\rangle\). Then, \(\rho_{\mathcal{B}^{\prime}}^{-1}\left(AUA^{-1}\right)\) fixes \(R^{\rho_{\mathcal{B}^{\prime}}^{-1}\left(A\right)}\) but not \(R\). In fact, if it fixes \(R\), then it must be the identity element, which is not true. Hence, we set a new basis \(\left\{R^{\rho_{\mathcal{B}^{\prime}}^{-1}\left(A\right)},R\right\}\) for \(E\left[\ell\right]\) and let \(\Gamma:=G\left(\left\{R^{\rho_{\mathcal{B}^{\prime}}^{-1}\left(A\right)},R \right\}\right)\). Figure 2 below shows the diagrams of these extensions over \(K\) and their corresponding Galois groups when \(K_{0}(R)\) is not a normal extension of \(K_{0}\). Since \(AUA^{-1}\) has order \(\ell\), we conclude that \(\Gamma\) and so \(\Gamma^{\circ}\) contains both \(U\) and \(U^{t}\).
Thus, \(\Gamma^{\circ}=\operatorname{SL}_{2}\left(\mathbb{F}_{\ell}\right)\) by Lemma 2.8, and \(2\mid\ell^{2}-1=\left[\Gamma^{\circ}:\Gamma_{01}^{\circ}\right]\mid\left[ \Gamma:\Gamma_{01}\right]=\left[K\left(R\right):K\right]\), which is a contradiction to our assumption. Therefore, \(K_{0}\left(R\right)\) is normal over \(K_{0}\).
Now, we consider two cases. First, if \(\ell\mid\left[K\left(E\left[\ell\right]\right):K\right]\), then there exists basis \(\mathcal{B}\) for \(E\left[\ell\right]\) such that \(U\in G:=G\left(\mathcal{B}\right)\). Referring to Figure 2, since \(\left\langle U\right\rangle=G_{01}^{\circ}\unlhd G^{\circ}\), we have that \(U\in G^{\circ}\subseteq\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell} \right)}\left(\left\langle U\right\rangle\right)=\mathcal{B}\). Since \(G^{\circ}\unlhd G\), we have that \(G\subseteq\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)} \left(G^{\circ}\right)\) and \(\mathscr{N}_{\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)}\left(G^{ \circ}\right)=\mathcal{B}\), since \(U\in G^{\circ}\).
Second, if \(\ell\nmid\left[K\left(E\left[\ell\right]\right):K\right]\), we fix a basis \(\mathcal{B}^{\prime}:=\left\{P,R\right\}\) for \(E\left[\ell\right]\) for some \(P\in E\left[\ell\right]-\left\langle R\right\rangle\) and let \(H:=G\left(\mathcal{B}^{\prime}\right)\). Then, \(\ell\nmid\left[H^{\circ}\right]\) and \(H_{01}^{\circ}=\left\{I_{2}\right\}\) by Lemma 2.3(d) since \(\left|H_{01}^{\circ}\right|\mid\ell\) but \(\ell\nmid\left[H\right]\). Hence, \(H^{\circ}\cap H_{01}=\left\{I_{2}\right\}\) and the semi-direct product \(H^{\circ}\rtimes H_{01}\) is a subgroup of
and \(\left|H^{\circ}\right|\) divides \(\left[H:H_{01}\right]\). Since \(\left[H:H_{01}\right]\) is odd, so is \(\left|H^{\circ}\right|\). So by Lemma 2.7, there exists \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that \(T^{-1}HT\) is contained in \(\mathscr{N}_{s}\) or \(\mathscr{N}_{ns}\). Therefore, there is a basis \(\mathcal{B}\) for \(E\left[\ell\right]\) such that \(G\left(\mathcal{B}\right)\) is contained in \(\mathscr{N}_{s}\) or \(\mathscr{N}_{ns}\).
Next, we observe a sufficient condition for the degree \(\left[K\left(R\right):K\right]\) to be divisible by \(2\) or \(3\) for a point \(R\in E\left[\ell\right]-\left\{O\right\}\) where \(\ell\geq 11\) is an unramified prime in \(K\), by applying Genao's result [3, Theorem 2] which describes the image of the inertia group based on the deduction types.
**Proposition 2.10**.: _Let \(K\) be a number field and \(E/K\) be an elliptic curve over \(K\). Suppose \(\ell\geq 11\) is a prime satisfying the following two conditions:_
1. \(\mathbb{Q}\left(\zeta_{\ell}\right)\cap K=\mathbb{Q}\)_._
2. \(\ell\) _is unramified in_ \(K\)_._
_If \(G\left(\mathcal{B}\right)\) is contained in \(\mathscr{N}_{s}\), or in \(\mathscr{N}_{ns}\) but not \(\mathscr{C}_{s}\) for some basis \(\mathcal{B}\) of \(E\left[\ell\right]\), then for any point \(S\in E\left[\ell\right]-\left\{O\right\}\), the degree \(\left[K\left(S\right):K\right]\) is divisible by \(2\) or \(3\)._
Proof.: First, suppose \(G:=G\left(\mathcal{B}\right)\subseteq\mathscr{N}_{ns}\). If \(A\in\mathscr{N}_{ns}\) has an eigenvalue \(1\), then the other eigenvalue of \(A\) is \(1\) or \(-1\). So for any non-zero row vector \(\left(c\,d\right)\in\mathbb{F}_{\ell}^{2}\), \(G_{c\,d}\) is contained in \(\left\{\left(\begin{smallmatrix}\pm 1&0\\ 0&1\end{smallmatrix}\right)\right\}\) or \(\left\{\left(\begin{smallmatrix}1&0\\ 0&\pm 1\end{smallmatrix}\right)\right\}\). Moreover, \(\mathscr{C}_{ns}\cap G_{c\,d}=\left\{I_{2}\right\}\). By Lemma 2.7, we know that \(\mathscr{C}_{ns}^{e}:=\left\{A^{e}:A\in\mathscr{C}_{ns}\right\}\) is a normal subgroup of \(G\) since \(G\subseteq\mathscr{N}_{ns}\), and by [3, Theorem 2] for some \(e\in\left\{1,2,3,4,6\right\}\),
\[2\ \big{|}\ (\ell^{2}-1)/e=\left|\mathscr{C}_{ns}^{e}\right|=\left[\mathscr{C}_{ ns}^{e}:\mathscr{C}_{ns}^{e}\cap G_{cd}\right]=\left[\mathscr{C}_{ns}^{e}G_{cd}:G_{cd }\right]\ \big{|}\ \left[G:G_{cd}\right].\]
This implies that \(\left[K\left(S\right):K\right]\) is divisible by \(2\), since \(\left[K\left(S\right):K\right]=\left[G:G_{cd}\right]\) for some nonzero vector \(\left(c\,d\right)\in\mathbb{F}_{\ell}^{2}\).
Now, suppose \(G\) is contained in \(\mathscr{N}_{s}\) but not \(\mathscr{C}_{s}\). The index of the subgroup \(\Delta:=G\cap\mathscr{C}_{s}\) in \(G\) is \(2\). By direct computation, we can show that \(G_{01}\) and \(G_{10}\) are contained in \(\Delta\). So \(\left[G:G_{10}\right]=\left[G:\Delta\right]\left[\Delta:G_{10}\right]\), which is even. For any non-zero \(s\in\mathbb{F}_{\ell}\), direct computation shows that \(\left|G_{1s}\right|\mid 2=\left[G:\Delta\right]\) and \(\left|\Delta\right|\mid\left[G:G_{1s}\right]\). Thus, it is enough to show that \(\left|\Delta\right|\) is divisible by \(2\) or \(3\). The assumption (i) implies that there exists \(A\in G\) such that \(\det A=\alpha\), where \(\alpha\) is a generator of \(\mathbb{F}_{\ell}^{\times}\).
If \(\ell\equiv 1\pmod{4}\), then \(2\mid(\ell-1)/2\) and \(\left|A^{2}\right|\ \big{|}\ \left|\Delta\right|\), so \(2\mid\left|\Delta\right|\).
Figure 2. The diagrams of subfields of \(K\left(E\left[\ell\right]\right)\) containing \(K\) and their corresponding Galois groups in \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) when \(K_{0}(R)\) is not normal over \(K_{0}\)
If \(\ell\equiv 3\pmod{4}\), we first claim that \(\ell=19\) or \(\rho\left(I_{K_{\mathfrak{p}}}\right)\subseteq\mathscr{C}_{s}\) where \(K_{\mathfrak{p}}\) is the completion of \(K\) at a prime \(\mathfrak{p}\) above \(\ell\), \(I_{K_{\mathfrak{p}}}\) is the inertia group of \(\mathfrak{p}\), and \(\rho=\rho_{\mathcal{B}}\), following the proof of [3, Theorem 2]. Suppose not, i.e., suppose that \(\ell\neq 19\) and \(\rho\left(I_{K_{\mathfrak{p}}}\right)\) is contained in \(\mathscr{N}_{s}\) but not \(\mathscr{C}_{s}\). Since \(I_{K_{\mathfrak{p}}}\) is cyclic, there exists an element \(B\in\mathscr{N}_{s}-\mathscr{C}_{s}\) which generates \(\rho\left(I_{K_{\mathfrak{p}}}\right)\). By [3, Theorem 2], \(\rho\left(I_{K_{\mathfrak{p}}}\right)\) contains \(T\operatorname{\mathrm{diagexp}}\left(0,e\right)T^{-1}\) for some \(e\in\left\{1,2,3,4,6\right\}\) and for some \(T\in\operatorname{\mathrm{GL}}_{2}\left(\mathbb{F}_{\ell}\right)\). If \(T\operatorname{\mathrm{diagexp}}\left(0,e\right)T^{-1}\in G\subseteq\mathscr{N }_{s}\), then \(T\operatorname{\mathrm{diagexp}}\left(0,e\right)T^{-1}\) is \(\operatorname{\mathrm{diagexp}}\left(0,e\right)\) or \(\operatorname{\mathrm{diagexp}}\left(e,0\right)\). If \(T\operatorname{\mathrm{diagexp}}\left(0,e\right)T^{-1}=\operatorname{\mathrm{ diagexp}}\left(0,e\right)\), we have that \(\operatorname{\mathrm{diagexp}}\left(e,0\right)=B\operatorname{\mathrm{diagexp}} \left(0,e\right)B^{-1}\in\rho\left(I_{K_{\mathfrak{p}}}\right)\) and \(\mathscr{C}_{s}^{e}=\left\{A^{e}:A\in\mathscr{C}_{s}\right\}\subseteq\rho \left(I_{K_{\mathfrak{p}}}\right)\). If \(T\operatorname{\mathrm{diagexp}}\left(0,e\right)T^{-1}=\operatorname{\mathrm{ diagexp}}\left(e,0\right)\), by the same argument, we can show that \(\mathscr{C}_{s}^{e}\subseteq\rho\left(I_{K_{\mathfrak{p}}}\right)\). Hence, since \(B^{2}\in\mathscr{C}_{s}\) and \(B^{2\left(\ell-1\right)}=I_{2}\), we have that
\[\left(\frac{\ell-1}{\gcd\left(\ell-1,e\right)}\right)^{2}=\left|\mathscr{C}_{s }^{e}\right|\ \big{|}\ \big{|}\rho\left(I_{K_{\mathfrak{p}}}\right)\big{|}=\left|B\right|\ \big{|}2\left(\ell-1\right),\]
which implies that \(\ell-1\) divides \(2e^{2}\). Thus, \(\ell-1\) divides \(32\) or \(72\). Since \(\ell-1\equiv 2\pmod{4}\), we have that \(\ell\in\left\{3,7,19\right\}\). But this contradicts that \(\ell\geq 11\) and \(\ell\neq 19\). Thus, \(\ell=19\) or \(\rho\left(I_{K_{\mathfrak{p}}}\right)\subseteq\mathscr{C}_{s}\). If \(\ell=19\), then \(3\ |\ \frac{\ell-1}{\gcd\left(\ell-1,e\right)}=\left|\operatorname{\mathrm{ diagexp}}\left(0,e\right)\right|\ \big{|}\ \left|\Delta\right|\). If \(\rho\left(I_{K_{\mathfrak{p}}}\right)\subseteq\mathscr{C}_{s}\), then we have that
\[2\ |\ \ell-1=\left|\mathbb{F}_{\ell}^{\times}\right|=\left|\left(\det\circ \rho\right)\left(I_{K_{\mathfrak{p}}}\right)\right|\ \big{|}\ \left|\Delta\right|,\]
since the assumption (i) implies that \(\left(\det\circ\rho\right)\left(I_{K_{\mathfrak{p}}}\right)=\mathbb{F}_{\ell}^ {\times}\). This completes the proof.
Proof of Proposition 1.3.: There is a constant \(N_{K}^{\prime}\) depending only on \(K\) such that any prime \(\ell>N_{K}^{\prime}\) satisfies the conditions (i) and (ii) of Proposition 2.10. Then, the proof follows from Proposition 2.9 and Proposition 2.10.
Now, we characterize when \(G\left(\mathcal{B}\right)\subseteq\mathscr{B}\) up to conjugacy for a sufficiently large prime \(\ell\) if \(K\) has no RCM. We recall the following result which gives a lower bound of such a prime \(\ell\) depending on \(K\).
**Theorem 2.11** ([12, Theorem 1]).: _Let \(K\) be a number field. Then, there exists a finite set \(S_{K}\) of primes depending only on \(K\) such that for a prime \(\ell\not\in S_{K}\), and an elliptic curve \(E/K\) for which \(E\left[\ell\right]\otimes\overline{\mathbb{F}}_{\ell}\) is reducible with associated character \(\psi\) of degree \(1\), one of the following holds:_
1. _There exists an elliptic curve_ \(\mathscr{E}/K\) _whose CM field is contained in_ \(K\)_, with an_ \(\ell\)_-adic degree_ \(1\) _associated character whose mod-_\(\ell\) _reduction_ \(\phi\) _satisfies:_ \[\psi^{12}=\phi^{12}\]
2. _GRH fails for_ \(K\left[\sqrt{-\ell}\right]\) _and_ \(\psi^{12}=\operatorname{cyc}_{\ell}^{6}\)_, where_ \(\operatorname{cyc}_{\ell}\) _denotes an_ \(\ell\)_-cyclotomic character of_ \(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\)_._
_(Refer to [12, SS1] for the associated characters of degree \(d\).)_
**Proposition 2.12**.: _Let \(K\) be a number field without RCM, \(S_{K}\) a set of primes given in Theorem 2.11, and \(\ell\geq 11\) a prime satisfying the following three conditions;_
1. \(\ell\not\in S_{K}\)_,_
2. \(\ell\) _is unramified in_ \(K\)_, and_
3. \(\mathbb{Q}\left(\zeta_{\ell}\right)\cap K=\mathbb{Q}\)_._
_Then, for any elliptic curve \(E\) over \(K\) with a point \(R\in E\left[\ell\right]-\left\{O\right\}\) such that \(2,3\nmid[K\left(R\right):K]\), the followings hold:_
1. _There is a basis_ \(\mathcal{B}\) _of_ \(E\left[\ell\right]\) _such that_ \(G:=G\left(\mathcal{B}\right)\) _is_ \(\left\langle\Delta,U\right\rangle\) _where_ \(\Delta:=G\cap\mathscr{C}_{s}\)_,_
2. \(\Delta\) _is either_ \(\Delta_{1}:=\operatorname{diagexp}\left(\left\langle\left(2\tau,2\tau\right), \left(0,\frac{\ell-1}{2\tau}\right)\right\rangle\right)\) _or_ \(\Delta_{2}:=\operatorname{diagexp}\left(\left\langle\left(2\tau,2\tau\right), \ \left(\frac{\ell-1}{2\tau},0\right)\right\rangle\right),\) _where the constant_ \(\tau\) _is defined by_ \[\tau=\begin{cases}1&\text{ if }\ell\equiv 2\pmod{3},\\ 3&\text{ if }\ell\equiv 1\pmod{3},\end{cases}\]
3. \(\frac{\ell-1}{2\tau}\) _divides_ \(\left[K\left(S\right):K\right]\) _for any_ \(S\in E\left[\ell\right]-\left\{O\right\}\)_, and_
4. \(\ell\equiv 3\pmod{4}\)_,_ \(\ell\not\equiv 1\pmod{9}\)_, and_ \(\ell\in\mathcal{PDI}_{2}\left(K\right)\)_._
Proof.: First, we show that there exists a basis \(\mathcal{B}\) of \(E\left[\ell\right]\) such that \(G:=G\left(\mathcal{B}\right)\) is \(\Delta\) or \(\left\langle\Delta,U\right\rangle\). By Proposition 1.3, there exists a basis \(\mathcal{B}^{\prime}\) for \(E\left[\ell\right]\) such that \(H:=G\left(\mathcal{B}^{\prime}\right)\subseteq\mathscr{B}\).
If \(\ell\nmid\left|H\right|\), then we note that \(A\in H\) has only one eigenvalue \(1\) if and only if \(A=I_{2}\). Also, since we can show that for any \(A,B\in H\subseteq\mathscr{B}\), \(ABA^{-1}B^{-1}\) has only one eigenvalue \(1\) by direct calculation, \(H\) is abelian. Since \(\ell\nmid\left|H\right|\) and all matrices in \(H\) have rational eigenvalues, all elements of \(H\) are simultaneously diagonalizable over \(\mathbb{F}_{\ell}\). So \(T^{-1}HT\subseteq\mathscr{C}_{s}\) for some \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\), i.e., \(G:=G\left(\mathcal{B}\right)\subseteq\mathscr{C}_{s}\) for some basis \(\mathcal{B}\) for \(E\left[\ell\right]\).
If \(\ell\ \bigm{|}\ \left|H\right|\), then \(H\) contains a subgroup of order \(\ell\) by Cauchy's theorem. Since \(\mathscr{B}\) has the unique subgroup \(\left\langle U\right\rangle\) of order \(\ell\), we have that \(U\in H\). Therefore, \(\begin{pmatrix}a&0\\ 0&d\end{pmatrix}=\begin{pmatrix}a&b\\ 0&d\end{pmatrix}U^{-ba^{-1}}\in H\) for any \(\begin{pmatrix}a&b\\ 0&d\end{pmatrix}\in H\). Hence, for \(G:=H\), we have that \(G=\left\langle G\cap\mathscr{C}_{s},U\right\rangle\).
Therefore, we have shown that for some basis \(\mathcal{B}=\left\{P,Q\right\}\) for \(E\left[\ell\right]\),
\[\text{the group }G:=G(\mathcal{B})\text{ is }\Delta\text{ or }\left\langle\Delta,U\right\rangle. \tag{3}\]
Next, we prove (b)-(d). To prove (b), first, we recall the situation so that we can apply Theorem 2.11 under the assumption (i): The representation \(E\left[\ell\right]\otimes\overline{\mathbb{F}_{\ell}}\) of \(\mathcal{G}:=\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\) has exactly two associated characters \(\psi_{1}\) and \(\psi_{2}\) of degree \(1\). They satisfy that \(\rho\left(\sigma\right)=\left(\begin{smallmatrix}\psi_{1}\left(\sigma\right)&* \\ 0&\psi_{2}\left(\sigma\right)\end{smallmatrix}\right)\) for all \(\sigma\in\mathcal{G}\) where \(\rho=\rho_{\mathcal{B}}\). Since \(K\) has no RCM, the second case of Theorem 2.11 holds, which implies that for each \(i=1,2\), \(\psi_{i}^{12}\left(\sigma\right)=\operatorname{cyc}_{\ell}^{6}\left(\sigma \right)=\det\left(\rho\left(\sigma\right)\right)^{6}\) for all \(\sigma\in\mathcal{G}\), and equivalently,
\[\text{for all }\operatorname{diagexp}\left(u,t\right)\in\Delta,\quad 12u\equiv 12t \equiv 6\left(u+t\right)\pmod{\ell-1}. \tag{4}\]
Since \(\mathbb{F}_{\ell}^{\times}\) is a cyclic group generated by \(\alpha\), \(\Delta^{\circ}:=\Delta\cap G^{\circ}\) is generated by a single matrix, say \(\operatorname{diagexp}\left(\frac{\ell-1}{m},\frac{1-\ell}{m}\right)\) for some positive divisor \(m\) of \(\ell-1\). Next, we claim that there exists \(A\in\Delta\) such that \(\det\left(A\right)=\alpha\). In fact, since \(\mathbb{Q}\left(\zeta_{\ell}\right)\cap K=\mathbb{Q}\) by (ii), \(\operatorname{Gal}\left(K\left(\zeta_{\ell}\right)/K\right)\cong\operatorname{ Gal}\left(\mathbb{Q}\left(\zeta_{\ell}\right)/\mathbb{Q}\right)\) and \(\operatorname{cyc}_{\ell}\left(\operatorname{Gal}\left(K\left(\zeta_{\ell}\right) \right)/K\right)=\operatorname{cyc}_{\ell}\left(\operatorname{Gal}\left( \mathbb{Q}\left(\zeta_{\ell}\right)/\mathbb{Q}\right)\right)=\mathbb{F}_{\ell} ^{\times}=\left\langle\alpha\right\rangle\). Hence, there exists \(A^{\prime}=\begin{pmatrix}a&b\\ 0&d\end{pmatrix}\in G\) such that \(\det A^{\prime}=\alpha\). If \(U\in G\), then \(A:=A^{\prime}U^{-ba^{-1}}\in\Delta\) and \(\det\left(A\right)=\alpha\). If \(U\notin G\), then \(A:=A^{\prime}\in G=\Delta\) by (3). So we let \(A=\operatorname{diagexp}\left(1-t,t\right)\in\Delta\) for some \(t\in\mathbb{Z}/\left(\ell-1\right)\mathbb{Z}\). Then, since \(\operatorname{diagexp}\left(1-t,t\right)^{-a-b}\operatorname{diagexp}\left(a,b \right)\in\Delta^{\circ}=\operatorname{diagexp}\left(\left\langle\left(\frac{ \ell-1}{m},\frac{1-\ell}{m}\right)\right\rangle\right)\) for any \(\operatorname{diagexp}\left(a,b\right)\in\Delta\), we see that
\[\Delta=\operatorname{diagexp}\left(\left\langle\left(\frac{\ell-1}{m},\frac{1- \ell}{m}\right),(1-t,t)\right\rangle\right).\]
Next, we show that \(m=1\) and so \(\Delta=\operatorname{diagexp}\left(\left\langle\left(1-t,t\right)\right\rangle\right)\).
By (4) for \(\operatorname{diagexp}\left(\frac{\ell-1}{m},\frac{1-\ell}{m}\right)\), \(12\frac{\ell-1}{m}\equiv 0\pmod{\ell-1}\), and equivalently, \(m\mid 12\).
Recalling (3), if \(G=\Delta\), then we have that \(G_{01}=\{\left(\begin{smallmatrix}a&0\\ 0&1\end{smallmatrix}\right)\in G\}\), \(G_{10}=\left\{\left(\begin{smallmatrix}1&0\\ 0&d\end{smallmatrix}\right)\in G\right\}\), and \(G_{1k}=\{I_{2}\}\) for all non-zero \(k\in\mathbb{F}_{\ell}\). Therefore, we can get
\[\left[G:G_{01}\right]=\left|\left\{d\in\mathbb{F}_{\ell}^{\times}:\left( \begin{smallmatrix}a&0\\ 0&d\end{smallmatrix}\right)\in\Delta\text{ for some }a\in\mathbb{F}_{\ell}^{ \times}\right\}\right|,\]
\[\left[G:G_{1k}\right]=\left|G\right|,\text{ and }\left[G:G_{10}\right]=\left| \left\{a\in\mathbb{F}_{\ell}^{\times}:\left(\begin{smallmatrix}a&0\\ 0&d\end{smallmatrix}\right)\in\Delta\text{ for some }d\in\mathbb{F}_{\ell}^{ \times}\right\}\right|.\]
By (ii), we have that \(\ell-1\mid\left|G\right|=\left[G:G_{1k}\right]\) for each non-zero \(k\in\mathbb{F}_{\ell}\), so \(2\mid\left[G:G_{1k}\right]\). Moreover, since \(\operatorname{diagexp}\left(\frac{\ell-1}{m},\frac{1-\ell}{m}\right)\in \Delta=G\), \(m\) divides \(\left[G:G_{01}\right]\) and \(\left[G:G_{10}\right]\). Since the degree \(\left[K\left(R\right):K\right]\) is equal to one of \(\left[G:G_{01}\right]\), \(\left[G:G_{01}\right]\), or \(\left[G:G_{1k}\right]\) for some non-zero \(k\in\mathbb{F}_{\ell}\) by Lemma 2.3 and \(2\nmid\left[K\left(R\right):K\right]\) by (iii), we conclude that \(2\nmid m\). If \(\ell\equiv 1\pmod{3}\), then since a similar argument shows that \(3\nmid m\) since \(3\nmid\left[K\left(R\right):K\right]\) by (iii). If \(\ell\equiv 2\pmod{3}\), then clearly \(3\nmid m\) since \(m\) is a divisor of \(\ell-1\). Therefore, since \(m\mid 12\) but \(2,3\nmid m\), we conclude that \(m=1\) and \(\Delta=\operatorname{diagexp}\left(\left\langle\left(1-t,t\right)\right\rangle\right)\).
If \(G=\left\langle\Delta,U\right\rangle\), then \(G_{01}=\left\{\left(\begin{smallmatrix}a&b\\ 0&1\end{smallmatrix}\right)\in G\right\}\) and \(G_{1k^{\prime}}=\left\{\left(\begin{smallmatrix}1&k^{\prime}(1-d)\\ 0&d\end{smallmatrix}\right)\in G\right\}\) for all \(k^{\prime}\in\mathbb{F}_{\ell}\). Hence,
\[\left[G:G_{01}\right]=\left|\left\{d\in\mathbb{F}_{\ell}^{\times}:\left( \begin{smallmatrix}a&0\\ 0&d\end{smallmatrix}\right)\in\Delta\text{ for some }a\in\mathbb{F}_{\ell}^{ \times}\right\}\right|\text{ and }\]
\[\left[G:G_{1k^{\prime}}\right]=\ell\cdot\left|\left\{a\in\mathbb{F}_{\ell}^{ \times}:\left(\begin{smallmatrix}a&0\\ 0&d\end{smallmatrix}\right)\in\Delta\text{ for some }d\in\mathbb{F}_{\ell}^{ \times}\right\}\right|.\]
Since \(\ell\geq 5\), we can show that \(\Delta=\operatorname{diagexp}\left(\left\langle\left(1-t,t\right)\right\rangle\right)\) by the same argument.
Note that one of \(t\) or \(1-t\) is even modulo \(\ell-1\). Suppose that \(t\) is even. Recalling the indices \(\left[G:G_{1k^{\prime}}\right]\) and \(\left[G:G_{01}\right]\) described in the above, we have that for all \(k^{\prime}\in\mathbb{F}_{\ell}\), \(\left[G:G_{1k^{\prime}}\right]\) is divisible by \(\left|1-t\right|=\frac{\ell-1}{\gcd\left(1-t,\ell-1\right)}\) which is even since \(1-t\) is odd, and \(\left[G:G_{01}\right]=\left|t\right|\). Since the degree \(\left[K\left(R\right):K\right]\) is equal to \(\left[G:G_{01}\right]\) or \(\left[G:G_{1k^{\prime}}\right]\) for some \(k^{\prime}\in\mathbb{F}_{\ell}\) by Lemma 2.3 again and \(2\nmid\left[K\left(R\right):K\right]\) by (iii), we conclude that \(\left[K\left(R\right):K\right]=\left[G:G_{01}\right]=\left|t\right|\), which should not be divisible by \(2\) or \(3\).
By (4) for \(\operatorname{diagexp}\left(1-t,t\right)\in G\), we have \(12t\equiv 6\pmod{\ell-1}\), so \(6t\equiv 3\pmod{\frac{\ell-1}{2}}\). If \(\ell\equiv 1\pmod{3}\), then \(2t\equiv 1\pmod{\frac{\ell-1}{6}}\) so \(t\equiv\frac{\ell+5}{12}\pmod{\frac{\ell-1}{6}}\). Therefore,
\[\gcd\left(t,\ell-1\right)\mid\gcd\left(\ell+5,\ell-1\right)=\gcd\left(6,\ell- 1\right)=6.\]
Thus, since \(\left|t\right|=\frac{\ell-1}{\gcd\left(t,\ell-1\right)}\), \(\left|t\right|\) is divisible by \(\frac{\ell-1}{6}\) and \(\left|t\right|\) divides \(\ell-1\). Moreover, since \(2,3\nmid\left|t\right|\), we have that \(\left|t\right|=\frac{\ell-1}{6}\) and \(\gcd\left(6,\frac{\ell-1}{6}\right)=1\). Hence, \(\gcd\left(t,\ell-1\right)=6\) and there is an \(u\in\left(\mathbb{Z}/\left(\ell-1\right)\mathbb{Z}\right)^{\times}\) satisfying \(tu\equiv 6\pmod{\ell-1}\). Moreover, \(\Delta=\operatorname{diagexp}\left(\left\langle\left(1-t,t\right)\right\rangle \right)=\operatorname{diagexp}\left(\left\langle\left(u-6,6\right)\right\rangle\right)\). By (4) for \(\operatorname{diagexp}\left(u-6,6\right)\in G\), we have that \(u\equiv 12\pmod{\frac{\ell-1}{6}}\). Since \(\gcd\left(u,\ell-1\right)=1\), we have that \(2,3\nmid u\) and that \(u\equiv 12\pm\frac{\ell-1}{6}\pmod{\ell-1}\). Hence \(\left(u-6,6\right)=\left(6\pm\frac{\ell-1}{6},6\right)\) in \(\left(\mathbb{Z}/\left(\ell-1\right)\mathbb{Z}\right)^{2}\). Since \(\left\langle\left(6+\frac{\ell-1}{6},6\right)\right\rangle\subseteq\left\langle \left(6,6\right),\left(\frac{\ell-1}{6},0\right)\right\rangle\) and both groups have the same order \(\ell-1=\operatorname{lcm}\left(\frac{\ell-1}{6},6\right)\), we have that
\[\left\langle\left(6-\frac{\ell-1}{6},6\right)\right\rangle=\left\langle\left(6,6 \right),\left(\frac{\ell-1}{6},0\right)\right\rangle=\left\langle\left(6+\frac{ \ell-1}{6},6\right)\right\rangle.\]
Finally, \(\Delta=\operatorname{diagexp}\left(\left\langle\left(6,6\right),\left(\frac{ \ell-1}{6},0\right)\right\rangle\right)\). If \(\ell\equiv 2\pmod{3}\), we can show by a similar argument that \(\Delta=\operatorname{diagexp}\left(\left\langle\left(2,2\right),\left(\frac{ \ell-1}{2},0\right)\right\rangle\right)\). Therefore, in either case, \(\Delta=\Delta_{2}\) if \(t\) is even. If \(1-t\) is even, we can show that \(\Delta=\Delta_{1}\) similarly.
For (c), we note that \(\operatorname{diagexp}\left(27,2\tau\right)\in\Delta\) by (b), so \(\left|2\tau\right|=\frac{\ell-1}{2\tau}\) divides \(\left[G:G_{01}\right]\) and \(\left[G:G_{1k^{\prime}}\right]\) for any \(k^{\prime}\in\mathbb{F}_{\ell}\) recalling those indices. Thus, \(\frac{\ell-1}{2\tau}\) divides \(\left[K\left(S\right):K\right]\) for any \(S\in E\left[\ell\right]-\left\{O\right\}\).
For (d), by (c) and the condition (iii), \(\frac{\ell-1}{2\tau}\) should not be divisible by \(2\) or \(3\), which implies that \(\ell\equiv 3\pmod{4}\) and \(\ell\nmid 1\pmod{9}\). Moreover, the isogeny character of \(\left(E,\left\langle Q\right\rangle\right)\in Y_{0}\left(\ell\right)\)
referring to Theorem 1.6 is \(\psi_{2}\), and by (b), \(\psi_{2}\) satisfies \(\psi_{2}^{6}=\operatorname{cyc}_{\ell}^{12}\) for \(\mathcal{B}=\left\{P,Q\right\}\) in (3). Hence \(\ell\in\mathcal{PDI}_{2}\left(K\right)\).
To complete the proof of (a), by (3), it is enough to show that \(U\in G\). Suppose \(U\not\in G\). Then by (3), \(G=\Delta\) and \(\ell\nmid\left|G\right|\). Hence, [3, Theorem 2] implies that there exist \(e\in\left\{1,2,3,4,6\right\}\) and \(T\in\operatorname{GL}_{2}\left(\mathbb{F}_{\ell}\right)\) such that either \(\operatorname{diagexp}\left(0,e\right)\in TGT^{-1}\) or \(\mathscr{C}_{ns}^{e}\subseteq TGT^{-1}\). If \(\operatorname{diagexp}\left(0,e\right)\in TGT^{-1}\), then \(e\in\left\langle\frac{\ell-1}{2\tau}\right\rangle\). So \(\ell-1\mid 2\tau e\). Therefore, \(\ell-1\) divides \(24\) or \(36\). Since \(\ell-1\equiv 2\pmod{4}\) and \(\ell\not\equiv 1\pmod{9}\) by (d), we have that \(\ell=3\) or \(7\), which contradicts that \(\ell\geq 11\). If \(TGT^{-1}\) contains \(\mathscr{C}_{ns}^{e}\), then
\[\frac{\ell^{2}-1}{e}=\left|\mathscr{C}_{ns}^{e}\right|\ \left|G\right|=\ell-1,\]
so \(\ell+1\mid e\), which contradicts that \(\ell\geq 11\) again. This completes the proof.
Proof of Proposition 1.8.: This follows from Proposition 2.12, by letting \(N_{K}=\max\left(S_{K}^{\prime}\right)\) where \(S_{K}^{\prime}:=\,S_{K}\cup\left\{\text{a prime }\ell\,:\,\ell\text{ is ramified in }K,\text{ or }\mathbb{Q}(\zeta_{\ell})\cap K\,\neq \mathbb{Q}\right\}\) which is a finite set.
**Remark 2.13**.: In Proposition 2.12(b), we note that \(\Delta_{1}\) and \(\Delta_{2}\) do appear mutually intensively. More precisely, for an elliptic curve \(E/K\) and a basis \(\mathcal{B}=\left\{P,Q\right\}\) of \(E\left[\ell\right]\), if \(G:=G\left(\mathcal{B}\right)=\left\langle\Delta,U\right\rangle\), then we can show there exist an elliptic curve \(E^{\prime}/K\) and a basis \(\mathcal{B}^{\prime}\) of \(E^{\prime}\left[\ell\right]\) such that \(G^{\prime}:=G\left(\mathcal{B}^{\prime}\right)=\left\langle\Delta_{\text{ flip}},U\right\rangle\) where \(\Delta_{\text{flip}}:=\left\{\operatorname{diag}\left(d,a\right): \operatorname{diag}\left(a,d\right)\in\Delta\right\}\) as follows:
Note that \(\Delta_{1}\) and \(\Delta_{2}\) are the flips of each other. Since the subspace \(\left\langle Q\right\rangle\subseteq E\left[\ell\right]\) is \(\operatorname{Gal}\left(K\left(E\left[\ell\right]\right)/K\right)\)-invariant, there is a \(K\)-rational isogeny \(\alpha:E\to E^{\prime}\) with kernel \(\left\langle Q\right\rangle\). We denote the dual isogeny of \(\alpha\) by \(\widehat{\alpha}\). Then, \(\widehat{\alpha}\circ\alpha=\left[\ell\right]\) and \(\ker\alpha=\left\langle Q\right\rangle\), so the point \(Q^{\prime}:=\alpha\left(P\right)\in E^{\prime}\left[\ell\right]\) is non-zero and it is in \(\ker\widehat{\alpha}\). Since the order of \(\widehat{\alpha}\) is \(\ell=\#\ker\alpha\), the kernel \(\widehat{\alpha}\) is generated by a \(Q^{\prime}\). Since \(\#\ker\widehat{\alpha}\) is of order \(\ell\), \(Q^{\prime}\) generates \(\ker\widehat{\alpha}\). Similarly, for any \(P^{\prime}\in E^{\prime}\left[\ell\right]-\left\langle Q^{\prime}\right\rangle\), \(\widehat{\alpha}\left(P^{\prime}\right)\) generates \(\ker\alpha\). Replacing \(P^{\prime}\) by a multiple of \(P^{\prime}\) of an appropriate scalar, we may assume that \(\widehat{\alpha}\left(P^{\prime}\right)=Q\). Then, the set \(\mathcal{B}^{\prime}=\left\{P^{\prime},Q^{\prime}\right\}\) is a basis of \(E^{\prime}\left[\ell\right]\). So far, we have shown that the matrix representation of the linear transformations \(\alpha:E\left[\ell\right]\to E^{\prime}\left[\ell\right]\) and \(\widehat{\alpha}:E^{\prime}\left[\ell\right]\to E\left[\ell\right]\) with respect to the bases \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\), respectively are both equal to \(\left(\begin{smallmatrix}0&1\\ 0&0\end{smallmatrix}\right)\). We may consider \(\sigma\in\operatorname{Gal}\left(\overline{K}/K\right)\) as automorphisms on the \(\mathbb{F}_{\ell}\)-vector spaces \(E\left[\ell\right]\) and \(E^{\prime}\left[\ell\right]\). We denote by \(\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\) and \(\left(\begin{smallmatrix}a^{\prime}&b^{\prime}\\ c^{\prime}&d^{\prime}\end{smallmatrix}\right)\) the matrix representations of automorphisms \(\sigma\) on \(E\left[\ell\right]\) and \(E^{\prime}\left[\ell\right]\) with respect to the bases \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\), respectively. Since \(\alpha\) and \(\sigma\) commute, we have the following commutative diagram,
and we have that
\[\begin{pmatrix}0&a\\ 0&c\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}0&1\\ 0&0\end{pmatrix}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}a^{\prime}&b^{\prime}\\ c^{\prime}&d^{\prime}\end{pmatrix}=\begin{pmatrix}c^{\prime}&d^{\prime}\\ 0&0\end{pmatrix}.\]
Thus, \(a=d^{\prime}\) and \(c=c^{\prime}=0\). Replacing \(\alpha\) by \(\widehat{\alpha}\), the same argument shows that \(a^{\prime}=d\). For \(G^{\prime}:=G\left(\mathcal{B}^{\prime}\right)\), we see that \(G^{\prime}\) is contained in \(\left\langle\Delta_{\text{flip}},U\right\rangle\). Moreover, as in the end of the proof of Proposition 2.12 we can show that \(U\in G^{\prime}\). Therefore, we conclude that \(G^{\prime}=\left\langle\Delta_{\text{flip}},U\right\rangle\).
## 3. The proofs of the main theorems
In this section, we give the proofs of our main theorems, Theorem 1.2 and Theorem 1.9. We will give a prime \(p_{K}\) depending on \(K\), more specifically, depending on \(N_{K}\) given in Proposition 1.8, \(\mathcal{R}\left(K\right)\) given in (2), and Merel's bound on the torsion points over \(K\) ([14]), and we prove Theorem 1.2 by dividing it into two cases; for primes \(\ell\leq p_{K}\) and for primes \(\ell>p_{K}\). For the latter case, we prove that under the assumption of the degree of the extension \(L\) over \(K\), the \(\ell\)-torsion subgroups over both \(K\) and \(L\) are the same as the trivial group by applying Proposition 1.8, and for the former case, we prove that \(\ell^{\infty}\)-torsion parts of the elliptic curves, upon base change, do not grow by applying Proposition 3.5 after proving it below.
We recall Merel's theorem ([14]) which gives an uniform upper bound of orders of torsion points over \(K\) of an elliptic curve \(E/K\), and [7, Lemma 3.2] which gives an equivalent condition for the \(N\)-torsion subgroup of \(E\left(K\right)\) to contain a certain type of subgroups.
**Theorem 3.1** ([14]).: _Let \(K\) be a number field. There is a positive integral constant \(M_{K}\) satisfying \(E\left(K\right)_{\mathrm{tors}}=E\left(K\right)\left[M_{K}\right]\) for all elliptic curves \(E/K\)._
**Definition 3.2** (the Merel constant).: _For a number field \(K\), we let \(M\left(K\right)\) be the smallest positive constant among \(M_{K}\) given in Theorem 3.1, and call it the Merel constant._
We start by proving the following basic lemma.
**Lemma 3.3**.: _For two abelian groups \(A\subseteq B\) and a prime \(\ell\), if \(B[\ell^{n^{\prime}}]=A[\ell^{n^{\prime}}]=A\left[\ell^{n}\right]\) for some non-negative integers \(n^{\prime}\) and \(n\) such that \(n^{\prime}>n\), then \(B\left[\ell^{\infty}\right]=A\left[\ell^{\infty}\right]\)._
Proof.: Since
\[B\left[\ell^{n}\right]=\left(B[\ell^{n^{\prime}}]\right)\left[\ell^{n}\right] =\left(A[\ell^{n^{\prime}}]\right)\left[\ell^{n}\right]\subseteq A\left[\ell ^{\infty}\right],\]
it is enough to show that \(B\left[\ell^{\infty}\right]=B\left[\ell^{n}\right]\). If \(B\left[\ell^{\infty}\right]\supsetneq B\left[\ell^{n}\right]\), then there exists \(b\in B\left[\ell^{\infty}\right]-B\left[\ell^{n}\right]\) and if \(m\) is the smallest non-negative integer \(m\) such that \(\ell^{m}b=0\) then \(m>n\). Hence,
\[\ell^{m-n-1}b\in B\left[\ell^{n+1}\right]=\left(B[\ell^{n^{\prime}}]\right) \left[\ell^{n+1}\right]=\left(A[\ell^{n^{\prime}}]\right)\left[\ell^{n+1} \right]=\left(A\left[\ell^{n}\right]\right)\left[\ell^{n+1}\right]=A\left[\ell ^{n}\right],\]
thus, \(\ell^{m-1}b=0\), which contradicts the minimality of \(m\).
Now we give a sufficient condition on the extension degree over \(K\) over which a given elliptic curve \(E/K\) has the growth of \(\ell^{\infty}\)-torsion subgroups for finitely many primes \(\ell\), simultaneously. We restate the following lemma which will be used to establish it.
**Lemma 3.4** ([7, Lemma 3.2]).: _Let \(K\) be a number field and \(E/K\) be an elliptic curve over \(K\). For positive integers \(m\), \(n\), and \(N\) satisfying \(m\mid n\mid N\),_
\[E\left(K\right)\left[N\right]\supseteq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/ n\mathbb{Z}\text{ if and only if for some basis }\mathcal{B}\text{ of }E\left[N\right],G\left(\mathcal{B}\right)\text{ is contained in}\]
\[\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right):a\equiv 1,b \equiv 0\pmod{m},\text{ and }c\equiv 0,d\equiv 1\pmod{n}\right\}.\]
**Proposition 3.5**.: _Let \(K\) be a number field, \(E/K\) an elliptic curve over \(K\), and \(p\) a prime which is greater than or equal to the maximal prime divisor of the Merel constant \(M\left(K\right)\). Let \(d\) be a positive integer whose minimal prime divisor is greater than \(p\). Then, for any extension \(L/K\) with \(\left[L:K\right]=d\) and for any prime \(\ell\leq p\),_
\[E\left(L\right)\left[\ell^{\infty}\right]=E\left(K\right)\left[\ell^{\infty} \right].\]
Proof.: Let \(N:=M\left(K\right)\)\(\prod\limits_{\text{a prime }\ell\leq p}\ell\). The extension degree \(\left[K\left(E\left[N\right]\right):K\right]\) divides the order \(\left|\operatorname{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right)\right|\). By the Chinese remainder theorem, \(\left|\operatorname{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right)\right|=\prod \nolimits_{i}\left|\operatorname{GL}_{2}\left(\mathbb{Z}/\ell_{i}^{e_{i}} \mathbb{Z}\right)\right|\) where \(\prod_{i}\ell_{i}^{e_{i}}\) is a prime factorization of \(N\). For any prime \(q\) and any positive integer \(e\), the natural projection \(\pi:\operatorname{GL}_{2}\left(\mathbb{Z}/q^{e}\mathbb{Z}\right)\to \operatorname{GL}_{2}\left(\mathbb{F}_{q}\right)\) is surjective. Thus,
\[\left|\operatorname{GL}_{2}\left(\mathbb{Z}/q^{e}\mathbb{Z}\right)\right|= \left|\operatorname{GL}_{2}\left(\mathbb{F}_{q}\right)\right|\left|\ker\pi \right|=\left(q^{2}-1\right)\left(q^{2}-q\right)q^{4e-4}=q^{4e}\left(1-q^{-2} \right)\left(1-q^{-1}\right),\]
and so we have that
\[\left|\operatorname{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right)\right|=N^{4} \prod\limits_{\text{a prime }\ell\leq p}\left(1-\ell^{-2}\right)\left(1-\ell^{-1}\right).\]
We note that every prime divisor of \(\left|\operatorname{GL}_{2}\left(\mathbb{Z}/N\mathbb{Z}\right)\right|\) is less than or equal to \(p\). Therefore, \(\left[K\left(E\left[N\right]\right):K\right]\) and \(\left[L:K\right]\) are relatively prime, so \(\operatorname{Gal}\left(L\left(E\left[N\right]\right)/L\right)\cong \operatorname{Gal}\left(K\left(E\left[N\right]\right)/K\right)\) since \(K\left(E\left[N\right]\right)\cap L=K\). For any positive divisors \(m\) and \(n\) of \(N\) such that \(m\mid n\), we have that \(E\left(L\right)\left[N\right]\supseteq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/ n\mathbb{Z}\) if and only if \(E\left(K\right)\left[N\right]\supseteq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/ n\mathbb{Z}\) by Lemma 3.4, i.e., \(E\left(L\right)\left[N\right]=E\left(K\right)\left[N\right]\). Then, this implies that \(E\left(K\right)\left[M\left(K\right)\right]=E\left(K\right)\left[N\right]=E \left(L\right)\left[N\right]\) recalling that the Merel constant \(M\left(K\right)\) divides \(N\). Since \(E\left(K\right)_{\text{tors}}\subseteq E\left(L\right)_{\text{tors}}\), Lemma 3.3 completes the proof.
### The proofs of our main theorems
Finally, we prove Theorem 1.9 and Theorem 1.2.
Proof of Theorem 1.9.: If \(\mathcal{R}\left(K\right)\) in (2) is finite, we let
\[p_{K}=\max\left(\mathcal{R}\left(K\right)\cup\left\{\text{a prime }p:p\mid M \left(K\right)\cdot N_{K}\right\}\right),\]
where \(N_{K}\) is given in Proposition 1.8.
Let \(d\) be a positive integer whose minimal prime divisor is greater than \(p_{K}\) and let \(L\) be an extension of \(K\) with \(\left[L:K\right]=d\).
For any prime \(\ell\leq p_{K}\), Proposition 3.5 implies that \(E\left(L\right)\left[\ell^{\infty}\right]=E\left(K\right)\left[\ell^{\infty}\right]\).
For any prime \(\ell>p_{K}\), we note that \(p_{K}\geq 7\), since \(7\mid M(\mathbb{Q})\mid M\left(K\right)\) by Mazur's classification of torsion subgroups over \(\mathbb{Q}\) ([13, Theorem 2]), so \(\ell\geq 11\). Also, since \(p_{K}\geq 7\), we know that \(2,3\nmid\left[L:K\right]\). If \(\left[K\left(S\right):K\right]\) is divisible by \(2\) or \(3\) for any point \(S\in E\left[\ell\right]-\left\{O\right\}\), then \(E\left(L\right)\left[\ell\right]=\left\{O\right\}\) since \(2,3\nmid\left[L:K\right]\). If there exists a point \(R\in E\left[\ell\right]-\left\{O\right\}\) such that \(2,3\nmid\left[K\left(R\right):K\right]\), then Proposition 1.8 implies that there exists a prime \(q\in\mathcal{R}\left(K\right)\) which divides \(\left[K\left(T\right):K\right]\) for all \(T\in E\left[\ell\right]-\left\{O\right\}\). Since the minimal prime divisor of \(\left[L:K\right]\) is greater than \(p_{K}\) and \(p_{K}\geq q\) from our choice of \(p_{K}\), we conclude that \(E\left(L\right)\left[\ell\right]=\left\{O\right\}\). So in either case, we have shown that \(E\left(K\right)\left[\ell\right]=\left\{O\right\}=E\left(L\right)\left[\ell\right]\) for \(\ell>p_{K}\). Hence, \(E\left(K\right)_{\text{tors}}=E\left(L\right)_{\text{tors}}\).
Theorem 1.9 implies Theorem 1.2.
Proof of Theorem 1.2.: [15, Theorem 4] implies that if \(\mathcal{K}\) is a quadratic field without RCM, the set \(\mathcal{PDI}_{2}\left(\mathcal{K}\right)\) is finite, and so is \(\mathcal{R}\left(\mathcal{K}\right)\). Hence, it follows from Theorem 1.9.
Moreover, our results imply Genao's result [3, Theorem 3] as well.
**Corollary 3.6** ([3, Theorem 3]).: _Let \(K\) be a number field without RCM. Assuming GRH, the answer to Question 1 is affirmative._
Proof.: Under GRH, the set \(\mathcal{PDI}_{2}\left(K\right)\) is finite (see [15, Remark 8]), and so is \(\mathcal{R}\left(K\right)\). Hence, it follows from Theorem 1.9.
**Remark 3.7**.: As mentioned in Remark 1.5 and Remark 1.10, the finiteness of \(\mathcal{PDI}_{2}\left(K\right)\) is an essential condition for obtaining our results and concerning Question 2. On the other hand, we can see in the proofs of Theorem 1.2 and Corollary 3.6 that the finiteness of \(\mathcal{PDI}_{2}\left(K\right)\) implies the finiteness of \(\mathcal{R}\left(K\right)\). But our final remark is that the converse is unknown to be true.
|
2304.00648 | Improving RF-DNA Fingerprinting Performance in an Indoor Multipath
Environment Using Semi-Supervised Learning | The number of Internet of Things (IoT) deployments is expected to reach 75.4
billion by 2025. Roughly 70% of all IoT devices employ weak or no encryption;
thus, putting them and their connected infrastructure at risk of attack by
devices that are wrongly authenticated or not authenticated at all. A physical
layer security approach -- known as Specific Emitter Identification (SEI) --
has been proposed and is being pursued as a viable IoT security mechanism. SEI
is advantageous because it is a passive technique that exploits inherent and
distinct features that are unintentionally added to the signal by the IoT Radio
Frequency (RF) front-end. SEI's passive exploitation of unintentional signal
features removes any need to modify the IoT device, which makes it ideal for
existing and future IoT deployments. Despite the amount of SEI research
conducted, some challenges must be addressed to make SEI a viable IoT security
approach. One challenge is the extraction of SEI features from signals
collected under multipath fading conditions. Multipath corrupts the inherent
SEI features that are used to discriminate one IoT device from another; thus,
degrading authentication performance and increasing the chance of attack. This
work presents two semi-supervised Deep Learning (DL) equalization approaches
and compares their performance with the current state of the art. The two
approaches are the Conditional Generative Adversarial Network (CGAN) and Joint
Convolutional Auto-Encoder and Convolutional Neural Network (JCAECNN). Both
approaches learn the channel distribution to enable multipath correction while
simultaneously preserving the SEI exploited features. CGAN and JCAECNN
performance is assessed using a Rayleigh fading channel under degrading SNR, up
to thirty-two IoT devices, and two publicly available signal sets. The JCAECNN
improves SEI performance by 10% beyond that of the current state of the art. | Mohamed k. Fadul, Donald R. Reising, Lakmali P. Weerasena, T. Daniel Loveless, Mina Sartipi | 2023-04-02T23:01:13Z | http://arxiv.org/abs/2304.00648v1 | Improving RF-DNA Fingerprinting Performance In An Indoor Multipath Environment Using Semi-Supervised Learning
###### Abstract
The number of Internet of Things (IoT) deployments is expected to reach 75.4 billion by 2025. Roughly 70% of all IoT devices employ weak or no encryption; thus, putting them and their connected infrastructure at risk of attack by devices that are wrongly authenticated or not authenticated at all. A physical layer-based security approach-known as Specific Emitter Identification (SEI)-has been proposed and is being pursued as a viable IoT security mechanism. SEI is advantageous because it is a passive technique that exploits inherent and distinct features that are unintentionally imparted upon the signal during its formation and transmission within and by the IoT device's Radio Frequency (RF) front-end. SEI's passive exploitation of unintentional signal features removes any need to modify the IoT device, which makes it ideal for existing and future IoT deployments. Despite the amount of SEI research conducted there remains challenges that must be addressed to make SEI a viable IoT security approach. One of these challenges is the extraction of SEI features from signals collected under multipath fading conditions. Multipath corrupts the inherent SEI exploited features that are used to discriminate one IoT device from another; thus, degrading authentication performance and increasing the chance of attack. This work presents two semi-supervised Deep Learning (DL) equalization approaches and compares their performance with the current state of the art. The two approaches are the Conditional Generative Adversarial Network (CGAN) and Joint Convolutional Auto-Encoder and Convolutional Neural Network (JCAECNN). Both approaches learn the channel distribution to enable multipath correction while simultaneously preserving the SEI exploited signal features. CGAN and JCAECNN performance is assessed using a Rayleigh fading channel under degrading SNR, up to thirty-two IoT devices, as well as two publicly available signal sets. The JCAECNN improves SEI performance by 10% beyond that of the current state of the art.
Deep Learning, Specific Emitter Identification (SEI), Internet of Things (IoT), Security
## I Introduction
It is estimated that the number of operational Internet of Things (IoT) devices will reach 75.4 billion by 2025 [1]. Disturbingly, roughly seventy-percent of IoT devices employ weak or no encryption at all due to limited onboard resources (e.g., memory, computation, etc.), manufacturing costs that prohibit adoption, or difficulties associated with key management and implementation at scale [2, 3, 4]. The lack of or limited encryption creates a security vulnerability that leaves individual IoT devices and corresponding infrastructure open to exploitation by nefarious actors [5, 6, 7, 8, 9, 10, 11, 12]. Based upon this observation and attacks there is a critical need for an effective IoT security solution.
IoT deployments conform to the Open Systems Interconnection (OSI) model, which states that the Physical (PHY) layer is responsible for completing point-to-point bit stream communication and is considered the lowest and first layer [13]. By design, traditional security techniques are implemented at higher OSI layers (e.g., Data Link or Network); thus, they do not consider the PHY layer despite the fact that attackers must traverse it to conduct their attacks [14]. Specific Emitter Identification (SEI) has been proposed as a PHY layer IoT security solution [15, 16, 17]. SEI has been shown capable of achieving serial number discrimination by exploiting immutable, unintentional coloration that is imparted upon a signal during its formation and transmission. The coloration's source is attributed to the tolerated, manufacturing variation(s) present within the individual components, sub-systems, and systems that comprise an emitter's Radio Frequency (RF) front-end [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. SEI is advantageous because it provides a passive authentication mechanism that does not require modification of the end device (i.e., the coloration is generated as part of normal operations), which makes it ideally suited to IoT applications since there is no additional resource demands placed on the device being identified. However, since its inception-almost thirty years ago-very little attention has been paid to the performance of SEI within contentious operating conditions such as multipath. SEI's viability-as an IoT security solution-rests on it remaining effective even under degrading operating conditions.
RF-Distinct Native Attributes (RF-DNA) fingerprinting is an SEI implementation that extracts exploitable features from portions of the transmitted signal corresponding to fixed, known symbol sequences such as the IEEE 802.11a Wireless-Fidelity (Wi-Fi) preamble. Prior RF-DNA fingerprinting research uses supervised and unsupervised learning algorithms that rely on expert knowledge and handcrafted features to facilitate IoT device discrimination [14, 17, 21, 23, 24, 25, 26, 34, 35, 36, 37, 38, 39, 40, 41]. These algorithms may results in sub-optimal models that limit RF-DNA fingerprinting performance, especially under
degraded operating conditions such as time-varying or lower Signal-to-Noise Ratio (SNR) channels. Over the past five years the SEI research community has pursued Deep Learning (DL) due to its successful application in spectrum management [42, 43], modulation and emitter identification [44, 45, 46], system design [47, 48, 49, 50, 44, 45], as well as its ability to thrive under increasing amounts of information while removing the need for handcrafted features [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. Based upon the success of these published DL works, our work-presented herein-shows that DL can provide an effective RF-DNA fingerprinting process capable of facilitating serial number discrimination of IoT devices under degraded operating conditions. Specifically, we show that up to thirty-two IoT devices-that only differ in serial number-can be successfully discriminated from one another within an indoor multipath channel environment.
The remainder of this paper is organized as follows. Sect. II summarizes current publications that are most pertinent to our work as well as providing a more detailed description of our work's contributions. Sect. III provides descriptions of the; signal of interest, signal collection and detection processes, multipath channel modeling, Nelder-Mead (N-M) channel estimator and Minimum Mean Square Error (MMSE) equalizer, as well as the deep learning architectures used herein. The methodology for the developed DL-driven RF-DNA fingerprinting approach for indoor multipath environments is described in Sect. IV. The results are presented in Sect. V and the article concluded in Sect. VI.
## II Related Work and Contribution
DL-based SEI has been shown to provide an optimal end-to-end solution that results in superior performance, however most of the published works do not conduct SEI using RF signals that have undergone time-varying fading. In fact, only a small subset of DL-based SEI works have assessed its performance under multipath conditions and most can be categorized as using a channel that is: (i) static (i.e., the channel coefficients or characteristics never change) [35], or (ii) time-varying, but the environments specific characteristics (e.g., number of reflectors, line-of-sight present or not, type of fading, etc.) are unknown, unstated, or not disclosed by the authors [35, 46, 53, 62, 63, 74, 76, 81]. Regardless, most DL-based SEI efforts assume the selected, modified, or developed algorithm can learn discriminatory features directly from the received signals; thus, eliminating the need for channel estimation and correction prior to RF signal classification. The fact is that the time-varying nature of multipath fading channels impedes or significantly hinders the DL algorithms' ability to learn discriminating signal features that are invariant to multipath fading.
The work in [41] is the exception to this trend in that the authors: (i) adopt the IEEE 802.11a Wi-Fi indoor, channel model and state the specific values used to configure it [86] as well as (ii) perform traditional (i.e., not DL-based) multipath channel estimation and correction prior to performing SEI using a Convolutional Auto-Encoder (CAE) initialized Convolutional Neural Network (CNN). The authors' use of traditional estimation and correction approaches results in sub-optimal SEI performance due to errors in the estimated channel coefficients and a dependence on knowing the channels' statistics. Lastly, the authors assess their CAE-CNN SEI approach using only four emitters, which does not reflect typical IoT deployments consisting of tens to hundreds of devices.
Our work-presented in detail in the remainder of this paper-advances DL-based SEI's current state of the art through the following contributions:
1. Combining label embedding with a Conditional Generative Adversarial Network (CGAN) to efficiently learn each emitter's conditional feature distribution.
2. Augmenting discriminatory feature learning through the use of the rectangular and polar signal representation [87].
3. Introduces and assesses two semi-supervised learning equalization approaches that facilitate superior CNN-based SEI under Rayleigh fading and degrading SNR conditions.
4. Introduces and assesses a scalable, semi-supervised learning architecture that jointly develops CAE-based generative models and a CNN-based discriminative model to correct for Rayleigh fading while preserving SEI discriminative features. This joint architecture-designated herein as JCAECNN-decomposes the multipath signal into individual scaled and delayed versions of the original transmitted signal prior to CNN classification.
5. Improving our JCAECNN's SEI performance through the use of exponentially decaying loss function weights.
6. SEI performance assessment of the CGAN and JCAECNN architectures as four, eight, sixteen, or thirty-two emitters communicate over a Rayleigh fading channel.
7. JCAECNN SEI performance assessment using the public signal sets generated by the authors of [88, 89]. These results serve as a benchmark to permit comparative assessment with current and future publications.
8. Directly compares the CGAN and JCAECNN architectures' SEI performance with our prior work in [38, 41].
These contributions result in an average percent correct classification performance of 94.35% or better for SNR values of 9 dB or higher, Rayleigh fading channels comprised of five reflections/paths and an IoT deployment consisting of sixteen devices.
## III Background
### _Signal of Interest_
This work makes use of IoT emitters that communicate using the IEEE 802.11a Wi-Fi protocol. The reasons for choosing the IEEE 802.11a protocol for the signal sub-layer are as follows: (i) 802.11a is based on Orthogonal Frequency Division Multiplexing (OFDM) signal, which is used in multiple wireless communication standards such as 802.11ac, 802.11ad, 802.11ax, Long Term Evolution (LTE), and Worldwide Interoperability for Microwave Access (WiMAX) [90], (ii) multiple research efforts within the SEI community have demonstrated success using 802.11a signals [18, 24, 27, 33, 34, 36, 37, 38, 39, 46, 56, 85, 87, 91], (iii) availability of the same data set used in our previous publications [36, 37, 38, 41, 85, 87] to facilitate comparative assessments, and (iV) 802.11a Wi-Fi has been adopted as an IoT communications protocol [92]. Consistent with our prior publications, this work performs SEI by extracting RF-DNA fingerprints from the IEEE 802.11a Wi-Fi preamble, which comprises the first 16 \(\mu\)s of every transmission [36, 37, 38, 41]. Use of the 802.11a preamble is ideal, because it is the portion of the signal used by the receiver to perform channel equalization. An 802.11a preamble consists of ten-designated \(t_{1}\) through \(t_{10}\)-Short Training Symbols (STS), a Guard Interval (GI), and two Long Training Symbols (LTS) that are designated \(T_{1}\) and \(T_{2}\) in Fig. 1[93].
### _Signal Collection & Detection_
The primary data set used in this work is comprised of 802.11a Wi-Fi signals transmitted by \(N_{D}=4\) Cisco AIRCB21G-A-K9 Wi-Fi emitters and collected using an Agilent E3238S-based spectrum analyzer. The spectrum analyzer can sample signals at rates up to 95 mega-samples per second (Msps), has an operating range from 20 MHz to 6 GHz, an RF bandwidth of 36 MHz, and a 12-bit analog-to-digital converter [94]. Upon collection, a total of \(N_{B}=2,000\) signals are selected for each of the \(N_{D}=4\) emitters using amplitude-based variance trajectory detection [23]. After that, each signal is filtered using a fourth order low-pass Butterworth filter with a cutoff frequency of 7.7 MHz. Following filtering, the signals are post-processed by correcting for carrier frequency offset and downsampled to 20 MHz [39].
### _Multipath Channel Modeling_
Multipath is a major concern for communications systems operating in indoor environments. It degrades system performance by limiting the receiver's ability to correctly recover the transmitted message. This degradation is due to multiple copies of the transmitted signal-associated with different delays and attenuation values-destructively interfere with one another at the receiver's antenna(s). These copies are reflections of the transmitted signal off of objects located throughout the propagation environment [86]. The combination of these reflected copies-at the receiver-randomly shifts the carrier frequency between the transmitter and receiver as well as changes the signal strength over short time intervals [86]. For the work presented in this article, the indoor, multipath channels are modeled using a Rayleigh distribution for two reasons: (i) this distribution is used to assess Wi-Fi modulation performance by the IEEE 802.11 working group [86], and (ii) prior SEI publications have used or assessed system performance using the same distribution [36, 37, 38, 41, 53, 74, 76, 81].
In multipath fading, a single reflection (a.k.a., path) is quantified using coefficient and a corresponding time delay known as a delay spread [86]. For the case of Rayleigh fading,
Fig. 1: Structure of the 16 \(\mu\)s duration preamble that is present at the start of every IEEE 802.11a Wi-Fi frame [93].
a Tap Delay Line (TDL) is used to mathematically describe the channel. In a TDL, each "tap" represents a single coefficient and delay. For each path \(k\), the coefficient is represented using a circularly symmetric complex Gaussian random variable,
\[\alpha_{k}=A+jB, \tag{1}\]
where \(k=1,\ldots,L\) for a total number of \(L\) paths comprising the channel, and \(\sigma^{2}\) is the variance of the zero mean independent and identically distributed Gaussian random variables \(A\) and \(B\)[86]. If the delay spread's Root-Mean-Squared (RMS) is \(T_{\text{r}}\) and the sampling period is \(T_{s}\), then the variance can be defined as,
\[\sigma^{2}=\frac{\sigma_{k}^{2}}{2}=\frac{1}{2}\left\{\left[1-\exp\left(\frac{ -T_{s}}{T_{\text{r}}}\right)\right]\exp\left(\frac{-kT_{s}}{T_{\text{r}}} \right)\right\}. \tag{2}\]
Finally, the TDL for a multipath environment consisting of \(L\) total reflecting paths is,
\[h(t)=\sum_{k=1}^{L}\alpha_{k}\delta(t-\tau_{k}T_{s}), \tag{3}\]
where \(\alpha_{k}\) and \(\tau_{k}\) are the coefficient and delay spread associated with the \(k^{\text{th}}\) path [86, 95]. A received signal with multipath is generated by,
\[r(t)=x(t)*h(t)+n(t), \tag{4}\]
where \(x(t)\) is a collected 802.11a Wi-Fi signal, \(h(t)\) is the TDL from (3), \(n(t)\) is complex, white Gaussian noise, and \(*\) denotes the convolution operation. The noise \(n(t)\) is filtered, scaled, and added to the result of \(x(t)*h(t)\) to generate multipath signals with SNR values from 9 dB to 30 dB in increments of 3 dB between consecutive values.
### _Traditional Channel Estimations & Equalization_
This section explains the Nelder-Mead (N-M) estimator and MMSE equalizer, which are used here to facilitate comparative assessment between our results in Sect. V and those presented in our published work [41].
#### Iii-D1 Nelder-Mead Channel Estimator
The N-M estimator is constructed using the N-M simplex algorithm, which minimizes unconstrained optimization problems using a direct search approach [96]. The N-M simplex algorithm iteratively determines a \(d\)-variable, non-linear function's minimum solution using only function values and four defined operations. The N-M simplex algorithm is computationally efficient, because it does not require computation of the function derivatives [97]. At the start of iteration \(j\), a \(d+1\) vertices simplex is defined, one or more of four operations-reflection, expansion, contraction, and shrinkage-are performed to calculate one or more new points, and the function's values calculated using the new point(s). If the calculated function values at vertices given by \(x_{1}\) through \(xd+1\) satisfy certain conditions, then the new point(s) replace the worst point(s) in the simplex that is denoted as \(x_{d+1}\)[96, 97]. If none of the first three operations' conditions are satisfied, then a new set of points-\(v2\) through \(vd+1\)-are calculated using the shrinkage operation using the following formula [97],
\[v_{i}=x_{i}+\varphi(x_{i}-x_{1}), \tag{5}\]
where \(1<i\leq d+1\), and the new simplex for the next iteration is \((x_{1},v_{2},\cdots v_{d+1})\)[97]. The N-M simplex algorithm stops when the calculated function value-at iteration \(j\)-satisfies a certain termination condition or conditions. The termination conditions used in estimating the Rayleigh fading channel's coefficients are the same as those used in [36, 37, 38], which are given by the following formulas,
\[\frac{1}{d}\sum_{i=1}^{d+1}[f(x_{i})-\bar{f}]^{2}<\epsilon_{1}, \tag{6}\]
\[\frac{1}{d}\sum_{i=1}^{d}\left\|x_{i}^{j}-x_{i}^{j+1}\right\|^{2}<\epsilon_{2}, \tag{7}\]
where \(\bar{f}\) is the average of the function values \(f(x_{i})\), \(\left\|\bullet\right\|\) is the \(l_{2}\)-norm, and \(\epsilon_{1}\) and \(\epsilon_{2}\) are tolerances based on the function values \(f(x_{i})\) and points \(x_{i}\) respectively. Both conditions are checked at the end of each iteration. The function to be minimized is,
\[f(h)=\sum_{m\in T}\biggl{|}r(m)-\sum_{k=1}^{L}x(m-\tau_{k})\alpha_{k}\biggr{|} ^{2}. \tag{8}\]
The function \(f(h)\) can be described as a square error function between the received signal and the weighted and delayed copies of the transmitted signal \(x(m)\)[37]. Equation (8) is minimized in two parts because \(r(m)\) and \(x(m)\) are complex-valued while the N-M simplex algorithm solves real-valued functions [96, 97]. The two parts are created by separating the complex-valued signals into their real and imaginary components; thus, resulting in two real-valued, square error functions [36]. The N-M simplex algorithm is applied to each function separately to calculate the coefficient values \(\alpha_{k}\) and the the estimation error reduced by using the average of the estimates from the two square error functions as the final coefficient estimates [36]. As in [36, 37, 38, 38], five "candidate" preambles are randomly selected from each of \(N_{D}=4\) IEEE 802.11a emitters' set of signals to represent the transmitted signal \(x(m)\). Using the N-M simplex algorithm to solve equation (8)-for each candidate preambles-results in a total of \(N_{C}=20\) channel impulse response estimates. The residual power formula is given as [25],
\[\hat{h}(m)=\operatorname*{arg\,min}_{c}\left\{\sum_{m}\left|r(m)-\hat{h}_{c}(m)* x_{c}(m)\right|^{2}\right\}, \tag{9}\]
and used to select the best channel estimate. Where \(\hat{h}_{c}(m)\) is the estimated channel impulse response corresponding to the candidate preamble \(x_{c}(m)\), and \(1\leq c\leq N_{C}\)[37].
#### Iii-A2 MMSE Channel Equalizer
After estimating the channel impulse response, using the N-M estimator, the MMSE algorithm is used to correct for the multipath channel effects. The MMSE equalizer attempts to reconstruct the transmitted signal from the received signal by integrating channel statistics-such as the noise power-along with the estimated multipath channel coefficients [36]. The MMSE's Inclusion of the channel statistics in the equalization process makes it a more robust approach under degrading SNR conditions. The MMSE equalizer aims to reduce the the squared error between the estimated \(\hat{x}(m)\) and original \(x(m)\) transmitted signals as follows,
\[\hat{x}(m)=\operatorname*{arg\,min}_{\hat{x}(m)}E\left[(x(m)-\hat{x}(m))^{2} \right]. \tag{10}\]
If the noise power or channel SNR is known, then the MMSE estimates the transmitted signal by solving,
\[\hat{\mathbf{x}}_{M}=\mathbf{A}^{H}\left(\mathbf{A}\mathbf{A}^{H}+\gamma^{-1} \mathbf{I}_{A}\right)^{-1}r \tag{11}\]
where \(\mathbf{A}\) is a 2D matrix representing the estimated channel impulse response, \(\gamma\) is the SNR, and \(I_{A}\) is an identity matrix with matching dimensions to \(\mathbf{A}\)[98].
### _Deep Learning Architectures_
This section provides brief explanations of each DL architecture and algorithm used to generated the results in Sect. V.
#### Iii-E1 Convolutional Neural Network
Convolutional Neural Networks (CNNs) are supervised learning Multi-Layer Perceptron (MLP)-based Neural Networks (NNs) designed to process multi-dimensional data in a grid architecture. It can be used to process a one-dimensional vector such as time-interval data, two-dimensional and three-dimensional grids of pixels such as images [99]. CNNs learn parameters \(\theta\) using the back-propagation algorithm to estimate a mapping function of a discriminative model by minimizing a loss function. The loss function computes the error between the model's prediction and the ground truth.
The CNN network is comprised of a MLP prepended with convolutional and pooling layers. Convolutional layers consist of multiple neurons in a grid structure where each neuron corresponds within an element-wise multiplication of a multi-dimensional kernel (a.k.a., filter) and a region in the input that is the same size as the kernel. The kernel's elements are called weights and they are shared between all neurons [100]. In a CNN, convolutional layer(s) are used to detect and extract features from multiple regions of the input data to generate feature maps. Each neuron applies an activation function within the convolutional layer to non-linearly transform the feature map's corresponding element [56, 100]. Pooling layers normally follow convolutional layers to reduce the dimensionality of the activated feature map by computing a statistic summary such as a maximum, minimum, and average of nearby outputs [99]. Max pooling is the most common pooling layer used in CNN networks. In this work, Max Pooling is adopted to extract the maximum-value features within rectangular frames of the activated feature maps. After one or more convolutional and pooling layer stages, fully connected layers (a.k.a., dense layers) are used to detect high level features and pass them onto the output layer [46]. The purpose of the output layer is to predict the label to which the extracted features belong. In this work CNNs are adopted to implement RF-DNA fingerprint classification using IEEE 802.11a Wi-Fi preambles that transverse a Rayleigh fading channel under degrading SNR.
#### Iii-E2 Auto-Encoder
An Auto-Encoder (AE) is an MLP-based NN that attempts to regenerate a multi-dimensional input at the output with as little error as possible [101, 102]. An AE is an unsupervised learning-based architecture used to learn the distribution and an efficient representation of the input data. Logically, an AE consists of two main parts: the encoder and decoder. Generally, the encoder attempts to estimate a mapping function that outputs an intermediate hidden representation \(\mathbf{h}=f(\mathbf{x})\). The decoder aims to reconstruct the input from the hidden representation by applying another mapping function \(g(\mathbf{h})\) so the final output is \(\mathbf{r}=g(\mathbf{h}(\mathbf{x}))\) is as close as possible to the input \(\mathbf{x}\)[99]. During training, the AE's loss function penalizes the decoder's output for being different from \(\mathbf{x}\). This work uses the Mean Square Error (MSE) as the loss function for all AEs. When the encoder is comprised of convolutional and pooling layers, the AE is designated a Convolutional AE (CAE). If the multi-dimensional data (a.k.a., tensor) at the CAE input is \(\mathbf{x}\), then the hidden representation corresponding to the \(i^{\text{th}}\) tensor is given by,
\[h_{i}=\beta(x_{i}*W+b), \tag{12}\]
where \(W\) are the elements (a.k.a., weights) of the convolutional kernel, \(b\) is the bias, \(*\) denotes the convolution operation, and \(\beta\) is the activation function [102]. The decoder's output \(r_{i}\) for \(h_{i}\) is given by,
\[r_{i}=\beta(h_{i}*\tilde{W}+\tilde{b}), \tag{13}\]
where \(\tilde{W}\) are the deconvolutional kernel weights, and \(\tilde{b}\) is the decoder's bias [102]. During training, the CAE parameters-including \(W\), \(\tilde{W}\), \(b\), and \(\tilde{b}\)-are adjusted using backpropagation to minimize the loss function given by,
\[J(\theta)=\sum_{i=1}^{m}{(x_{i}-r_{i})^{2}}, \tag{14}\]
where \(m\) is the number of training samples.
#### Iii-E3 Generative Adversarial Networks
A Generative Adversarial Network (GAN) is a DL-based architecture that aims to estimate a generative model by simultaneously training two deep network models using adversarial training. GANs can be used to estimate generative models for multiple applications including: image editing, style transfer, and image synthesis [103]. A GAN is comprised of two models referred to as the Generator, \(G\), and the Discriminator, \(D\). The \(G\) is a generative network that learns the training data distribution so it can generate new samples with the same distribution. The \(D\) is a discriminative network tasked with determining whether an input sample belongs to the training data set or was
generated by the \(G\). GAN training can be viewed as a minimax two-player game in which the \(G\) attempts to learn the training data distribution and estimate a mapping function capable of generating new samples that increase the \(D\)'s probability of making the wrong decision, while \(D\) tries to maximize its probability of making the right decision (i.e., differentiating a training sample from a generated one) [104, 99]. The training results in a unique solution when the \(D\)'s output is uniformly distributed with a probability of one-half everywhere.
When MLP networks are used for both the \(G\) and \(D\) networks, backpropagation can be used to train the entire system where the NN representing the \(G\) recovers the data distribution without access to the training data and \(D\) maximizes the probability of making the right binary decision [103, 104]. If the prior probability of \(G\) input \(\mathbf{z}\) is \(P_{\mathbf{z}}(z)\), then the generator mapping function can be given by \(G(\mathbf{z};\ \theta_{g})\) where \(\theta_{g}\) is the MLP parameters of \(G\). The \(D\) function is given by \(D(\mathbf{x};\ \theta_{d})\) for input \(\mathbf{x}\) and a single output \(D(\mathbf{x})\), which represents the probability that \(\mathbf{x}\) came from the training data and not the \(G\)[104]. During GAN training, the \(G\) network intends to learn the distribution \(P_{g}\) over the training data \(\mathbf{X}\) by simultaneously maximizing the correct decision probability \(D(\mathbf{x})\) and minimizing the term \((1-D(G(\mathbf{z})))\) corresponding to the \(G\) network [104]. The GAN minimax optimization problem can be described by the following objective function,
\[\min_{G}\ \max_{D}\ V(D,G) =E_{x\sim P_{\mathbf{z}}(x)}\{\log[D(x)]\}\] \[+E_{z\sim P_{z}(z)}\{\log[1-D(G(z))]\}, \tag{15}\]
where \(E\) is the expected value. The optimum point is reached when the \(G\) perfectly recovers the training data distribution (i.e. \(P_{g}=P_{data}\)) [104, 99].
## IV Methodology
This section describes a nontraditional, pre-processing approach that uses semi-supervised DL to correct for multipath channel effects while simultaneously preserving the SEI exploited RF-DNA fingerprints. In fact, two semi-supervised learning-based channel equalization approaches are investigated. The first approach leverages the GAN architecture's adversarial relationship to train the \(G\) such that it learns the distribution of the multipath channel effects as well as the signals' RF-DNA fingerprints to create a mapping function capable of estimating and correcting the multipath channel effects. The second approach jointly trains a CAE and CNN architecture-designated herein as JCAECNN-that corrects the multipath channel effects while simultaneously improving the system's ability to extract more discriminative SEI features.
A Rayleigh fading channel comprised of \(L\) paths is applied to each collected IQ preamble to simulate a multipath environment. Each semi-supervised learning approach is used to equalize the resulting multipath preamble set with the goal of maximizing SEI performance. Motivated by the results in [87], semi-supervised learning is conducted using the raw IQ preamble samples as well as their Natural Logarithm (NL) representation. The NL of the raw IQ samples is given by,
\[\tilde{r}(n) =\ln[r_{\mathscr{R}}(n)+jr_{\mathscr{I}}(n)]=\ln\left[\rho_{r} \exp(j\phi_{r})\right]\] \[=\ln[\rho_{r}]+j\phi_{r}=\tilde{\rho}_{r}+j\phi_{r}, \tag{16}\]
where \(r_{\mathscr{R}}\) is the signal's real (In-phase) component, \(r_{\mathscr{I}}\) is the imaginary (Quadrature) component, \(\rho_{r}\) is the magnitude, and \(\phi_{r}\) is the phase angle [87]. The \(i^{\text{th}}\), equalized preamble's augmented representation-denoted here in as IQ+NL-is,
\[R^{i}(m,n)=\begin{bmatrix}r^{i}_{\mathscr{R}}(1,1)&r^{i}_{\mathscr{R}}(1,2)& \cdots&r^{i}_{\mathscr{R}}(1,320)\\ r^{i}_{\mathscr{R}}(2,1)&r^{i}_{\mathscr{R}}(2,2)&\cdots&r^{i}_{\mathscr{R}}(2, 320)\\ \tilde{\rho}^{i}_{r}(3,1)&\tilde{\rho}^{i}_{r}(3,2)&\cdots&\tilde{\rho}^{i}_{r} (3,320)\\ \phi^{i}_{r}(4,1)&\phi^{i}_{r}(4,1)&\cdots&\phi^{i}_{r}(4,320)\\ \end{bmatrix}, \tag{17}\]
where \(i=1,2,\ldots,N_{B}\) for a total of \(N_{B}\) preambles, \(m=1,2,\ldots,N_{D}\), and \(n=1,2,\ldots,320\) for a IEEE 802.11a Wi-Fi preamble sampled at \(20\) MHz. Combining the preamble's phase behavior-captured by \(\phi_{r}\)-with the IQ features provides a computationally efficient mechanism that improves SEI performance under degrading SNR [87]. The rest of this section provides detailed descriptions fo the two semi-supervised learning approaches that enable SEI using signal's collected under Rayleigh fading and degrading SNR conditions.
### _Multipath Equalization using a Conditional GAN_
Prior to RF-DNA fingerprint generation, all signals undergo channel equalization using a Conditional GAN (CGAN), which is introduced by the authors of [105]. The CGAN is constructed using a CAE and CNN for the \(G\) and \(D\), respectively. The use of semi-supervised training enables estimation of a generative function \(G(z|y)\) capable of reconstructing a Wi-Fi preamble without Rayleigh fading effects despite those effects being present within the preamble input to the \(G\). Fig. 2 shows the CGAN training and RF signal equalization process developed to generate the results presented in Sect. V. Table II provides the configuration and parameters used to construct the CGAN's \(G\) and \(D\).
Multipath channel effects are induced by filtering each collected preamble using a unique TDL-given in equation (3)-configured to represent a Rayleigh fading channel consisting of \(L=5\) reflecting paths. A unique instance of scaled and like-filtered Additive White Gaussian Noise (AWGN) is then added to each preamble to achieve a SNR of 9 dB to 30 dB in 3 dB increments. For each SNR, a total of ten, unique AWGN realizations are generated to augment the data set and facilitate Monte Carlo analysis. The resulting received signal \(r[n]\)-as expressed in equation (4)-is represented using IQ+NL and each feature is re-scaled to be within the range [0, 1] using Min-Max normalization. The resulting normalized set of \(N_{B}\) preambles are then randomly assigned to either the training or test set. The training set is comprised of \(N_{R}=1800\) (i.e., 90% of the total available at the chosen SNR and noise realization) preambles and the test set consists of \(N_{T}=200\) (i.e., 10% of the total available at the chosen SNR and noise realization) preambles where \(N_{T}=N_{B}-N_{R}\).
The CGAN is first proposed in [105] and is an extension of the traditional GAN introduced in [104]. In CGAN, the \(G\) and \(D\) networks-shown in Fig. 2-accept the class label \(y\) as an additional input; thus, both mapping functions \(G(z)\) and \(D(X)\) are conditioned on the variable \(y\) and the traditional GAN's minimax optimization equation is rewritten as [105],
\[\min_{G}\;\max_{D}\;V(D,G) =E_{x\sim P_{\rm{z}}(x)}\{\log[D(x|y)]\}\] \[+E_{z\sim P_{\rm{z}}(z)}\{\log[1-D(G(z|y))]\}, \tag{18}\]
where \(P_{\rm{d}}(x)\) is the training data distribution learned by the \(G\). In this work, the class label is combined with the input to the \(G\) and \(D\) using a hidden representation that enables the GAN to estimate a one-to-many generative function conditioned by \(y\) instead of the traditional one-to-one mapping. The hidden representation is generated by an embedding layer that maps each emitter's class label to a length fifty vector. It is important to note that the length of the vector is empirically chosen based on [106]. The label vector size is expanded to a length of 1,280 using a dense layer and reshaped into a 4\(\times\)320 tensor to match the dimensionality of its assigned preamble. The 4\(\times\)320 label is appended to the corresponding emitter's normalized IQ+NL preamble representations to form a 4\(\times\)320\(\times\)2 labeled preamble representation denoted as \(\mathbf{R}_{e}\).
During CGAN training, the \(G\)'s input is the preamble \(\mathbf{R_{e}}\) and the \(D\)'s input is a set of _AWGN-only_ preambles \(\mathbf{\tilde{X}}_{e}\) (i.e., there are no multipath effects present within this signal set). The \(G\) attempts to learn the multipath impacted preambles' distribution and estimate a function \(G(z)\) that maps them to \(\mathbf{\hat{X}}_{e}\) such that the distribution of \(\mathbf{\hat{X}}_{e}\) matches the distribution associated with the _AWGN-only_ preambles \(\mathbf{\tilde{X}}_{e}\) used to train the \(D\). The \(D\) outputs a single-value \(D(x)\) representing the probability that the input \(x\) is from the training data \(\mathbf{\tilde{X}}_{e}\) rather than \(\mathbf{\hat{X}}_{e}\). The CGAN is trained using backpropagation with a minibatch size of 256 tensors, 10,000 epochs, and an alternating scheme in which the \(D\) is trained for \(S\) steps based upon a given \(G\). Training for \(S\) steps results in the best version of the \(D\). The authors of [104] treat \(S\) as a hyperparameter with \(S=1\) being the least computationally complex; thus, \(S\) is set to one for the results presented in Sect. V. The \(D\) is trained using forward- and backpropagation with the goal of maximizing \(V(D,G)\) to achieve the highest correct decision probability. After \(S=1\) steps. The \(G\) is trained using Stochastic Gradient Descent (SGD) to minimize \(V(D,G)\) with the goal of reducing the \(D\)'s ability to make a correct decision. This training process continues until \(D(x)=0.5\) everywhere or the total number of training iterations equals the empirically chosen value of 10,000.
Fig. 2: _CGAN Training: Flowchart illustrating the process used to train the \(G\) to facilitate channel equalization that preserves the exploited, emitter-specific SEI features._
Once the CGAN is trained, the resulting \(G\)-that provides the mapping function \(G(z|y,\theta_{g})\)-is disconnected from the \(D\) and used to equalize the multipath preambles \(r[n]\) generated from the \(N_{T}\) preambles comprising the test set of \(x[n]\). Each test preamble \(r[n]\) is combined with each of the \(N_{D}=4\) labels using the hidden representation-described earlier in this section-to create a total of four labeled preambles \(\mathbf{R}_{e}^{i}\) where \(i=1,2,3,4\). The trained \(G\) generates an equalized output \(\mathbf{\hat{X}}_{e}^{i}\) corresponding to each labeled preamble \(\mathbf{R}_{e}^{i}\). Finally, the second channel (a.k.a., the hidden representation label) is removed from the \(G\)'s outputs \(\mathbf{\hat{X}}_{e}^{i}\) prior to classification using a trained CNN. This process is illustrated in Fig. 3.
The CNN is trained using: (i) \(4\times 320\) tensors formed using the IQ+NL preamble representations generated from the _AWGN-only_ training set \(\tilde{x}[n]\), (ii) backpropagation, (iii) SGD to minimize the categorical, cross-entropy loss function, (iv) \(l_{2}\)-norm regularization to reduce overfitting, and (v) Adam optimization for the adjustment of the network's weights [107]. The CNN's output layer uses a softmax decision to assign the \(i^{\text{th}}\) equalized representation \(\mathbf{\hat{X}}_{e}^{i}\) a label of \(y_{i}\) according to,
\[y_{i}=\max_{j}(Q_{ij}), \tag{19}\]
where \(Q_{ij}\) is the \(j^{\text{th}}\) CNN output for the \(i^{\text{th}}\) equalized representation input \(\mathbf{\hat{X}}_{e}^{i}\), \(j=[1,2,3,4]\) is the index of the softmax layer outputs, and \(i=[1,2,3,4]\) for a given received preamble \(r[n]\). Lastly, a confidence score decision is used to assign the received multipath preamble \(r[n]\) a label \(\hat{y}\) that satisfies,
\[\hat{y}=\max_{i}(y_{i})=\max_{i}\max_{j}(Q_{ij}). \tag{20}\]
The best possible SEI performance is achieved by training the CNN at SNRs lower than those in the test set. A grid search is performed to determine the best SNR at which to train the CNN and achieve the highest SEI performance across all SNR values. For example, the CNN is trained using preambles collected at an SNR of 12 dB and once trained that CNN classifies preambles collected at each SNR in the range from 9 dB to 30 dB with 3 dB increments between consecutive SNR values. The 'best' training SNR is the SNR whose corresponding trained CNN results in the highest average percent correct classification across all SNR values. The CGAN SEI process can be summarized using the following algorithm,
### _Multipath Equalization using JCAECNN_
This equalization approach uses a joint CAE and CNN (a.k.a., JCAECNN) architecture similar to that used in [61]; however, the approach described here differs from that in [61] in that the CAE architecture is modified to consist of multiple decoder heads instead of one. The decoder heads decompose a received preamble-that undergoes Rayleigh fading-into its \(L\) weighted and delayed copies of the transmitted signal, \(x[n]\). As illustrated in Fig. 4, the JCAECNN equalization process is built on a Single Input Multiple Output (SIMO) system that includes: signal collection and detection (Sect. III-A), multipath fading and AWGN scaling (Sect. III-C), and signal preparation-in accordance with our prior work [87]-prior to equalization and classification.
Equalization is performed using a CAE consisting of a single encoder and \(L\) decoder heads (i.e., one for each path of the Rayleigh fading channel) using the preambles' IQ+NL representations at the selected SNR. Motivated by the results presented in [108], a Densely Connected Convolutional Network (DenseNet) is used to implement the "shared" encoder shown in Fig. 4. DenseNet is comprised of multiple Dense blocks where each block is created by connecting each convolutional layer to all preceding convolutional layers of the same size. These dense connections grant each convolutional layer access to all previously generated feature maps to enable feature reuse. In this work, two densely connected blocks are used to construct the encoder. The DenseNet encoder generates compressed representation \(h(\mathbf{R}[m,n])\), which is the input shared with each of the following NN architectures: the \(L\) decoder heads and the classifier The resulting JCAECNN architecture is derived from the square error function of equation (8). The optimization goal is to minimize the error between the received signal \(r(m)\) and its delayed and scaled copies \(C_{k}=\alpha_{k}x(m-\tau_{k})\) generated using the TDL of equation (3). The target of each decoder head is to reconstruct a single copy \(C_{k}\) corresponding to the \(k^{\text{th}}\) path of the Rayleigh fading channel. In addition to the decoder heads, a CNN classifier (CNN\({}_{I}\)) is used to assign the compressed representation \(h(\mathbf{R}[m,n])\) to any of the \(N_{D}=4\) emitters using a softmax decision. The NN configurations for the shared encoder, decoder heads, and CNN\({}_{I}\) classifier are provided in Table III.
The JCAECNN is trained iteratively to jointly optimize the combined loss function given by,
\[F(\theta)=\sum_{k=1}^{L}\left[\lambda_{k}\times M(m,\tau_{k})+\lambda_{c}\times C (y_{c},\tilde{y_{c}})\right], \tag{21}\]
where \(\lambda_{k}\) are the \(k^{\text{th}}\) decoder head's loss weights corresponding to each Rayleigh fading path, \(\lambda_{c}\) are CNN's loss weights, \(M(m,\tau_{k})\) is the MSE loss given by,
\[M(m,\tau_{k})=\frac{1}{N_{s}}\sum_{m=1}^{N_{s}}(x(m-\tau_{k})-\hat{x}(m-\tau_{ k}))^{2} \tag{22}\]
Fig. 3: Illustration of the SEI process that uses a trained CGAN’s \(G\) network to perform multipath channel equalization while simultaneously preserving emitter specific RF-DNA fingerprinting exploited features prior to CNN classification.
where \(\hat{x}(m-\tau_{k})=\hat{C}_{k}\) is the decoder's output corresponding to the \(k^{\text{th}}\) path, and \(C(y_{c},\hat{y}_{c})\) is the categorical cross entropy function used by the CNN classifier to compute the difference between the ground-truth label \(y_{c}\) and estimated label \(\hat{y}_{c}\). If the ground-truth label is one-hot encoded where \(y_{c}=[y_{c,1},y_{c,2},\ldots,y_{c,N_{D}}]\), the categorical cross entropy can be calculated by the following formula,
\[C(y_{c},\hat{y}_{c})=-\sum_{l=1}^{N_{D}}(y_{c,l}\times\log(\hat{y}_{c,l})) \tag{23}\]
where \(l\) is the index of the class at both the one-hot encoded ground truth and the output layer of the classifier [109]. An individual fading path's MSE loss is optimized by updating the weights of the shared encoder and decoder head assigned to the chosen fading path.
Decoder head training allows the shared encoder to learn compressed and delay-invariant SEI features. The CNN classifier head is trained to minimize the \(C\) loss between the actual label \(y_{c}\) and the estimated label \(\tilde{y}_{c}\), which allows the shared encoder to learn more discriminating SEI features than those learned while training the decoder heads.
As shown in Fig. 5, the trained JCAECNN supplies the equalized preambles \(\hat{C}_{k}\)-for \(k=1,2,\cdots,L\)-along with the CNN\({}_{I}\) decision \(\tilde{y}_{c}\) for every normalized IQ+NL preamble representation \(\mathbf{R}[m,n]\) to the discriminating CNN, which is denoted as CNN\({}_{D}\). CNN\({}_{D}\) assigns \(\hat{C}_{k}\) label \(y_{k}\) for all \(k\) and decides that the received preamble was transmitted by the emitter whose class label achieves the highest vote out of \(y_{k}\) and \(y_{c}\). The CNN\({}_{D}\) architecture is similar to the RF-DNA fingerprint classifier architecture shown in Table II and trained using the same data set used to train the JCAECNN with only AWGN present (i.e., there are no multipath effects present within the signal set).
## V Results
This section presents the experimental results and analysis for the RF-DNA fingerprint-based SEI of IEEE 802.11a Wi-Fi emitters whose signals transverse a Rayleigh fading channel and degrading SNR using the CGAN and JCAECNN pro
Fig. 4: _JCAECNN Training:_ Flowchart illustrating signal pre-processing and the joint CAE multipath equalization.
Fig. 5: _JCAECNN Testing:_ Flowchart illustrating the RF-DNA fingerprinting process using the trained JCAECNN system that performs channel equalization and supplies the CNN\({}_{I}\) decision for an aggregated CNN classification decision.
cesses described in Sect. IV-A and Sect. IV-B, respectively. Results for the following experiments are presented.
1. _Experiment #1:_ Comparative assessment is conducted between the CGAN and JCAECNN approaches and our RF-DNA fingerprinting of emitters under Rayleigh fading published in [38, 41].
2. _Experiment #2:_ Scalability analysis of the CGAN and JCAECNN approaches. This experiments assesses SEI performance for both approaches using data sets consisting of signals collected from: four, eight, sixteen, and thirty-two commercial emitters to represent larger IoT device deployments.
3. _Experiment #3:_ Assessment of JCAECNN SEI performance using the sixteen emitter signal set of Experiment #2 as well as the publicly available data sets associated with the published results presented in [88, 89]. This experiment evaluates the JCAECNN architecture's effectiveness in learning SEI discriminating features from multipath signals whose transmitting emitters differ in manufacturer, model, or Size, Weight, and Power-Cost (SWaP-C) with respect the to the emitters described in Sect. III-B. Furthermore, the results of this experiment serve as a benchmark to facilitate the evaluation of future work.
4. _Experiment #4:_ This experiment assesses JCAECNN SEI performance using a loss function whose weights are selected based upon the Rayleigh fading channel's path variances. This experiment optimizes the MSE loss weights to maximize JCAECNN performance.
The specifics of the data set(s), training, and testing approaches are explained within each experiment's section.
### _Results for Experiment #1_
The IEEE 802.11a Wi-Fi signal set used in this experiment is comprised of preambles extracted from the four Cisco AIR-CB21G-A-K9 Wi-Fi emitters described in Sect. III-B and used to generate the results presented in our prior publications [38, 41]. A total of 2,000 preambles are extracted for each of the four emitters, which is subdivided into a training and testing set comprised of \(N_{R}=1,800\) and \(N_{T}=200\) randomly selected preambles per emitter. The training set is duplicated to permit creation of a _AWGN-only_ set-that is used in the training of the CGAN-and a Rayleigh fading plus AWGN set. The test set is comprised of only Rayleigh fading plus AWGN impacted preambles. It is important to note that the Rayleigh fading channel consists of \(L=5\) paths, is generated and applied to each signal as described in Sect. III-C, and is unique to each preamble (i.e., the channel coefficients change every transmission). A specific SNR is achieved by adding scaled and like-filtered AWGN to every preamble and the process repeated ten times to permit Monte Carlo simulation. SEI performance is assessed using average percent correct classification performance at SNR values ranging from 9 dB to 30 dB in steps of 3 dB between consecutive values.
Training of the CGAN and JCAECNN approaches is conducted in accordance with Sect. IV-A and Sect. IV-B, respectively. Recall that CGAN SEI performance is maximized by training it using SNR values lower than those comprising the test set. The results of the grid search-described in Sect. IV-A are presented in Table. IV in which SNR\({}_{R}\) designates the SNR of the preambles used to train the CGAN and SNR\({}_{T}\) is the SNR of the preambles being classified by the CGAN trained at SNR\({}_{R}\). The JCAECNN approach's "best" training SNR is determined using the same approach as that of the CGAN and resulted in a SNR\({}_{R}\) value of 9 dB for all SNR\({}_{T}\) values. The Partitioned Time RF Fingerprinting (PTRFF) approach from [41] and traditional Time-Frequency Feature-Engineered RF Fingerprinting (TFRFF) approach from [38] both perform channel correction using the N-M channel estimator and MMSE channel equalizer described in Sect. III-D1 and Sect. III-D2, respectively. The PTRFF approach uses a pre-trained one-dimensional CNN to classify partitioned, time IQ preambles. The time partitions are generated by slicing the IQ samples of each equalized preamble using a \(N_{w}=64\) length, sliding window. The TFRFF approach performs feature extraction from the normalized, Gabor coefficients calculated from a preamble's IQ samples and classification performed using the Multiple Discriminant Analysis/Maximum Likelihood (MDA/ML) classifier. The reader is directed to [41] and [38] for the specific methodologies, results, analysis, and conclusions of the PTRFF and TFRFF approaches, respectively.
Average percent correct classification performance for the CGAN, JCAECNN, PTRFF, and TFRFF SEI approaches are presented in Fig. 6. Each data point is associated with \(N_{T}\times N_{z}\times N_{D}=8,000\) individual classification decisions. The results of the JCAECNN are generated by setting \(\lambda_{k}\) and \(\lambda_{c}\) to one. Compared to the PTRFF, and TFRFF approaches, the results in Fig. 6 show that the CGAN and JCAECNN approaches achieve superior average percent correct classification performance for SNR values of 18 dB and lower. The CGAN approach achieves the best average percent correct classification performance for all SNR values with a maximum of 99.85% at an SNR of 30 dB and a minimum of 95% at an SNR of 9 dB. Based on the results in Fig. 6, DL-based equalization using the semi-supervised learning approaches presented in Sect. IV provides a better mechanism for compensating for multipath channel effects while simultaneously preserving SEI features.
### _Results for Experiment #2_
The CGAN and JCAECNN equalization and classification approaches are assessed under an increased number of emitters to reflect larger IoT device deployments. The number of to be identified emitters is: \(N_{D}=[4,8,16,32]\). This additional assessment is conducted using the preambles extracted from the signals transmitted by thirty-two TP-Link AC1300 USB Wi-Fi adapters. The assessment is conducted using the following four scenarios.
* _Scenario #1:_\(N_{B}=\) 10,000 preambles extracted from the signals transmitted by Emitter #1 through Emitter #4.
* _Scenario #2:_\(N_{B}=\) 10,000 preambles extracted from the signals transmitted by Emitter #1 through Emitter #8.
* _Scenario #3:_\(N_{B}=\) 10,000 preambles extracted from the signals transmitted by Emitter #1 through Emitter #16.
* _Scenario #4:_\(N_{B}=\) 10,000 preambles extracted from the signals transmitted by Emitter #1 through Emitter #32.
For each scenario, a unique \(L=5\) Rayleigh fading channel is convolved with each collected preamble prior to adding scaled and like-filtered AWGN to achieve SNR values of 9 dB to 30 dB with 3 dB steps between consecutive SNR values and ten noise realizations per SNR. CGAN and JCAECNN approaches equalize and classify the preambles of each scenario using the same procedure described earlier in this section, but with \(N_{R}=8,000\) and \(N_{T}=2,000\) for each SNR, noise realization, and emitter. The average percent correct classification of the CGAN and JCAECNN approaches for each of the four scenarios is shown in Fig. 7 where CGAN\({}_{i}\) and JCAECNN\({}_{i}\) denote the CGAN and JCAECNN performance results for the \(i^{\text{th}}\) scenario, respectively.
Fig. 7 shows the average percent correct classification results for all four scenarios of Experiment #2. When discriminating four emitters, both approaches result in similar performance as that shown in Fig. 6 with the CGAN approach achieving superior average percent correct classification performance for all SNRs. As the number of to be identified emitters increases, the performance of the JCAECNN approach begins to surpass that of the CGAN approach. When a total of sixteen emitters are used (a.k.a., Scenario #3), the JCAECNN performance exceeds that of the CGAN approach by a minimum of 1% for all SNRs. When the entire set of thirty-two emitters is used (a.k.a., Scenario #4), the JCAECNN exceeds the performance of the CGAN approach for all SNRs. The smallest improvement is 5% at an SNR of 18 dB and the largest improvement is 11% at an SNR of 9 dB.
Fig. 7 shows that JCAECNN performance suffers less degradation as the number of to be identified emitters increases from four to thirty-two; thus, this approach scales better for larger IoT deployments. The poorer performance of the CGAN approach is attributed to the increasing number of conditional distributions that the \(G\) must learn as the number of emitters goes from four to thirty-two. This is exacerbated by the decreasing efficiency of the hidden label representation. In terms of decision complexity, the JCAECNN performs a total of \(L+1\) CNN decisions corresponding to the reconstructed \(\hat{C}_{k}\) and the CNN classifier output \(\hat{y}_{c}\) for each received preamble \(r[n]\). The number of JCAECNN classifications does not depend on the number of emitters (i.e., labels), which is not the case for the CGAN. In the CGAN approach, assigning preamble \(r[n]\) to a label \(y\) requires \(N_{D}\) total equalization actions by the \(G\) that is followed by an additional \(N_{D}\) sequential classifications. This is to due the fact that the identity of the transmitting emitter is unknown; thus, the preamble must be compared to every known emitter's class. So, as the number of emitters linearly increases so does the number of conditional distributions learned by the \(G\) as well as the number CGAN equalizations and classification decisions.
### _Results for Experiment #3_
Based upon the results presented in Sect. V-B, the results in this section are generated using only the JCAECNN approach. Fig. 8 shows the average percent correct classification per
Fig. 6: _Experiment #1 Results:_ Average percent correct classification performance across the \(N_{D}=4\) IEEE 802.11a Wi-Fi emitters using the CGAN, JCAECNN, PTRFF, and TFRFF approaches for SNR\(\in\)[9, 30] dB in steps of 3 dB.
Fig. 7: _Experiment #2 Results:_ Average percent correct classification performance across up to \(N_{D}=32\) IEEE 802.11a TP-Link adapters using CGAN approach, JCAECNN, for SNR\(\in\)[9, 30] dB in steps of 3 dB. Note that subscripts denote scenario number.
formance of the JCAECNN approach for the following three different data sets that each contain signals collected from at least sixteen IEEE 802.11a Wi-Fi compliant emitters.
* _Data Set #1:_ This data set consists of the IEEE 802.11a Wi-Fi preambles used to generate the Scenario #3 results presented in Fig. 7 and explained in Sect. V-C. This data set's results are designated using "TP-Link".
* _Data Set #2:_ This data set was collected by the authors of [88] to evaluate the Optimized Radio clAssification through Convolutional neuraL nEtworks (ORACLE) approach. The data set contains IEEE 802.11a Wi-Fi signals collected from 16 USRP X310 Software-Defined Radios (SDRs) using a stationary Ettus Research USRP B210 SDR as the receiver. It is important to note that the B210 receiver's sampling rate was set to 5 MHz, which is lower than the 20 MHz sampling rate of the collected signals used to generate the results presented up to this point. This data set's results are designated using "ORACLE".
* _Data Set #3:_ This data set was collected by the authors of [89] and designated the WiFi Signal (WiSig) data set. The WiSig data set contains IEEE 802.11a Wi-Fi signals captured from a total of 174 transmitting emitters-including USRP B210, X310, and N210 SDRs-over a four day period and using forty-one USRP receivers. In order to maintain consistency with the other two data sets and the rest of this paper's results only a single day's and receiver's worth of WiSig signals are used. The chosen WiSig signals are detected using auto-correlation performed using the preamble's STS portion and then re-sampled to rate of 20 MHz. This data set's results are designated using "WiSig".
Similar to the TP-Link data of Scenario #3, each signal in the Oracle and WiSig data sets is filtered using a unique \(L=5\) Rayleigh fading channel prior to adding scaled and like-filtered AWGN to achieve SNR values of 9 dB up to 30 dB with 3 dB steps between consecutive SNR values and ten noise realizations per SNR. After that, the JCAECNN approach performs equalization and classification as described earlier in this section. For the sake of consistency, a total of sixteen emitters are randomly selected from the WiSig data set to match the number of emitters represented in the TP-Link and ORACLE data sets. For TP-Link and Oracle data sets, \(N_{R}=8,000\) and \(N_{T}=2,000\). For the single day and receiver portion of the WiSig data, the total number of signals collected per emitter is 1,000; thus, \(N_{R}=800\) and \(N_{T}=200\) to provide a ratio consistent with the other two data sets. The average percent correct classification results presented in Fig. 8 show that JCAECNN equalization and classification of the WiSig preambles achieves superior performance to that of the Oracle data for all SNRs despite the limited number of training samples. That can be attributed to the lower sampling rate used to collect the ORACLE data, which represents only 25% of the sampling rate of the TP-Link and ORACLE data. Additionally, the WiSig data set is comprised of multiple USRP models while the ORACLE signals are transmitted by USRPs that are all the same model. The latter is serial number discrimination, which is the most challenging SEI case. Additionally, ORACLE's USRP X310 is a $8,500 high-performance SDR made with low variability components, which means less feature variability across SDRs making SEI more difficult [88].
### _Results for Experiment #4_
Up to this point, all results generated by the JCAECNN approach set \(\lambda_{k}\) and \(\lambda_{c}\) to one. In an effort to optimize \(L=5\) Rayleigh fading channel equalization and classification performance the loss function weights \(\lambda_{k}\) for \(k=[1,2,3,4,5]\) and \(\lambda_{c}\) are set to: thirty-two, sixteen, eight, four, two, and thirty-two, respectively. Selection of these values is based on the fact that the variances of the corresponding Rayleigh channel paths given in equation (2) decay exponentially with \(k\). Fig. 9 shows the average percent correct classification for the JCAECNN approach with decaying loss weights and is designated as O-JCAECNN in which the 'O' denotes 'optimized'. Comparative assessment is conducted using results generated by: (i) the JCAECNN implementation in which the loss weights are the same (i.e., they are all set to a value of one) and (ii) the CGAN results presented in Fig. 6.
Superior average percent correct classification performance is achieved using the O-JCAECNN approach for all SNR values of 9 dB to 30 dB using a 3 dB step between consecutive values. It is the only SEI approach to achieve an average percent correct classification performance of 100% for \(N_{D}=4\) case, which occurs at an SNR of 27 dB and 30 dB. Fig. 10 is the confusion matrix corresponding to the 9 dB O-JCAECNN results presented in Fig. 9(b) and is included to show percent correct classification performance for each of the \(N_{D}=16\) emitters. It can be seen that eleven of the sixteen emitters are represented by dark blue diagonal entries, which indicates a percent correct classification performance of at least 95%.
Fig. 8: _Experiment #3 Results:_ Average percent correct classification performance for the TP-Link, WiSig, and ORACLE data sets–that each contain signals collected from \(N_{D}=16\) IEEE 802.11a Wi-Fi emitters–using the JCAECNN approach for SNR\(\in\)[9, 30] dB in steps of 3 dB between consecutive values.
The exceptions being Emitter #3, Emitter #4, Emitter #5, Emitter #6, and Emitter #12 of which only Emitter #4 is not classified correctly at least 90% of the time.
The results show that the performance of the JCAECNN model is dominated by: (i) the loss weights in equation (21) that correspond to the first few Rayleigh fading paths, and (ii) the loss weight \(\lambda_{c}\) corresponding to the \(\mathrm{CNN}_{I}\) classifier head. This suggests that channel-invariant RF-DNA fingerprinting is possible by optimizing the first few and most dominant weights of the JCAECNN's loss function.
## VI Conclusion and Future Work
This work presents and analyzes two semi-supervised DL approaches-designated herein as CGAN and JCAECNN-to enhance IoT security using SEI under Rayleigh fading and degrading SNR. These approaches are capable of extracting discriminating RF-DNA fingerprint features from signals transmitted by up to thirty-two IEEE 802.11a Wi-Fi emitters while traversing a \(L=5\) Rayleigh fading channel. The two architectures can reconstruct the transmitted preamble from its received, multipath corrupted version while preserving the SEI discriminating features using the SNR robust, IQ+NL preamble representation. The two semi-supervised learning approaches are compared with the traditional RF-DNA fingerprinting approaches in [38, 41]. The JCAECNN approach shows better scalability as the number of emitters increases from \(N_{D}=4\) to \(N_{D}=32\), which is not the case for the CGAN approach due to the number of preamble reconstructions and classification decisions increasing linearly with the number of emitters. The O-JCAECNN results in superior average percent correct classification performance that exceeds 94% for sixteen IEEE 802.11a Wi-Fi emitters at SNR values of 9 dB and higher.
Future research is focused on increasing the viability of SEI-based IoT security by modifying the CGAN and JCAECNN architectures to: (i) allow detection of emitters that are not represented within the training signals set, and (ii) leverage simultaneous training by integrating the collaborative learning scheme presented in [110]. [111] Goldsmith, A. (2005). Wireless Communications. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511841224
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2308.11490 | Can Authorship Representation Learning Capture Stylistic Features? | Automatically disentangling an author's style from the content of their
writing is a longstanding and possibly insurmountable problem in computational
linguistics. At the same time, the availability of large text corpora furnished
with author labels has recently enabled learning authorship representations in
a purely data-driven manner for authorship attribution, a task that ostensibly
depends to a greater extent on encoding writing style than encoding content.
However, success on this surrogate task does not ensure that such
representations capture writing style since authorship could also be correlated
with other latent variables, such as topic. In an effort to better understand
the nature of the information these representations convey, and specifically to
validate the hypothesis that they chiefly encode writing style, we
systematically probe these representations through a series of targeted
experiments. The results of these experiments suggest that representations
learned for the surrogate authorship prediction task are indeed sensitive to
writing style. As a consequence, authorship representations may be expected to
be robust to certain kinds of data shift, such as topic drift over time.
Additionally, our findings may open the door to downstream applications that
require stylistic representations, such as style transfer. | Andrew Wang, Cristina Aggazzotti, Rebecca Kotula, Rafael Rivera Soto, Marcus Bishop, Nicholas Andrews | 2023-08-22T15:10:45Z | http://arxiv.org/abs/2308.11490v2 | # Can Authorship Representation Learning Capture Stylistic Features?
###### Abstract
Automatically disentangling an author's style from the content of their writing is a longstanding and possibly insurmountable problem in computational linguistics. At the same time, the availability of large text corpora furnished with author labels has recently enabled learning authorship representations in a purely data-driven manner for authorship attribution, a task that ostensibly depends to a greater extent on encoding writing style than encoding content. However, success on this surrogate task does not ensure that such representations capture writing style since authorship could also be correlated with other latent variables, such as topic. In an effort to better understand the nature of the information these representations convey, and specifically to validate the hypothesis that they chiefly encode writing style, we systematically probe these representations through a series of targeted experiments. The results of these experiments suggest that representations learned for the surrogate authorship prediction task are indeed sensitive to writing style. As a consequence, authorship representations may be expected to be robust to certain kinds of data shift, such as topic drift over time. Additionally, our findings may open the door to downstream applications that require stylistic representations, such as style transfer.
## 1 Introduction
Knowing something about an author's writing style is helpful in many applications, such as predicting who the author is, determining which passages of a document the author composed, rephrasing text in the style of another author, and generating new text in the style of a particular author. The trouble is that fully characterizing something as complex as writing style has proven too unwieldy to admit fine-grained human annotations, which leaves the possibility of directly learning explicit and interpretable representations of writing style practically beyond reach. Instead, research in this area has largely focused on specific stylistic attributes, such as formality, toxicity, politeness, gender, simplicity, and humor, which are more straightforward to annotate (Rao and Tetreault, 2018; Pavlopoulos et al., 2020; Madaan et al., 2020; Li et al., 2018; Jin et al., 2022). Unfortunately, the reliance on human labels and the narrow focus of such stylistic distinctions severely limit the utility of such representations in tasks related to authorship, such as those listed above.
In this paper, we focus instead on the _authorship prediction task_, which enjoys the benefit of not requiring manually-elicited labels, since metadata in many corpora include either explicit author labels or usernames that may serve as proxies for latent authorship. As a result, the vast scale of data available for training authorship prediction models opens the door to learning _generalizable_ authorship representations using deep learning. We specifically consider similarity learning approaches that aim to produce vector representations of documents, where the distance between two vectors is inversely related to the likelihood that the corresponding documents were composed by the same author (Boenninghoff et al., 2019; Andrews and Bishop, 2019).
However, achieving high accuracy in the authorship prediction task does not necessarily imply that stylistic features have been successfully learned. For example, in a given corpus, correctly predicting that two writing samples were composed by the same author may be possible on the basis of non-stylistic signal, such as the topic of conversation. Therefore, this work is concerned with obtaining a better understanding of the na
ture of representations learned for the authorship prediction task.
Unfortunately, because deep learning models behave like black boxes, we cannot directly interrogate a model's parameters to determine what information such representations contain. For example, one might hope to employ attention-based approaches that provide post hoc explanations through token saliency maps Sundararajan et al. (2017). However, such methods provide no guarantee of the fidelity of their explanations to the underlying model. Furthermore, the subjective interpretation required to deduce the reasons that such methods highlight certain spans of text makes it nearly impossible to systematically draw conclusions about the model.
Instead, we propose targeted interventions to probe representations learned for the surrogate authorship prediction task. First, we explore masking content words at training time in SS5, an operation intended to gauge the degree to which a representation relies on content. Then we explore automatic paraphrasing in SS6, an operation intended to preserve meaning while modifying how statements are expressed. Finally, in SS7 we explore the capacity of these representations to generalize to unseen tasks, specifically topic classification and coarse style prediction.
Taken together, and despite approaching the research question from various points of view, our experiments suggest that representations derived from the authorship prediction task are indeed substantially stylistic in nature. In other words, success at authorship prediction may in large part be explained by having successfully learned discriminative features of writing style. The broader implications of our findings are discussed in SS8.
## 2 Related Work
Perhaps the work most closely related to our study is that of Wegmann and Nguyen (2021) and Wegmann et al. (2022) who propose measuring the stylistic content of authorship representations through four specific assessments, namely formality, simplicity, contraction usage, and number substitution preference. Our work differs in two main respects. First, we regard style as an abstract constituent of black-box authorship representations rather than the aggregate of a number of specific stylistic assessments. Second, the works above deal with stylistic properties of _individual sentences_, whereas we use representations that encode longer spans of text. Indeed, we maintain that the writing style of an author manifests itself only after observing a sufficient amount of text composed by that author. For example, it would be difficult to infer an author's number substitution preferences after observing a single sentence, which is unlikely to contain multiple numbers. The same is true of other stylometric features, such as capitalization and punctuation choices, abbreviation usage, and characteristic misspellings.
In another related work, Sari et al. (2018) find that although content-based features may be suitable for datasets with high topical variance, datasets with lower topical variance benefit most from style-based features. Like the works mentioned above, Sari et al. explicitly identify a number of style-based features, so writing style is more of a premise than the object of study. In addition, experiments in this previous work are limited to datasets featuring a small number of authors, with the largest dataset considered containing contributions of only 62 authors.
A number of end-to-end methods have been proposed to learn representations of authorship Andrews and Bishop (2019); Boenninghoff et al. (2019); Saedi and Dras (2021); Hay et al. (2020); Huertas-Tato et al. (2022). A common thread among these approaches is their use of _contrastive learning_, although they vary in the particular objectives used. They also differ in the the domains used in their experiments, the numbers of authors considered for training and evaluation, and their open- or closed-world perspectives. As discussed in SS3, we use the construction introduced in Rivera-Soto et al. (2021) as a representative neural method because it has shown evidence of capturing stylistic features through both its success in the challenging open-world setting in multiple domains and its performance in zero-shot domain transfer.
## 3 Authorship Representations
In this article, an _authorship representation_ is a function mapping documents to a fixed Euclidean space. The fact that such representations are useful for a number of authorship-related tasks is generally attributed to their supposed ability to encode author-specific style, an assertion we aim to validate in this paper. In this section, we describe how these representations arise and how they are
intended to be used.
Our analysis centers around representations \(f\) implemented as deep neural networks and trained using a _supervised contrastive objective_Khosla et al. (2020). At training time this entails sampling pairs of documents \(x,x^{\prime}\) composed by the same author (resp. by _different_ authors) and minimizing (resp. _maximizing_) the distance between \(f\left(x\right)\) and \(f\left(x^{\prime}\right)\). Therefore, we may assume at inference time that \(f\left(x\right),f\left(x^{\prime}\right)\) are closer together if \(x,x^{\prime}\) were composed by the same author than they would be if \(x,x^{\prime}\) were composed by different authors. No meaning is ascribed to any attribute of \(f\left(x\right)\), such as its coordinates, its length, or its direction. Rather, \(f\left(x\right)\) is meaningful only in relation to other vectors.
In all the experiments of this paper \(f\) is an instance of the _Universal Authorship Representation_ (UAR) introduced in Rivera-Soto et al. (2021).1 Notwithstanding the merits of a number of other recent approaches discussed in SS2, we argue that because of its typical neural structure and typical contrastive training objective, UAR serves as a representative model. The same paper also illustrates that UAR may be used for zero-shot transfer between disparate domains, suggesting a capacity to learn generalizable features, perhaps of a stylistic nature, thereby making it a good candidate for our experiments.
Footnote 1: An open-source implementation is available at [https://github.com/LLNL/LUAR](https://github.com/LLNL/LUAR).
Note that UAR defines a _recipe_ consisting of an architecture and a training process that must be carried out in order to arrive at a representation, with the understanding that care must be taken in assembling appropriate training datasets. Specifically, we consider a diverse set of authors at training time in an effort to promote representations that capture _invariant_ authorship features, chiefly writing style, rather than time-varying features, such as topic. Invariance is a desirable feature of authorship representations because it improves the likelihood of _generalization_ to novel authors or even to novel domains. However, there is no guarantee that invariant features are exclusively stylistic in any given corpus, or that any training process we might propose will result in representations capturing exclusively invariant features. Therefore, this work is concerned with estimating the _degree_ to which authorship representations are capable of capturing stylistic features, with the understanding that completely disentangling style from topic may be beyond reach.
## 4 Experimental Setup
Mirroring Rivera-Soto et al. (2021), we conduct experiments involving three datasets, each consisting of documents drawn from a different domain. For the reader's convenience, we present further details and some summary statistics of these datasets in SSA.1.
To evaluate an authorship representation, we use the common experimental protocol described below. The objective is to use the representation to retrieve documents by a given author from among a set of candidate documents, which are known as the _targets_, on the basis of the distances between their representations and the representation of a document by the desired author, which is known as the _query_. To this end, each evaluation corpus has been organized into queries and targets, which are used to calculate the _mean reciprocal rank (MRR)_. Following is a friendly description of this metric, with a more elaborate formulation presented in SSA.2.
An authorship representation may be used to sort the targets according to the distances between their representations and that of any fixed query. In fact, this ranking is often seen as the primary outcome of an authorship representation. Because one would need to manually inspect the targets in the order specified by the ranking, it would be desirable for any target composed by the same author as the query to appear towards the beginning of this list. The MRR is the expectation of the reciprocal of the position in the ranked list of the first target composed by the same author as a randomly chosen query. This metric ranges from 0 to 1, with higher values indicating a greater likelihood of finding documents composed by an author of interest within a large collection in the first few search results.
Following Rivera-Soto et al. the queries and targets in all our experiments are _episodes_, each consisting of 16 comments or product reviews contiguously published by the same author in the Reddit or Amazon domains, respectively, or 16 contiguous paragraphs of the same story in the fanfic domain.
In order to conduct a wide variety of experiments in a time-efficient manner, we train all representations on one GPU for 20 epochs, although
we acknowledge that better results may be obtained by training with more data, on multiple GPUs, or for longer than 20 epochs.
## 5 Masking Content Words
Our first series of experiments aims to illustrate through a simple training modification that authorship representations are capable of capturing style. Specifically, the strategy of _masking_ training data in a way that preserves syntactic structure, something which is known to relate to style, while removing thematic or topical information, has been effective, particularly in cross-domain authorship experiments Stamatatos (2018). To this end, we propose training authorship representations with restricted access to topic signal by masking varying proportions of content-related words in the training data. Evaluating each of these representations and comparing its ranking performance with that of a representation trained on the same _unmasked_ data reveals the capacity of the representation to capture style.
Words may be roughly divided into two categories: _content words_ and _function words_. Content words primarily carry topic signal. They tend to include nouns, main verbs, adjectives, and adverbs. Function words serve syntactic roles and convey style through their patterns of usage. They tend to include auxiliary verbs, prepositions, determiners, pronouns, and conjunctions Mosteller and Wallace (1964). These observations suggest masking words according to their parts of speech (POS), a process we call _Perturbing Linguistic Expressions_ or _PertLE_.
### The PertLE schema
In our PertLE masking schema, we replace all words belonging to certain POS categories with a distinguished masking token. This approach stems from the observation that content words may often be distinguished from function words on the basis of POS. However, this is simply a heuristic and there are many exceptions. For instance, although many adverbs may be categorized as content words, such as _happily_, others play a functional role, such as _instead_. Because masking on the basis of POS is an imperfect strategy to eliminate content, we introduce the following _levels_ of the PertLE schema. In our _PertLE Grande_ schema we mask all nouns, main verbs, adjectives, and adverbs. This is a greedy approach intended to mask words that could possibly convey content, at the expense of occasionally masking some function words. In contrast, in our _PertLE_ schema we mask only nouns, which are most likely to carry content information.2 In a follow-up reported in SSA.3 we repeat the main experiment below using a masking schema based on TF-IDF scores rather than POS.
Footnote 2: We also tried masking every word belonging to certain POS categories with a distinguished _pseudoword_ specific to its POS. These pseudowords were selected to be morphologically similar to other words in their POS categories but not appear in our corpora. However, we adopt the simpler masking approach described above because it surprisingly produced very similar ranking results.
ProcedureTo identify POS categories, we use the Stanford NLP Group's Stanza tokenizer and POS tagger Qi et al. (2020) due to their efficiency, flexibility, versatility, and capacity for handling other languages. We use the Universal POS (UPOS) tagset because it distinguishes between main verbs (VERB) and auxiliary verbs (AUX), labels _not_ and _-n't_ as particles rather than adverbs, and tags many foreign language words with their correct POS category rather than labeling them as foreign words. For both masking levels, we replace each word to be masked with SBERT's masking token, \(<\)mask\(>\), preserving any contracted particles (e.g. gonna \(\rightsquigarrow\)\(<\)mask\(>\)na). As an example, Figure 1 illustrates both levels of the PertLE schema applied to the same statements.
Using the procedure described in Rivera-Soto et al. (2021), for each domain we train multiple authorship representations on that domain's training corpus: one with the training corpus masked according to PertLE Grande, one masked according to PertLE Lite, and one unmasked to serve as a baseline. We evaluate each representation on each _unmasked_ evaluation corpus to afford a fair comparison of the effects of the masking level for each combination of training and evaluation domain.
Note that for representations trained on _masked_ data, this evaluation introduces a _mismatch_ between training and evaluation datasets, although the baseline representations remain unaffected. In cases where masking results in a large _degradation_ in performance, this setup makes it impossible to distinguish between our interventions and the train-test mismatch as the cause of the degradation. On the other hand, this distinction is immaterial in the case that masking does _not_ degrade performance, and in fact, this case is the desired out
come of the experiment, as it would suggest that the corresponding representation does not benefit significantly from the information withheld by masking content words.
The results of the experiment are shown in Figure 2. For each corpus and each masking level, we independently trained _three_ representations in an effort to reduce variance. Each number reported is the sample mean of the MRR according to each of the three independent representations, where 0.014 is the maximum sample standard deviation over all experiments reported in the figure.
DiscussionA one-way ANOVA was performed for each combination of training and evaluation domain, showing that the masking schema had a statistically significant impact on the mean values of the MRR reported in Figure 2 with \(p<0.01\) in all cases except the case with training domain Amazon and evaluation domain fanfic. In that case, we conclude that masking words at training time had no significant effect on ranking performance, as desired. In the other cases the change in performance _was_ significant but relatively minor.
In cases where performance _improved_, we believe the most likely explanation is that, deprived of content words at training time, the model was forced to discover other authorship features, which turned out to be more useful than content words in the corresponding evaluation domains. In cases where performance _dropped_, it appears that the model was unable to compensate for the loss of content words. However, we emphasize that in these cases the drop in performance is surprisingly small in light of the fact that PertLE Grande masks nearly _half_ of all training tokens. See Table 4 in SSA.3 for the exact proportions of tokens
Figure 1: Various levels of the PertLE masking schema applied to the same statements.
Figure 2: MRR results for models trained on unmasked data (U), or data masked according to the PertLE Grande (G), the PertLE Lite (L), or additionally for fanfic, the PertLE Xtra-Lite (X) schema.
masked in each domain or Figure 1 for a qualitative example. Another possibility is that, as discussed above, PertLE Grande obscures writing style to some extent, which could also account for the small drop in ranking performance if the representations were primarily style-focused.
For all three training and evaluation domains, the MRR of the Lite model is quite close to that of the unmasked model. This suggests that masking words most likely to convey content changes ranking performance very little, and even improves it in some settings. We also observe that although the MRR of each Grande model is generally less than that of the corresponding Lite model, it is not dramatically so. This suggests that increasing the proportion of tokens masked appears to eventually impair ranking performance, but not to the degree one might expect given the considerable proportion of words masked.
We know of no way to _completely_ redact the content of a document while retaining its writing style. We doubt that this is even possible, least of all in an automated fashion. It follows that the representations trained on data masked according to the PertLE schema (as well as the PertLE schema discussed SSA.3) probably _do_ encode a small amount of content. Being trained to distinguish authors on the basis of such masked text, these models are therefore likely to learn to use that information to their advantage when appropriate, which would mean that the representations considered in this paper _do_ convey a small amount of topical signal, an observation which is corroborated by the experiments in SS6 and SS7.
Nevertheless, the experiment shows that PertLE obscures _much_ of the content of a training corpus, which in turn affects ranking performance only marginally. We argue that those representations are therefore likely to have learned to avail of features other than content, thereby illustrating their _capacity_ to avail of writing style.
### PertLE Xtra-Lite
As observed in Rivera-Soto et al. (2021) the representation trained on the fanfic corpus generalizes poorly to the other two domains, something which is probably due to the comparatively small size and lack of topical diversity of that dataset. This suggests that representations trained on fanfic stand to improve the most by a targeted inductive bias. Indeed, the Lite model trained on the fanfic dataset improves performance in all three evaluation domains. This may be explained by the observation that the fanfic domain may contain more jargon and specialized language appearing in the form of proper nouns representing names, places, and things. This is borne out by the observation that in the Reddit, Amazon, and fanfic domains, around 22%, 20%, and 35% respectively of all nouns are proper.
To further explore this observation we introduce the _Xtra-Lite_ level of the PertLE schema, in which we mask only proper nouns, the POS category most likely to convey content information. Repeating the same procedure as before, we train a PertLE Xtra-Lite (X) representation on the fanfic domain and evaluate it on each unmasked evaluation dataset. The results in Figure 2 show that the Xtra-Lite model not only outperforms the unmasked and Grande models in all three domains, but also outperforms the Lite model in the Reddit and fanfic domains and performs nearly as well in the Amazon domain, confirming that representations trained on fanfic benefit from a targeted inductive bias.
## 6 Removing Style by Paraphrasing
In contrast with the experiments in SS5 that aim to assess the ability of authorship representations to capture style by removing _content_, our next group of experiments aims to make the same assessment by instead removing _style_. For this purpose we consider automatic paraphrasing, which ideally introduces stylistic changes to a document while largely retaining its original content. If an authorship representation avails of stylistic features, then we expect paraphrasing a document to _impair_ its ability to match the document with other documents by the same author.
### Implementation details
To generate paraphrases, we use the STRAP paraphraser developed by Krishna et al. (2020), which consists of a fine-tuned GPT-2 language model trained on ParaNMT-50M Wieting and Gimpel (2018), a large dataset consisting of paraphrases of English sentences.
Because automatic paraphrasing models provide no guarantee that the proposed paraphrases of a document retain its meaning, we need to check this explicitly. For this purpose we adopt the BERTScore Zhang et al. (2019) as our pri
mary similarity metric, rescaled to the unit interval. Unlike \(n\)-gram-matching metrics like Bleu, BERTScore leverages contextual embeddings to make predictions about semantic similarity.
Because STRAP acts on _sentences_ rather than _episodes_, we apply it independently to each sentence comprising an episode, with the following caveat. Preliminary experiments revealed that the degree to which automatic paraphrasing retained meaning varied widely, an issue that we mitigate as follows. For each of the sixteen constituent documents \(x\) of an episode, we paraphrase the sentences within \(x\) to obtain \(x^{\prime}\) and calculate the mean BERTScore of each sentence of \(x\) with its paraphrase in \(x^{\prime}\). We discard the eight \(x^{\prime}\) with lowest mean BERTScore to form the paraphrased episode, and also drop the eight corresponding \(x\) from the original episode for comparability.
### Impact of paraphrasing on ranking
For each domain, we train an authorship representation in the usual way, perform the primary ranking experiment described in SS4, and repeat the experiment after paraphrasing all the queries in the manner described in SS6.1. In Table 1 we report the MRR for the original experiment (Orig), the MRR for the paraphrased variation (Para), and the change in performance (\(\Delta\)) for each domain. To serve as a baseline, we repeat the entire experiment with the SBERT model in place of the trained authorship representation, which is denoted by UAR in Table 1.
For each domain and each model, the MRR substantially decreased for the paraphrased queries relative to the original queries, which confirms that both models rely to some extent on author style. However, the performance degradation was much more pronounced for UAR than for SBERT, suggesting that UAR captures style to a much greater extent than SBERT.
For each domain and each model, a paired \(t\) test shows that the decrease in MRR of the paraphrased queries relative to that of the original queries is significant with \(p<0.01\). Additionally, for each domain, a further \(t\) test shows that the difference between the two models of the differences in MRR between the original and paraphrased queries is significant with \(p<0.01\).
We also present the results of this experiment in a more qualitative way in Figure 3. Recall from SSA.2 that for each query \(q_{i}\) and its corresponding target \(t_{i}\), our primary ranking experiment entails ranking all the targets \(t_{1},t_{2},\ldots,t_{N}\) according to their similarity to \(q_{i}\) and reporting the position \(r_{i}\) of \(t_{i}\) in the ranked list. In Figure 3 we plot \(r_{i}\) against \(r^{\prime}_{i}\) for each \(1\leq i\leq M\), where \(r^{\prime}_{i}\) is the position of \(t_{i}\) in the list ranked according to similarity to \(q^{\prime}_{i}\), the paraphrase of \(q_{i}\).
Examples for which \(r_{i}\approx r^{\prime}_{i}\) correspond to points near the diagonal line shown, whereas examples for which \(r_{i}>r^{\prime}_{i}\) correspond to points above this line. Note that for the UAR model, most points lie above the diagonal line, while for SBERT, the points are more evenly distributed across both sides of the line. This again suggests that paraphrasing impairs the ranking performance of UAR much more dramatically than that of SBERT.
### Quality of paraphrases
If the premise that our paraphrases retain meaning but alter style were not satisfied, then we would not be able to infer that paraphrasing the queries in SS6.2 is responsible for the drop in ranking performance. To confirm that the premise is largely satisfied, we propose the following metrics, both averaged over all the sentences comprising a query. To assess the degree of content preservation between a query \(q\) and its paraphrase \(q^{\prime}\), we calculate their BERTScore. To confirm that \(q^{\prime}\) significantly modifies the style of \(q\), rather than making only minor changes, we calculate their _normalized edit distance_. While neither metric is perfect, together they provide a reasonable estimate of paraphrase quality.
As a baseline, we calculate the same metrics for the Microsoft Research Paraphrase Corpus (MRPC) Dolan and Brockett (2005). Figures 4
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Domain** & **Model** & **Orig** & **Para** & \(\Delta\) \\ \hline \multirow{2}{*}{**fanfic**} & **UAR** & 0.325 & 0.139 & **0.186** \\ & **SBERT** & 0.203 & 0.167 & 0.036 \\ \hline \multirow{2}{*}{**Reddit**} & **UAR** & 0.263 & 0.026 & **0.237** \\ & **SBERT** & 0.043 & 0.026 & 0.017 \\ \hline \multirow{2}{*}{**Amazon**} & **UAR** & 0.266 & 0.025 & **0.241** \\ & **SBERT** & 0.165 & 0.069 & 0.096 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The impact on ranking performance of paraphrasing queries. Paraphrasing drastically impairs ranking performance of the UAR model relative to the baseline SBERT model, suggesting a reliance on stylistic features.
and 5 show that the distributions of both scores overlap substantially with those for the MRPC corpus restricted to sentence pairs deemed to constitute paraphrases by human annotators, which is labeled by MRPC\(+\). As a further baseline, Figure 4 also shows the distribution of MRPC scores restricted to pairs deemed _not_ to constitute paraphrases, which is labeled by MRPC\(-\). Therefore we may rule out the possibility that the drop in ranking performance in SS6.2 might be due to low-quality paraphrases.
### Impact of content overlap
As a final illustration, in Figure 6 we plot the BERTScore\(b_{i}\) of \(q_{i}\) with \(q^{\prime}_{i}\) against the change in rank \(\Delta r_{i}=r^{\prime}_{i}-r_{i}\) for all \(1\leq i\leq M\). If authorship representations were significantly influenced by content, then we might expect to see a strong negative relationship between \(b_{i}\) and \(\Delta r_{i}\). Instead, we observe little correlation, with Kendall's \(\tau\) values of \(-0.092\), \(-0.019\), and \(-0.015\) for the fanfic, Reddit, and Amazon domains respectively, suggesting that the ranking performance degradation in SS6.2 cannot be well-explained by content overlap between \(q_{i}\) and \(q^{\prime}_{i}\).
### A further application
Although beyond the scope of this paper, we remark that a broader research problem is to determine whether the capacity of an authorship representation to encode style is _correlated_ with its performance on the authorship attribution task. For example, if a new representation were introduced that performed better than UAR on attribution, would it necessarily encode style to a greater degree than UAR? Conversely, if a new approach were proposed to learn representations that encode style to a greater degree than UAR, would such representations perform better on attribution?
Addressing these questions will require assessing the degree to which a representation encodes
Figure 4: Distribution of BERTScores comparing documents to their paraphrases.
Figure 5: Distribution of edit distances between documents and their paraphrases.
Figure 3: Rankings \(r_{i}\) against \(r^{\prime}_{i}\). UAR has more points above the diagonal line \(r=r^{\prime}\) than SBERT, which correspond with queries for which paraphrasing hurts ranking performance.
style. We submit that the experiments presented in this paper are well-suited to making such assessments. As an illustration, we repeat the primary experiment described in SS6.2 using two further instances of the UAR architecture introduced in SS3, but trained on the Reddit histories of around 100K and 5M authors respectively, in contrast with the version used throughout this paper, which was trained on the histories of around 1M authors.3
Footnote 3: We trained the smaller model with the dataset released by Andrews and Bishop (2019). For the larger model we queried Baumgartner et al. (2020) for comments published between January 2015 and October 2019 by authors publishing at least 100 comments during that period. All three models were trained using the default hyperparameter settings of [https://github.com/LLML/LUAR](https://github.com/LLML/LUAR). The MRRs of UAR19, UAR, and UAR23 evaluated on a test set comprised of comments published future to those comprising the three training corpora are 0.482, 0.592, and 0.682 respectively.
A paired \(t\) test shows that the difference in rank induced by paraphrasing is significant with \(p<0.01\) for all three models. These differences are positively correlated with the MRR scores of the corresponding models, which are shown in footnote 3: We trained the smaller model with the dataset released by Andrews and Bishop (2019). For the larger model we queried Baumgartner et al. (2020) for comments published between January 2015 and October 2019 by authors publishing at least 100 comments during that period. All three models were trained using the default hyperparameter settings of [https://github.com/LLML/LUAR](https://github.com/LLML/LUAR). The MRRs of UAR19, UAR, and UAR23 evaluated on a test set comprised of comments published future to those comprising the three training corpora are 0.482, 0.592, and 0.682 respectively.
## 7 Generalization to Novel Tasks
Our experiments have thus far focused on authorship prediction, a task which is presumably best addressed with a model capable of distinguishing writing styles. We now use authorship representations to _directly_ distinguish writing styles using a corpus of documents furnished with style annotations, namely the CDS dataset Krishna et al. (2020). This consists of writings in disparate styles, including writings by two classical authors, namely Shakespeare and Joyce, historical American English from different eras, social media messages, lyrics, poetry, transcribed speech, and the Bible. With the notable exception of the two classical authors, most styles in CDS are not author-specific, but rather, represent broad stylistic categories. This means that identifying CDS styles is not the same problem as authorship prediction, an important observation we revisit below.
In addition, we repeat the experiment with a corpus furnished instead with _topic_ annotations, namely the Reuters21578 dataset Lewis (1997). This is a popular benchmark in text classification consisting of financial news articles, each annotated by one or more topics. We note that certain topics may be associated with particular authors and editors, and therefore style could be a spurious correlate, although we nevertheless expect the authorship representation to perform worse _relative_ to the semantic baseline described below.
For each corpus, the experiment consists of simply applying an authorship representation trained on the Reddit dataset to two randomly chosen documents from the corpus. We used Reddit because it has been shown to yield representations that generalize well to other domains Rivera-Soto et al. (2021). We record the dot product of the resulting vectors and pair this score with a binary indicator specifying whether the two documents carry the same labels. Noting that predicting the binary indicator from the dot product is a highly imbalanced problem, with most document pairs bearing _non-matching_ rather than _matching_ labels, we simply construct the corresponding receiver op
\begin{table}
\begin{tabular}{l c c c c} \hline \hline & \multicolumn{2}{c}{**Training**} & & \\
**Model** & **Examples** & **Orig** & **Para** & \(\Delta\) \\ \hline
**UAR23** & 5M & 0.293 & 0.032 & 0.261 \\
**UAR** & 1M & 0.263 & 0.026 & 0.237 \\
**UAR19** & 100K & 0.188 & 0.019 & 0.169 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Impact of paraphrasing on attribution performance for authorship representations trained on varying numbers of Reddit users.
Figure 6: BERTScore against change in rank \(\Delta R\). BERTScore is minimally correlated with \(\Delta R\), suggesting that \(\Delta R\) is not a function of content overlap.
erating characteristic (ROC) curve, an illustrative device intended to explore the tradeoffs in making that prediction by thesholding. We report the equal error rate (EER), a simple summary statistic of the ROC curve. Smaller values of this metric are preferable. For good measure, we also report the area under that curve (AUC) in SSA.4, another summary statistic of the ROC curve.
Finally, because writing style may be difficult to assess without sufficient text content, we also vary the amount of text contributing to the dot products mentioned above. Specifically, rather than predicting whether the label of a _single_ document matches that of another on the basis of the dot product of their representations, we more generally predict whether a _group_ of randomly chosen documents of the same label shares that label with another group of randomly chosen documents sharing another label on the basis of the dot product of the means over each group of the representations of their constituents.
As a baseline we repeat both experiments using the general purpose document embedding SBERT in place of the authorship representation. SBERT is commonly regarded as a semantic embedding, but is not typically used to discriminate writing styles without further training.
The rationale for the experiment is the following. If the authorship representation primarily encodes stylistic features, then we would expect _poor performance_ relative to SBERT on the topic discrimination task since the task presumably does not involve stylistic distinction. However, we would expect _better performance_ from the authorship representation than SBERT on the style discrimination task.
These expectations are borne out in the results reported in Figures 7 and 8, which show EER against the number of input documents for each task and each model. The generalization performance of UAR on these novel tasks relative to SBERT is consistent with a representation that is sensitive to stylistic information. Namely, SBERT consistently outperforms UAR on topic classification, while UAR consistently outperforms SBERT on style classification. We present 95% confidence intervals for each curve as lighter regions of the same color surrounding the curve. Although these were calculated using a bootstrap approach, the confidence intervals of the corresponding AUC results shown in Figures 10 and 11 of SSA.4 were calculated using a bootstrap-free calculation.
Also shown in Figures 7 to 11 are the results of the same experiments using the two variations of UAR introduced in SS6.5. These additional models were included to support an auxiliary argument raised in SS8, but also afford an interesting but subtle insight about the current task. Namely, although UAR19 performs _strictly worse_ and UAR23 _strictly better_ than UAR on the authorship attribution task, classifying style into broad categories is a different problem than authorship attribution, the latter dealing with fine-grained stylometric features. This accounts for the seemingly contradictory results in Figure 8, in which UAR19 performs _better_ than UAR, which in turn performs better than UAR23. Indeed, training UAR on _more_ authors produces representations that are more discriminative of individual authors, something which is at odds with identifying broad stylistic categories for the simple reason that being exposed to more authors affords more opportunities for UAR to discover stylistic features that distinguish authors.
Notwithstanding these observations, we remark that within the CDS dataset, certain styles are likely to be correlated with particular topics, while in the Reuters dataset, certain authors are likely to often write about particular topics. This would suggest that SBERT might perform better on CDS and UAR better on Reuters than one might expect, so the absolute performance on both tasks is not particularly informative.
## 8 Discussion
FindingsWe have examined properties of an exemplary authorship representation construction, finding consistent evidence that the success of the representations it engenders at distinguishing authors may be attributed in large part to their sensitivity to style. First, the masking experiments of SS5 show that for sufficiently large training corpora, masking a large fraction of content words at training time does not significantly affect ranking performance on held-out data, suggesting that these representations are largely invariant to the presence of content words. On the other hand, the paraphrasing experiments of SS6, which seek to alter writing style while preserving content, confirm that paraphrasing drastically impairs ranking performance. Taken together, these experiments suggest that the authorship representations considered are indeed sensitive to stylistic features. This conclusion is corroborated in SS7 where we see poor generalization of one of these representations to a topic discrimination task, but better generalization to a style discrimination task, both assessments relative to a semantic baseline.
LimitationsAll of the experiments in this paper involve instances of the UAR construction. Since our primary research question involves testing the capacity of representations trained for the authorship prediction task to capture stylistic features, we select this construction because there is prior evidence that the representations it engenders perform well at zero-shot cross-domain transfer, for certain combinations of source and target domains, which likely requires some degree of stylistic sensitivity Rivera-Soto et al. (2021).
By design, our analysis is focused on aggregate model behavior. While this addresses the high-level research questions we pose in the introduction, such _global_ analysis does not enable predictions about which specific _local_ features are involved in model predictions. As such, an important avenue for future work lies in developing methods that can faithfully identify local authorship features. To this end, frameworks for evaluating the quality of explanations, such as Pruthi et al. (2022), are essential. Beyond the usual challenges of explaining the decisions of deep neural networks, explaining author style may pose further challenges, such as the need to identify groups of features that in combination predict authorship.
We emphasize that completely disentangling style from content may not be attainable, since certain aspects of writing likely blur the line between style and content Jafaritazehjani et al. (2020). For example, we notice degradation in ranking performance of the SBERT model in Table 1, suggesting that to some extent, SBERT features are also stylistic. Nonetheless, UAR exhibits a markedly larger degradation in performance, suggesting a greater degree of sensitivity to writing style.
Broader ImpactThis work contributes to the broader goal of formulating notions of _content_ and _style_ that constitute mutually exclusive and collectively exhaustive aspects of writing that may be embedded in orthogonal subspaces of a Euclidean space. Not knowing whether this ambition is fully realizable, but hopeful others will explore the question in future research, we resign ourselves in this paper to exhibiting two embeddings that accomplish the objective to a limited extent. Specifically, we focus on UAR, which we show to mostly encode style, and SBERT, which is widely assumed to encode content. This being an imperfect decomposition, the primary goal of this paper is to qualitatively assess the _degree_ to which UAR encodes style rather than content.
Authorship attribution is likely to be a task that benefits from a representation that is relatively stable over time, specifically an encoding capturing primarily writing style. To this end, another open question is whether a representation may be constructed that encodes style to a _greater_ degree than
Figure 8: Equal error rate (EER) for UAR and SBERT on style distinction as the size of the writing sample is varied. Smaller values of EER correspond with better performance.
UAR, and if so, whether the representation improves performance on the attribution task. If such a representation were proposed in the future, the experiments we propose in this paper could be used to validate the assertion that it encodes style to a greater degree than UAR.
On the other hand, content features constitute perfectly legitimate discriminators of authorship in some cases. For example, an author who discusses only a narrow range of topics on a particular forum may easily be distinguished from other authors on the basis of topic features. Not knowing under which circumstances and to what degree content plays a role in authorship attribution, we maintain that the relationship between the performance of a representation on the attribution task and the degree to which the representation encodes content should be explored fully, something that will again require an estimate of the degree to which a representation encodes style.
Another promising application of authorship representations is _style transfer_, where one hopes to rephrase a given statement in the style of a given author. This has been analogously accomplished in the domain of speech, resulting in the ability to have a given statement recited in the voice of a given speaker (see e.g. Kim et al. (2021)). The primary ingredient in this task is a speaker embedding, which is analogous to an authorship representation. However, by construction, a speaker embedding encodes almost exclusively _acoustic_ features, but encodes content features, namely the specific words spoken, to a very limited degree. The fact that this observation might be the primary reason for the success of speech transfer portends possible difficulties for the style transfer task. However, as with authorship attribution, the relationship between the success of using a representation for style transfer and the degree to which the representation encodes style should also be fully explored, and again, the experiments proposed in this paper would constitute a natural assessment of that degree.
## Acknowledgments
We thank the TACL reviewers and action editors for their insightful comments. We also thank Carina Kauf for the initial masking idea and Hope McGovern for early discussions on PertLE. Part of this work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DEAC52-07NA27344.
|
2307.00318 | Calculation of asymptotic charges at the critical sets of null infinity | The asymptotic structure of null and spatial infinities of asymptotically
flat spacetimes plays an essential role in discussing gravitational radiation,
gravitational memory effect, and conserved quantities in General Relativity.
Bondi, Metzner and Sachs established that the asymptotic symmetry group for
asymptotically simple spacetimes is the infinite-dimensional BMS group. Given
that null infinity is divided into two sets: past null infinity
$\mathscr{I}^{-}$ and future null infinity $\mathscr{I}^{+}$, one can identify
two independent symmetry groups: $\text{BMS}^{-}$ at $\mathscr{I}^{-}$ and
$\text{BMS}^{+}$ at $\mathscr{I}^{+}$. Associated with these symmetries are the
so-called BMS charges. A recent conjecture by Strominger suggests that the
generators of $\text{BMS}^{-}$ and $\text{BMS}^{+}$ and their associated
charges are related via an antipodal reflection map near spatial infinity. To
verify this matching, an analysis of the gravitational field near spatial
infinity is required. This task is complicated due to the singular nature of
spatial infinity for spacetimes with non-vanishing ADM mass. Different
frameworks have been introduced in the literature to address this singularity,
e.g., Friedrich's cylinder, Ashtekar-Hansen's hyperboloid and Ashtekar-Romano's
asymptote at spatial infinity. This paper reviews the role of Friedrich's
formulation of spatial infinity in the investigation of the matching of the
spin-2 charges on Minkowski spacetime and in the full GR setting. | Mariem Magdy Ali Mohamed | 2023-07-01T12:07:22Z | http://arxiv.org/abs/2307.00318v2 | # The calculation of the asymptotic charges at the critical sets of null infinity
###### Abstract
The studies of the asymptotic structure at null or spatial infinity in General Relativity play different roles in the discussion of gravitational radiation, gravitational memory effect, and conserved quantities. Bondi, Metzner, and Sachs established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity \(\mathcal{I}^{-}\) and future null infinity \(\mathcal{I}^{+}\), one can identify two independent symmetry groups: BMS\({}^{-}\) at \(\mathcal{I}^{-}\) and BMS\({}^{+}\) at \(\mathcal{I}^{+}\). Associated with these symmetries are the so-called BMS charges. Recently, it was suggested that the generators of BMS\({}^{-}\) and BMS\({}^{+}\) and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks are introduced to address this singularity. This paper reviews two formulations and their relationship: Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity. It also reviews the role of Friedrich's cylinder at spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting.
## 1 Introduction
In classical General Relativity (GR), isolated systems are generally described as asymptotically flat spacetimes. The BMS group [1], named after Bondi, Metzner and Sachs, is the infinite-dimensional symmetry group for asymptotically flat spacetimes. A conjecture by Strominger [2] suggests that the asymptotic symmetries and charges for asymptotically flat spacetimes can be linked to soft theorems [3, 2, 4] and the gravitational memory effect [5, 6, 7]. This link is based on the idea that the BMS groups at past and future null infinities (BMS\({}^{-}\) at \(\mathcal{I}^{-}\) and BMS\({}^{+}\) at \(\mathcal{I}^{+}\)) can be matched by an antipodal reflection map near spatial infinity. The matching of these symmetries leads to a global diagonal symmetry group in GR. Recently, Capone et al. [8] derived the map relating the asymptotic data and their charges in the limits of spatial infinity. This map was used in [8] to show that the different BMS charges defined in the literature match with the conserved charges at spatial infinity.
Generically, the matching of BMS\({}^{+}\) and BMS\({}^{-}\) and their associated charges requires an analysis of the gravitational field and the charges near spatial infinity. However, the standard conformal representation of asymptotically flat spacetimes is not well suited for this discussion, mainly due to the singular nature of the conformal structure near spatial infinity \(i^{0}\). To this end, different formulations are used to overcome this singular behaviour [9, 10, 11, 12], and in recent years, numerous articles discussed the asymptotic symmetry group at spatial infinity [13, 14, 15, 16, 17, 18] and their matching with the asymptotic charges at null infinities [13, 14, 15, 19, 20, 21].
The notion of asymptotic flatness mentioned earlier classifies spacetimes that resemble Minkowski at large null distances, and it is not concerned with the behaviour of the gravitational field at spatial infinity.
In an attempt to rectify this, the notion of asymptotically Minkowskian spacetimes at spatial infinity was introduced in [11]. The central idea behind this definition is that the standard conformal representation of Minkowski forces all points at spatial infinity to be mapped to a single point \(i^{0}\). To resolve the structure of spatial infinity, one is compelled to give up on the idea of a conformal rescaling of the spacetime metric in order to blow up \(i^{0}\) to a \(3-\)dimensional unit timelike hyperboloid. This hyperboloid is known as the hyperboloid at spatial infinity. This idea can be carried over to more general spacetimes by introducing the asymptote at spatial infinity \(\mathcal{H}\) which acts as a timelike boundary of a 4-dimensional manifold; this is similar to Penrose's definition of asymptotic simplicity in which \(\mathcal{I}\) acts as a null boundary for the conformal manifold. Then a spacetime is said to be asymptotically Minkowskian at spatial infinity (AMSI) if the asymptote \(\mathcal{H}\) satisfies the following: i) the boundary \(\mathcal{H}\) is at infinity with respect to the physical metric, ii) the boundary \(\mathcal{H}\) is timelike, the intrinsic metric \(q_{ab}\) on \(\mathcal{H}\) and the normal \(n^{a}\) admit smooth limits to \(\mathcal{H}\), iii) the vacuum Einstein equations holds in the limits of \(\mathcal{H}\), and finally iv) the boundary \(\mathcal{H}\) have the topology \(\mathbb{R}\times\mathbb{S}^{2}\) and is geodesically complete.
Another formulation used in literature [22, 23, 24, 25, 26, 27] to resolve the structure of spatial infinity is Friedrich's formulation of spatial infinity originally introduced in [12]. The motivation for Friedrich's formulation of spatial infinity is to obtain a regular initial value problem at spatial infinity for the conformal Einstein field equations. This representation of spatial infinity is linked to the conformal properties of spacetimes, and it introduces a blow-up of the spatial infinity point \(i^{0}\) to a cylinder \((-1,1)\times\mathbb{S}^{2}\) commonly known as the cylinder at spatial infinity \(\mathcal{I}\). The cylinder \(\mathcal{I}\) touches the endpoints of past and future null infinities at the critical sets \(\mathcal{I}^{\pm}=\{\pm 1\}\times\mathbb{S}^{2}\). This representation of spatial infinity is useful for relating quantities at the critical sets \(\mathcal{I}^{\pm}\) to initial data on a Cauchy hypersurface -- See [28, 26]. In [29], we show that Ashtekar's representation of spatial infinity can be related to Friedrich's formulation, a brief discussion of this relation will be provided in this work.
The purpose of this paper is to provide a streamlined presentation of the calculation of the asymptotic charges at the critical sets using Friedrich's formulation of spatial infinity. The full analysis of the asymptotic charges in a full GR setting using Friedrich's formulation will be presented elsewhere. However, the main results can be summarised as follows:
_For the generic initial data set given in [30], the asymptotic charges associated with supertranslation symmetries at \(\mathcal{I}^{\pm}\) are well-defined if and only if the initial data satisfy extra regularity conditions. The regularity conditions can be imposed on the free conformal initial data. Finally, given initial data that satisfy the regularity conditions, the asymptotic charges at \(\mathcal{I}^{+}\) are equal to the charges at \(\mathcal{I}^{-}\)._
The structure of this paper will be as follows: In Section 2, a brief discussion of the relation between Ashtekar's and Friedrich's formulations of spatial infinity is presented. In Section 3, the calculation of the spin-2 asymptotic charges on Minkowski spacetime using Friedrich's formulation is reviewed. Finally, the tools and techniques used in the analysis of the asymptotic charges in full GR are presented in Section 4.
### Notations and conventions
This article will use tensors and spinors separately in various calculations. The following indices will be used:
* \(a,\,b,\,c,\ldots\): spacetime abstract tensorial indices.
* \(i,\,j,\,k,\ldots\): spatial abstract indices
* \(\mu,\,\nu,\ldots\): spacetime coordinate indices.
* \(\alpha,\,\beta,\ldots\): spatial coordinate indices.
* \(\mathcal{A},\mathcal{B},\mathcal{C},\ldots\): coordinate indices on a 2-sphere.
* \(A,\,B,\,C,\ldots\): abstract spinorial indices.
The components of a tensor \(T_{ab}\) with respect to a tensorial frame \(\{\mathbf{e_{a}}\}\) are defined as
\[T_{\mathbf{ab}}=T_{ab}\mathbf{e_{a}}^{a}\mathbf{e_{b}}^{b}.\]
Similarly, if \(\{\mathbf{o},\mathbf{\iota}\}\) is a spin basis defined by
\[o^{A}\equiv\mathbf{\epsilon_{0}}^{A},\qquad\iota^{A}\equiv\mathbf{\epsilon_{1}}^{A},\]
then the components of a spinor \(\xi_{A}\) with respect to the spin frame \(\{\mathbf{\epsilon_{A}}\}\) are given by
\[\xi_{\mathbf{A}}=\xi_{A}\mathbf{\epsilon_{A}}^{A}.\]
The spin basis \(\{\mathbf{o},\mathbf{\iota}\}\) satisfies \(\llbracket o,\mathbf{\iota}\rrbracket=1\), where \(\llbracket.,.\rrbracket\) is the antisymmetric product defined by
\[\llbracket\zeta,\lambda\rrbracket=\zeta_{B}\lambda^{B}=\epsilon_{AB}\zeta^{A} \lambda^{B}.\]
Here, \(\epsilon_{AB}\) is the antisymmetric \(\epsilon\)-spinor that can be regarded as a raising/lowering object for spinor indices.
## 2 Ashtekar's and Friedrich's formulations of spatial infinity
In this section, we will briefly explore the connection between Ashtekar's and Friedrich's approaches to resolving the singular behaviour of the conformal structure near spatial infinity, following [29]. First, the relationship between these two formulations on Minkowski spacetime will be examined. This relationship will demonstrate key differences between these formulations, and it establishes the significance of Friedrich's formulation in the calculation of the asymptotic charges near spatial infinity in the context of an initial value problem.
### Representations of spatial infinity in Minkowski spacetime
To establish the relationship between Ashtekar's hyperboloid and Friedrich's cylinder on Minkowski spacetime, we explore different representations of spatial infinity in Minkowski spacetime, starting with the standard conformal representation of spatial infinity.
#### Point compactification of Minkowski spacetime
Assume \((\mathbb{R}^{4},\tilde{\mathbf{\eta}})\) denote the Minkowski spacetime and let \((\tilde{x}^{\mu})\) denotes the standard Cartesian coordinates. Then the metric \(\tilde{\mathbf{\eta}}\) can be written as
\[\tilde{\mathbf{\eta}}=\tilde{\eta}_{\mu\nu}\mathbf{d}\tilde{x}^{\mu}\otimes\mathbf{d} \tilde{x}^{\nu},\]
where \(\tilde{\eta}_{\mu\nu}=\text{diag}(1,-1,-1,-1)\). In standard spherical coordinates \((\tilde{t},\tilde{\rho},\tilde{x}^{A})\), one has that
\[\tilde{\mathbf{\eta}}=\mathbf{d}\tilde{t}\otimes\mathbf{d}\tilde{t}-\mathbf{d}\tilde{\rho} \otimes\mathbf{d}\tilde{\rho}-\tilde{\rho}^{2}\mathbf{\sigma},\]
where \(\mathbf{\sigma}\) is the standard round metric on \(\mathbb{S}^{2}\).
Given that \(\tilde{x}^{0}\equiv\tilde{t}\) and \(\tilde{\rho}^{2}\equiv(\tilde{x}^{1})^{2}+(\tilde{x}^{2})^{2}+(\tilde{x}^{3})^ {2}\), define \(\tilde{X}^{2}\equiv\tilde{\eta}_{\mu\nu}\tilde{x}^{\mu}\tilde{x}^{\nu}=\tilde{ t}^{2}-\tilde{\rho}^{2}\), then it is clear to see that spatial infinity is contained in the domain \(\tilde{\mathcal{D}}\) (See Figure 1) defined as
\[\tilde{\mathcal{D}}\equiv\{p\in\mathbb{R}^{4}|\;\tilde{\eta}_{\mu\nu}\tilde{ x}^{\mu}(p)\tilde{x}^{\nu}(p)<0\}.\]
The conformal metric \(\tilde{\mathbf{\eta}}=\Xi^{2}\tilde{\mathbf{\eta}}\), with \(\Xi=\tilde{X}^{-2}\) implies a point compactification of the physical spacetime \((\mathbb{R}^{4},\tilde{\mathbf{\eta}})\), and it can be written explicitly as
\[\tilde{\mathbf{\eta}}=\mathbf{d}\mathbf{\iota}\otimes\mathbf{d}\mathbf{t}-\mathbf{d}\rho\otimes\mathbf{d} \rho-\rho^{2}\mathbf{\sigma},\]
where
\[t=-\frac{\tilde{t}}{\tilde{t}^{2}-\tilde{\rho}^{2}},\quad\rho=-\frac{\tilde{ \rho}}{\tilde{t}^{2}-\tilde{\rho}^{2}},\]
In this conformal representation, the conformal boundary defined by \(\Xi=0\) can be decomposed into different sets given by
\[\tilde{\mathscr{I}}^{+} \equiv\{p\in\mathbb{R}^{4}|\;t(p)>0,t(p)^{2}-\rho(p)^{2}=0\},\] \[\tilde{\mathscr{I}}^{-} \equiv\{p\in\mathbb{R}^{4}|\;t(p)<0,t(p)^{2}-\rho(p)^{2}=0\},\]
\[i^{0}\equiv\{p\in\mathbb{R}^{4}|\ (t(p),x^{1}(p),x^{2}(p),x^{3}(p))=(0,0,0,0)\},\]
where the inverse Cartesian coordinates \(\{x^{\mu}\}\) are related to \(\{\tilde{x}^{\mu}\}\) by
\[x^{\mu}=-\frac{\tilde{x}^{\mu}}{\tilde{X}^{2}}.\]
We will refer to \(\tilde{\mathscr{I}}^{+},\tilde{\mathscr{I}}^{-}\) and \(i^{0}\) as future null infinity, past null infinity and spatial infinity, respectively. However, note that \(\tilde{\mathscr{I}}^{\pm}\) only denotes the parts of null infinity close to spatial infinity \(i^{0}\). As mentioned earlier, this conformal representation maps all the points at infinite spatial distances in the physical spacetime \((\mathbb{R}^{4},\tilde{\mathbf{\eta}})\) to the spatial infinity point \(i^{0}\) in \((\mathbb{R}^{4},\tilde{\mathbf{\eta}})\).
#### Ashtekar's hyperboloid at spatial infinity
To obtain the hyperboloid at spatial infinity, start with the Minkowski metric in hyperbolic coordinates \((\psi,\chi,\theta,\phi)\)
\[\tilde{\mathbf{\eta}}=-\mathbf{d}\psi\otimes\mathbf{d}\psi+\psi^{2}\mathbf{\ell}.\]
Here, \(\mathbf{\ell}\) is the \(3-\)metric on a unit timelike hyperboloid defined as
\[\mathbf{\ell}\equiv\mathbf{d}\chi\otimes\mathbf{d}\chi-\cosh^{2}\chi\ \mathbf{\sigma}.\]
Introduce the inverse radial coordinate \(\zeta\equiv 1/\psi\) so that
\[\tilde{\mathbf{\eta}}=-\frac{1}{\zeta^{4}}\mathbf{d}\zeta\otimes\mathbf{d}\zeta+\frac{1}{ \zeta^{2}}\mathbf{\ell}.\]
Then, consider the conformal factor \(H=\zeta\) and define \(\tilde{\mathbf{\eta}}=H^{2}\tilde{\mathbf{\eta}}\). Then, \(\tilde{\mathbf{\eta}}\) can be written as
\[\tilde{\mathbf{\eta}}=-\frac{1}{\zeta^{2}}\mathbf{d}\zeta\otimes\mathbf{d}\zeta+\mathbf{\ell}\]
Observe that \(\tilde{\mathbf{\eta}}\) is singular at \(\zeta=0\) while the intrinsic \(3-\)metric \(\bar{\mathbf{q}}\) of the unit timelike hyperboloid at \(\zeta=0\) is well-defined and given by
\[\bar{\mathbf{q}}=\mathbf{\ell}.\]
Figure 1: (a) The domain \(\tilde{\mathcal{D}}\) containing spatial infinity, (b) The domain \(\tilde{\mathcal{D}}\) on the conformal diagram of Minkowski spacetime.
In the following, the timelike hyperboloid with \(\zeta=0\) will be denoted by \(\mathcal{H}_{\mathbb{R}^{4}}\). Then \((\mathcal{H}_{\mathbb{R}^{4}},\tilde{\mathbf{q}})\) will be referred to as the hyperboloid at spatial infinity.
#### Friedrich's cylinder at spatial infinity
A different representation of spatial infinity can be obtained by defining a new time coordinate \(\tau=t/\rho\) and the rescaling
\[\mathbf{\eta}=\frac{1}{\rho^{2}}\tilde{\mathbf{\eta}},\]
so that
\[\mathbf{\eta}=\mathbf{d}\tau\otimes\mathbf{d}\tau+\frac{\tau}{\rho}(\mathbf{d}\tau\otimes\mathbf{ d}\rho+\mathbf{d}\rho\otimes\mathbf{d}\tau)-\frac{(1-\tau^{2})}{\rho^{2}}\mathbf{d}\rho \otimes\mathbf{d}\rho-\mathbf{\sigma}. \tag{2}\]
From this, we can also write
\[\mathbf{\eta}=\Theta^{2}\tilde{\mathbf{\eta}},\qquad\Theta=\rho(1-\tau^{2}).\]
Again, one sees that the spacetime metric \(\mathbf{\eta}\) is singular at \(\rho=0\) while the intrinsic metric on the \(\rho=0\) hypersurface is well-defined and given by
\[\mathbf{q}=\mathbf{d}\tau\otimes\mathbf{d}\tau-\mathbf{\sigma}.\]
Given the above, define the conformal extension \((\mathcal{M},\mathbf{\eta})\) with
\[\mathcal{M}\equiv\{p\in\mathbb{R}^{4}|-1\leq\tau(p)\leq 1,\rho(p)\geq 0\},\]
then introduce the following subsets of the conformal boundary \((\Theta=0)\) -- see Figure 2.
\[\mathscr{I}^{\pm}\equiv\big{\{}p\in\mathcal{M}\,\mid\,\tau(p)=\pm 1\big{\}}, \qquad\text{past and future null infinity}\]
\[\mathcal{I}\equiv\big{\{}p\in\mathcal{M}\,\mid\,\,\,|\tau(p)|<1,\,\,\rho(p)=0 \big{\}},\qquad\text{the cylinder at spatial infinity}\]
\[\mathcal{I}^{\pm}\equiv\big{\{}p\in\mathcal{M}\,\mid\,\tau(p)=\pm 1,\,\,\rho(p)=0 \big{\}},\qquad\text{the critical sets at of null infinity}\]
and
\[\mathcal{I}^{0}\equiv\big{\{}p\in\mathcal{M}\,\mid\,\tau(p)=0,\,\,\rho(p)=0 \big{\}},\]
where \(\mathcal{I}^{0}\) is the intersection of \(\mathcal{I}\) with the initial hypersurface \(\mathcal{S}_{*}\equiv\{\tau=0\}\). In subsequent discussions, we will refer to \((\mathcal{I},\mathbf{q})\) as the cylinder at spatial infinity.
Figure 2: A diagram of the neighbourhood of spatial infinity in Friedrich’s representation. In this representation, the spatial infinity point \(i^{0}\) is blown up to a cylinder \(\mathcal{I}\) connecting past null infinity \(\mathscr{I}^{+}\) and future null infinity \(\mathscr{I}^{-}\). The critical sets \(\mathcal{I}^{\pm}\) represents the sets where \(\mathcal{I}\) touches \(\mathscr{I}^{\pm}\). The set \(\mathcal{I}^{0}\) represents the intersection of the cylinder \(\mathcal{I}\) with the initial hypersurface \(\{\tau=0\}\).
#### The relation between Ashtekar's hyperboloid and Friedrich's cylinder
This section aims to show that Ashtekar's and Friedrich's formulations are conformally related. To achieve this, we need to demonstrate that \((\mathcal{H}_{\mathbb{R}^{4}},\bar{\mathbf{q}})\) and \((\mathcal{I},\mathbf{q})\) are conformally related, as well as the spacetime metrics \(\bar{\mathbf{\eta}}\) and \(\mathbf{\eta}\). Since the expressions for these metrics are explicit, obtaining a conformal factor relating these two constructions is straightforward. Nevertheless, dismissing this discourse as insignificant would be unwise, as it offers valuable insights into similar computations in more general spacetimes.
Introducing the coordinate transformation
\[\mathbf{d}\chi=\cosh\chi\mathbf{d}\tau,\]
and substituting in \(\mathbf{\ell}\), we get
\[\mathbf{\ell}=\cosh^{2}\chi\left(\mathbf{d}\tau\otimes\mathbf{d}\tau-\mathbf{\sigma}\right).\]
This immediately indicates that \(\bar{\mathbf{q}}\equiv\mathbf{\ell}\) and \(\mathbf{q}\) are conformally related, and the conformal factor is given by \(\omega=\cosh\chi\). Thus, we have
\[\bar{\mathbf{q}}=\omega^{2}\mathbf{q}.\]
In order to relate Ashtekar's and Friedrich's constructions in a neighbourhood of spatial infinity, introduce the coordinate transformation
\[\rho=\zeta\cosh\chi,\qquad\tau=\tanh\chi, \tag{4}\]
Then, given that \(\bar{\mathbf{\eta}}=H^{2}\bar{\mathbf{\eta}}\) and \(\mathbf{\eta}=\Theta^{2}\bar{\mathbf{\eta}}\), the conformal relation between \(\bar{\mathbf{\eta}}\) and \(\mathbf{\eta}\) is given by
\[\bar{\mathbf{\eta}}=\varpi^{2}\mathbf{\eta},\]
where
\[\varpi=H\Theta^{-1}=\cosh\chi.\]
Given that \(\chi=\operatorname{arctanh}\tau\), it can be shown that
\[\varpi=\frac{1}{\sqrt{1-\tau^{2}}}.\]
From the above discussion, we conclude
* The conformal factor \(\varpi\) gives rise to a conformal representation that does not reach null infinity. This is demonstrated by the fact that \(\varpi\to\infty\) as \(\tau\to\pm 1\).
* The hyperboloids of constant \(\zeta\) never reaches the conformal boundary. This can be seen by considering the hyperboloids of constant \(\zeta\) on the \((\tau,\rho)\) plane. Given (4), we have \[(\tau,\rho)\to(\pm 1,\infty)\qquad\text{as }\chi\to\pm\infty.\]
In [29], the relation between Ashtekar's and Friedrich's formulations of spatial infinity is analysed for spacetimes that satisfy Ashtekar's AMSI definition [11]. As mentioned earlier, this definition introduces the concept of an asymptote as spatial infinity \(\mathcal{H}\) which is a generalisation of the hyperboloid at spatial infinity \(\mathcal{H}_{\mathbb{R}^{4}}\). Specifically, we show that in order to obtain a conformal factor relating \(\mathcal{H}\) and \(\mathcal{I}\), and the spacetime metrics in a neighbourhood of spatial infinity, one has to impose extra conditions to ensure the regularity of the spacetime metric in the sets where spatial infinity touches null infinity. Given these assumptions, the main steps are somewhat similar to the analysis on Minkowski spacetime. First, for the intrinsic metrics on \(\mathcal{H}\) and \(\mathcal{I}\), we have the following result
**Proposition 1**.: _The metric \(\bar{\mathbf{q}}\) of an asymptote \(\mathcal{H}\) satisfying Ashtekar's AMSI definition is conformally related to the standard metric of Friedrich's cylinder \(\mathbf{q}\)._
For the spacetime metrics, one can construct a conformal Gaussian coordinate system in a small neighbourhood of \(\mathcal{H}\). This conformal Gaussian gauge defines a conformal factor that gives the relation between Ashtekar's and Friedrich's representations of spatial infinity.
The spin-2 asymptotic charges on Minkowski spacetime
In this section, a brief review of the calculations of the asymptotic charges of the spin-2 field at the critical of null infinity on Minkowski spacetime is presented following [27].
### The Minkowski spacetime in the F-gauge
It will be convenient to introduce a frame basis \(\{\boldsymbol{e_{a}}\}\) adapted to Friedrich's cylinder at spatial infinity on Minkowski spacetime, the so-called F-gauge frames. Start with the Minkowski metric \(\boldsymbol{\eta}\) given by (2). It is straightforward to see that the metric on the hypersurfaces \(\mathcal{Q}_{\tau,\varrho}\) of constant \(\rho\) and \(\tau\) is the standard metric on \(\mathbb{S}^{2}\). Then, introduce the complex null frame \(\{\boldsymbol{\partial}_{+},\boldsymbol{\partial}_{-}\}\) on \(\mathcal{Q}_{\tau,\rho}\) and propagate \(\{\boldsymbol{\partial}_{+},\boldsymbol{\partial}_{-}\}\) off \(\mathcal{Q}_{\tau,\rho}\) by imposing
\[[\boldsymbol{\partial}_{\tau},\boldsymbol{\partial}_{\pm}]=0,\qquad[ \boldsymbol{\partial}_{\rho},\boldsymbol{\partial}_{\pm}]=0.\]
Now, the F-gauge frames \(\{\boldsymbol{e_{AA^{\prime}}}\}\) and their dual \(\{\boldsymbol{\omega^{AA^{\prime}}}\}\) can be defined as follows
\[\boldsymbol{e_{00^{\prime}}}=\frac{\sqrt{2}}{2}\left((1-\tau)\boldsymbol{ \partial}_{\tau}+\rho\boldsymbol{\partial}_{\rho}\right),\qquad\boldsymbol{ \omega^{00^{\prime}}}=\frac{\sqrt{2}}{2}\left(\boldsymbol{d}\tau-\frac{1}{ \rho}(1-\tau)\boldsymbol{d}\rho\right),\]
\[\boldsymbol{e_{11^{\prime}}}=\frac{\sqrt{2}}{2}\left((1+\tau)\boldsymbol{ \partial}_{\tau}-\rho\boldsymbol{\partial}_{\rho}\right),\qquad\boldsymbol{ \omega^{11^{\prime}}}=\frac{\sqrt{2}}{2}\left(\boldsymbol{d}\tau+\frac{1}{ \rho}(1+\tau)\boldsymbol{d}\rho\right),\]
\[\boldsymbol{e_{01^{\prime}}}=\frac{\sqrt{2}}{2}\boldsymbol{\partial}_{+}, \qquad\boldsymbol{\omega^{01^{\prime}}}=\sqrt{2}\boldsymbol{\omega}^{+},\]
\[\boldsymbol{e_{10^{\prime}}}=\frac{\sqrt{2}}{2}\boldsymbol{\partial}_{-}, \qquad\boldsymbol{\omega^{10^{\prime}}}=\sqrt{2}\boldsymbol{\omega}^{-},\]
where \(\boldsymbol{e_{AA^{\prime}}}\) is obtained from \(\boldsymbol{e_{a}}\) by contraction with the Infeld-van der Waerden symbols \(\sigma^{\boldsymbol{a}}{}_{\boldsymbol{AA^{\prime}}}\). So, \(\boldsymbol{e_{AA^{\prime}}}=\sigma^{\boldsymbol{a}}{}_{\boldsymbol{AA^{ \prime}}}\boldsymbol{e_{a}}\). The dual frames \(\boldsymbol{\omega}^{\pm}\) satisfy
\[\langle\boldsymbol{\omega}^{+},\boldsymbol{\partial}_{+}\rangle=1,\qquad \langle\boldsymbol{\omega}^{-},\boldsymbol{\partial}_{-}\rangle=1.\]
In terms of the above frame fields, the metric \(\boldsymbol{\eta}\) can be written as
\[\boldsymbol{\eta}=\epsilon_{\boldsymbol{AB}}\boldsymbol{e_{A^{\prime}B^{ \prime}}}\boldsymbol{\omega^{AA^{\prime}}}\otimes\boldsymbol{\omega^{BB^{ \prime}}}.\]
### The spin-2 charges in the F-gauge
The goal of this section is to obtain an expression of the charges that can be evaluated at the critical sets \(\mathcal{I}^{\pm}\). Generally, the expressions for the asymptotic charges are written in terms of frames adapted to null infinity and that satisfy the Newman Penrose gauge conditions. In subsequent discussions, these frames will be referred to as the NP-gauge frames and will be denoted by \(\{\boldsymbol{e_{a}^{\bullet}}\}\). Now, introduce the NP null tetrad \((l^{a},n^{a},m^{a},\bar{m}^{a})\) satisfying the NP gauge conditions and define \(\boldsymbol{e_{AA^{\prime}}^{\bullet}}=\sigma^{\boldsymbol{a}}{}_{\boldsymbol{ AA^{\prime}}}\boldsymbol{e_{a}^{\bullet}}\) such that
\[l^{a}\equiv\boldsymbol{e_{00^{\prime}}^{\bullet}},\qquad n^{a}\equiv \boldsymbol{e_{11^{\prime}}^{\bullet}},\qquad m^{a}\equiv\boldsymbol{e_{01^{ \prime}}^{\bullet}},\qquad\bar{m}^{a}\equiv\boldsymbol{e_{10^{\prime}}^{ \bullet}}. \tag{6}\]
Let \(W^{\bullet}_{abcd}\) denotes a Weyl-like tensor and define \(\mathcal{W}^{\bullet}_{abcd}\) as
\[\mathcal{W}^{\bullet}_{abcd}\equiv W^{\bullet}_{abcd}+i(^{*}W)^{\bullet}_{abcd},\]
where \((^{*}W)^{\bullet}_{abcd}\) is the left Hodge dual of \(W^{\bullet}_{abcd}\). Then, the spinorial counterpart of \(W^{\bullet}_{abcd}\) can be decomposed in terms of the symmetric spin-2 spinor \(\psi^{\star}_{ABCD}\) as
\[W^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=-\psi^{\bullet}_{ ABCD}\boldsymbol{e_{A^{\prime}B^{\prime}}^{\bullet}}\boldsymbol{e_{C^{\prime}D^{ \prime}}^{\star}}-\bar{\psi}^{\star}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime }}\boldsymbol{e_{AB}^{\bullet}}\boldsymbol{e_{CD}^{\bullet}}. \tag{7}\]
Now consider the asymptotic charges associated with smooth functions \(\lambda\) on \(\mathbb{S}^{2}\). For each \(\lambda\), the asymptotic charges on some cross-section \(\mathcal{C}\) of \(\mathscr{I}^{\pm}\) is given by
\[\mathscr{Q}=\int_{\mathcal{C}}\lambda\mathcal{W}^{\bullet}_{abcd}l^{a}n^{b}m^{c }\bar{m}^{d}\boldsymbol{dS}.\]
From (6) and (7), it can be shown that the charges \(\mathscr{Q}\) can be written as
\[\mathscr{Q}=-2\int_{\mathcal{C}}\lambda\bar{\psi}_{2}^{\bullet}\mathbf{d}S,\]
where \(\bar{\psi}_{2}^{\bullet}\equiv\bar{\psi}_{\mathbf{0}^{\prime}\mathbf{0}^{\prime}\mathbf{1}^ {\prime}\mathbf{1}^{\prime}}^{\bullet}\).
To evaluate the charges at the critical sets, one must obtain an expression for \(\mathscr{Q}\) in terms of the F-gauge frames. The transformation from the NP-gauge frames to the F-gauge frames in Minkowski spacetime [27, 26] implies that
\[\psi_{2}^{\bullet}=\psi_{2}.\]
Thus, the final expression of the charges in the F-gauge is given by
\[\mathscr{Q}=-2\int_{\mathcal{C}}\lambda\bar{\psi}_{2}\mathbf{d}S. \tag{8}\]
To evaluate this expression of the charges at \(\mathcal{I}^{\pm}\), the next step is to obtain a solution for \(\bar{\psi}_{2}\) using the field equations.
### The spin-2 field equations
The spin-2 field equation can be written as a wave equation
\[\Box\psi_{ABCD}=0, \tag{9}\]
where \(\Box\equiv\nabla_{AA^{\prime}}\nabla^{AA^{\prime}}\) is the D'Alembertian operator. To analyse the solutions for this equation in a neighbourhood of spatial infinity, assume that the components \(\psi_{n}\) of the spin-2 spinor can be expanded near \(\rho=0\) in terms of spin-weighted spherical harmonics \({}_{n}Y_{l,m}\) as
\[\psi_{n}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(\tau)_{2-n}Y_{l,m}+o_ {1}(\rho),\qquad\text{for }n=0,\ldots,4. \tag{10}\]
where \(a_{n;l,m}:\mathbb{R}\to\mathbb{C}\). Using (10) and substituting in (9), one obtains second order ordinary differential equations for the coefficients \(a_{n;l,m}(\tau)\)
\[(1-\tau^{2})\ddot{a}_{0;l,m}+2(2-\tau)\dot{a}_{0;l,m}+l(l+1)a_{0 ;l,m}=0, \tag{11a}\] \[(1-\tau^{2})\ddot{a}_{1;l,m}+2(1-\tau)\dot{a}_{1;l,m}+l(l+1)a_{1 ;l,m}=0,\] (11b) \[(1-\tau^{2})\ddot{a}_{2;l,m}-2\tau\dot{a}_{2;l,m}+l(l+1)a_{2;l,m}=0,\] (11c) \[(1-\tau^{2})\ddot{a}_{3;l,m}-2(1+\tau)\dot{a}_{3;l,m}+l(l+1)a_{3 ;l,m}=0,\] (11d) \[(1-\tau^{2})\ddot{a}_{4;l,m}-2(2+\tau)\dot{a}_{4;l,m}+l(l+1)a_{4 ;l,m}=0. \tag{11e}\]
Now, assume that the initial data \((\psi_{n})|_{\mathcal{S}_{*}}\) on \(\mathcal{S}_{*}\equiv\{\tau=0\}\) can be expanded near \(\rho=0\) as
\[\psi_{n}|_{\mathcal{S}_{*}}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(0 )_{2-n}Y_{l,m}+o(\rho).\]
Given that the expression of the charges (8) is written in terms of \(\psi_{2}\), one only requires the solution for (11c) in order to evaluate \(\mathscr{Q}\) at \(\tau=\pm 1\). It can be shown that for \(l\geq 0\) and \(-l\leq m\leq l\), the solution \(a_{2;l,m}\) is given by
\[a_{2;l,m}=A_{l,m}P_{l}(\tau)+B_{l,m}Q_{l}(\tau), \tag{12}\]
where \(P_{l}(\tau)\) is the Legendre polynomial of order \(l\) and \(Q_{l}(\tau)\) is the Legendre function of the second kind of order \(l\). The constants \(A_{l,m}\) and \(B_{l,m}\) can be expressed in terms of the initial data for \(a_{2;l,m}\). It is obvious that the solution (12) diverges logarithmically near \(\tau=\pm 1\), unless \(B_{l,m}=0\).
To obtain a regular solution for \(a_{2;l,m}\) at the critical sets, the constant \(B_{l,m}\) is required to vanish. In fact, the following regularity conditions ensure that the charges \(\mathscr{Q}\) are well-defined at the critical sets
**Lemma 1**.: _The solution (12) is regular at \(\mathcal{I}^{\pm}\) if and only if:_
1. \(a_{2;l,m}(0)=0\) _for even_ \(l\)_, and_
2. \(\dot{a}_{2;l,m}(0)=0\) _for odd_ \(l\)_._
These regularity conditions can be expressed in terms of freely specifiable data as shown in [27].
Making use of (8), (10) and (12) and by choosing initial data satisfying Lemma 1 and \(\lambda=Y_{l,m}\), the charges at \(\mathcal{I}^{\pm}\) can be written as
\[\mathscr{Q}|_{\mathcal{I}^{\pm}}=\begin{cases}2(l+1)Q_{l+1}(0)(a_{2})_{*}& \quad\text{for even $l\geq 0$},\\ \pm\sqrt{l(l+1)}Q_{l}(0)\left((a_{1})_{*}-(a_{3})_{*}\right)&\quad\text{for odd $l$}.\end{cases} \tag{13}\]
The main conclusions from the above discussion are
1. For generic boosted initial data, the charges \(\mathscr{Q}\) are not well-defined in the limits of spatial infinity, i.e., at the critical sets \(\mathcal{I}^{\pm}\).
2. A generic boosted initial data satisfying Lemma 1 allows us to obtain regular expressions for \(\mathscr{Q}\) at the critical sets.
3. The matching of the charges is obtained naturally in this formalism. In particular, we have \(\mathscr{Q}|_{\mathcal{I}^{+}}=(-1)^{l}\mathscr{Q}|_{\mathcal{I}^{-}}\).
## 4 The asymptotic charges in full GR
The process of the calculation of the asymptotic charges for the spin-2 field at the critical sets presented in the previous section can be extended to the full GR setting. For this, assume that \((\tilde{\mathcal{M}},\tilde{\mathbf{g}})\) is a spacetime satisfying the vacuum Einstein field equations i.e.
\[\tilde{R}_{ab}=0, \tag{14}\]
where \(\tilde{R}_{ab}\) is the Ricci tensor associated with the Levi-Civita connection \(\tilde{\mathbf{\nabla}}\) of \(\tilde{\mathbf{g}}\). The conformal rescaling
\[\mathbf{g}=\Xi^{2}\tilde{\mathbf{g}}, \tag{15}\]
implies transformation laws for the physical fields e.g. the curvature tensor \(\tilde{R}^{a}{}_{bcd}\), the Schouten tensor \(\tilde{L}_{ab}\) etc. It follows that the vacuum Einstein field equations are not conformally invariant and that the field equations implied by (15) can not be analysed at the conformal boundary \(\Xi=0\) since the conformal Ricci tensor \(R_{ab}\) is singular at the points where \(\Xi=0\). If \(\tilde{C}^{a}{}_{bcd}\) denotes the Weyl tensor, then the Bianchi identity can be written in terms of the Levi-Civita connection associated with \(\mathbf{g}\) as
\[\nabla_{a}(\Xi^{-1}\tilde{C}^{a}{}_{bcd})=0. \tag{16}\]
If one defines the rescaled Weyl tensor \(d^{a}{}_{bcd}\equiv\Xi^{-1}\tilde{C}^{a}{}_{bcd}\), then equation (16) can be written as
\[\nabla_{a}d^{a}{}_{bcd}=0. \tag{17}\]
Exploiting the symmetries of the rescaled Weyl tensor implies
\[\nabla_{[e}d^{a}{}_{|b|cd]}=0. \tag{18}\]
Our calculations of the asymptotic charges rely on the extended conformal field equations written in terms of a Weyl connection \(\tilde{\mathbf{\nabla}}\) satisfying
\[\hat{\nabla}_{a}g_{bc}=-2f_{a}g_{bc},\]
where \(f_{a}\) is an arbitrary 1-form. The explicit form of these equations will not be necessary for this article, interested readers can refer to Chapter 8 in [31]. The extended conformal field equations yield differential equations to be solved for the \(\mathbf{g}\)-orthonormal frame fields \(\{\mathbf{e}_{\mathbf{a}}\}\), the components of the Weyl connection
coefficients \(\tilde{\Gamma}_{\boldsymbol{a}}{}^{\boldsymbol{b}}{}_{\boldsymbol{c}}\), the Schouten tensor \(\tilde{L}_{\boldsymbol{a}\boldsymbol{b}}\) and the rescaled Weyl tensor \(d^{\boldsymbol{a}}{}_{\boldsymbol{b}\boldsymbol{c}\boldsymbol{d}}\). One significant feature of the extended conformal field equations is that they exhibit gauge freedom indicated by the fact that there are no equations to fix the conformal factor \(\Xi\) and the Weyl connection \(\boldsymbol{\nabla}\). To fix this gauge freedom, one can make use of the so-called conformal Gaussian gauge, based on conformal geodesics, that allows us to write the field equations as a symmetric hyperbolic system in which the evolution equations reduce to a transport system along the conformal geodesics. Given the field equations in this gauge, it is possible to obtain a spinorial version of these equations to be analysed near spatial infinity.
The above discussion highlights the essence of conformal methods in GR. The following section will introduce Friedrich's regular initial value problem for the conformal field equations.
### Friedrich's regular initial value problem
The purpose of this section is to briefly introduce Friedrich's formulation in full GR. As mentioned in the introduction, the aim of Friedrich's formulation is to introduce a regular initial value problem for the conformal field equations near spatial infinity. An extensive discussion of this framework is provided in [12, 28]. In this framework, the spacetime \((\tilde{\mathcal{M}},\tilde{\boldsymbol{q}})\) is assumed to be the development of some asymptotically Euclidean and regular [32, 31] initial data \((\tilde{\mathcal{S}},\tilde{\boldsymbol{h}},\tilde{\boldsymbol{K}})\). In the following, we specify initial data satisfying the Hamiltonian and momentum constraints as introduced in [30]. In particular, we have
**Proposition 2**.: _For any \(\alpha,\beta\in C^{2}(\mathbb{S}^{2})\), there exists a vacuum initial data set \((\tilde{\boldsymbol{h}},\tilde{\boldsymbol{K}})\) such that the components of \(\tilde{\boldsymbol{h}}\) and \(\tilde{\boldsymbol{K}}\) with respect to the standard Euclidean coordinate chart \(\{x^{\alpha}\}\) have the following asymptotics:_
\[\tilde{h}_{\alpha\beta}=-\delta_{\alpha\beta}-\frac{1}{r}\left[\left(A-\frac{ \alpha}{2}\right)\delta_{\alpha\beta}+\alpha\frac{x_{\alpha}x_{\beta}}{r^{2}} \right]+O_{2}(r^{-2}), \tag{19a}\] \[\tilde{K}_{\alpha\beta}=\frac{1}{r^{2}}\left[-\frac{1}{2}\beta\delta_{\alpha \beta}+\frac{1}{r}\left(-B_{\alpha}x_{\beta}-B_{\beta}x_{\alpha}+(B^{\gamma}x_ {\gamma})\delta_{\alpha\beta}\right)+\beta\frac{x_{\alpha}x_{\beta}}{r^{2}} \right]+O_{1}(r^{-3}) \tag{19b}\]
_where \(A\), \(\{B_{\alpha}\}_{\alpha=1}^{3}\) are some constants and \(r=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}\)._
Now, let \((\mathcal{S}^{\prime},\boldsymbol{h}^{\prime})\) be a 3-dimensional smooth compact manifold with an asymptotic end \(i\in\mathcal{S}^{\prime}\). Then \((\tilde{\mathcal{S}},\tilde{\boldsymbol{h}})\) is an asymptotically Euclidean and regular manifold if there exists a diffeomorphic map \(\Phi\) from \(\mathcal{S}^{\prime}\setminus\{i\}\) onto \(\tilde{\mathcal{S}}\) and a conformal factor \(\Omega^{\prime}\) which is analytic on \(\mathcal{S}^{\prime}\) and satisfying i) \(\Omega^{\prime}=0,\boldsymbol{d}\Omega^{\prime}=0\) and \(\text{Hess}(\Omega^{\prime})=-2\boldsymbol{h}^{\prime}\) at \(i\), ii) \(\Omega^{\prime}>0\) on \(\mathcal{S}^{\prime}\setminus\{i\}\), iii) \(\boldsymbol{h}^{\prime}=\Omega^{\prime 2}\Phi_{*}\tilde{\boldsymbol{h}}\) on \(\mathcal{S}^{\prime}\setminus\{i\}\).
To apply this, define the inverse coordinates \(\{y^{\alpha}\}\) and the conformal factor \(\Omega^{\prime}\) as
\[y^{\alpha}=-\frac{x^{\alpha}}{r^{2}},\qquad\Omega^{\prime}=\frac{\varrho^{2}} {\sqrt{1+A\varrho}},\]
so that the components of the conformal initial data \(\boldsymbol{h}^{\prime}=\Omega^{\prime 2}\tilde{\boldsymbol{h}}\) and \(\boldsymbol{K}^{\prime}=\Omega^{\prime}\tilde{\boldsymbol{K}}\) can be expanded around \(\varrho=\sqrt{(y^{1})^{2}+(y^{2})^{2}+(y^{3})^{2}}=0\) as
\[h^{\prime}_{\alpha\beta}=-\delta_{\alpha\beta}-\alpha\varrho\left(\frac{y_{ \alpha}y_{\beta}}{\varrho^{2}}-\frac{1}{2}\delta_{\alpha\beta}\right)+O_{2}( \varrho^{2}), \tag{20a}\] \[K^{\prime}_{\alpha\beta}=-\frac{\beta}{2}\delta_{\alpha\beta}-\frac{1}{\varrho }\left(B_{\alpha}y_{\beta}+B_{\beta}y_{\alpha}+\frac{1}{2}(B^{\gamma}y_{\gamma })\delta_{\alpha\beta}\right)+\left(\beta-4\frac{(B^{\gamma}y_{\gamma})}{ \varrho}\right)\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}+O_{1}(\varrho), \tag{20b}\]
The \(O(\varrho)\) term in (20a) can be made to vanish by performing a coordinate transformation from \(\{y^{\alpha}\}\) to normal coordinates \(\{z^{\alpha}\}\)[33]. Then, the term \(O(\varrho^{2})\) can be removed by performing a further conformal transformation
\[\Omega^{\prime}\rightarrow\Omega\equiv\varpi\Omega^{\prime},\]
where
\[\varpi\equiv e^{f},\quad\text{ with }f=\frac{1}{2}l^{\prime}_{\alpha\beta}(i)z^{ \alpha}z^{\beta}.\]
Here, \(\mathbf{l}^{\prime}_{\alpha\beta}(i)\) denotes the components of the Schouten tensor associated with \(\mathbf{h}^{\prime}\) in normal coordinates \(\{z^{\alpha}\}\) evaluated at \(i(\varrho=0)\). If \(h^{\prime(0)}_{\alpha\beta}\) is the metric at \(i\) and \(|z|^{2}\equiv h^{\prime(0)}_{\alpha\beta}z^{\alpha}z^{\beta}\), then the components of conformal initial data \(\bar{\mathbf{h}}=\varpi^{2}\mathbf{h}^{\prime}\) and \(\bar{\mathbf{K}}=\varpi\mathbf{K}^{\prime}\) can be written as
\[\bar{h}_{\alpha\beta}=-\delta_{\alpha\beta}+O(|z|^{3}).\]
\[\bar{K}_{\alpha\beta}=-\frac{\beta}{2}\delta_{\alpha\beta}-\frac{1}{2}\left(B _{\alpha}\vartheta_{\beta}+B_{\beta}\vartheta_{\alpha}+\frac{1}{2}(B^{\gamma }\vartheta_{\gamma})\delta_{\alpha\beta}\right)+\beta\vartheta_{\alpha} \vartheta_{\beta}-4(B^{\gamma}\vartheta_{\gamma})\vartheta_{\alpha}\vartheta_ {\beta}+O(|z|).\]
where \(\vartheta^{\alpha}=z^{\alpha}/|z|\). The initial data \((\bar{\mathbf{h}},\bar{\mathbf{K}})\) will be referred to as the conformal normal initial data. It can be shown, using the conformal constraint equations, that the initial data for the components of the conformal Schouten tensor \(\bar{L}_{\alpha\beta}\) and the electric and magnetic parts of the Weyl tensor, \(\tilde{d}_{\alpha\beta}\) and \(\bar{d}_{\alpha\beta\gamma}\), respectively, are singular at \(|z|=0\). To introduce regular initial data, one must introduce a conformal rescaling as suggested in [12]
\[\Omega\rightarrow\kappa^{-1}\Omega, \tag{22}\]
with \(\kappa=O(|z|)\). Let \(\rho=|z|\), then the conformal factor can be expanded around \(\rho=0\) as
\[\Omega=\rho^{2}+\frac{1}{6}\Pi_{3}[\Omega]\rho^{3}+O(\rho^{4}),\]
where \(\Pi_{3}[\Omega]\) is written in terms of the angular coordinates \(\vartheta^{\alpha}\), the constant \(A\), the function \(\alpha\) and its angular derivatives.
The conformal rescaling (22) introduces the conformal metric \(\mathbf{h}=\kappa^{-2}\bar{\mathbf{h}}\). Then, if \(\{\mathbf{e_{i}}\}\) is an \(\mathbf{h}\)-orthonormal frame, one can show
\[h_{\mathbf{i}\mathbf{j}}=-\delta_{\mathbf{i}\mathbf{j}}+O(|z|^{3}),\]
and
\[K_{\mathbf{i}\mathbf{j}}=O(|z|),\qquad L_{\mathbf{i}\mathbf{j}}=O(|z|),\qquad d_{\mathbf{i}\mathbf{j}} =O(1),\qquad d_{\mathbf{i}\mathbf{j}\mathbf{k}}=O(1).\]
Hence, the conformal initial data for the conformal field equations are regular at \(|z|=0\).
One of the advantages of using the conformal Gaussian gauge mentioned in the last section is that it implies a conformal factor \(\Theta\) that can be written in terms of initial data. Following [34, 12], we have
\[\Theta=\kappa^{-1}\Omega\left(1-\tau^{2}\frac{\kappa^{2}}{\omega}\right), \tag{23}\]
where \(\tau\) refers to the parameter along the conformal geodesics used to construct the conformal Gaussian system and
\[\omega=\frac{2\Omega}{\sqrt{|\mathbf{h}(\mathbf{d}\Omega,\mathbf{d}\Omega)|}}.\]
In Friedrich's formulation, the blow-up of the point \(i\) to a 2-dimensional sphere is achieved by considering the bundle of normalised spin frames \(\mathrm{SU}(\mathcal{S}^{\prime})\) with structure group \(\mathrm{SU}(2,\mathbb{C})\) at \(i\). The spin frames \(\mathbf{e_{A}}(\mathbf{t})\), with \(\mathbf{t}\in\mathrm{SU}(2,\mathbb{C})\), can be extended to an open ball \(B_{a}(i)\) in \(\mathcal{S}^{\prime}\) of radius \(a\) centered at \(i\) by parallel propagation along an \(\mathbf{h}\)-geodesic starting at \(i\). If \(\rho\) is the affine parameter along the geodesic, then for a fixed \(\mathbf{t}\), the propagated spin frame can be written as \(\mathbf{e_{A}}(\rho,\mathbf{t})\). Now, define \(\mathcal{M}_{a,\kappa}\), a submanifold of \(\mathbb{R}\times\mathbb{R}\times\mathrm{SU}(2,\mathbb{C})\) as
\[\mathcal{M}_{a,\kappa}=\{(\tau,\rho,\mathbf{t})\in\mathbb{R}\times\mathbb{R}\times SU (2,\mathbb{C})|\ 0\leq\rho<a,-\frac{\omega}{\kappa}\leq\tau\leq\frac{\omega}{\kappa}\},\]
with the following subsets
\[\mathscr{I}_{a}^{\pm}=\{(\tau,\rho,\mathbf{t})\in\mathcal{M}_{a,\kappa}|\ 0< \rho<a,\tau=\pm\frac{\omega}{\kappa}\},\qquad\text{past and future null infinity} \tag{24a}\] \[\mathcal{I}=\{(\tau,\rho,\mathbf{t})\in\mathcal{M}_{a,\kappa}|\ \rho=0,-1<\tau<1\}, \qquad\text{the cylinder at spatial infinity}\] (24b) \[\mathcal{I}^{\pm}=\{(\tau,\rho,\mathbf{t})\in\mathcal{M}_{a,\kappa}|\ \rho=0,\tau=\pm 1\}, \qquad\text{the critical sets of null infinity} \tag{24c}\]
\[\mathcal{I}_{0}=\{(\tau,\rho,\mathbf{t})\in\mathcal{M}_{a,\kappa}|\;\rho=0,\tau=0\}. \tag{25}\]
To relate the structures on the fibre bundle to the spacetime manifold \((\tilde{\mathcal{M}},\tilde{\mathbf{g}})\) satisfying (14). Let \((\mathcal{M},\mathbf{g})\) denote a smooth conformal extension such that i) \(\Theta>0\) and \(\mathbf{g}=\Theta^{2}\tilde{\mathbf{g}}\) on \(\tilde{\mathcal{M}}\), ii) \(\Theta=0\) and \(d\Theta\neq 0\) on \(\mathscr{I}_{a}^{\pm}\). Now let \(\mathcal{N}\subset\mathcal{M}\) denote the domain of influence of \(B_{a}(i)\setminus\{i\}\), then the projection map \(\pi^{\prime}\) from \(\mathcal{M}_{a,\kappa}\) to \(\mathcal{N}\) can be factored as
\[\mathcal{M}_{a,\kappa}\overset{\pi^{\prime}_{1}}{\longrightarrow}\mathcal{M }_{a,\kappa}^{\prime}\overset{\pi^{\prime}_{2}}{\longrightarrow}\mathcal{N},\]
where \(\mathcal{M}_{a,\kappa}^{\prime}\equiv\mathcal{M}_{a,\kappa}/U(1)\) is implied by the action of \(U(1)\) on \(SU(2,\mathbb{C})\).
Finally, note that the spin frames \(\mathbf{e_{A}}(\rho,\mathbf{t})\) can be extended to the spacetime \(\mathcal{M}_{a,\kappa}\) by a certain propagation law along the conformal geodesics orthogonal to the \(\mathcal{S}_{a}\), where \(\mathcal{S}_{a}\) can be thought of as the initial hypersurface on \(\mathcal{M}_{a,\kappa}\), i.e.
\[\mathcal{S}_{a}=\{(\rho,\mathbf{t})\in\mathbb{R}\times SU(2,\mathbb{C})|\;0\leq \rho<a\}.\]
The propagated spin frames \(\mathbf{\epsilon_{A}}(\tau,\rho,\mathbf{t})\) are determined at a \(p\in\mathcal{M}_{a,\kappa}\setminus(\mathcal{I}\cup\mathcal{I}^{+}\cup \mathcal{I}^{-})\) up to a multiplication factor corresponding to the action of \(U(1)\) on \(SU(\mathcal{M})\).
### The supertranslation asymptotic charges in full GR
Let \(d^{\bullet}_{abcd}\) denote the rescaled Weyl tensor in the NP-gauge, the spinorial counterpart \(d^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}\) can be decomposed as follows
\[d^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=-\phi^{\bullet}_{ ABCD}\mathbf{\epsilon^{\bullet}_{A^{\prime}B^{\prime}}}\mathbf{\epsilon^{\bullet}_{C^{ \prime}D^{\prime}}}-\bar{\phi}^{\bullet}_{A^{\prime}B^{\prime}C^{\prime}D^{ \prime}}\mathbf{\epsilon^{\bullet}_{AB}}\mathbf{\epsilon^{\bullet}_{CD}}, \tag{26}\]
where \(\phi^{\bullet}_{ABCD}\) is a symmetric valence 4 spinor. Given the above, the asymptotic charges associated with smooth functions \(f\) on \(\mathbb{S}^{2}\) can be written as
\[\mathcal{Q}(f;\mathcal{C})\equiv\oint_{\mathcal{C}}\mathbf{\epsilon}_{2}f( \mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})+\tfrac{1}{2}\sigma^{\bullet ab }N^{\bullet}_{ab}), \tag{27}\]
where \(\mathcal{C}\) is some cross-section of \(\mathscr{I}^{\pm}\) and \(\mathbf{\epsilon}_{2}\) is its area element, \(\sigma^{\bullet ab}\) is the shear tensor, \(N^{\bullet}_{ab}\) is the news tensor and
\[\mathcal{P}^{\bullet}\equiv d^{\bullet}_{abcd}l^{a}n^{b}l^{c}n^{d},\]
\[(*\mathcal{P}^{\bullet})\equiv(*d^{\bullet})_{abcd}l^{a}n^{b}l^{c}n^{d}.\]
Using (26) and (6), one gets
\[\mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})=-2\bar{\phi}^{\bullet}_{2}. \tag{29}\]
Moreover, the term involving \(\sigma^{\bullet ab}N^{\bullet}_{ab}\) can be written in terms of the NP-connection coefficients [35, 36, 37]. The explicit form depends on whether we're considering the asymptotic charges at \(\mathscr{I}^{+}\) or \(\mathscr{I}^{-}\). In particular,
\[\sigma^{\bullet ab}N^{\bullet}_{ab}=2\Delta|\sigma^{\bullet}|^{2} -|\sigma^{\bullet}|^{2}\big{(}3\mu^{\bullet}+3\bar{\mu}^{\bullet}+\gamma^{ \bullet}+\bar{\gamma}^{\bullet}\big{)},\qquad\text{on }\mathscr{I}^{+} \tag{30a}\] \[\sigma^{\bullet ab}N^{\bullet}_{ab}=2\Delta|\lambda^{\bullet}|^{2} -|\lambda^{\bullet}|^{2}\big{(}3\rho^{\bullet}+3\bar{\rho}^{\bullet}+\mathbf{ \epsilon}^{\bullet}+\bar{\epsilon}^{\bullet}\big{)},\qquad\text{on }\mathscr{I}^{-}. \tag{30b}\]
Here, \(\Delta\equiv n^{a}\nabla^{\bullet}_{a}\) and \(\sigma^{\bullet},\mu^{\bullet},\gamma^{\bullet},\lambda^{\bullet},\rho^{ \bullet},\mathbf{\epsilon}^{\bullet}\) are the NP-connection coefficients defined as
\[\sigma^{\bullet}\equiv-\Gamma^{\bullet}_{\mathbf{0}1^{\prime}}\,{}^{ \mathbf{1}}\,{}_{\mathbf{0}},\qquad\mu^{\bullet}\equiv-\Gamma^{\bullet}_{\mathbf{0}1^{ \prime}}\,{}^{\mathbf{0}}\,{}_{\mathbf{1}},\qquad\gamma^{\bullet}\equiv\Gamma^{ \bullet}_{\mathbf{1}1^{\prime}}\,{}^{\mathbf{0}}\,{}_{\mathbf{0}}, \tag{31a}\] \[\lambda^{\bullet}\equiv\Gamma^{\bullet}_{\mathbf{1}0^{\prime}}\,{}^{ \mathbf{0}}\,{}_{\mathbf{1}},\qquad\rho^{\bullet}\equiv-\Gamma^{\bullet}_{\mathbf{1}0^{ \prime}}\,{}^{\mathbf{1}}\,{}_{\mathbf{0}},\qquad\epsilon^{\bullet}\equiv\Gamma^{ \bullet}_{\mathbf{0}0^{\prime}}\,{}^{\mathbf{0}}\,{}_{\mathbf{0}}. \tag{31b}\]
To evaluate the expression of the charges (27) at the critical sets \(\mathcal{I}^{\pm}\), one must find a transformation between the NP-gauge frames and the F-gauge frames in full GR. Following [28], a general transformation between a NP-gauge spin frame \(\{\mathbf{\epsilon}^{\bullet}_{\mathbf{A}}\}\) and an F-gauge spin frame \(\{\mathbf{\epsilon}_{\mathbf{A}}\}\) is parameterised by a conformal factor \(\theta\) and an \(\mathrm{SL}(2,\mathbb{C})\) transformation matrix \(\Lambda^{\mathbf{B}}{}_{\mathbf{A}}\)
\[\mathbf{\epsilon}^{\bullet}_{\mathbf{A}}=\theta^{-1/2}\Lambda^{\mathbf{B}}{}_{\mathbf{A}}\mathbf{ \epsilon}_{\mathbf{B}},\]
implying transformations for \(\tilde{\phi}_{2}^{\bullet}\) and the NP-connection coefficients (31). The expressions for these will not be presented here.
As we are interested in evaluating the expressions of the charges at \(\mathcal{I}^{\pm}\), an asymptotic solution for the conformal field equations is analysed, given the initial data prescribed in the previous section. Given the zero-order solution, asymptotic expansions for the conformal factor \(\theta\) and the transformation matrices \(\Lambda^{\mathbf{B}}_{\mathbf{A}}\) are obtained, following [28].
If \(\phi_{0},\phi_{1},\phi_{2},\phi_{3},\phi_{4}\) denote the components of the rescaled Weyl tensor in the F-gauge. The explicit transformation from the NP-gauge to the F-gauge implies
1. Contributions to \(\mathcal{Q}|_{\mathcal{I}^{\pm}}\) from \(\phi_{0},\phi_{1},\phi_{3},\phi_{3}\) are at higher order.
2. The background term \(\sigma^{\bullet ab}N^{\star}_{ab}\) does not contribute to \(\mathcal{Q}|_{\mathcal{I}^{\pm}}\) at zero order.
Hence, the asymptotic charges at \(\mathcal{I}^{\pm}\) are determined by \(f\) and the zero-order solution of \(\phi_{2}\), i.e.,
\[\mathcal{Q}|_{\mathcal{I}^{\pm}}=\mathcal{Q}|_{\mathcal{I}^{\pm}}(f,\phi_{2}^ {(0)}). \tag{32}\]
## 5 Conclusions
This article addresses the matching of the asymptotic charges associated with supertranslation symmetries in the context of an initial value problem using Friedrich's formulation of spatial infinity. The results in this paper demonstrate that the zero-order solution of \(\phi_{2}\) develops logarithmic singularities at \(\mathcal{I}^{\pm}\) given the prescribed initial data in Section 4.1. Therefore, \(\mathcal{Q}|_{\mathcal{I}^{\pm}}\) are only well-defined if extra regularity conditions are imposed on our initial data. An upcoming article will present the explicit form of these regularity conditions. A significant consequence of this result is that identifying a global symmetry group for generic asymptotically flat spacetimes is not feasible unless these spacetimes are the development of initial data satisfying certain regularity conditions.
|
2306.10222 | Coherent two-dimensional THz magnetic resonance spectroscopies for
molecular magnets: Analysis of Dzyaloshinskii-Moriya interaction | To investigate the novel quantum dynamic behaviors of magnetic materials that
arise from complex spin-spin interactions, it is necessary to probe the
magnetic response at a speed greater than the spin-relaxation and dephasing
processes. Recently developed two-dimensional (2D) terahertz magnetic resonance
(THz-MR) spectroscopy techniques use the magnetic components of laser pulses,
and this allows investigation of the details of the ultrafast dynamics of spin
systems. For such investigations, quantum treatment -- not only of the spin
system itself but also of the environment surrounding the spin system -- is
important. In our method, based on the theory of multidimensional optical
spectroscopy, we formulate nonlinear THz-MR spectra using an approach based on
the numerically rigorous hierarchical equations of motion. We conduct numerical
calculations of both linear (1D) and 2D THz-MR spectra for a linear chiral spin
chain. The pitch and direction of chirality (clockwise or anticlockwise) are
determined by the strength and sign of the Dzyaloshinskii-Moriya interaction
(DMI). We show that not only the strength but also the sign of the DMI can be
evaluated through the use of 2D THz-MR spectroscopic measurements, while 1D
measurements allow us to determine only the strength. | Jiaji Zhang, Yoshitaka Tanimura | 2023-06-17T01:15:46Z | http://arxiv.org/abs/2306.10222v1 | Coherent two-dimensional THz magnetic resonance spectroscopies for molecular magnets: Analysis of Dzyaloshinskii-Moriya interaction
###### Abstract
To investigate the novel quantum dynamic behaviors of magnetic materials that arise from complex spin-spin interactions, it is necessary to probe the magnetic response at a speed greater than the spin-relaxation and dephasing processes. Recently developed two-dimensional (2D) terahertz magnetic resonance (THz-MR) spectroscopy techniques use the magnetic components of laser pulses, and this allows investigation of the details of the ultrafast dynamics of spin systems. For such investigations, quantum treatment--not only of the spin system itself but also of the environment surrounding the spin system--is important. In our method, based on the theory of multidimensional optical spectroscopy, we formulate nonlinear THz-MR spectra using an approach based on the numerically rigorous hierarchical equations of motion. We conduct numerical calculations of both linear (1D) and 2D THz-MR spectra for a linear chiral spin chain. The pitch and direction of chirality (clockwise or anticlockwise) are determined by the strength and sign of the Dzyaloshinskii-Moriya interaction (DMI). We show that not only the strength but also the sign of the DMI can be evaluated through the use of 2D THz-MR spectroscopic measurements, while 1D measurements allow us to determine only the strength.
## I Introduction
Electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) have a long history as typical examples of two-dimensional (2D) spectroscopy techniques that measure the varying time intervals of magnetic pulse trains applied to electron or nuclear spin systems.[1; 2] Although these techniques are powerful means for structural analysis of organic and inorganic materials, it is difficult to apply them to the investigation of spin dynamics because the time resolution of the magnetic pulses involved is limited to the order of microseconds.
Recently, coherent terahertz (THz) magnetic resonance (MR) spectroscopy was developed using magnetic pulses in the subpicosecond range generated by the magnetic-field component of THz light.[3; 4; 5] Just as ultrafast laser spectroscopy has made it possible to study the electronic excitation-state dynamics and intra- and intermolecular vibrational motions of complex molecular systems,[6] THz-MR spectroscopy has opened up the possibility of investigating strongly correlated spin dynamics in molecular magnets; this depends on the complex spin-spin interactions and the configurations of spins in the molecular environment, which cause dephasing and relaxation of the spin system.
The quasi-ferromagnetic (FM) and antiferromagnetic (AFM) precession modes in YFeO\({}_{3}\) (YFO) that arise from the antisymmetric spin-orbit coupling[7] have been observed as free induction decay (FID) signals.[8; 9] Spin-wave excitation in AFM NiO has been detected with the aid of the Faraday effect.[10; 11] The skyrmion, which is a quasiparticle composed of vortex-like spin orientations, has been detected in several FM and AFM materials with thin-film structures.[12; 13; 14] THz electron spin resonance (ESR) and Hall conductivity spectroscopy have been employed to investigate the topological Hall effect and phase transitions.[3; 15; 16; 17]
Although THz-MR and THz-ESR spectroscopic approaches based on linear response theory have been successful for the classification of complex spin states in condensed phases,[18; 19; 20] it is unclear whether the spin states that are investigated are quantum-mechanically entangled, and whether the width of the peaks that are measured arise from the relaxation or dephasing processes, because these peaks are usually broad and overlap. Thus, an extension to coherent 2D THz-MR spectroscopy was developed. The observed 2D THz-MR spectrum for a YFO crystal made it possible to illustrate characteristic nonlinear spin responses, such as double-quantum (2Q) coherence and second-harmonic generation (SHG).[21]
As already illustrated in 2D optical spectroscopy,[22; 23] 2D THz-MR spectroscopy is not only useful for identifying complex spin interactions but also for monitoring the complex quantum spin dynamics under the influence of relaxation and dephasing arising from the environment at the femtosecond scale. Theoretical input regarding the complex profiles of spin-spin interactions is important for analyzing these 2D spectra under ultrafast nonlinear processes. In particular, this includes those in molecular magnets, which play a central role in spintronics and next-generation information technologies.[24; 25; 26]
In this paper, we provide a comprehensive theoretical framework for both 1D and 2D THz-MR measurements based on the response-function theory developed for nonlinear optical spectroscopy techniques. To illustrate our approach more closely, we employ a chiral spin
chain described as a Heisenberg model with exchange coupling and Dzyaloshinskii-Moriya interaction (DMI) arising from the antisymmetric spin-orbit coupling. [27; 28; 29] Such anisotropy of the spin system leads to a series of unique magnetic and optical properties. [30] Unlike with conventional FM materials, in addition to the magnetic anisotropy, a non-centrosymmetric structure arises as the result of the chiral ordering of spins. As a result, a series of new phenomena arising from the nonlinear magnetic response, such as magneto-chiral dichroism and magnetization-induced SHG, were observed. [31; 32; 33; 34]
To investigate the properties of such materials as devices, it is necessary to investigate ultrafast spin dynamics under time-dependent external fields, in which quantum coherence and entanglement--not only among spins but also between spins and the environment--play a significant role. [35; 2] Thus, we include a harmonic heat bath of finite temperature in the spin system. The number of degrees of freedom of the heat bath is then reduced to obtain a time-irreversible equation of motion describing the effects of thermal fluctuations and dissipation. [36; 37] Because the motion of the spins is much faster than the thermal noise arising from the heat bath, the heat bath must be treated in a non-Markovian manner. Thus, the reduced equations of motion derived using Markovian approximations and other assumptions--such as the Bloch, Lindblad, and Redfield equations--are not suitable for the description of such ultrafast spin dynamics. We then employ the numerically "exact" hierarchical equations of motion (HEOM) approach, which can be used to treat non-Markovian and non-perturbative system-bath interactions at finite temperatures. [38; 39; 40; 41]
To demonstrate the applicability of the present theory, 1D and 2D THz-MR spectra were calculated for the chiral spin model with different DMI strengths describing the pitch and direction of chirality (clockwise or counterclockwise). We show that in 1D spectra, only the absolute strength of the DMI can be evaluated, whereas in 2D spectra, the sign of the DMI, which determines the direction of the chirality, can also be determined. While neutron-scatting techniques have been used to determine the structures of chiral materials, the present results indicate the possibility of determining the DMI through spin-dynamic processes. This finding should be valuable for the design of spintronic materials.
The rest of this paper is organized as follows. In Sec. II, we formulate linear (1D) and 2D THz-MR spectra based on the response-function theory. In Sec. III, we introduce the Heisenberg model with the DMI coupled to the harmonic heat bath. The HEOM are then presented. Numerical results and some discussion are presented in Sec. IV. Finally, Sec. V is devoted to our conclusions.
## II Magnetic susceptibilities in 2D THz-MR spectroscopy
We consider a spin system coupled to a bath system driven by an external magnetic field \(B(s)\). The total Hamiltonian is expressed as
\[\hat{H}^{\prime}(s)=\hat{H}_{tot}+B(s)\,\hat{M}, \tag{1}\]
where \(\hat{H}_{tot}\) is the Hamiltonian of a composite system and \(\hat{M}\) is the polarization operator for magnetic fields.
The observable in a magnetic measurement at time \(t\) is expressed as \(M(t)\equiv\langle\hat{M}(t)\rangle-\langle\hat{M}\rangle\), where \(\hat{M}(t)\) is the Heisenberg representation of \(\hat{M}\) for \(\hat{H}^{\prime}(s)\). The thermal average for any operator \(\hat{A}\) is defined as \(\langle\hat{A}\rangle\equiv\mathrm{tr}\{\hat{A}\hat{\rho}_{tot}^{eq}\}\), with the equilibrium density operator expressed as \(\hat{\rho}_{tot}^{eq}\). We can also express this as \(M(t)=\mathrm{tr}\{\hat{M}\mathcal{G}^{\prime}(t)\hat{\rho}_{tot}^{eq}\}- \mathrm{tr}\{\hat{M}\hat{\rho}_{tot}^{eq}\}\), where the Liouville operator is defined as
\[\mathcal{G}^{\prime}(t)\hat{\rho}_{tot}(0)\equiv \underset{\longleftarrow}{\exp}\left[-\frac{i}{\hbar}\int_{0}^{t} \mathrm{d}s^{\prime}\hat{H}^{\prime}(s^{\prime})\right]\hat{\rho}_{tot}(0)\] \[\times \underset{\longrightarrow}{\exp}\left[\frac{i}{\hbar}\int_{0}^{t} \mathrm{d}s^{\prime}\hat{H}^{\prime}(s^{\prime})\right]. \tag{2}\]
Here, the arrows indicate time-ordered exponentials.
Experiments such as 2D NMR and 2D EPS measurements [1; 2] are conducted using multiple magnetic pulses with finite time widths. In such experiments, the excitation by the external field is non-perturbative, and the desired spin dynamics are investigated by designing the profiles of pulse trains, as in the cases of spin-echo and correlation spectroscopy measurements. Theoretically, these signals are obtained by integrating the equations of motion for a spin system, such as the Bloch, [42] Redfield, [43] or stochastic Liouville equations, [44] or the HEOM, [45; 46; 40] under a sequence of magnetic pulses.
As in the case of coherent optical laser spectroscopies, the excitations of coherent THz-MR spectroscopy are assumed to be impulsive, which allows us to measure the signals in different orders of the field-system interactions separately. Thus, we can employ a response-function theory developed for ultrafast nonlinear laser spectroscopy. [6] Up to the third order, the signal is then expressed as
\[M(t) =\int_{0}^{t}\mathrm{d}s\,\chi_{1}(s)\,B(t-s)\] \[+\int_{0}^{t}\mathrm{d}s_{1}\int_{0}^{s_{1}}\mathrm{d}s_{2}\, \chi_{2}(s_{1},s_{2})\,B(t-s_{1})\,B(t-s_{2})\] \[+\int_{0}^{t}\mathrm{d}s_{1}\int_{0}^{s_{1}}\mathrm{d}s_{2}\, \int_{0}^{s_{2}}\mathrm{d}s_{3}\,\chi_{3}(s_{1},s_{2},s_{3})\] \[\times B(t-S_{1})\,B(t-s_{2})\,B(t-s_{3}), \tag{3}\]
where the linear, second-order, and third-order response functions are defined as [40]
\[\chi_{1}(s) \equiv-\frac{i}{\hbar}\langle[\hat{M}(s),\hat{M}]\rangle\] \[=-\frac{i}{\hbar}\mathrm{tr}\left\{\hat{M}\mathcal{G}(s)\,\hat{M} ^{\times}\hat{\rho}_{tot}^{eq}\right\}, \tag{4}\]
\[\chi_{2}(s_{1},s_{2}) \equiv-\frac{1}{\hbar^{2}}\langle[[\hat{M}(s_{1}+s_{2}),\hat{M}(s_{1})],\,\hat{M}]\rangle\] \[=-\frac{1}{\hbar^{2}}\text{tr}\left\{\hat{M}\,\mathcal{G}(s_{2})\, \hat{M}^{\times}\mathcal{G}(s_{1})\hat{M}^{\times}\hat{\rho}^{eq}_{tot}\right\}, \tag{5}\]
and
\[\chi_{3}(s_{1},s_{2},s_{3})\] \[\equiv\frac{i}{\hbar^{3}}\langle[[[\hat{M}(s_{1}+s_{2}+s_{3}), \hat{M}(s_{1}+s_{2})],\,\hat{M}(s_{1})],\,\hat{M}]\rangle\] \[=\frac{i}{\hbar^{3}}\text{tr}\left\{\hat{M}\,\mathcal{G}(s_{3})\, \hat{M}^{\times}\,\mathcal{G}(s_{2})\,\hat{M}^{\times}\mathcal{G}(s_{1})\hat{ M}^{\times}\hat{\rho}^{eq}_{tot}\right\}. \tag{6}\]
Here, \(\mathcal{G}(s)\) is the Liouvillian without magnetic interaction, which is defined from \(\mathcal{G}^{\prime}(s)\) with \(B(s)=0\), and we have introduced the superoperator \(\hat{M}^{\times}\hat{\rho}\equiv[\hat{M},\hat{\rho}]\). The right-hand side (rhs) of the second line in each of the above equations allows us to employ the equations of motion to calculate the response functions and give us an intuitive picture of the higher-order optical processes. [40; 41] For example, the rhs of the second line of Eq. (II) can be read from right to left as follows. The total system is initially in the equilibrium state \(\hat{\rho}^{eq}_{tot}\). The initial state is then modified by the first magnetic pulses via the dipole operator as \((\hat{M}^{\times}\hat{\rho}^{eq}_{tot})=[\hat{M},\hat{\rho}^{eq}_{tot}]\) at \(t=0\) and is propagated for time \(t=s\) by \(\mathcal{G}(s)\). The expectation value is then obtained by calculating the trace of \(\hat{M}\).
Actual 2D experiments have been conducted using a pair of magnetic pulses \(a\) and \(b\) with inter-pulse delay \(\tau\). [47; 21] Under the impulsive approximation, the magnetic field is expressed as
\[B(s)=B_{a}\delta(s)+B_{b}\delta(s-\tau), \tag{7}\]
where \(B_{a}\) and \(B_{b}\) are the magnetic field strengths. The nonlinear element of the signal \(M_{NL}(t,\tau)\) at \(t+\tau\) is then evaluated as
\[M_{NL}(t,\tau)=M_{ab}(t+\tau)-M_{a}(t+\tau)-M_{b}(t+\tau), \tag{8}\]
where \(M_{ab}(t+\tau)\), \(M_{a}(t)\), and \(M_{b}(t)\) are the total signal and the linear elements for pulses \(a\) and \(b\), respectively.
From Eqs. (II) and (7), the nonlinear element is evaluated as
\[M_{NL}(t,\tau) =B_{a}B_{b}\,\chi_{2}(\tau,t)+B_{a}B_{b}^{2}\,\chi_{3}(\tau,0,t)\] \[+B_{a}^{2}B_{b}\,\chi_{3}(0,\tau,t). \tag{9}\]
Thus, the characteristic features of THz-MR spectroscopy are described by the nonlinear susceptibilities. Because each term on the rhs has different proportionality with respect to \(B_{a}\) and \(B_{b}\), they can be evaluated separately by changing their respective field strengths.
In nonlinear optical spectroscopies, \(\chi_{2}\) is used to analyze 2D THz-Raman [48; 49] and 2D THz-IR-Raman signals [50; 51] that involve a 2Q transition process, while \(\chi_{3}(s_{3},s_{2},s_{1})\) is used to analyze nonlinear 2D THz spectroscopies [52; 53] including 2D THz rotational spectroscopy. [54; 55] As illustrated in nonlinear optical spectroscopies, the \(s_{1}\) and \(s_{3}\) periods describe the time evolutions of coherent states, while \(s_{2}\) describes the evolution of population states. [22; 23] While \(\chi_{3}(\tau,0,t)\) is used for 2D spectroscopies with waiting time \(s_{2}=0\), \(\chi_{3}(0,\tau,t)\) is used for transient absorption spectra in which the time evolution of the excited-state population is measured.
Using \(\chi_{1}(s)\) in Eq. (II), the linear absorption spectrum is defined as
\[\chi_{1}(\omega)=\text{Im}\int_{0}^{\infty}\text{d}se^{-i\omega s}\chi_{1}(s), \tag{10}\]
where Im denotes the imaginary part. Following the experimental setup, we consider two kinds of 2D spectrum for different time configurations of \(\tau\) and \(t\) in Eq. (II), expressed as: (II) \(\chi_{3}(\tau,0,t)\) and (II) \(\chi_{3}(0,\tau,t)\). They are then expressed in the Fourier translation form as
\[\chi_{3}^{(1)}(\omega_{\tau},\omega_{t}) =\text{Im}\int_{0}^{\infty}\text{d}\tau\int_{o}^{\infty}\text{d}t \,e^{-i\omega_{\tau}\tau-i\omega_{t}t}\chi_{3}(\tau,0,t), \tag{11}\] \[\chi_{3}^{(2)}(\omega_{\tau},\omega_{t}) =\text{Im}\int_{0}^{\infty}\text{d}\tau\int_{o}^{\infty}\text{d}t \,e^{-i\omega_{\tau}\tau-i\omega_{t}t}\chi_{3}(0,\tau,t), \tag{12}\]
where \(\omega_{\tau}\) and \(\omega_{t}\) represent the excitation and detection frequencies, respectively. [21]
## III Chiral spin model and Heom approach
To describe the novel phenomena of chiral magnets, we consider a linear chain system consisting of \(L\) spins. The system Hamiltonian is defined as
\[\hat{H}_{S} =J\sum_{n=1}^{L}\left[\hat{\sigma}_{n}^{z}\,\hat{\sigma}_{n+1}^{ z}+\Delta\left(\hat{\sigma}_{n}^{x}\,\hat{\sigma}_{n+1}^{x}+\hat{\sigma}_{n}^{y}\, \hat{\sigma}_{n+1}^{y})\right]\] \[+D_{y}\sum_{n=1}^{L}\left[\hat{\sigma}_{n}^{z}\,\hat{\sigma}_{n+1}^ {x}-\hat{\sigma}_{n}^{x}\,\hat{\sigma}_{n+1}^{z}\right], \tag{13}\]
where \(\hat{\sigma}_{n}^{\alpha}\) (\(\alpha=x,y,z\)) denotes the Pauli operator at the \(n\)-th site in the \(\alpha\)-th direction. Here, the first term with \(J>0\) represents the AFM Heisenberg exchange coupling, and \(\Delta\) is the anisotropic parameter. The second term in Eq. (II) is the DMI that is perpendicular to the \(xz\) plane, with coupling strength \(D_{y}\). The sign of \(D_{y}\) determines the direction (i.e., clockwise or anticlockwise) or handedness (i.e., right- or left-handed circular configuration) of chirality, which is not easily determined by experimental measurements. [3; 9] The magnetization operator is defined as
\[\hat{M}=\sum_{l=1}^{L}\hat{\sigma}_{l}^{x}. \tag{14}\]
A very important aspect of investigating ultrafast dynamics in magnetic materials is the inclusion of a
quantum-mechanically consistent relaxation and dephasing mechanism. This can be achieved by including a harmonic heat bath in the spin system. The Hamiltonian of this heat bath is defined as
\[\hat{H}_{B}=\sum_{j}\left[\frac{\hat{p}_{j}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2 }}{2}\hat{x}_{j}^{2}\right], \tag{15}\]
where \(\hat{p}_{j}\), \(\hat{x}_{j}\), \(\omega_{j}\), and \(m_{j}\) represent the momentum, position, frequency, and mass of the \(j\)-th oscillator, respectively. The system-bath interaction is defined by
\[\hat{H}_{I}=\hat{V}\,\sum_{j}g_{j}\,\hat{x}_{j}, \tag{16}\]
where \(\hat{V}\) represents the spin part of the system-bath interaction function, defined as
\[\hat{V}=\sum_{n}^{L}\hat{\sigma}_{n}^{z}, \tag{17}\]
and \(g_{j}\) is the coupling strength with the \(j\)-th oscillator. The total Hamiltonian is then given by
\[\hat{H}_{tot}=\hat{H}_{S}+\hat{H}_{I}+\hat{H}_{B}. \tag{18}\]
In open quantum dynamics theory, the time-irreversible process can be described using a reduced set of equations of motion. After reducing the number of degrees of freedom of the heat bath, the noise effects are characterized by the correlation function,
\[C(t)=\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d}\omega\,J(\omega)\left[\coth\left( \frac{\beta\hbar\omega}{2}\right)\cos(\omega t)-i\sin(\omega t)\right], \tag{19}\]
where \(J(\omega)\) is the spectral density function (SDF), \(\beta=1/k_{B}T\) is the inverse temperature, and \(k_{B}\) is the Boltzmann constant. For the heat bath, we assume the Drude SDF, which is expressed as,
\[J(\omega)=\frac{\zeta\gamma^{2}\omega}{\gamma^{2}+\omega^{2}}, \tag{20}\]
where \(\zeta\) is the coupling strength and \(\gamma\) is the inverse correlation time.
In non-perturbative and non-Markovian conditions, the time evolution can be described by the HEOM approach.[38; 39] We rewrite Eq. (19) in a linear-summation form,
\[C(t)=\sum_{k}^{K}c_{k}\,e^{-\nu_{k}|t|}, \tag{21}\]
where \(c_{k}\) and \(\nu_{k}\) are complex-valued coefficients. The HEOM can then be expressed as[40; 41]
\[\begin{split}\frac{\partial}{\partial t}\hat{\rho}_{[\vec{n}]}( t)=&-\left[i\hat{H}_{S}^{\times}+\sum_{k}^{N}n_{k}\nu_{k}\right]\hat{ \rho}_{[\vec{n}]}(t)\\ &-i\hat{V}^{\times}\sum_{k}^{K}\hat{\rho}_{[\vec{n}+\vec{e}_{k}] }(t)\\ &-i\sum_{k}^{K}\left[c_{k}\hat{V}\hat{\rho}_{[\vec{n}-\vec{e}_{k}] }(t)-c_{k}^{*}\hat{\rho}_{[\vec{n}-\vec{e}_{k}]}(t)\hat{V}\right],\end{split} \tag{22}\]
where \(\vec{n}=\{n_{1},n_{2},\cdots,n_{K}\}\) denotes the index vector, and \(n_{k}\) are non-negative integers. Among all the \(\hat{\rho}_{[\vec{n}]}(t)\), the 0-th, with \(\vec{0}=\{0,0,\cdots,0\}\), corresponds to the density operator of the reduced system, while all the others are introduced for ancillary purposes.
In the HEOM approach, the response functions Eqs. (4)-(6) are evaluated as the time evolution of the system under external excitation. The density matrix is replaced by the HEOM elements, and the Liouvillian \(\mathcal{G}(t)\) is replaced using Eq. (22).
For example, we can evaluate \(\chi_{3}(\tau,0,t)\) using the expression of the second line in Eq. (6) as follows.[40; 41] We first run the HEOM program for a sufficiently long period from a temporally initial condition at \(t=-t_{i}\) (such as the factorized initial condition, \(\hat{\rho}_{[\vec{0}]}(-t_{i})=\exp[-\beta\hat{H}_{S}]\) with all the other hierarchy elements set to zero) to time \(t=0\) to reach a true thermal equilibrium, denoted by \(\hat{\rho}_{[\vec{n}]}^{eq}(0)\). All the hierarchy elements have to be used to define a correlated initial condition. The system is next excited by the first magnetic interaction \(\hat{M}\) at \(t=0\) as \(\hat{\rho}_{[\vec{n}]}^{eq}(0)=[\hat{M},\hat{\rho}_{[\vec{n}]}^{eq}(0)]\). The perturbed elements \(\hat{\rho}_{[\vec{n}]}^{\prime}\) then evolve in time by numerically integrating Eq. (22) up to \(t\). At \(t\), the system is excited by the second and third magnetic interactions as \(\hat{\rho}_{[\vec{n}]}^{\prime\prime}(t)=[\hat{M},[\hat{M},\hat{\rho}_{[\vec{ n}]}^{\prime}(t)]]\). Then, after \(\hat{\rho}_{[\vec{n}]}^{\prime\prime}\) evolves in time with the initial condition \(\hat{\rho}_{[\vec{n}]}^{\prime\prime}(t)\) to \(t+\tau\), the response function is calculated from the expectation value of the magnetic moment as \(\chi_{3}(\tau,0,t)=tr_{A}\{\hat{M}\hat{\rho}_{[\vec{0}]}^{\prime\prime}(t+\tau)\}\).
Notice that to take the system-bath coherence (or bath entanglement[41]) into account during the external perturbation, it is important to operate \(\hat{M}\) to all of the hierarchy elements \(\hat{\rho}_{[\vec{n}]}^{\prime\prime}(t)\). Although we only use \(\hat{\rho}_{[\vec{0}]}(t)\) to calculate an expectation value, the other elements are essential to obtain an echo signal for non-Markovian noise in 2D spectroscopy.[40; 41]
## IV Numerical results
In the following, we set the exchange-coupling strength as the base unit \(J=1\) and calculate 1D (or linear absorption) and 2D spectra for \(\Delta=0.1\) as a model for typical AFM metal oxides.[56; 57] The number of spin sites was \(L=6\), with periodic boundary conditions \(\hat{\sigma}_{n}^{\alpha}=\hat{\sigma}_{n+L}^{\alpha}\)
Here, we consider cases without the DMI (\(D_{y}=0\)) and with the weak DMI (\(|D_{y}|\leq 0.05\)), which is appropriate for a small model system. Although the HEOM used in this study can investigate cases in which the system interacts strongly with the heat bath, here, we focus on the characterization of the chiral spin system and keep the coupling weak. Thus, the bath parameters were chosen as \(\zeta=0.05\), \(\gamma=1.0\), and \(\beta\hbar=10\). The hierarchy parameters were chosen as \(K=50\) and \(\sum_{k}^{K}n_{k}\leq 10\), and the time step for numerical integration was set to \(\delta t=0.01\).
### Energy eigenstates and transition elements of \(\hat{M}\)
To illustrate the origin of the peaks in the 1D and 2D spectra that will be shown later, in Table 1, we present some representative energy eigenstates \(|n_{m}^{(\prime)}\rangle\) that are necessary to explain THz-MR spectra. Here, \(n=0\), \(1\), and \(2\) represent the ground, first, and second excited states, respectively, and \(m\) and \({}^{\prime}\) are introduced to signify the energy splitting arising from the anisotropic coupling \(\Delta\) and the DMI, respectively. Thus, for example, the states \(|1_{0}\rangle\) and \(|1_{0}^{\prime}\rangle\) are degenerate when \(D_{y}=0\). Note that to conduct numerical simulations, we employed all of the spin-\(z\) basis states to maintain the numerical accuracy.
In Fig. 1, we depict the energy eigenstates (a) without the DMI (\(D_{y}=0\)) and (b) with the DMI (\(D_{y}=\pm 0.05\)). The possible magnetic transitions that arise from \(\hat{M}\) are denoted by vertical arrows. Here, depending on the role of \(D_{y}\), we classify the eigenstates as (I) independent of the DMI (black lines), (II) dependent on the square of \(D_{y}\) (blue lines), and (III) independent of the DMI but the phases of their wave functions change depending on the sign of \(D_{y}\) (red lines). For a finite \(D_{y}\) in Fig. 1(b), the degeneracy of eigenenergy is resolved because the inversion symmetry is broken, and we observe the finite energy gap in (II) (blue dashed lines).
### 1D THz-MR spectrum
In Fig. 2, linear absorption (1D) spectra \(\chi_{1}(\omega)\) calculated from Eq. (10) with Eq. (4) are presented for different \(D_{y}\). Here, we only focus on \(D_{y}>0\) because the spectral profiles for \(-|D_{y}|\) are identical to those for \(+|D_{y}|\). The tiny symmetrical peaks around the main peak arise as an artifact of numerical Fourier transformation; these can be suppressed by increasing the time interval and introducing window functions.
Under current low-temperature conditions, the spin system is almost in the ground equilibrium states \(|0_{0}\rangle\) and \(|0_{1}\rangle\). The populations of other excited states are less than \(0.1\%\), and we cannot observe the transitions from
Figure 1: Schematic view of the energy states of the spin Hamiltonian, \(\hat{H}_{S}\), (a) without the DMI (\(D_{y}=0\)) and (b) with the DMI (\(D_{y}=\pm 0.05\)). The black, blue, and red lines represent the states that are (I) independent of the DMI, (II) dependent on the square of \(D_{y}\), and (III) independent of the DMI but the phases of their wave functions change depending on the sign of \(D_{y}\), respectively. The purple arrows denote the possible transitions that arise from \(\hat{M}\).
\begin{table}
\begin{tabular}{c|c|c} \hline \hline & (a) \(D_{y}=0\) & (b) \(D_{y}=\pm 0.05\) \\ \hline \hline \(|0_{0}\rangle\) & \(0\) & \(0\) \\ \hline \(|0_{1}\rangle\) & \(0.001\) & \(0.001\) \\ \hline \(|1_{0}\rangle\) & \(1.02\) & \(0.99\) \\ \hline \(|1_{0}^{\prime}\rangle\) & \(-\) & \(1.14\) \\ \hline \(|1_{1}\rangle\) & \(0.87\) & \(0.81\) \\ \hline \(|1_{1}^{\prime}\rangle\) & \(-\) & \(0.98\) \\ \hline \(|1_{2}\rangle\) & \(0.93\) & \(0.93\) \\ \hline \(|1_{3}\rangle\) & \(1.15\) & \(1.15\) \\ \hline \(|2_{0}\rangle\) & \(2.02\) & \(1.99\) \\ \hline \(|2_{1}\rangle\) & \(2.03\) & \(2.03\) \\ \hline \(|2_{2}\rangle\) & \(1.85\) & \(1.86\) \\ \hline \(|2_{3}\rangle\) & \(2.02\) & \(2.02\) \\ \hline \(|2_{4}\rangle\) & \(2.12\) & \(2.12\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Eigenenergies of spin Hamiltonian, \(\hat{H}_{S}\) (a) without the DMI and (b) with the DMI.
the excited states in the 1D spectrum. The main peak \(A_{1}\) arises from the transition \(|0_{0}\rangle\rightarrow|1_{0}\rangle\). Because the DMI resolves the degeneracy of states \(|1_{0}\rangle\) and \(|1_{1}\rangle\) (blue lines), the energy states \(|1^{\prime}_{0}\rangle\) and \(|1^{\prime}_{1}\rangle\) (blue dashed lines) appear for finite DMI, as illustrated in Fig. 1(b). Accordingly, we observe the adjoint peak \(A_{2}\) that arises from the transition \(|0_{0}\rangle\rightarrow|1^{\prime}_{0}\rangle\). The energy eigenvalues of \(|1_{0}\rangle\) and \(|1^{\prime}_{0}\rangle\) are (II) dependent upon the square of \(D_{y}\), and we found that, for our system with \(\Delta/J=0.1\), these peak positions are evaluated as \(\omega_{A_{1}}\approx-8.34\,D_{y}^{2}+1.02\) and \(\omega_{A_{2}}\approx 8.34\,D_{y}^{2}+1.12\). Thus, from the position of the \(A_{2}\) peak, we can estimate the magnitude of \(D_{y}\), but we cannot determine the direction of chirality, which is described as the sign of \(D_{y}\).
### 2D THz-MR spectra
To elucidate the chirality of the system further, we next present numerical results for 2D THz-MR spectrum evaluated from \(\chi_{3}^{(1)}(\omega_{\tau},\omega_{t})\) and \(\chi_{3}^{(2)}(\omega_{\tau},\omega_{t})\), expressed as Eqs. (11) and (12) with Eq. (6). In the present case, the rephasing parts (\(\omega_{t}<0\) and \(\omega_{\tau}>0\)) in both 2D spectra have no signal, indicating that there is no spin echo.[22; 23] Thus, we only present the non-rephasing parts of the spectra (\(\omega_{t}>0\) and \(\omega_{\tau}>0\)).
#### iii.3.1 Contribution from \(\chi_{3}^{(1)}(\omega_{\tau},\omega_{t})\)
In Fig. 3, we present contour maps of \(\chi_{3}^{(1)}(\omega_{\tau},\omega_{t})\) in the (a) \(D_{y}=0\), (b) +0.05, and (c) \(-0.05\) cases, in which the red and blue areas represent positive and negative intensities. To analyze the physical origin of each peak, we present a double-sided Feynman diagram corresponding to each labeled peak in Fig. 4(a). Here, the \(B_{3}\) peak consists of two Liouville paths, which we denote as \(B_{3}^{(i)}\) and \(B_{3}^{(ii)}\).
These diagrams describe the time evolution of the left and right wave functions of the density operators involved in the response function from bottom to top under the magnetic excitations and deexcitations. For example, diagram \(B_{3}^{(i)}\) in Fig. 4(a) describes the time evolution of the density operator undergoing magnetic interactions described by the purple arrows in Fig. 1 with the left wave function at time \(s=0\), \(s=\tau\), and \(s=\tau+t\) with the right wave function at time \(s=\tau\). The left wave function evolves with time \(|0_{1}\rangle\rightarrow|2_{1}\rangle\rightarrow|2_{4}\rangle\rightarrow|1_{3}\rangle\), while the right one evolves with time \(\langle 0_{1}|\rightarrow\langle 1_{3}|^{\text{,6,4}}\).
The peaks labeled as \(B_{1}\) and \(B_{2}\) along \(\omega_{t}=0.98\) correspond to peaks \(A_{1}\) and \(A_{2}\) in the 1D spectra, and the frequency difference \(\omega_{A_{2}}-\omega_{A_{1}}\) is equal to \(\omega_{B_{2}}-\omega_{B_{1}}\). As illustrated in Fig. 4(a), the diagrams \(B_{1}\) and \(B_{2}\) only involve the eigenstates that are (I) independent of the DMI and (II) dependent on the square of \(D_{y}\). Thus, the intensities of the \(B_{1}\) and \(B_{2}\) peaks do not depend on the sign of \(D_{y}\).
Figure 2: Linear absorption (1D) spectra, \(\chi_{1}(\omega)\) calculated for \(D_{y}=\) (a) 0, (b) +0.025, and (c) +0.05. The signal intensities are normalized with respect to the absolute values of the peak intensities.
However, the \(B_{3}^{(ii)}\) diagram in Fig. 4 involves states (II) and (III), while the \(B_{3}^{(i)}\) diagram contains the state \(|2_{4}\rangle\) that is (III) independent of the DMI but the phase of its wave function changes depending on the sign of \(D_{y}\). As a result, the sign of the peak intensity changes for \(D_{y}=+0.05\) and \(-0.05\). However, the peak profiles in Figs. 3 and 3 are not positively and negatively symmetric due to the contribution from \(B_{3}^{(ii)}\).
#### iii.2.2 Contribution from \(\chi_{3}^{(2)}(\omega_{\tau},\omega_{t})\)
We next depict \(\chi_{3}^{(2)}(\omega_{\tau},\omega_{t})\) in Fig. 5. Double-sided Feynman diagrams corresponding to each labeled peak are presented in Figs. 4 and 4.
Peaks \(C_{1}\)-\(C_{4}\) appear in the case \(D_{y}\neq 0\) because they involve the states classified in (II), as depicted in Fig. 4. Moreover, because peaks \(C_{1}\)-\(C_{4}\) involve the \(|1_{2}\rangle\) and \(|2_{3}\rangle\) states classified in (III) in each diagram in Fig. 4, the sign of the peak intensity changes in correspondence with the sign of \(D_{y}\). Note that the peak profiles of \(C_{1}\) for \(D_{y}=0\) and \(D_{y}>0\) are identical because the \(|1_{2}\rangle\) state does not vanish even for \(D_{y}=0\). We also find that the positive peak near \((\omega_{\tau},\omega_{t})=(1.05,1)\) in Fig. 5 arises as the consequence of the heat-bath-induced coherence \(|0_{0}\rangle\langle 1_{0}|\).
In Figs. 4 and 4, we observe that peaks \(C_{5}\)-\(C_{9}\) arise from the 2Q coherence. Although the \(C_{9}\) diagram involves the \(|2_{3}\rangle\) state in (III), it overlaps with \(C_{7}\) for \(D_{y}\neq 0\), and the signs of the peak intensities cannot be easily evaluated. Note that here we chose a small system (\(L=6\)) with periodic boundary conditions, so energy states higher than the second excited state are lowered, and the profiles of peaks \(C_{5}\)-\(C_{9}\) are distorted. Hence, within the calculations of the present model, it is more reliable to use the profiles of \(C_{1}\)-\(C_{4}\) to determine the sign of \(D_{y}\).
Figure 4: Double-sided Feynman diagrams for all labeled peaks. Two different contributions of \(B_{3}\) are shown separately.
Figure 3: 2D THz-MR spectrum calculated from \(\chi_{3}^{(1)}(\omega_{\tau},\omega_{t})\) for three values of \(D_{y}\): (a) 0, (b) +0.05, and (c) \(-0.05\). The intensity of each spectrum is normalized with respect to its maximum peak intensity. The red and blue areas represent the positive and negative intensities of the spectra.
## V Conclusion
A recently developed 2D THz-MR spectroscopic technique has created new possibilities for measuring complex molecular magnetic systems. In the present work, we have illustrated the key features of this technique based on nonlinear response-function theory and have described a method for simulating 2D THz-MR spectra through the use of the HEOM formalism. Using simulated 1D and 2D THz-MR spectra for a chiral magnetic material, we demonstrated that the 2D technique allows us to evaluate the pitch and direction of chirality determined from the strength and sign of the DMI, while only the absolute amplitude of the DMI can be determined from 1D measurements.
The reason the sign of the DMI was detected by 2D spectroscopy in this study is that there are eigenstates in which the phase of the wavefunction changes following the sign of the DMI, which is categorized as (III). Although there have been studies of eigenstates of spin systems with the DMI,[58] the existence of eigenstates that change phase with the sign of the DMI, as found in this study, has not been explored. It is important to know the causes of these eigenstates because they may lead to the appearance of novel phenomena.
To make a direct comparison between the results of our simulations and those obtained experimentally, however, we must increase the number of spins in accordance with those available in experimental systems. In investigating spin dynamics in the condensed phase, it is also important to consider the non-perturbative and non-Markovian system-bath interactions. Nevertheless, we believe that the present results elucidate the key features of 2D THz-MR spectroscopic methods with regard to probing the fundamental nature of a magnetic spin system.
For further investigations to monitor the ultrafast dynamical aspects of the spin system, such as the dynamics of spin waves, it is necessary to conduct a variety of advanced nonlinear spectroscopic approaches, such as pump-probe and transient absorption measurements to foster the development of this spectroscopic method. We leave such extensions to future studies, depending on progress in experimental and simulation techniques.
###### Acknowledgements.
The authors are thankful to Professor Jun-ichiro Kishine and Professor Keisuke Tominaga for helpful discussions. Y.T. was supported by JSPS KAKENHI (Grant No. B21H01884). J.Z. was supported by JST SPRING (Grant No. JPMJSP2110).
### Conflict of Interest
The authors have no conflicts to disclose.
## Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
2303.15584 | Dealing with large gaps in asteroseismic time series | With long data sets available for asteroseismology from space missions, it is
sometimes necessary to deal with time series that have large gaps. This is
becoming particularly relevant for TESS, which is revisiting many fields on the
sky every two years. Because solar-like oscillators have finite mode lifetimes,
it has become tempting to close large gaps by shifting time stamps. Using
actual data from the Kepler Mission, we show that this results in artificial
structures in the power spectrum that compromise the measurements of mode
frequencies and linewidths. | Timothy R. Bedding, Hans Kjeldsen | 2023-03-27T20:28:20Z | http://arxiv.org/abs/2303.15584v1 | # Dealing with large gaps in asteroseismic time series
###### Abstract
With long data sets available for asteroseismology from space missions, it is sometimes necessary to deal with time series that have large gaps. This is becoming particularly relevant for _TESS_, which is revisiting many fields on the sky every two years. Because solar-like oscillators have finite mode lifetimes, it has become tempting to close large gaps by shifting time stamps. Using actual data from the _Kepler_ Mission, we show that this results in artificial structures in the power spectrum that compromise the measurements of mode frequencies and linewidths.
Asteroseismology
## Introduction
Asteroseismology involves studying the oscillations of stars by analysing the power spectra of their flux or radial-velocity variations. With long data sets now available from space missions such as BRITE, CoRoT, _Kepler_, K2 and _TESS_, it is sometimes necessary to deal with time series that have large gaps. This is becoming particularly relevant for _TESS_, which is revisiting many fields on the sky every two years (Ricker et al., 2015).
The Fourier transform of a long time series results in a very large array. This is because the power spectrum is calculated with a step size that must sample the frequency resolution, and this is inversely proportional to the total duration of the time series (including the gap). Because solar-like oscillators have finite mode lifetimes, it has become tempting to close large gaps by shifting time stamps and we are aware of several cases in which this has been done, both in published works (e.g., Hekker et al., 2010; Nielsen et al., 2022) and in papers in preparation. The justification is that mode lifetimes are much shorter than the gap, so shifting segments should not have much effect on the power spectrum. However, we argue that even if the modes are completely incoherent between the two segments, an arbitrary shift will introduce artificial structures in Fourier space that will compromise the profile fitting. We demonstrate this with a simple test on a star observed by _Kepler_ (we have tested other stars using _TESS_ data and found similar results).
## Analysis and Results
We used the subgiant star KIC 11137075 (also known as 'Zebedee'), whose _Kepler_ light curve contains more than one year of short-cadence data (1-minute sampling) that show high signal-to-noise solar-like oscillations centered at about \(1700\,\mu\)Hz (Tian et al., 2015). For this test we defined two segments, as shown in the top panel of Fig 1, where the boundaries coincided with the short breaks each month during which data were downloaded from the spacecraft (Haas et al., 2010). The segment lengths were \(30.1\,\)d and \(26.0\,\)d, and the gap was \(412.1\,\)d.
The other panels in the figure show power spectra in regions \(2.5\,\mu\)Hz wide that are centred on four of the strongest modes (the first two are \(l=0\) modes and the others are \(l=1\)). For each mode, the vertical dashed line marks the frequency measured by Tian et al. (2015) from the full _Kepler_ time series. The thin blue line shows the power spectrum calculated from the two segments when using the correct time stamps, that is, with the gap left in place. The rapid oscillations are due to the spectral window from the large gap. If desired, this spectrum could be smoothed to lower resolution and then re-sampled to produce a smaller array. Another way to achieve the same result is to compute the power spectra of the two segments separately (using exactly the same frequency resolutions) and take the average.
This is shown by the black line in each figure. We see that it follows quite closely the upper envelope of the blue line, although not exactly because of differences in the mode amplitudes between the two segments. Such differences reflect the well-know stochastic nature of solar-like oscillations.
The red line in Fig. 1 shows the power spectrum of a time series formed by shifting the segment 2 so that it follows immediately after segment 1. The peaks in this spectrum are narrower than in the average of two segments (black line) but this does not indicate a real improvement in frequency resolution. Rather, it is a natural consequence of placing the two segments side-by-side. Furthermore, the structures in the red curve have been artificially introduced by the shifting procedure, and they greatly reduce the precision with the mode frequencies (and linewidths) can be measured.
## 2 Discussion and Conclusions
To understand why the line profiles are modified by shifting time stamps, it helps to consider a simple scenario. Suppose a pure sine wave with a single frequency is present in both segments, but with an arbitrary phase difference. If the two segments are shifted so that they are adjacent, the power spectrum at the chosen frequency will depend very sensitively on the phase difference. If the two signals happen to be in phase then the peak will be strong, but if they are out of phase then the peak will be very weak (and the power will be spread into nearby frequency bins). Indeed, we have confirmed that the power spectrum in our test (red lines in Fig. 1) changes greatly if the position of the shifted segment is slightly adjusted. Leaving a gap of only five minutes (about half the typical oscillation period) results in very different line profiles.
The procedure described above for averaging the individual power spectra of the segments (black lines in Fig. 1) is only appropriate if the data segments have similar lengths. In that case, segments with different noise levels--which often occur with _TESS_ data--can be combined by calculating a weighted average of the two power spectra. In this case, the weights are easily calculated as the inverse of the square of the mean power level in a region of the power spectrum that is dominated by noise (usually at high frequencies). If the segments do _not_ have similar lengths then it is best to stay with the standard power spectrum, using the full data set, and to smooth to lower resolution if desired (see above). In the data have non-uniform quality then the scatter in the time series can be used to calculate a _weighted_ power spectrum (e.g., Kjeldsen et al., 2005 and references therein).
Finally, we always prefer to deal with power spectra that are over-sampled, typically by a factor of 5 or so. This is because a critically-sampled spectrum does not contain the full information because it discards the phases, which means that genuine fine structure near the limit of the resolution is often not revealed.
## Acknowledgments
We thank the _Kepler_ and _TESS_ team for providing such wonderful data. We gratefully acknowledge support from the Australian Research Council through Discovery Project DP210103119, and from the Danish National Research Foundation (Grant DNRF106) through its funding for the Stellar Astrophysics Centre (SAC).
_Facilities:_ Kepler,TESS
|
2304.00890 | MIMO Radars and Massive MIMO Communication Systems can Coexist | In this paper, we investigate the coexistence of a single cell massive MIMO
communication system with a MIMO radar. We consider the case where the massive
MIMO BS is aware of the radar's existence and treats it as a non-serviced user,
but the radar is unaware of the communication system's existence and treats the
signals transmitted by both the BS and the communication users as noise. Using
results from random matrix theory, we derive the rates achievable by the
communication system and the radar. We then use these expressions to obtain the
achievable rate regions for the proposed joint radar and communications system.
We observe that due to the availability of a large number of degrees of freedom
at the mMIMO BS, results in minimal interference even without co-design.
Finally we corroborate our findings via detailed numerical simulations and
verify the validity of the results derived previously under different settings. | Aparna Mishra, Ribhu Chopra | 2023-04-03T11:23:43Z | http://arxiv.org/abs/2304.00890v2 | # MIMO Radars and Massive MIMO Communication Systems can Coexist
###### Abstract
In this paper, we investigate the coexistence of a single cell massive MIMO communication system with a MIMO radar. We consider the case where the massive MIMO BS is aware of the radar's existence and treats it as a non-served user, but the radar is unaware of the communication system's existence and treats the signals transmitted by both the BS and the communication users as noise. Using results from random matrix theory, we derive the rates achievable by the communication system and the radar. We then use these expressions to obtain the achievable rate regions for the proposed joint radar and communications system. We observe that due to the availability of a large number of degrees of freedom at the mMIMO BS, results in minimal interference even without co-design. Finally we corroborate our findings via detailed numerical simulations and verify the validity of the results derived previously under different settings.
Joint Radar and Communication, MIMO Radar, massive MIMO, Performance Analysis, Radar Communication Co-existence.
## I Introduction
Recently, the design of joint radar and communications (JRC) systems, especially that of jointly designed communication and sensing (JCAS) systems has become an active area of research [1, 2, 3]. This is due to the improved spectral efficiencies and hardware costs offered by these systems. Moreover, these systems are seen as key enablers for the paradigm of intelligent transportation systems (ITS), popularly known as smart vehicles. In general, there exist three approaches towards the design of JRC systems, viz. coexistence, cooperation, and co-design [4]. As apparent from the name, the coexistence based approach considers the performance of the two subsystems designed separately treating the signals from each other as interference. In this case the two subsystems may or may not be aware of each other's existence. In the cooperation based approach, the two systems might be designed separately but are aware of each others presence and cooperate to mitigate interference. Finally, a co-designed JRC system considers a scenario where the two subsystems are designed jointly to maximize each other's performance. It is apparent that while the co-design based approach optimizes the performances of both the underlying subsystems it is unsuited for most legacy hardware and necessitates a hard reboot of the system architecture. On the other hand, the coexistence based approach is the least disruptive approach for legacy hardware. Therefore, in this paper we focus on the coexistence based design of a JRC system [5].
Over the last decade, the idea of using a large number of antennas in both radar and communications systems, dubbed massive multiple input multiple output (MIMO), has also gained much traction [6, 7, 8, 9]. While the literature dealing with radar technologies has focused more on conventional MIMO radars [10], with limited focus on massive MIMO [9]; massive MIMO has been established as a front runner technology for next generation wireless communication systems. Massive MIMO systems have been shown to be resilient to jamming [11] and other forms of in-band interference making them ideal candidates for sharing spectrum with radars. These features, coupled with the fact a BS with a large antenna array can easily form a null to minimize interference to a co-existant radar subsystem make massive MIMO communication systems ideal candidates for coexistance based JRC systems. Consequently, in this paper we study the performance of a JRC system where a MIMO radar co-exists with massive MIMO communication system.
### _Related Work_
In the context of coexistence based JRC systems, the authors in [12] and [13] have experimentally demonstrated the detrimental effects of the presence of a proximal in-band radar on communications systems. In [14] the idea of opportunistic spectrum sharing between a rotating radar, and a cognitive communication system is analyzed. The authors in [15, 16] have considered the use of null space projection (NSP) for achieving radar communication coexistence. Under this approach the system with a larger number of degrees of freedom projects its signal onto the null space of the interference channel to the system with smaller number of degrees of freedom. The present literature pertaining to NSP based JRC mostly focuses on MIMO radars having a larger number of degrees of freedom preventing interference to SISO systems [15, 17].
Since coexistence based JRC systems witness a performance degradation in both radar and communication subsystems, it becomes important to quantify these losses on the same scale in order to appropriately identify the underlying trade-offs. Consequently, authors in [18] have introduced the idea of "radar information rate" as an analogue of the achievable rate of a communication system. The formulation of the radar information rate models the target as an unwilling source of information, with the radar receiver acting as the sink. The radar information rate is then defined as the mutual
information between the unwilling source and the sink. Consequently, a JRC system can be viewed as multiple access channel (MAC) [19] consisting of a radar subsystem and a communication subsystem whose overall performance can be analyzed in terms of the achievable rate regions of these two subsystems [4, 20, 21, 22, 18]. Alternatively, the authors in [23] have considered each resolution cell of the radar as a constellation point and defined the "channel capacity" of the radar as the maximum information contained in the echo signal. However, in this work we evaluate the performance of the JRC system in terms of rate regions between the achievable communication rate and the radar information rate due to the simplicity and intuitiveness of this approach. The performance of massive MIMO communications systems is mostly quantified in terms of the per user achievable rates [24] that are well characterized by the logarithms of the deterministic equivalents (DEs) of their signal to interference-plus-noise ratios (SINRs) [25]. In this paper as well, we analyse the performance of the underlying massive MIMO system in terms of achievable rates obtained via DE analysis.
It is also important to note that despite the recent advances in millimetre wave communication technologies, the sub-6 GHz spectrum remains preferred for long distance communications, much of which has been dedicated for radar usage [26]. Therefore, our system model is built around the sub-6 GHz rich scattering channel model [27].
The idea of massive MIMO enabled JRC has recently been explored in the literature [28, 11, 29]. Out of these only [11] discusses a coexistence based JRC system. However, the underlying analysis is limited to studying the effect of radar interference on the uplink of a massive MIMO system. In contrast, in this paper, we analyse the problem from perspectives of both the communication system and the radar, using the achievable rate regions as a performance metric over an entire communication frame (i.e. both uplink and downlink). Also, instead of the use-and-then-forget bounds [8] used in [11] for evaluating the performance of massive MIMO systems, we use deterministic equivalent analysis [30], that results in better approximations of achievable SINRs for MMSE type receivers.
### _Contributions_
In this paper, we use rate regions to characterize the performance of a coexistence based JRC system comprising a single cell massive MIMO communication system and a static MIMO radar over a full communication frame. Our specific contributions are enumerated as follows:
1. We first determine the channel estimation performance of the massive MIMO system with uplink training in the presence of radar generated interference, and obtain expressions for consequent channel estimation mean squared error (MSE). Similarly, we obtain the MSE of the angle of arrival estimate at the radar in the presence of the interference generated by the communication system. These expressions are then used to characterize a trade-off between the pilot powers employed by the users and the radar transmit power (See Section III).
2. Following this, using DE analysis, we obtain expressions for uplink achievable rates using MMSE combining at the BS in the presence of radar generated interference. We then derive the Cramer Rao Bound (CRB) on the angle of arrival (AoA) estimate at the radar, in the presence of interference caused due to uplink transmission by the users, and use it to form an upper bound on the radar rate (See Section IV).
3. We then derive the DEs for the downlink SINR at the users, assuming regularized zero forcing (RZF) beamforming at the BS, in the presence of radar generated interference. We assume that the RZF beamforming at the BS also forms a null in the direction of the available estimate of the radar channel, and use this information to calculate the CRB on the AoA estimation performance and the corresponding radar rates (See Section V).
4. Via extensive numerical simulations we validate our derived results, and plot the achievable rate regions for our JRC system for various use cases. We find that the availability of a large number of degrees at the BS results in minimal interference to both the constituents of the JRC systems, resulting in a significantly convex rate regions (See Section VI).
We can thus conclude that coexistence based design of JRC systems is possible in the massive MIMO regime, thus allowing for the addition of sensing capabilities to legacy systems, without the need for an extensive redesign. We next describe the system model considered in this work.
## II System Model
We consider an RCC system comprising a single cell massive MIMO subsystem coexisting with an in-cell MIMO radar as shown in Fig. 1. The two subsystems are assumed to transmit over the same time frequency resources with a full bandwidth overlap, but are assumed to have only non line of sight (NLoS), rich scattering interference channels. We next describe the system and signal models for the two subsystems individually.
### _The Radar Subsystem_
We consider a mono-static pulsed MIMO radar equipped with collocated transmit and receiver antenna arrays, as shown in Fig. 1. The radar subsystem consists of \(N_{t}\) transmit antennas and \(N_{r}\) receive antennas. Since the radar is mono-static, the transmit and receive antenna arrays can safely be assumed to be synchronized, allowing for coherent processing of transmit and receive signals. We assume that a single target is present in the LOS of the radar, at an angle \(\theta\), such that the array response vectors of the transmit and receive arrays are respectively given by \(\mathbf{a}_{t}^{T}(\theta)\) and \(\mathbf{a}_{r}^{T}(\theta)\). We also let \(h_{rr}\) denote the reflection coefficient of the said target that accumulates the effects of propagation attenuation, phase shifts, and the radar cross section of the target. Now, let \(\mathbf{s}[n]\in\mathbb{C}^{N_{t}\times 1}\) be the signal transmitted by the radar at the \(n\)th instant, such that \(E[\mathbf{s}[n]\mathbf{s}^{H}[n]]=\mathbf{R}_{ss}=\sigma_{r}^{2}\mathbf{I}_{N_ {t}}\), with \(\mathbf{I}_{K}\) representing the order \(K\) identity matrix. Then, the signal received by the radar, denoted by \(\mathbf{z}[n]\in\mathbb{C}^{N_{r}\times 1}\), in the absence of any communication
interference and in multipath free propagation [31], can be expressed as,
\[\mathbf{z}[n]=h_{rr}\mathbf{A}(\theta)\mathbf{s}[n]+\sqrt{N}_{0}\mathbf{w}_{r}[n], \tag{1}\]
where \(\mathbf{w}_{r}[n]\) denotes the temporally and spatially white, zero mean circularly symmetric complex Gaussian (ZMCSCG) additive noise, and \(\mathbf{A}(\theta)=\mathbf{a}_{r}(\theta)\mathbf{a}_{t}^{T}(\theta)\). We also assume that the target is moving slowly with respect to the radar, and we can ignore the Doppler shift within a pulse [32].
### _The Communication Subsystem_
We consider a single cell massive MIMO communication system operating in the time division duplexed (TDD) mode with a BS equipped with \(M\) antenna elements serving \(K\) single antenna user equipments (users). We assume the channels between the BS and users to be frequency flat rich scattering. We let \(\sqrt{\beta_{k}}\mathbf{h}_{k}\in\mathbb{C}^{M\times 1}\) denote the channel vector between the BS and the \(k\)th user with \(\beta_{k}\) and \(\mathbf{h}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})\) representing the large scale and small scale fading coefficients, respectively.
The communication frame is divided into three sub-frames, viz. channel estimation, uplink data transmission, and downlink data transmission. In the first sub-frame, spanning \(K\) channel uses, the users transmit orthogonal pilot signals that are received by the BS and are used to form MMSE estimates of the BS to user channels. Following this, during the second sub-frame, spanning \(\tau_{u}\) channel uses, the users transmit uplink data, and the BS uses the available channel estimates to effectively decode this data via MMSE combining [24]. Finally, during the downlink data transmission sub-frame spanning \(\tau_{d}\) channel uses, the BS, under the assumption of channel reciprocity [33], uses the available channel estimates to appropriately beamform and transmit data to the users. We next describe the signal models for these sub-frames.
#### Ii-B1 Channel Estimation
Let the \(k\)th user transmit a pilot signal \(\psi_{k}[n]\) for \(n\in[1,K]\), with an energy \(\epsilon_{u,p,k}\) such that \(\sum_{n=1}^{K}\psi_{k}[n]\psi_{t}^{*}[n]=\delta[k-l]\), with \(\delta[n-k]\) representing the Kronecker delta function. Then, the signal vector received by the BS at the \(n\)th instant without accounting for radar interference can be expressed as
\[\mathbf{y}[n]=\sum_{k=1}^{K}\sqrt{\beta_{k}\epsilon_{u,p,k}}\mathbf{h}_{k} \psi_{k}[n]+\sqrt{N_{0}}\mathbf{w}_{b}[n], \tag{2}\]
with \(\mathbf{w}_{b}[n]\) representing the temporally and spatially white ZMCSCG additive noise with unit variance.
#### Ii-B2 Uplink Transmission
Letting the \(k\)th user transmit the data symbol \(x_{k}[n]\) (\(E[x_{k}[n]x_{l}[m]^{*}]=\delta[n-m]\delta[k-l]\)) at the \(n\)th instant with an energy \(\epsilon_{u,s,k}\), we can write the signal received at the BS in the absence of any radar interference as,
\[\mathbf{y}[n]=\sum_{k=1}^{K}\sqrt{\beta_{k}\epsilon_{u,s,k}}\mathbf{h}_{k}x_{ k}[n]+\sqrt{N_{0}}\mathbf{w}_{b}[n]. \tag{3}\]
#### Ii-B3 Downlink Transmission
Letting \(\mathbf{Q}\in\mathcal{C}^{M\times K}\) denote the preceding matrix at the BS, \(\epsilon_{d,s,k}\) the downlink symbol energy for the \(k\)th user such that the corresponding symbol sent to the \(k\)th user at the \(n\)th instant is \(p_{k}[n]\), we can write the downlink signal received at the \(k\)th user at the \(n\)th instant as
\[r_{k}[n]=\mathbf{h}_{k}^{T}\mathbf{Q}\text{diag}(\sqrt{\epsilon_{d,s}}) \mathbf{p}[n]+\sqrt{N_{0}}w_{k}[n], \tag{4}\]
where \(\mathbf{p}[n]=[p_{1}[n],p_{2}[n],\ldots,p_{K}[n]]^{T}\), and \(\epsilon_{d,s}=[\epsilon_{d,s,1},\ldots,\epsilon_{d,s,K}]^{T}\), and \(w_{k}[n]\) is the ZMCSCG noise with unit variance at the \(k\)th user.
### _Interference Channel Models_
As stated earlier, the MIMO radar and massive MIMO communication sub-systems do not have line of sight interference channels. Since both the BS and Radar are fixed, the interference channels between them, denoted by \(\mathbf{G}_{rb}\in\mathbb{C}^{M\times N_{t}}\) and \(\mathbf{G}_{br}\in\mathbb{C}^{N_{r}\times M}\), respectively, for the radar transmit and receive arrays are also assumed to be time invariant. Now, in accordance with the rich scattering assumption, the entries of \(\mathbf{G}_{rb}\) and \(\mathbf{G}_{rb}\) are assumed to be independent and identically distributed (i.i.d.) ZMCSCG with a variance \(\eta_{I}=\min(d_{br}^{-\alpha},1)\) with \(d_{br}\) being the distance between the BS and the radar. Also, their MMSE estimates, respectively, given by \(\tilde{\mathbf{G}}_{rb}\) and \(\tilde{\mathbf{G}}_{br}\) are assumed to be available at the BS such that,
\[\mathbf{G}_{rb}=\hat{\mathbf{G}}_{rb}+\tilde{\mathbf{G}}_{rb},\quad\mathbf{G} _{br}=\hat{\mathbf{G}}_{br}+\tilde{\mathbf{G}}_{br}, \tag{5}\]
where \(\tilde{\mathbf{G}}_{rb}\) and \(\tilde{\mathbf{G}}_{br}\) are estimation errors orthogonal to \(\hat{\mathbf{G}}_{rb}\) and \(\hat{\mathbf{G}}_{br}\), respectively, and their entries have a variance \(\eta_{e}\). During the training and uplink data transmission phases, the BS uses \(\hat{\mathbf{G}}_{rb}\) to form nulls in the directions containing the radar interfering signals to minimize their effects on channel estimation and uplink performances. Similarly, during the downlink transmission, the BS forms nulls in the direction of \(\hat{\mathbf{G}}_{br}\) to minimize the interference to the radar.
Similarly, the interference channel between the radar transmit array and the \(k\)th user is represented by \(\mathbf{g}_{rk}\in\mathbb{C}^{N_{t}\times 1}\), \(k\in\{1,2,\ldots,K\}\), and the interference channel between the \(k\)th user and the radar receive array is represented by \(\mathbf{g}_{kr}\in\mathbb{C}^{N_{r}\times 1}\), \(k\in\{1,2,\ldots,K\}\). Both \(\mathbf{g}_{rk}\) and \(\mathbf{g}_{kr}\) are assumed to consist of i.i.d. ZMCSCG entries having a variance \(\eta_{rk}\) that is equal to the large scale fading coefficient between the radar and the \(k\)th user.
Figure 1: The System Model
## III The Channel Estimation Sub-frame
### _The Communication Subsystem_
Considering the effect of radar generated interference, the signal received by the \(i\)th BS antenna at the \(n\)th instant, denoted by, \(y_{i}[n]\) can be expressed as,
\[y_{i}[n]=\sum_{k=1}^{K}\sqrt{\beta_{k}\epsilon_{u,p,k}}h_{ik}\psi_{k}[n]+\mathbf{ g}_{rb,i}^{H}\mathbf{s}[n]+\sqrt{N_{0}}w_{i}[n], \tag{6}\]
where \(\mathbf{g}_{rb,i}^{H}\) represents the \(i\)th row of \(\mathbf{G}_{rb}\). Defining \(y_{il}\triangleq\sum_{n=1}^{K}y_{i}[n]\psi_{l}^{*}[n]\), we obtain
\[y_{il}=\sqrt{\beta_{l}\epsilon_{u,p,l}}h_{il}+\sum_{n=1}^{K} \mathbf{\hat{g}}_{rb,i}^{H}\mathbf{s}[n]\psi_{l}^{*}[n]\\ +\sum_{n=1}^{K}\mathbf{\tilde{g}}_{rb,i}^{H}\mathbf{s}[n]\psi_{l }^{*}[n]+\sum_{n=1}^{K}\sqrt{N_{0}}w_{i}[n]\psi_{l}^{*}[n]. \tag{7}\]
Now, the BS can either have the phase synchronization information of \(\mathbf{s}[n]\) or it may not have this information. In the former case \(y_{ll}^{\prime}\) can be formed by subtracting \(\sum_{n=1}^{K}\mathbf{\hat{g}}_{rb,i}^{H}\mathbf{s}[n]\psi_{l}^{*}[n]\) from \(y_{il}\) as
\[y_{il}^{\prime}=\sqrt{\beta_{l}\epsilon_{u,p,l}}h_{il}+\sum_{n=1}^{K}\mathbf{ \hat{g}}_{rb,i}^{H}\mathbf{s}[n]\psi_{l}^{*}[n]+\sum_{n=1}^{K}\sqrt{N_{0}}w_{i }[n]\psi_{l}^{*}[n]. \tag{8}\]
Clearly, in this case, the LMMSE estimate \(\hat{h}_{il}\) of \(h_{il}\) can be written as \(a_{il}^{\prime}y_{il}^{\prime}\), with \(a_{il}^{\prime}=\frac{E[h_{il}y_{il}^{\prime}]}{E[h_{il}^{\prime}][j]}\), such that \(E[h_{il}y_{il}^{\prime}]=E[h_{il}y_{il}^{\prime}]=\sqrt{\beta_{l}\epsilon_{u,p,l}}\), and \(E[|y_{il}^{\prime}|^{2}]=\beta_{l}\epsilon_{u,p,l}+N_{l}\eta_{e}\sigma_{r}^{2} +N_{0}\). Similarly, in case the synchronization information about \(\mathbf{s}[n]\) is not available at the BS, \(\hat{h}_{il}=a_{il}y_{il}\), such that
\[a_{il}=\frac{E\left[h_{il}y_{il}^{\prime}|\mathbf{\hat{g}}_{rb,i} \right]}{E\left[|y_{il}|^{2}|\mathbf{\hat{g}}_{rb,i}\right]}\\ =\frac{\sqrt{\beta_{l}\epsilon_{u,p,l}}}{\beta_{l}\epsilon_{u,p, l}+\sigma_{r}^{2}\|\mathbf{\hat{g}}_{rb,i}\|^{2}+N_{t}\eta_{e}\sigma_{r}^{2}+N_{0}}. \tag{9}\]
Now, letting \(\tilde{h}_{il}\) represent the estimation error orthogonal to \(\hat{h}_{il}\), it is easy to show that \(h_{il}\) can be represented as
\[h_{il}=\hat{h}_{il}+\tilde{h}_{il}, \tag{10}\]
with \(\hat{h}_{il}\) and \(\tilde{h}_{il}\) both being ZMCSCG random variables having variances \(b_{il}^{2}\) and \(\tilde{b}_{il}^{2}\) respectively. Here,
\[b_{il}^{2}=\frac{\beta_{l}\epsilon_{u,p,l}}{\beta_{l}\epsilon_{u,p,l}+N_{t} \eta_{e}\sigma_{r}^{2}+N_{0}}\]
and
\[\tilde{b}_{il}^{2}=\frac{N_{t}\eta_{e}\sigma_{r}^{2}+N_{0}}{\beta_{l}\epsilon _{u,p,l}+N_{t}\eta_{e}\sigma_{r}^{2}+N_{0}},\]
when the radar signal synchronization information is known at the BS, and
\[b_{il}^{\prime 2}=\frac{\beta_{l}\epsilon_{u,p,l}}{\beta_{l}\epsilon_{u,p,l}+ \sigma_{r}^{2}\|\mathbf{\hat{g}}_{rb,i}\|^{2}+N_{t}\eta_{e}\sigma_{r}^{2}+N_ {0}}\]
and
\[\tilde{b}_{il}^{\prime 2}=\frac{\sigma_{r}^{2}\|\mathbf{\hat{g}}_{rb,i}\|^{2}+N_{t} \eta_{e}\sigma_{r}^{2}+N_{0}}{\beta_{l}\epsilon_{u,p,l}+\sigma_{r}^{2}\| \mathbf{\hat{g}}_{rb,i}\|^{2}+N_{t}\eta_{e}\sigma_{r}^{2}+N_{0}},\]
when radar signal information is not known at the BS.
### _DoA Estimation at the Radar_
The received signal at the radar, in the presence of interference caused duster to the pilots transmitted by the users can be expressed as
\[\mathbf{z}[n]=h_{rr}\mathbf{A}(\theta)\mathbf{s}[n]+\sum_{k=1}^{K}\sqrt{ \epsilon_{u,p,k}}\mathbf{g}_{kr}\psi_{k}[n]+\sqrt{N_{0}}\mathbf{w}_{r}[n]. \tag{11}\]
The signal received by the radar can now be reduced to the standard desired signal+noise form, where a standard signal processing technique such as the Multiple Signal Classification (MUSIC) algorithm can be used to extract the AoA [34]. Letting \(\mathbf{Z}=[\mathbf{z}[1],\mathbf{z}[2],\ldots,\mathbf{z}[N]]\), we can express it as,
\[\mathbf{Z}=h_{rr}\mathbf{A}(\theta)\mathbf{S}+\sum_{k=1}^{K}\sqrt{\epsilon_{u, p,k}}\mathbf{g}_{kr}\boldsymbol{\psi}_{k}[N]+\sqrt{N_{0}}\mathbf{W}_{r}, \tag{12}\]
with \(\mathbf{S}=[\mathbf{s}[1],\mathbf{s}[2],\ldots,\mathbf{s}[N]]\in\mathcal{C}^{N_ {t}\times N}\), \(\mathbf{W}=[\mathbf{w}[1],\mathbf{w}[2],\ldots,\mathbf{w}[N]]\in\mathcal{C}^{N_ {r}\times N}\), and \(\boldsymbol{\psi}_{k}[N]=[\psi_{k}[1],\ldots,\psi_{k}[N]]\in\mathcal{C}^{1 \times N}\). Consequently, we can write the sample covariance matrix of the received radar signal, \(\mathbf{\hat{R}}_{zz}\), as,
\[\mathbf{\hat{R}}_{zz}=\mathbf{Z}\mathbf{Z}^{H}=|h_{rr}|^{2}\mathbf{ A}(\theta)\mathbf{S}\mathbf{S}^{H}\mathbf{A}^{H}(\theta)\\ +\sum_{k=1}^{K}\sum_{l=1}^{K}\sqrt{\epsilon_{u,p,k}}\sqrt{\epsilon _{u,p,l}}\mathbf{g}_{kr}\boldsymbol{\psi}_{k}[N]\boldsymbol{\psi}_{l}^{H}[N] \mathbf{g}_{lr}^{H}\\ +N_{0}\mathbf{W}_{r}\mathbf{W}_{r}^{H}+2\Re\left\{h_{rr}\mathbf{A}( \theta)\mathbf{S}\left(\sum_{k=1}^{K}\sqrt{\epsilon_{u,p,k}}\mathbf{\psi}_{k}^{H}[N] \mathbf{g}_{kr}^{H}\right)\\ +\left(\sum_{k=1}^{K}\sqrt{\epsilon_{u,p,k}}\mathbf{g}_{kr} \boldsymbol{\psi}_{k}[N]\right)\left(\sqrt{N_{0}}\mathbf{W}_{r}^{H}\right)\\ +\sqrt{N_{0}}h_{rr}^{*}\mathbf{W}_{r}\mathbf{S}^{H}\mathbf{A}^{H}( \theta)\right\}, \tag{13}\]
where \(\Re\{.\}\) represents the real part of a complex number. Considering \(N=N_{t}\), we can reduce \(\mathbf{S}\mathbf{S}^{H}=\sigma_{r}^{2}\mathbf{I}_{N_{r}}\). Consequently, it is easy to show that the actual covariance matrix of \(\mathbf{z}[n]\) takes the form
\[\mathbf{R}_{zz}=\sigma_{r}^{2}|h_{rr}|^{2}\mathbf{A}(\theta)\mathbf{A}^{H}( \theta)+\left(\sum_{k=1}^{K}\epsilon_{u,p,k}\eta_{rk}+N_{0}\right)\mathbf{I}_{N_ {r}} \tag{14}\]
Now, \(\mathbf{A}(\theta)\) is a rank-1 matrix, and so is \(\mathbf{A}(\theta)\mathbf{A}^{H}(\theta)\), therefore, \(\mathbf{R}_{zz}\) can still be viewed as the sum of a rank-1 matrix and a multiple of the identity matrix. Consequently, the noise subspace of \(\mathbf{R}_{zz}\) consists of the eigenvectors corresponding to the \(N_{r}-1\) smallest eigenvalues (each being equal to \(\sum_{k=1}^{K}\epsilon_{u,p,k}\eta_{rk}+N_{0}\)). We let the noise subspace of \(\mathbf{R}_{zz}\) be represented by the matrix \(\mathbf{V}\), and can express the estimate \(\hat{\theta}\) of \(\theta\) as [34],
\[\hat{\theta}=\arg\max_{\phi}\frac{1}{\mathbf{S}^{H}\mathbf{A}(\phi)^{H}\mathbf{V} \mathbf{V}^{H}\mathbf{A}(\phi)\mathbf{S}}. \tag{15}\]
Since the MUSIC algorithm is intractable for closed form performance analysis,
## IV The Uplink Sub-frame
In this section, we analyse the performance of the JRC system during the uplink sub-frame. For this purpose, we first evaluate the rates achievable by the communications subsystem via DE analysis [35], and then derive the radar rate via the CRLB on the MSE performance of the radar subsystem.
We can write the received signal at the communication BS as
\[\mathbf{y}[n]=\sum_{k=1}^{K}\sqrt{\beta_{k}\epsilon_{u,s,k}}\mathbf{ \hat{h}}_{k}x_{k}[n]+\sum_{k=1}^{K}\sqrt{\beta_{k}\epsilon_{u,s,k}}\mathbf{ \hat{h}}_{k}x_{k}[n]\\ +\mathbf{\hat{G}}_{rb}\mathbf{s}[n]+\mathbf{\tilde{G}}_{rb} \mathbf{s}[n]+\sqrt{N_{0}}\mathbf{w}_{b}[n]. \tag{16}\]
We use MMSE combining at the BS, with the matrix \(\mathbf{C}\triangleq\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}^{ -1}\mathbf{\hat{H}}\) being the combining matrix, such that, \(\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}\) represents the covariance matrix of \(\mathbf{y}[n]\) given the availability of the channel estimates \(\mathbf{\hat{G}}_{rb}\) and \(\mathbf{\hat{H}}\), and can be expressed as,
\[\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}=\sum_{k=1 }^{K}\beta_{k}\epsilon_{u,s,k}\mathbf{\hat{h}}_{k}\mathbf{\hat{h}}_{k}^{H}+ \sigma_{r}^{2}\mathbf{\hat{G}}_{rb}\mathbf{\hat{G}}_{rb}^{H}\\ +\left(\sum_{k=1}^{K}\beta_{k}\epsilon_{u,s,k}\bar{b}_{k}^{2}+ \sigma_{r}^{2}\eta_{e}+N_{0}\right)\mathbf{I}_{M}. \tag{17}\]
Consequently, the processed signal vector, \(\mathbf{r}[n]\in\mathcal{C}^{K\times 1}\), at the BS at the \(n\)th instant is given by \(\mathbf{r}[n]=\mathbf{C}^{H}\mathbf{y}[n]=\mathbf{\hat{H}}^{H}\mathbf{R}_{ \text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}^{-1}\mathbf{\mathbf{Y}}[n]\).
Now, letting \(r_{k}[n]\) be the \(k^{th}\) component of \(\mathbf{r}[n]\), we can write,
\[r_{k}[n]=\sqrt{\beta_{k}\epsilon_{u,s,k}}\mathbf{\hat{h}}_{k}^{H }\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}^{-1}\mathbf{\hat{h}} _{k}x_{k}[n]\\ +\sum_{\begin{subarray}{c}l=1\\ l\neq k\end{subarray}}^{K}\sqrt{\beta_{l}\epsilon_{u,s,l}}\mathbf{\hat{h}}_{k}^{H }\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}^{-1}\mathbf{\hat{h}} _{k}x_{l}[n]\\ +\sum_{l=1}^{K}\sqrt{\beta_{l}\epsilon_{u,s,l}}\mathbf{\hat{h}}_{k}^{H }\mathbf{R}_{\text{wj}|\mathbf{G}_{rb},\mathbf{\hat{H}}}^{-1}\mathbf{\hat{h}} _{k}x_{l}[n]\\ +\sum_{i=1}^{N_{t}}\mathbf{\hat{h}}_{k}^{H}\mathbf{R}_{\text{wj}| \mathbf{\hat{G}}_{rb},\mathbf{\hat{H}}}^{-1}\mathbf{\hat{G}}_{rb}\mathbf{\hat {H}}_{rb}\mathbf{\hat{H}}_{rb}\mathbf{\hat{H}}_{rb}\mathbf{\hat{H}}_{rb}[n]\\ +\mathbf{\hat{h}}_{k}^{H}\mathbf{R}_{\text{wj}|\mathbf{G}_{rb}, \mathbf{\hat{H}}}^{-1}\mathbf{\hat{G}}_{rb}\mathbf{\hat{S}}_{rb}\mathbf{s}[n]+ \sqrt{N_{0}}\mathbf{\hat{h}}_{k}^{H}\mathbf{R}_{\text{wj}|\mathbf{\hat{G}}_{rb },\mathbf{\hat{H}}}^{-1}\mathbf{w}_{b}[n]. \tag{18}\]
Here, the first term corresponds to the desired signal, the second to the cancellable inter user interference, the third to the interference due to the channel estimation errors at the BS, the fourth to the cancellable interference from the radar subsystem, the fifth to the non-cancellable interference from the radar subsystem and the last term to the additive white Gaussian noise.
_Theorem 1_.: The rate achievable by the \(k\)th user in the uplink of the communication subsystem of a massive MIMO based JRC subsystem can be expressed as
\[R_{k}=\log_{2}(1+\gamma_{u,k}), \tag{19}\]
where \(\gamma_{u,k}\) is the SINR for the \(k\)th user's signal at the BS, and is given as,
\[\gamma_{u,k}=\frac{\zeta_{s,k}}{\zeta_{I,k}+\zeta_{E,k}+\zeta_{RC,k}+\zeta_{RE, k}+\zeta_{w,k}}. \tag{20}\]
Here \(\zeta_{s,k}\) corresponds to the desired signal's power and is given by,
\[\zeta_{s,k}=\beta_{k}\epsilon_{u,s,k}\frac{|b_{k}^{2}\mu_{k}|^{2}}{|1+b_{k}^{ 2}\mu_{k}|^{2}}, \tag{21}\]
such that \(\mu_{k}=\text{Tr}\{\mathbf{T}_{k}(\rho)\}\), and
\[\mathbf{T}_{k}(\rho)=\left(\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\frac{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M }}{1+\delta_{k,m}(\rho)}+\mathbf{S}+\rho\mathbf{I}_{M}\right)^{-1}, \tag{22}\]
with \(\delta_{k,m}(\rho)=\lim_{t\rightarrow\infty}\delta_{k,m}^{(t)}(\rho)\), such that
\[\delta_{k,m}^{(t)}(\rho)=\text{Tr}\Bigg{\{}\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}\\ \times\left(\sum_{l=1,\neq k}^{K}\frac{\beta_{l}\epsilon_{u,s,l}b_{ m}^{2}\mathbf{I}_{M}}{1+\delta_{k,l}^{t-1}(\rho)}+\mathbf{S}+\rho\mathbf{I}_{M} \right)^{-1}\Bigg{\}}, \tag{23}\]
having initial values \(\delta_{k,m}^{(0)}(\rho)=\frac{1}{\rho}\)\(\forall m\).
Similarly, \(\zeta_{I,k}\) corresponds to the inter user interference power, and is given as,
\[\zeta_{I,k}=\sum_{\begin{subarray}{c}l=1,\\ l\neq k\end{subarray}}^{K}\frac{\beta_{l}\epsilon_{u,s,l}}{|1+b_{k}^{2}\mu_{k}| ^{2}}\Bigg{(}b_{k}^{2}b_{l}^{2}\mu_{k,l}^{{}^{\prime}}+\frac{(b_{k}^{2}b_{l}^{2 }\mu_{k,l}^{{}^{\prime}})^{2}}{|1+b_{l}^{2}\mu_{k,l}|^{2}}\\ -2\Re\left\{\frac{(b_{k}^{2}b_{l}^{2}\mu_{k,l}^{{}^{\prime}})^{3/2}}{ 1+b_{l}^{2}\mu_{k,l}}\right\}\Bigg{)}. \tag{24}\]
where \(\mu_{k,l}=\text{Tr}\{\mathbf{T}_{k,l}(\rho)\}\),
\[\mathbf{T}_{k,l}(\rho)=\left(\sum_{\begin{subarray}{c}m=1,\\ m\neq k,l\end{subarray}}^{K}\frac{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}}{1+ \delta_{k,l,m}(\rho)}+\mathbf{S}+\rho\mathbf{I}_{M}\right)^{-1}, \tag{25}\]
with \(\delta_{k,l,m}(\rho)=\lim_{t\rightarrow\infty}\delta_{k,l,m}^{(t)}(\rho)\), such that
\[\delta_{k,l,m}^{(t)}(\rho)=\text{Tr}\Bigg{\{}\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}\\ \times\left(\sum_{\begin{subarray}{c}p=1,\\ p\neq k,l\end{subarray}}^{K}\frac{\beta_{p}\epsilon_{u,s,p}b_{p}^{2}\mathbf{I}_{M}}{1+ \delta_{k,l,p}^{t-1}(\rho)}+\mathbf{S}+\rho\mathbf{I}_{M}\right)^{-1}\Bigg{\}}, \tag{26}\]
having initial values \(\delta_{k,l,m}^{(0)}(\rho)=\frac{1}{\rho}\)\(\forall m\), and \(\mu_{k,l}^{\prime}=\text{Tr}\{\mathbf{T}_{k,l}^{{}^{\prime}}(\rho)\}\) where \(\mathbf{T}_{k,l}^{{}^{\prime}}(\rho)\in\mathcal{C}^{M\times M}\) is given by
\[\mathbf{T}_{k,l}^{{}^{\prime}}(\rho)=\mathbf{T}_{k,l}(\rho) \mathbf{T}_{k,l}(\rho)\\ +\mathbf{T}_{k,l}(\rho)\sum_{\begin{subarray}{c}m=1\\ m\neq k,l\end{subarray}}^{K}\frac{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M} \delta_{m}^{{}^{\prime}}(\rho)}{(1+\delta_{k,l,
and \(\boldsymbol{\delta}^{{}^{\prime}}(\rho)=[\delta_{1}^{{}^{\prime}}(\rho)\dots \delta_{K}^{{}^{\prime}}(\rho)]^{T}\) such that \(\boldsymbol{\delta}^{{}^{\prime}}(\rho)=(\mathbf{I}-\mathbf{J}(\rho))^{-1} \mathbf{v}(\rho),\) with
\[[\mathbf{J}(\rho)]_{pq}=\frac{\text{Tr}\{\beta_{p}\epsilon_{u,s,p}b_{p}^{2} \mathbf{I}_{M}\mathbf{T}_{k,l}(\rho)\beta_{q}\epsilon_{u,s,q}b_{q}^{2}\mathbf{ I}_{M}\mathbf{T}_{k,l}(\rho)\}}{(1+\delta_{k,l,p}(\rho))^{2}},\]
and \([\mathbf{v}(\rho)]_{p}=\text{Tr}\{\beta_{p}\epsilon_{u,s,p}b_{p}^{2}\mathbf{ I}_{M}\mathbf{T}_{k,l}(\rho)\mathbf{T}_{k,l}(\rho)\}.\)
The term \(\zeta_{E,k}\) corresponds to the interference power due to channel estimation error and is given by
\[\zeta_{E,k}=\sum_{l=1}^{K}\beta_{l}\epsilon_{u,s,l}\frac{b_{k}^{2}b_{l}^{2} \mu_{k}^{{}^{\prime}}}{|1+b_{k}^{2}\mu_{k}|^{2}}, \tag{28}\]
where \(\mu_{k}^{{}^{\prime}}=\text{Tr}\{\mathbf{T}_{k}^{{}^{\prime}}(\rho)\}\) and \(\mathbf{T}_{k}^{{}^{\prime}}(\rho)\in\mathcal{C}^{M\times M}\) is given by
\[\mathbf{T}_{k}^{{}^{\prime}}(\rho) =\mathbf{T}_{k}(\rho)\mathbf{T}_{k}(\rho)\] \[\quad+\mathbf{T}_{k}(\rho)\sum_{\begin{subarray}{c}m=1\\ m\neq k\end{subarray}}^{K}\frac{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M }\delta_{m}^{{}^{\prime}}(\rho)}{(1+\delta_{k,m}(\rho))^{2}}\mathbf{T}_{k}( \rho), \tag{29}\]
with \(\mathbf{T}_{k}(\rho)\) and \(\boldsymbol{\delta}^{{}^{\prime}}(\rho)\) as defined in (25) and (27) respectively.
\(\zeta_{RC,k}\) corresponds to the interference at the \(k\)th user due to the cancellable component of radar interference and is given by
\[\zeta_{RC,k}=\sum_{i=1}^{N_{t}}\sigma_{r}^{2}\frac{1}{|1+b_{k}^{2 }\mu_{k}|^{2}}\Bigg{\{}b_{k}^{2}(\eta_{I}-\eta_{e})\mu_{k,i}^{{}^{\prime}}\\ +\frac{(b_{k}^{2}(\eta_{I}-\eta_{e})\mu_{k,i}^{{}^{\prime}})^{3/2} }{|1+\mu_{k,i}|^{2}}-2\Re\Bigg{\{}\frac{b_{k}^{2}(\eta_{I}-\eta_{e})\mu_{k,i}^ {{}^{\prime}}}{1+\mu_{k,i}}\Bigg{\}}\Bigg{\}}, \tag{30}\]
where \(\text{Tr}\{\mathbf{T}_{k,i}^{{}^{\prime}}(\rho)\}=\mu_{k,i}^{{}^{\prime}}\), \(\text{Tr}\{\mathbf{T}_{k,i}(\rho)\}=\mu_{k,i}\) and \(\mathbf{T}_{k,i}^{{}^{\prime}}(\rho)\in\mathcal{C}^{M\times M}\) is given by
\[\mathbf{T}_{k,i}^{{}^{\prime}}(\rho)=\mathbf{T}_{k,i}(\rho) \mathbf{T}_{k,i}(\rho)\\ +\mathbf{T}_{k,i}(\rho)\sum_{m=1,\neq k}^{K}\frac{\beta_{m} \epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}\delta_{k,i,m}^{{}^{\prime}}(\rho)}{(1 +\delta_{k,i,m}(\rho))^{2}}\mathbf{T}_{k,i}(\rho), \tag{31}\]
and \(\mathbf{T}_{k,i}(\rho)\) is given by
\[\mathbf{T}_{k,i}(\rho)=\left(\sum_{m=1,\neq k}^{K}\frac{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}}{1+\delta_{k,i,m}(\rho)}+\mathbf{S}_{i}+\rho \mathbf{I}_{M}\right)^{-1}, \tag{32}\]
with \(\delta_{k,i,m}(\rho)=\lim_{t\to\infty}\delta_{k,i,m}^{(t)}(\rho)\) being obtained iteratively from
\[\delta_{k,i,m}^{(t)}(\rho) =\text{Tr}\left\{\beta_{m}\epsilon_{u,s,m}b_{m}^{2}\mathbf{I}_{M}\right.\] \[\left.\times\left(\sum_{\begin{subarray}{c}u=1\\ u\neq k\end{subarray}}^{K}\frac{\eta_{int}\mathbf{I}_{M}}{1+\delta_{k,i,u}^{-1 }(\rho)}+\mathbf{S}_{i}+\rho\mathbf{I}_{M}\right)^{-1}\right\}, \tag{33}\]
after being initialized as \(\delta_{k,i,m}^{(0)}(\rho)=\frac{1}{\rho}\ \forall j\), and \(\boldsymbol{\delta}_{k,i}^{{}^{\prime}}(\rho)=[\delta_{k,i,1}^{{}^{\prime}}( \rho)\dots\delta_{k,i,K}^{{}^{\prime}}(\rho)]^{T}\) such that \(\boldsymbol{\delta}_{k,i}(\rho)=(\mathbf{I}_{K}-\mathbf{J}_{k,i}(\rho))^{-1} \mathbf{v}_{k,i}(\rho),\) with
\[[\mathbf{J}_{k,i}(\rho)]_{pq}=\frac{\text{Tr}\{\beta_{p}\epsilon_{u,s,p}b_{p}^{ 2}\mathbf{I}_{M}\mathbf{T}_{k,i}(\rho)\beta_{q}\epsilon_{u,s,q}b_{q}^{2} \mathbf{I}_{M}\mathbf{T}_{k,i}(\rho)\}}{(1+\delta_{k,i,p}(\rho))^{2}},\]
and \([\mathbf{v}_{k,i}(\rho)]_{p}=\text{Tr}\{\beta_{p}\epsilon_{u,s,p}b_{p}^{2} \mathbf{I}_{M}\mathbf{T}_{k,i}(\rho)\mathbf{T}_{k,i}(\rho)\}.\)
\(\zeta_{RE,k}\) corresponds to the interference at the \(k\)th user due to error in the estimate of inter-system interference channel between BS and radar and is given by
\[\zeta_{RE,k}=\sum_{i=1}^{N_{t}}\sigma_{r}^{2}\frac{b_{k}^{2}\eta_{e}\mu_{k}^{{}^{ \prime}}}{|1+b_{k}^{2}\mu_{k}|^{2}}. \tag{34}\]
where \(\mu_{k}^{{}^{\prime}}=\text{Tr}\{\mathbf{T}_{k}^{{}^{\prime}}(\rho)\}\) and is given by (29).
\(\zeta_{w,k}\) corresponds to the interference at \(k\)th user due to additive Gaussian noise and is given by
\[\zeta_{w,k}=N_{0}\frac{b_{k}^{2}\mu_{k}^{{}^{\prime}}}{|1+b_{k}^{2}\mu_{k}|^{2}}. \tag{35}\]
Proof.: See Appendix A.
We can observe that the use of MMSE combining, along with the effects of channel hardening offered by the large number of antennas available at the BS effectively cancels the radar generated interference in the uplink. This effect is illustrated in the results obtained in Figs 7 and 8.
We next quantify the performance of the radar subsystem in terms of the achievable radar rate according the notion developed in [18]. We note that the received signal at the radar takes the form
\[\mathbf{z}[n]=h_{rr}\mathbf{A}(\theta)\mathbf{s}[n]+\sum_{k=1}^{K}\sqrt{ \epsilon_{u,p,k}}\text{g}_{kr}\psi_{k}[n]+\sqrt{N}_{0}\mathbf{w}_{r}[n]. \tag{36}\]
_Theorem 2_.: The radar rate can be expressed as [18],
\[R_{\text{radar},u}=\log\left(1+\frac{1}{\text{CRB}(\theta)}\right), \tag{37}\]
where \(\text{CRB}(\theta)\) corresponds to the Cramer-Rao bound on the MSE of the AoA estimate at the radar, given as,
\[CRB(\theta)=\frac{N_{0}+\sum_{k=1}^{K}\epsilon_{u,p,k}\eta_{r}k}{2\sigma_{r}^{2}|h_{ rr}|^{2}}\frac{1}{\Re\{\text{Tr}(\dot{\mathbf{A}}(\theta)\dot{\mathbf{A}}^{H}( \theta))\}}, \tag{38}\]
with \(\dot{\mathbf{A}}(\theta)\) representing the derivative of \(\mathbf{A}(\theta)\) wrt \(\theta\).
Proof.: See Appendix B.
We can observe in the expression for the CRLB that the numerator contains the noise term as well as the interferences from all the communications users, limiting the performance of the radar subsystem. We can obtain the rate regions for the overall JRC system during the uplink communication frame by using Theorems 1 and 2. We next look at the performance of the JRC system during the downlink communication subframe.
## V The Downlink Sub-frame
In this section, we analyse the performance of the JRC system during the downlink sub-frame. Letting \(\bar{p}_{k}[n]\) denote the data symbol to be transmitted to the \(k\)th user, we can write the data vector to be transmitted by the massive MIMO BS in the downlink as \(\mathfrak{P}[n]=[\bar{p}_{1}[n],\bar{p}_{2}[n],\dots,p_{K}[n],0,\dots,
this, we can define the precoder matrix at the BS as, \(\mathbf{Q}=(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{ \mathbf{H}}^{*}\), with \(\alpha\) being the regularization parameter. This is equivalent to forming a null in the direction of interference channel to the radar. Consequently, the precoded downlink signal transmitted by the BS is expressed as \(\mathbf{p}[n]=\mathbf{Q}\text{diag}(\sqrt{\mathbf{\epsilon}_{d,s}})\bar{\mathbf{p} }[n]\), with \((\mathbf{\epsilon}_{d,s})=[\sqrt{\epsilon_{d,s,1}},\ldots,\sqrt{\epsilon_{d,s,K}}]^ {T}\), and \(\sqrt{\epsilon_{d,s,k}}\) representing the downlink energy allocated to the \(k\)th user after power control. We can now write the signal received at the \(k\)th user as,
\[r_{k}[n]=\sqrt{\beta_{k}}\hat{\mathbf{h}}_{k}^{T}\mathbf{p}[n]+\sqrt{\beta_{k} }\hat{\mathbf{h}}_{k}^{T}\mathbf{p}[n]+\mathbf{g}_{rk}^{T}\mathbf{s}[n]+w_{k} [n]. \tag{39}\]
This can also be written as
\[r_{k}[n]=\sqrt{\beta_{k}\epsilon_{d,s,k}}\hat{\mathbf{h}}_{k}^{ T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{ \mathbf{h}}_{k}^{*}\bar{p}_{k}[n]\\ +\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\sqrt{\beta_{k}\epsilon_{d,s,m}}\hat{\mathbf{h}}_{k}^ {T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{ \mathbf{h}}_{m}^{*}\bar{p}_{m}[n]\\ +\sqrt{\beta_{k}}\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}\bar{ \mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{H}}^{*}\bar{\mathbf{p} }[n]+\mathbf{g}_{rk}^{T}\mathbf{s}[n]+w_{k}[n]. \tag{40}\]
where the first term indicates the desired signal corresponding to the \(k\)th user, the second term is the cancelable inter user interference from the data meant for the other users, the third term corresponds to the interference due to the channel estimation error at the BS, the fourth term is due to the interference from the radar subsystem and the last term is due to AWGN.
_Theorem 3_.: The rate achievable by the \(k\)th user in the downlink of the communication subsystem of a JRC system can be expressed as
\[R_{d,k}=\log_{2}(1+\gamma_{d,k}), \tag{41}\]
where \(\gamma_{d,k}\) is the SINR for the \(k\)th user's signal at the BS, and is given as,
\[\gamma_{d,k}=\frac{\zeta_{r,k}}{\zeta_{r,I,k}+\zeta_{r,E,k}+\zeta_{r,RC,k}+ \zeta_{r,w,k}}. \tag{42}\]
Here \(\zeta_{r,k}\) corresponds to the desired signal power and is given by
\[\zeta_{r,k}=\beta_{k}\epsilon_{d,s,k}\left|\frac{b_{k}^{2}\mu_{k,\alpha}}{1+b _{k}^{2}\mu_{k,\alpha}}\right|^{2} \tag{43}\]
with \(\mu_{k,\alpha}=\text{Tr}\{\mathbf{T}_{k}(\alpha)\}\) such that
\[\mathbf{T}_{k}(\alpha)=\left(\sum_{\begin{subarray}{c}l=1,\\ l\neq k\end{subarray}}^{K}\frac{b_{l}^{2}\mathbf{I}_{M}}{1+\delta_{k,l}( \alpha)}+\sum_{l=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M}}{1+ \delta_{k,l}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}, \tag{44}\]
and \(\delta_{k,l}(\alpha)=\lim_{t\rightarrow\infty}\delta_{k,l}^{(t)}(\alpha)\), which is iteratively computed as
\[\delta_{k,l}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}b_{l}^{2}\mathbf{I} _{M}\left(\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\frac{b_{m}^{2}\mathbf{I}_{M}}{1+\delta_{k,m}^{(t-1)}( \alpha)}\right.\] \[\left.+\sum_{m=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{ M}}{1+\delta_{k,m}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}\Bigg{\}} \ \ 1\leq l\leq K\] \[\delta_{k,l}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}(\eta_{I}-\eta_{e}) \mathbf{I}_{M}\left(\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\frac{b_{m}^{2}\mathbf{I}_{M}}{1+\delta_{k,m}^{(t-1)}( \alpha)}\right.\] \[\left.+\sum_{m=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{ M}}{1+\delta_{k,m}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}\Bigg{\}} \ \ \ K+1\leq l\leq K+N_{r} \tag{45}\]
with initial values \(\delta_{k,l}^{(0)}(\alpha)=\frac{1}{\alpha}\ \forall\ l\). The term \(\zeta_{r,I,k}\) corresponds to the inter user interference power, and is given as,
\[\zeta_{r,I,k}=\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\beta_{k}\epsilon_{d,s,m}\frac{1}{|1+\beta_{k}^{2}\mu_{ k,\alpha}|^{2}}\\ \times\Bigg{\{}(\beta_{k}^{2}\beta_{m}^{2}\mu_{k,m,\alpha}^{{}^{ \prime}})+\frac{(\beta_{k}^{2}\beta_{m}^{2}\mu_{k,m,\alpha}^{{}^{\prime}})^{2} }{|1+\beta_{k}^{2}\mu_{k,\alpha}|^{2}}\\ -2\Re\left(\frac{(\beta_{k}^{2}\beta_{m}^{2}\mu_{k,m,\alpha}^{{}^{ \prime}})^{3/2}}{1+\beta_{k}^{2}\mu_{k,\alpha}}\right)\Bigg{\}}, \tag{46}\]
where \(\mu_{k,m,\alpha}^{{}^{\prime}}=\text{Tr}\{\mathbf{T}_{k,m}^{{}^{\prime}}(\alpha)\}\), such that
\[\mathbf{T}_{k,m}^{{}^{\prime}}(\alpha)=\mathbf{T}_{k,m}(\alpha) \mathbf{T}_{k,m}(\alpha)\\ +\mathbf{T}_{k,m}(\alpha)\left(\sum_{\begin{subarray}{c}l=1,\\ l\neq k,m\end{subarray}}^{K}\frac{b_{l}^{2}\mathbf{I}_{M}\delta_{k,m,l}^{{}^{ \prime}}(\alpha)}{1+\delta_{k,m,l}(\alpha)}\right.\] \[\left.+\sum_{l=K+1}^{K+N_{r}}\frac{(1-\eta_{e})\mathbf{I}_{M} \delta_{k,m,l}^{{}^{\prime}}(\alpha)}{1+\delta_{k,m,l}(\alpha)}\right)\mathbf{T}_ {k,m}(\alpha) \tag{47}\]
with
\[\mathbf{T}_{k,m}(\alpha)=\left(\sum_{\begin{subarray}{c}l=1,\\ l\neq k,m\end{subarray}}^{K}\frac{b_{l}^{2}\mathbf{I}_{M}}{1+\delta_{k,m,l}(\alpha)}\right.\] \[\left.+\sum_{l=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{ M}}{1+\delta_{k,m,l}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}, \tag{48}\]
and \(\delta_{k,m,l}(\alpha)=\lim_{t\rightarrow\infty}\delta_{k,m,l}^{(t)}(\alpha)\),
\[\delta_{k,m,l}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}b_{l}^{2}\mathbf{I} _{M}\left(\sum_{\begin{subarray}{c}p=1,\\ p\neq k,m\end{subarray}}^{K}\frac{b_{p}^{2}\mathbf{I}_{M}}{1+\delta_{k,m,p}^{(t-1)}( \alpha)}\right.\] \[\left.+\sum_{p=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{ I}_{M}}{1+\delta_{k,m,p}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}\Bigg{\}} \ \ 1\leq l\leq K\] \[\delta_{k,m,l}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}(\eta_{I}-\eta_{e}) \mathbf{I}_{M}\left(\sum_{\begin{subarray}{c}p=1,\\ p\neq k,m\end{subarray}}^{K}\frac{b_{p}^{2}\mathbf{I}_{M}}{1+\delta_{k,m,p}^{(t-1)}( \alpha)}\right.\] \[\left.+\sum_{p=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{ M}}{1+\delta_{k,m,p}^{(t-1)}(\alpha)}+
\[K+1\leq l\leq K+N_{r} \tag{49}\]
with initial values \(\delta_{k,m,l}^{(0)}(\alpha)=\frac{1}{\alpha}\ \forall\ l\), and \(\mathbf{\delta}_{l,d}^{{}^{\prime}}(\alpha)=(\mathbf{I}_{K+N_{r}-1}-\mathbf{J}( \alpha))^{-1}\mathbf{v}(\alpha)\),
\([\mathbf{J}(\alpha)]_{pq}=\frac{\text{Tr}\{b_{l}^{2}\mathbf{I}_{M}\mathbf{T}_{ k,m}(\alpha)b_{l}^{2}\mathbf{I}_{M}\mathbf{T}_{k,m}(\alpha)\}}{(1+\delta_{k,l }(\alpha))^{2}},\ [\mathbf{v}(\alpha)]_{p}=\text{Tr}\{b_{l}^{2}\mathbf{I}_{M} \mathbf{T}_{k,m}(\alpha)\mathbf{T}_{k,m}(\alpha)\}\), where \(p,q=1,2,\ldots,K+N_{r},\neq l\).
The term \(\zeta_{r,E,k}\) corresponds to the interference power due to the channel estimation error and is given by
\[\zeta_{r,E,k}=\sum_{l=1}^{K}\beta_{k}\epsilon_{d,s,l}\frac{b_{k}^{2}b_{l}^{2} \mu_{l,\alpha}^{{}^{\prime}}}{|1+b_{l}^{2}\mu_{l,\alpha}|^{2}}, \tag{50}\]
where \(\mu_{l,\alpha}=\text{Tr}\{\mathbf{T}_{l}(\alpha)\}\) and \(\mu_{l,\alpha}^{{}^{\prime}}=\text{Tr}\{\mathbf{T}_{l}^{{}^{\prime}}(\alpha)\}\) such that
\[\mathbf{T}_{l}^{{}^{\prime}}(\alpha)=\mathbf{T}_{l}(\alpha) \mathbf{T}_{l}(\alpha)+\mathbf{T}_{l}(\alpha)\left(\sum_{d=1,\neq l}^{K} \frac{b_{d}^{2}\mathbf{I}_{M}\delta_{l,d}^{{}^{\prime}}(\alpha)}{1+\delta_{l,d}(\alpha)}\right.\\ +\sum_{d=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M} \delta_{l,d}^{{}^{\prime}}(\alpha)}{1+\delta_{l,d}(\alpha)}\right)\mathbf{T}_ {l}(\alpha), \tag{51}\]
and
\[\mathbf{T}_{l}(\alpha)=\left(\sum_{\begin{subarray}{c}d=1\\ d\neq l\end{subarray}}^{K}\frac{b_{d}^{2}\mathbf{I}_{M}}{1+\delta_{l,d}( \alpha)}+\sum_{d=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M}}{1+ \delta_{l,d}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1} \tag{52}\]
where \(\delta_{l,d}(\alpha)=\lim_{t\to\infty}\delta_{l,d}^{(t)}(\alpha)\),
\[\delta_{l,d}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}b_{d}^{2}\mathbf{I} _{M}\left(\sum_{\begin{subarray}{c}m=1\\ m\neq l\end{subarray}}^{K}\frac{b_{m}^{2}\mathbf{I}_{M}}{1+\delta_{l,m}^{(t-1 )}(\alpha)}\right.\\ +\sum_{m=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M}}{1 +\delta_{l,m}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\Bigg{)}^{-1}\Bigg{\}}\ \ 1\leq d\leq K\] \[\delta_{l,d}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}(\eta_{I}-\eta_{e}) \mathbf{I}_{M}\left(\sum_{\begin{subarray}{c}m=1\\ m\neq l\end{subarray}}^{K}\frac{b_{m}^{2}\mathbf{I}_{M}}{1+\delta_{l,m}^{(t-1)}( \alpha)}\right.\\ +\sum_{m=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M}}{1 +\delta_{l,m}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\Bigg{)}^{-1}\Bigg{\}}\ \ \ \ K+1\leq d\leq K+N_{r} \tag{53}\]
with initial values \(\delta_{l,d}^{(0)}(\alpha)=\frac{1}{\alpha}\ \forall\ l\), and \(\mathbf{\delta}_{l,d}^{{}^{\prime}}(\alpha)=(\mathbf{I}_{K+N_{r}-1}-\mathbf{J}( \alpha))^{-1}\mathbf{v}(\alpha)\),
\([\mathbf{J}(\alpha)]_{pq}=\frac{\text{Tr}\{b_{l}^{2}\mathbf{I}_{M}\mathbf{T}_{ k,m}(\alpha)b_{l}^{2}\mathbf{I}_{M}\mathbf{T}_{k,m}(\alpha)\}}{(1+\delta_{k,l }(\alpha))^{2}},\ [\mathbf{v}(\alpha)]_{p}=\text{Tr}\{b_{p}^{2}\mathbf{I}_{M} \mathbf{T}_{k,m}(\alpha)\mathbf{T}_{k,m}(\alpha)\}\), where \(p,q=1,2,\ldots,K+N_{r},\neq l\).
\(\zeta_{r,RC,k}\) corresponds to the interference power due to the presence of radar subsystem is,
\[\zeta_{r,RC,k}=\sigma_{r}^{2}\eta_{r,k} \tag{54}\]
\(\zeta_{r,W,k}\) corresponds to the interference due to the presence of AWGN and is given by
\[\zeta_{r,W,k}=N_{0}. \tag{55}\]
Proof.: See Appendix C.
Here it is important to note that the RZF beamforming by the BS only nulls the interference from the BS to the radar and does not affect the interference from the radar to the users. The radar to user interference, denoted by \(\zeta_{r,RC,k}\) in the SINR expression remains unmitigated.
We next analyze the performance of radar subsystem in the downlink subframe. We note that the received signal at the radar is given by
\[\mathbf{z}[n]=h_{rr}\mathbf{A}(\theta)\mathbf{s}[n]+\mathbf{\hat{G }}_{br}^{H}\mathbf{Q}\text{diag}(\sqrt{\mathbf{\epsilon}_{d,s}})\mathbf{\bar{p}}[n]\\ +\mathbf{\tilde{G}}_{br}^{H}\mathbf{Q}\text{diag}(\sqrt{\mathbf{ \epsilon}_{d,s}})\mathbf{\bar{p}}[n]+\sqrt{N}_{0}\mathbf{w}_{r}[n]. \tag{56}\]
_Theorem 4_.: The radar rate can be expressed as,
\[R_{\text{radar},d}=\log\left(1+\frac{1}{\text{CRB}(\theta)} \right), \tag{57}\]
where
\[\text{CRB}(\theta)=\frac{\sigma_{wr,d}^{2}}{2\sigma_{r}^{2}|h_{rr}|^{2}}\frac{1}{ \Re\{\text{Tr}(\mathbf{A}(\theta)\mathbf{\hat{A}}^{H}(\theta))\}}, \tag{58}\]
with
\[\sigma_{wr,d}^{2}=N_{0}+\frac{\left(\sum_{i=1}^{K}\epsilon_{d,s,i}b_{i}^{2} \right)\mu_{i,\alpha}^{{}^{\prime}}}{|1+(\eta_{I}-\eta_{e})\mu_{\mathbf{\mathfrak{g}}_{ wr,m}}|^{2}}+\left(\sum_{i=1}^{K}\epsilon_{d,s,i}b_{i}^{2}\right)\mu_{ \alpha}^{{}^{\prime}} \tag{59}\]
such that \(\mu_{\alpha}^{{}^{\prime}}=\text{Tr}\{\mathbf{T}^{{}^{\prime}}(\alpha)\}\),
\[\mathbf{T}^{{}^{\prime}}(\alpha)=\mathbf{T}(\alpha)(\eta_{e}\mathbf{I}_{M}) \mathbf{T}(\alpha)+\mathbf{T}(\alpha)\left(\sum_{l=1}^{K}\frac{b_{l}^{2} \mathbf{I}_{M}\delta_{l}^{{}^{\prime}}(\alpha)}{1+\delta_{l}(\alpha)}\right.\\ \left.+\sum_{l=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I }_{M}\delta_{l}^{{}^{\prime}}(\alpha)}{1+\delta_{l}(\alpha)}\right)\mathbf{T}( \alpha) \tag{60}\]
with
\[\mathbf{T}(\alpha)=\left(\sum_{l=1}^{K}\frac{b_{l}^{2}\mathbf{I}_{M}}{1+\delta_ {l}(\alpha)}+\sum_{l=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M}}{1+ \delta_{l}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}, \tag{61}\]
and \(\delta_{l}(\alpha)=\lim_{t\to\infty}\delta_{l}^{(t)}(\alpha)\),
\[\delta_{l}^{(t)}(\alpha)=\text{Tr}\Bigg{\{}b_{l}^{2}\mathbf{I}_{M} \left(\sum_{p=1}^{K}\frac{b_{p}^{2}\mathbf{I}_{M}}{1+\delta_{p}^{(t-1)}( \alpha)}\right.\\ \left.+\sum_{p=K+1}^{K+N_{r}}\frac{(\eta_{I}-\eta_{e})\mathbf{I}_{M} }{1+\delta_{p}^{(t-1)}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}\Bigg{\}}\ \ 1\leq l\leq K\] \[\delta_{l}^{(
such that \(\delta^{(0)}_{l}(\alpha)=\frac{1}{\alpha}\ \forall\ l\), \(\boldsymbol{\delta}^{{}^{\prime}}(\alpha)=[\delta^{{}^{\prime}}_{1}(\alpha)\dots \delta^{{}^{\prime}}_{K+N_{r}-2}(\alpha)]^{T}\), \(\boldsymbol{\delta}^{{}^{\prime}}(\alpha)=(\mathbf{I}_{K+N_{r}}-\mathbf{J}( \rho))^{-1}\mathbf{v}(\rho)\), with \([\mathbf{J}(\alpha)]_{pq}=\frac{\mathrm{Tr}(b_{p}^{2}\mathbf{I}_{M}\mathbf{T}( \alpha)b_{q}^{2}\mathbf{I}_{M}\mathbf{T}(\alpha))}{(1+\delta_{q}(\alpha))^{2}},[\mathbf{v}(\alpha)]_{p}=\mathrm{Tr}\{b_{p}^{2}\mathbf{I}_{M}\mathbf{T}( \alpha)\mathbf{T}(\alpha)\},\) where \(p,q=1,2,\dots,K+N_{r}\). Also,
\[\mu^{{}^{\prime}}_{i,\alpha}=\mathrm{Tr}\{\mathbf{T}^{\prime}_{i}(\alpha)\} \tag{63}\]
where,
\[\mathbf{T}^{{}^{\prime}}_{i}(\alpha)=\mathbf{T}_{\bar{\mathbf{B} }^{x,m}}(\alpha)(\eta_{l}-\eta_{e})\mathbf{I}_{M}\mathbf{T}_{\bar{\mathbf{B} }^{x,m}}(\alpha)\\ +\mathbf{T}_{\bar{\mathbf{B}}^{x,m}}(\alpha)\left(\sum_{l=1}^{K} \frac{b_{l}^{2}\mathbf{I}_{M}\delta^{{}^{\prime}}_{l}(\alpha)}{1+\delta_{l}( \alpha)}\right.\\ +\sum_{l=K+1}^{K+N_{r}}\frac{(\eta_{l}-\eta_{e})\mathbf{I}_{M} \delta^{{}^{\prime}}_{l}(\alpha)}{1+\delta_{l}(\alpha)}\right)\mathbf{T}_{\bar {\mathbf{B}}^{x,m}}(\alpha) \tag{64}\]
with
\[\mathbf{T}_{\bar{\mathbf{B}}^{x,m}}(\alpha)\\ =\left(\sum_{l=1}^{K}\frac{b_{l}^{2}\mathbf{I}_{M}}{1+\delta_{l}( \alpha)}+\sum_{\begin{subarray}{c}l=K+1,\\ l\neq m\end{subarray}}^{K+N_{r}}\frac{(\eta_{l}-\eta_{e})\mathbf{I}_{M}}{1+ \delta_{l}(\alpha)}+\alpha\mathbf{I}_{M}\right)^{-1}, \tag{65}\]
and \(\mu_{\bar{\mathbf{B}}^{x,m}}=\mathrm{Tr}\{\mathbf{T}_{\bar{\mathbf{B}}^{x,m} }(\alpha)\}\).
Proof.: See Appendix D.
Again, the rate regions quantifying the performance of the proposed JRC system can be obtained by using Theorems 3 and 4 in conjunction. In the next section, we present simulation results to better visualize the ideas presented by these theorems.
## VI Simulation Results
In this section we validate our derived results using Monte-Carlo simulations and prescribe parameter values for optimized system operation. Here, the communication subsystem consists of single cell massive MIMO system having \(M=128\) antennas at a BS that is located at the cell centre serving \(K=8\) users distributed uniformly across the cell at a carrier frequency \(f_{c}\)= 3 GHz. For simplicity, we assume the cell to be circular having a radius 100 m. The communication frame consists of \(1024\) channel uses with the first \(K=8\) channel uses dedicated for training, and the remaining divided equally for uplink and downlink data transmission. The performance of the communication system, unless stated otherwise is measured in terms of the average per user achievable rate. For the purpose of these experiments, we consider the interference channels to be known at the BS with a \(10\%\) error. We also apply channel inversion based power control in both uplink and downlink to better quantify SNR.
On the other hand, the radar subsystem consists of a MIMO radar with \(N_{t}=N_{r}=8\), transmit and receive antennas, respectively. Both the radar system and the target are assumed to be located randomly within the cell and their location follows the same distribution as the users. Unless stated otherwise, the performance of the radar is quantified in terms of the radar rate derived in the previous sections. All the performance metrics presented in this section are generated by averaging over 10,000 realizations of the system.
### _CSI Acquisition_
Fig. 2 plots the MSE of uplink channel estimate as a function of the received pilot SNR for different values of radar transmit power, \(\sigma_{r}^{2}\). We observe that a higher radar transmit power does result in a saturation of the channel estimation MSE, however, the impact of radar interference is minimal when both the communication system and the radar are operating at SNRs around 10 dB, which results in a fair channel estimation MSE.
In Fig. 3 we plot the mean square estimation error of the AoA at the radar as a function of radar received SNR for three different values of pilot powers transmitted by the communication subsystem.
### _Uplink Data Transmission_
Fig. 4 plots the achievable communication rate in the uplink subframe as a function of received BS SNR for different levels of interference caused by radar subsystem. We observe that even a radar SNR of 30 dB results in a negligible loss in the average achievable per user rate for the communication system. This is because of the effective cancellation of the radar
Figure 3: Radar AoA MSE versus radar SNR for different values of transmit pilot power.
Figure 2: Channel estimation MSE as a function of the received pilot SNR for different values of radar transmit power.
generated interference at the BS, as postulated in Theorem 1.
Fig. \(5\) illustrates the achievable radar rate in the uplink communication subframe as a function of the received SNR for different levels of interference by the communication subsystem. We observe a loss of about 3 bits per channel use when the communication subsystem is operating at a received SNR of 10 dB. This result is in line with Theorem 2, where the radar is shown to face unmitigated interference.
Fig. \(6\) plots the rate region of the JRC system in the uplink communication sub-frame. The dotted line represents the achievable rates when both radar and communication sub-systems share the spectrum in the time division duplexed mode. The convexity of the rate region assures the acceptable performance of both radar and communication sub-systems under fully shared spectrum.
### _Downlink Data Transmission_
Fig. \(7\) illustrates the achievable communication rate in the downlink communication subframe as a function of receive BS SNR for different levels of interference caused by radar subsystem. We observe that in line with the uplink case, the performance degradation is minimal.
Fig. \(8\) plots the achievable radar rate in the downlink communication subframe as a function of the received SNR for different levels of interference by the communication subsystem. It is clearly visible that the radar rate achievable during the downlink sub-frame is better than that achievable during the uplink subframe, indicating the efficacy of the null being formed in the direction of the radar by the massive MIMO BS, as stated in Theorem 4.
In Fig. \(9\) we plot the achievable rate region for JRC system during the downlink subframe. We observe this rate region to be significantly convex, thus validating our hypothesis about the ability of massive MIMO systems to coexist with radars.
## VII Conclusions and Future Work
In this paper, we investigated the performance of JRC system in which both communication and radar sub-systems were operating simultaneously over same spectrum. To evaluate the performance of this system, we modelled it as a multiple
Figure 4: Average uplink communication rate versus received data SNR for different values of radar transmit power.
Figure 5: Average uplink radar rate as a function of the radar SNR for different values of uplink data power.
Figure 6: Consolidated Rate Region during the uplink sub-frame.
Figure 7: Average communication rate in downlink as a function of the received data SNR for different values of radar transmitted power.
access channel with both the subsystems non-cooperatively contending for the available resources. Following this, using results from random matrix theory, and via extensive simulations, we obtained the achievable rate regions for the system considering both uplink and downlink data transmission in the communication subsystem. We have observed these rate regions to be sufficiently convex and can safely conclude that massive MIMO systems can coexist with MIMO radars without any significant co-design. Future work may include the extension of this work to a cell free setting.
### _Proof of Theorem 1_
Treating interference as noise [36], we can write the SINR of the received signal as the ratio of the mean squared value of the desired component to the sum of the mean squared values of all other components. In this context, we can write,
\[\zeta_{s,k}=\beta_{k}\epsilon_{u,s,k}E[\{\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{ yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}}^{-1}\hat{\mathbf{h}}_{k}|^{2}\} \tag{66}\]
Now,
\[\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}} }^{-1}\hat{\mathbf{h}}_{k}=\frac{\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{ \mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}^{-1}\hat{\mathbf{h}}_{k}}{1+\hat{ \mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}} ^{-1}\hat{\mathbf{h}}_{k}} \tag{67}\]
where \(\hat{\mathbf{H}}_{k}\in\mathcal{C}^{M\times(K-1)}\) contains all columns of \(\widehat{\mathbf{H}}\) except \(\hat{\mathbf{h}}_{k}\). Thus, \(\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}\) can be expressed as
\[\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}=\sum_ {\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\beta_{m}\epsilon_{u,s,m}\hat{\mathbf{h}}_{m}\hat{ \mathbf{h}}_{m}^{H}+\sigma_{r}^{2}\hat{\mathbf{G}}_{rb}\hat{\mathbf{G}}_{rb}^{H} \\ +\left(\sum_{m=1}^{K}\beta_{m}\epsilon_{u,s,m}b_{m}^{2}+\sigma_{r} ^{2}\eta_{e}+N_{0}\right)\mathbf{I}_{M}. \tag{68}\]
Now, since \(\hat{\mathbf{h}}_{k}\) and \(\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}^{-1}\) are independent,
\[\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}} _{k}}^{-1}\hat{\mathbf{h}}_{k}\triangleq\frac{a.s.}{M\rightarrow\infty}\text{ Tr}\{\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}^{-1}b_{k}^{2} \mathbf{I}_{M}\}. \tag{69}\]
Letting \(\rho=\sum_{k=1}^{K}\beta_{k}\epsilon_{u,s,k}b_{k}^{2}+\sigma_{r}^{2}\eta_{e}+N _{0}\), \(\mathbf{S}=\sigma_{r}^{2}\hat{\mathbf{G}}_{rb}\hat{\mathbf{G}}_{rb}^{H}\) and \(\mathbf{D}_{k}\) be a diagonal matrix, such that its \(m\)th diagonal element \(d_{k,m}\) is given by \(d_{k,m}=\beta_{m}\epsilon_{u,s,m}\) where \(m\in\{1,2,\ldots,k-1,k+1,\ldots,K\}\). We can now write,
\[\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}=\hat{\mathbf{H}}_{ k}\mathbf{D}_{k,\beta\epsilon_{u,s}}\hat{\mathbf{H}}_{k}^{H}+\mathbf{S}+\rho \mathbf{I}_{M}. \tag{70}\]
Since \(\mathbf{S}\in\mathcal{C}^{M\times M}\) is a non negative definite matrix, \(\hat{\mathbf{H}}_{k}\in\mathcal{C}^{M\times(K-1)}\) is a random matrix and \(\rho>0\), [35],
\[\text{Tr}\{\hat{\mathbf{H}}_{k}\mathbf{D}_{\beta\epsilon_{u,s}}\hat{\mathbf{H }}_{k}^{H}+\mathbf{S}+\rho\mathbf{I}_{M}\}^{-1}\xrightarrow[M\rightarrow\infty] {a.s.}\text{Tr}\{\mathbf{T}_{k}(\rho)\}. \tag{71}\]
Back substituting the expression for \(\zeta_{s,k}\) results in (21). Now,
\[\zeta_{I,k}=\left(\sum_{\begin{subarray}{c}l=1,\\ l\neq k\end{subarray}}^{K}\beta_{l}\epsilon_{u,s,l}E\{|\hat{\mathbf{h}}_{k}^{H} \mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}^{-1}\hat{\mathbf{ h}}_{l}|^{2}\}\right) \tag{72}\]
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}} }^{-1}\hat{\mathbf{H}}_{l}|^{2}=\frac{|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{ yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k}}^{-1}\hat{\mathbf{G}}_{r+},\hat{ \mathbf{H}}_{k}}{1+\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+ },\hat{\mathbf{H}}_{k}}^{-1}\hat{\mathbf{h}}_{k}|^{2}}, \tag{73}\]
and,
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+}, \hat{\mathbf{H}}_{k}}^{-1}\hat{\mathbf{h}}_{l}|^{2}=\left|\hat{\mathbf{h}}_{k}^ {H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,k}}^{-1}\hat{ \mathbf{h}}_{l}-\right.\\ \left.\frac{\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G }}_{r+},\hat{\mathbf{H}}_{k,l}}^{-1}\hat{\mathbf{h}}_{l}\hat{\mathbf{h}}_{l}^ {H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,l}}^{-1}\hat{ \mathbf{h}}_{k}}{1+\hat{\mathbf{h}}_{l}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r +},\hat{\mathbf{H}}_{k,l}}^{-1}\hat{\mathbf{h}}_{l}}\hat{\mathbf{h}}_{l}\right|^{ 2}, \tag{74}\]
where \(\hat{\mathbf{H}}_{k,l}\in\mathcal{C}^{M\times(K-2)}\) contains all columns of \(\hat{\mathbf{H}}\) except \(\hat{\mathbf{h}}_{k}\) and \(\hat{\mathbf{h}}_{l}\). But,
\[\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,l}}=\hat{\mathbf{H}}_{ k,l}\mathbf{D}_{k,\beta\epsilon_{u,s}}\hat{\mathbf{H}}_{k,l}^{H}+\mathbf{S}+\rho\mathbf{I}_{M}, \tag{75}\]
where \(\mathbf{D}_{k,l,\beta\epsilon_{u,s}}\) of order \((K-2)\) having the \(m\)th diagonal entry as \(\beta_{m}\epsilon_{u,s,m}\), \(m\neq\{k,l\}\). Since, \(\hat{\mathbf{h}}_{k}^{H},\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,l}}^{-1}\) and \(\hat{\mathbf{h}}_{l}\) are independent, we have
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}} _{k,l}}^{-1}\hat{\mathbf{h}}_{l}|^{2}\xrightarrow[M\rightarrow\infty]{a.s.}b_{k}^ {2}b_{l}^{2}\text{Tr}\{\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,l}}^{- 1}\}. \tag{76}\]
Now,
\[\text{Tr}\{\mathbf{R}_{yy|\hat{\mathbf{G}}_{r+},\hat{\mathbf{H}}_{k,l}}^{-2}\} \xrightarrow[M\rightarrow\infty]{a.s.}\text{Tr}\{\mathbf{T}_{k,l}^{\prime}( \rho)\}, \tag{77}\]
Figure 8: Average radar rate in the downlink as a function of the radar SNR for different values of downlink data power.
Figure 9: Consolidated Rate Region during the uplink sub-frame.
\[\hat{\mathbf{h}}_{l}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k,l}}^{-1} \hat{\mathbf{h}}_{l}\xrightarrow[M\rightarrow\infty]{}b_{l}^{2}\text{Tr}\{ \mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k,l}}^{-1}\}, \tag{78}\] \[\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k,l}}^{-1}\}_{M \rightarrow\infty}\text{Tr}\{\mathbf{T}_{k,l}(\rho)\}. \tag{79}\]
Hence,
\[\hat{\mathbf{h}}_{l}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k,l}}^{-1}\hat{ \mathbf{h}}_{l}\xrightarrow[M\rightarrow\infty]{}b_{l}^{2}\mu_{k,l}, \tag{80}\]
we can hence write,
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}}^{- 1}\hat{\mathbf{h}}_{l}|^{2}=\frac{1}{|1+b_{k}^{2}\mu_{k}|^{2}}\Bigg{(}b_{k}^{2 }b_{l}^{2}\mu_{k,l}^{{}^{\prime}}+\frac{(b_{k}^{2}b_{l}^{2}\mu_{k,l}^{{}^{ \prime}})^{2}}{|1+b_{l}^{2}\mu_{k,l}|^{2}}\] \[\qquad\qquad-2\Re\Bigg{\{}\frac{(b_{k}^{2}b_{l}^{2}\mu_{k,l}^{{}^ {\prime}})^{3/2}}{1+b_{l}^{2}\mu_{k,l}}\Bigg{\}}\Bigg{)}, \tag{81}\]
back substituting the expressions in (72) to obtain (24).
\[\zeta_{E,k}=\sum_{l=1}^{K}\beta_{l}\epsilon_{u,s,l}E\{|\hat{\mathbf{h}}_{k}^{ H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}}^{-1}\mathbf{\tilde{h}}_{l}x_{l}[n]|^{2}\}, \tag{82}\]
where,
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}}^{- 1}\mathbf{\tilde{h}}_{l}|^{2}=\Bigg{|}\frac{\hat{\mathbf{h}}_{k}^{H}\mathbf{R }_{yy|\hat{G}_{r,b},\hat{H}_{k}}^{-1}\mathbf{\tilde{h}}_{l}}{1+\hat{\mathbf{h} }_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k}}^{-1}\mathbf{\tilde{h}}_{k}} \Bigg{|}^{2}, \tag{83}\] \[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k} }^{-1}\mathbf{\tilde{h}}_{l}|^{2}\xrightarrow[M\rightarrow\infty]{}b_{k}^{2}b _{l}^{2}\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k}}^{-2}\},\] (84) \[\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r,b},\hat{H}_{k}}^{-2}\} \xrightarrow[M\rightarrow\infty]{}\text{Tr}\{\mathbf{T}_{k}^{{}^{\prime}}( \rho)\}, \tag{85}\]
we can use these values to obtain (28).
\[\zeta_{RC,k}=\sum_{i=1}^{N_{t}}\sigma_{r}^{2}|\hat{\mathbf{h}}_{k}^{H} \mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}}^{-1}\mathbf{\hat{g}}_{rb,i}|^{2}, \tag{86}\]
where
\[\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{- 1}\mathbf{\hat{g}}_{rb,i}=\frac{\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G} _{r+b},\hat{H}_{k}}^{-1}\mathbf{\hat{g}}_{rb,i}}{1+\hat{\mathbf{h}}_{k}^{H} \mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\mathbf{\hat{h}}_{k}}, \tag{87}\]
and,
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k} }^{-1}\mathbf{\hat{g}}_{rb,i}|^{2}=\left|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{ yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\mathbf{\hat{g}}_{rb,i}-\right.\] \[\left.\frac{\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b}, \hat{H}_{k}}^{-1}\mathbf{\hat{g}}_{rb,i}}{1+\hat{\mathbf{h}}_{rb,i}^{H}\mathbf{ R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\mathbf{\hat{g}}_{rb,i}}\mathbf{\hat{h}}_{k}}{1+ \hat{\mathbf{g}}_{rb,i}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1} \mathbf{\hat{g}}_{rb,i}}\right|^{2}, \tag{88}\]
\[\mathbf{R}_{yy|\hat{G}_{r+b,i},\hat{H}_{k}}=\hat{\mathbf{H}}_{k}\mathbf{R}_{b, \hat{g}_{e_{u,s}}}\hat{\mathbf{H}}_{k}^{H}+\mathbf{S}_{i}+\rho\mathbf{I}_{M}, \tag{89}\]
where \(\mathbf{S}_{i}=\sigma_{r}^{2}\hat{\mathbf{G}}_{rb,i}\hat{\mathbf{G}}_{rb,i}^{H}\) and \(\hat{\mathbf{G}}_{rb,i}\) contains all the columns of \(\hat{\mathbf{G}}_{rb}\) except \(\mathbf{\hat{g}}_{i}\). Since, \(\hat{\mathbf{h}}_{k}^{H},\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\) and \(\hat{\mathbf{g}}_{rb,i}\) are independent, we have
\[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{- 1}\mathbf{\hat{g}}_{rb,i}|^{2}\xrightarrow[M\rightarrow\infty]{}b_{k}^{2}(\eta_{I }-\eta_{e})\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-2}\}, \tag{90}\] \[\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-2}\} \xrightarrow[M\rightarrow\infty]{}\text{Tr}\{\mathbf{T}_{k,i}^{{}^{\prime}}( \rho)\},\] (91) \[\hat{\mathbf{g}}_{rb,b}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k} }^{-1}\mathbf{\hat{g}}_{rb,i}\xrightarrow[M\rightarrow\infty]{}(\eta_{I}-\eta_{e })\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\},\] (92) \[\text{Tr}\{\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k}}^{-1}\mathbf{ \hat{g}}_{rb,i}\xrightarrow[M\rightarrow\infty]{}\text{Tr}\{\mathbf{T}_{k,i}^{{}^{ \prime}}(\rho)\},\] (93) \[|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}_{k} }^{-1}\mathbf{\hat{g}}_{rb,i}|^{2}=b_{k}^{2}(\eta_{I}-\eta_{e})\mu_{k,i}^{{}^{ \prime}}\] \[+\frac{(b_{k}^{2}(\eta_{I}-\eta_{e})\mu_{k,i}^{{}^{\prime}})^{2}}{|1+( \eta_{I}-\eta_{e})\mu_{k,i}|^{2}}-2\Re\Bigg{\{}\frac{(b_{k}^{2}(\eta_{I}-\eta_{e })\mu_{k,i}^{{}^{\prime}})^{3/2}}{1+(\eta_{I}-\eta_{e})\mu_{k,i}}\Bigg{\}}, \tag{94}\]
we can use these values to obtain (30). Finally,
\[\zeta_{RE,k}=\sum_{i=1}^{N_{t}}E\left[|\hat{\mathbf{h}}_{k}^{H} \mathbf{R}_{yy|\hat{G}_{r+b},\hat{H}}^{-1}\mathbf{\tilde{g}}_{rb,i}s_{i}[n]|^{2} \right], \tag{95}\]
and
\[\zeta_{w,k}=N_{0}E\left\{|\hat{\mathbf{h}}_{k}^{H}\mathbf{R}_{yy|\hat{G}_{r+b}, \hat{H}}^{-1}\mathbf{w}_{b}|^{2}\right\}, \tag{96}\]
can be simplified using techniques similar to the ones used for \(\zeta_
### _Proof of Theorem 3_
\[\zeta_{r,k}=\beta_{k}\epsilon_{d,s,k}E[|\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}} \bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{k}^{*}|^{2}] \tag{100}\]
with, \(|\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I }_{M})^{-1}\hat{\mathbf{h}}_{k}^{*}|^{2}=\frac{|\hat{\mathbf{h}}_{k}^{T}(\bar{ \mathbf{H}}_{k}\bar{\mathbf{H}}^{H}_{k}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{ h}}_{k}^{*}|^{2}}{|1+\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}_{k}\bar{ \mathbf{H}}^{H}_{k}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{k}^{*}|^{2}}\), where
\[\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}_{k}\bar{\mathbf{H}}_{k}^{H}+\alpha \mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{k}^{*}\xrightarrow[M\to\infty]{a.s}b_{k }^{2}\text{Tr}\{(\bar{\mathbf{H}}_{k}\bar{\mathbf{H}}_{k}^{H}+\alpha\mathbf{I }_{M})^{-1}\}, \tag{101}\]
and
\[\text{Tr}\{(\bar{\mathbf{H}}_{k}^{H}\bar{\mathbf{H}}_{k}+\alpha\mathbf{I}_{M })^{-1}\}\xrightarrow[M\to\infty]{a.s}\text{Tr}\{\mathbf{T}_{k}(\alpha)\}. \tag{102}\]
Back substituting the results into (100), we get (43).
\[\zeta_{r,l,k}=\sum_{\begin{subarray}{c}m=1,\\ m\neq k\end{subarray}}^{K}\beta_{k}\epsilon_{d,s,m}E\{|\hat{\mathbf{h}}_{k}^{T}( \bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{h }}_{m}^{*}|^{2}\}. \tag{103}\]
Now, \(\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I }_{M})^{-1}\hat{\mathbf{h}}_{m}^{*}=\frac{\hat{\mathbf{h}}_{k}^{T}(\bar{ \mathbf{H}}_{k}\bar{\mathbf{H}}_{k}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{ h}}_{k}^{*}}{1+\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}_{k}\bar{ \mathbf{H}}_{k}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{k}^{*}}\), with,
\[\hat{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}_{k}\bar{\mathbf{H}}_{k}^{H}+\alpha \mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{m}^{*}=\hat{\mathbf{h}}_{k}^{T}(\bar{ \mathbf{H}}_{k,m}\bar{\mathbf{H}}_{k,m}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{ \mathbf{h}}_{m}^{*}\]
and
\[\text{Tr}\{(\bar{\mathbf{H}}_{k,m}\bar{\mathbf{H}}_{k,m}^{H}+\alpha\mathbf{I }_{M})^{-1}\hat{\mathbf{h}}_{m}^{*}\}\]
and
\[\text{Tr}\{(\bar{\mathbf{H}}_{k,m}\bar{\mathbf{H}}_{k,m}^{H}+\alpha\mathbf{I }_{M})^{-1}\}\xrightarrow[M\to\infty]{a.s}\text{Tr}\{\mathbf{T}_{k,m}^{{}^{ \prime}}(\alpha)\}. \tag{106}\]
Substituting these into (103), we obtain (46).
\[\zeta_{r,E,k}=\sum_{l=1}^{K}\beta_{k}\epsilon_{d,s,l}E\{|\bar{\mathbf{h}}_{k}^ {T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{ \mathbf{h}}_{l}^{*}|^{2}\}, \tag{107}\]
where, \(\bar{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I }_{M})^{-1}\hat{\mathbf{h}}_{l}^{*}=\frac{\hat{\mathbf{h}}_{k}^{T}(\bar{ \mathbf{H}}_{k}\bar{\mathbf{H}}_{l}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{ h}}_{l}^{*}}{1+\hat{\mathbf{h}}_{l}^{T}(\bar{\mathbf{H}}_{k}\bar{ \mathbf{H}}_{l}^{H}+\alpha\mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{l}^{*}}\). Now,
\[\hat{\mathbf{h}}_{l}^{T}(\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H}+\alpha \mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{l}^{*}\xrightarrow[M\to\infty]{a.s}b_{l }^{2}\text{Tr}(\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H}+\alpha\mathbf{I} _{M})^{-1}, \tag{108}\]
\[\text{Tr}\{\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H}+\alpha\mathbf{I}_{M} \}^{-1}\xrightarrow[M\to\infty]{a.s}\text{Tr}\{\mathbf{T}_{l}(\alpha)\}, \tag{109}\]
\[|\bar{\mathbf{h}}_{k}^{T}(\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H}+\alpha \mathbf{I}_{M})^{-1}\hat{\mathbf{h}}_{l}^{*}|^{2}\xrightarrow[M\to\infty]{a.s} \bar{b}_{k}^{2}b_{l}^{2}\text{Tr}\{(\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H} +\alpha\mathbf{I}_{M})^{-2}\}, \tag{110}\]
and
\[\text{Tr}\{(\bar{\mathbf{H}}_{l}\bar{\mathbf{H}}_{l}^{H}+\alpha\mathbf{I}_{M})^{ -2}\}\xrightarrow[M\to\infty]{a.s}\text{Tr}\{\mathbf{T}_{l}^{{}^{\prime}}( \alpha)\}. \tag{111}\]
Substituting these values to (107), we get (50).
### _Proof of Theorem 4_
We can show that,
\[E[(\bar{\mathbf{G}}_{br}^{H}\mathbf{Qdiag}(\sqrt{\epsilon_{d,s}}) \bar{\mathbf{p}}[n])(\bar{\mathbf{G}}_{br}^{H}\mathbf{Qdiag}(\sqrt{\epsilon_{d, s}})\bar{\mathbf{p}}[n])^{H}]\] \[\xrightarrow[M\to\infty]{a.s}\text{Tr}\{(\bar{\mathbf{H}}\bar{ \mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}(\eta_{e}\mathbf{I}_{M})\] \[(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1} \left(\sum_{i=1}^{K}\epsilon_{d,s,i}b_{i}^{2}\right)\mathbf{I}_{M}\bigg{\}} \mathbf{I}_{N_{r}} \tag{112}\]
and
\[\text{Tr}\left\{(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M})^{-1}( \eta_{e}\mathbf{I}_{M})(\bar{\mathbf{H}}\bar{\mathbf{H}}^{H}+\alpha\mathbf{I}_{M}) ^{-1}\right\}\xrightarrow[M\to\infty]{a.s}\mu_{\alpha}^{{}^{\prime}}. \tag{113}\]
Similarly,
\[E[(\bar{\mathbf{g}}_{br,m}^{H}\mathbf{Q}_{\bar{\mathbf{g}}b,m}\text{ diag}(\sqrt{\epsilon_{d,s}})\] \[\times\bar{\mathbf{p}}[n])(\bar{\mathbf{g}}_{br,m}^{H}\mathbf{Q}_{\bar{ \mathbf{g}}b,m}\text{diag}(\sqrt{\epsilon_{d,s}})\bar{\mathbf{p}}[n])^{H}]\] \[\xrightarrow[M\to\infty]{a.s}\text{Tr}\{(\bar{\mathbf{H}}_{\bar{ \mathbf{g}}b,m}\bar{\mathbf{H}}_{\bar{\mathbf{g}}b,m}^{H}+\alpha\mathbf{I}_{M}) ^{-1}(\eta_{I}-\eta_{e})\mathbf{I}_{M}\] \[(\bar{\mathbf{H}}_{\bar{\mathbf{g}}b,m}\bar{\mathbf{H}}_{\bar{ \mathbf{g}}b,m}^{H}+\alpha\mathbf{I}_{M})^{-1}\left(\sum_{i=1}^{K}\epsilon_{d, s,i}b_{i}^{2}\right)\mathbf{I}_{M}\bigg{\}} \tag{114}\]
and
\[\text{Tr}\left\{(\bar{\mathbf{H}}_{\bar{\mathbf{g}}b,m}\bar{\mathbf{H}}_{\bar{ \mathbf{g
where \(\mathbf{d}(\theta)=\text{vec}(\mathbf{A}(\theta))\), and \(\mathbf{v}_{d}=\text{vec}\left(\sum_{n=1}^{N}(\mathbf{\hat{G}}_{br}^{H}\mathbf{ Qdiag}(\sqrt{\epsilon_{d,s}})\tilde{\mathbf{p}}[n]+\mathbf{\hat{G}}_{br}^{H} \mathbf{Qdiag}(\sqrt{\epsilon_{d,s}})\tilde{\mathbf{p}}[n])\mathbf{s}^{H}[n]+ \sum_{n=1}^{N}\sqrt{N}_{0}\mathbf{w}_{r}[n]\mathbf{s}^{H}[n]\right)\), such that \(\mathbf{v}_{d}\sim\mathcal{CN}(\mathbf{0},\sigma_{wr,d}^{2}\sigma_{r}^{2} \mathbf{I}_{N_{r}N_{t}})\).
Therefore, the Fisher information [37] for estimating \(\theta\) is given as
\[J_{\theta\theta}=\frac{2\sigma_{r}^{2}|h_{rr}|^{2}}{\sigma_{wr}^{2}}\Re\{ \text{Tr}\{\mathbf{\hat{A}}(\theta)\mathbf{\hat{A}}^{H}(\theta)\}\}, \tag{122}\]
and hence the CRB for \(\theta\) takes the form \(\text{CRB}(\theta)=\frac{1}{J_{\theta\theta}}\).
|
2304.11745 | GACER: Granularity-Aware ConcurrEncy Regulation for Multi-Tenant Deep
Learning | As deep learning continues to advance and is applied to increasingly complex
scenarios, the demand for concurrent deployment of multiple neural network
models has arisen. This demand, commonly referred to as multi-tenant computing,
is becoming more and more important. However, even the most mature GPU-based
computing systems struggle to adequately address the significant heterogeneity
and complexity among concurrent models in terms of resource allocation and
runtime scheduling. And this usually results in considerable resource
utilization and throughput issues. To tackle these issues, this work proposes a
set of optimization techniques that advance the granularity of computing
management from both the spatial and temporal perspectives, specifically
tailored to heterogeneous model compositions for deep learning inference and
training. These techniques are further integrated as GACER -- an automated
optimization framework that provides high-utilization, high-throughput, and
low-latency multi-tenant computing support. And our experiments demonstrate
that GACER significantly improves the overall resource utilization and
consistently achieves outstanding speedups compared to native GPU computing
frameworks and existing state-of-the-art optimization works. | Yongbo Yu, Fuxun Yu, Mingjia Zhang, Di Wang, Tolga Soyata, Chenchen Liu, Xiang Chen | 2023-04-23T20:34:36Z | http://arxiv.org/abs/2304.11745v1 | # GACER: Granularity-Aware ConcurrEncey Regulation for Multi-Tenant Deep Learning
###### Abstract.
As deep learning continues to advance and is applied to increasingly complex scenarios, the demand for concurrent deployment of multiple neural network models has arisen. This demand, commonly referred to as multi-tenant computing, is becoming more and more important. However, even the most mature GPU-based computing systems struggle to adequately address the significant heterogeneity and complexity among concurrent models in terms of resource allocation and runtime scheduling. And this usually results in considerable resource utilization and throughput issues. To tackle these issues, this work proposes a set of optimization techniques that advance the granularity of computing management from both the spatial and temporal perspectives, specifically tailored to heterogeneous model compositions for deep learning inference and training. These techniques are further integrated as _GACER_ -- an automated optimization framework that provides high-utilization, high-throughput, and low-latency multi-tenant computing support. And our experiments demonstrate that _GACER_ significantly improves the overall resource utilization and consistently achieves outstanding speedups compared to native GPU computing frameworks and existing state-of-the-art optimization works.
## 1. Introduction
The explosive success of deep learning techniques in various cognitive tasks, such as image classification and speech recognition, has made neural network models the hot spot of computing systems research. One of the primary drivers behind this trend is the support provided by GPUs, which offer excellent computing capacity and parallelism capability [1, 2]. Despite of the active emergence of various ASIC- and FPGA-based accelerators or systems, GPUs remain the most widely adopted platform in practice, accounting for >85% market share in both cloud and edge applications [3].
While, for a long time in the past, due to substantial parameters of neural network models, many research works tended to adopt a generic setting: one GPU instance could only host a single model. And even many recent outstanding efforts have not stepped out of this rut (e.g., MetaFlow [4], IOS [5]). However, cutting-edge developments have gradually revolutionized this setting: With the miniaturization of neural networks and the magnification of GPUs, it is now possible for a single GPU to host multiple models simultaneously [6, 7]. Furthermore, the need for concurrent model processing also scales up with multi-task or multi-modality intelligence integration, such as in autonomous driving [8, 9]. Along with this trend, GPU manufacturers like NVIDIA actively promote related software and hardware techniques [10, 11, 12]. Thus, **"multi-tenant"** deep learning computing becomes more and more prominent, especially for GPU-based systems.
However, multi-tenant deep learning presents a greater challenge than conventional single-tenant deployment regarding computing management. This is due to the increased heterogeneity and complexity of the concurrency of multiple neural network models, which can vary in operator composition, structural design, and model scale [13, 14]. Unfortunately, these issues are not well addressed in current GPU computing, even with emerging supporting techniques from manufacturers: (1) When it comes to resource allocation, current techniques for issuing concurrent models into the GPU either result in using fixed hardware resource budgets or competing models for resources in a greedy manner. These approaches at the model level can lead to wastage of hardware capabilities or resource contention and corresponding overhead [15, 16, 17]. (2) When it comes to runtime scheduling, though many recent works have dived into the operator level, most of them either cannot handle multi-tenant scenarios or have overlooked the multi-tenant coordination overhead, leading to ineffective and unscalable runtime management [5, 18].
Given these observations, the primary motivation for multi-tenant deep learning optimization is to develop a fine-grained and feasible coordination computing framework to resolve resource contention and enhance complementary resource utilization across different model tenants with minimal regulation overhead. Therefore, several optimization expectations for multi-tenant deep learning can be derived: (1) From a spatial perspective, the resource allocation scheme requires unprecedented operator-level granularity to adapt to the dynamics of intra-model operators and therefore complement
intra-model resource requirements [18; 19]. (2) From a temporal perspective, the granularity of multi-tenant runtime scheduling should not only deepen to the operator level, but also improve the manageability to regulate the runtime overhead and balance the overall performance given different deployment complexities [14; 18]. These two optimization expectations from both the spatial and temporal perspectives point to the same optimization focus of this work: _To advance the current multi-tenant concurrency regulation granularity and co-optimize the spatial and temporal domains for optimal computing performance_.
Centering on this research focus, our work extensively examines the entire GPU-based system stacks and conducts a comprehensive analysis on current multi-tenant computing issues with state-of-the-art techniques. Subsequently, we propose a set of optimization techniques that advance the granularity of computing management in both the spatial and temporal management domains, significantly improving runtime performance for both deep learning inference and training. The contributions of this work are as follows:
* We reveal computing issues in multi-tenant deep learning and formulate corresponding processes for GPU-based systems and deep neural network (DNN) tenants. And this allows us to further identify multi-tenant optimization objectives and approaches.
* In the spatial domain, we propose a set of novel operator resizing and decomposition methods combined with emerging GPU techniques to enhance the granularity and flexibility of multi-tenant resource allocation. It fully resolves the model contention and exploits resource utilization.
* In the temporal domain, we improve multi-tenant scheduling at the operator level by optimizing related operator issuing and CPU-GPU synchronization mechanisms. Additionally, we identify specific scheduling overheads and regulate corresponding optimization trade-offs to improve the runtime performance comprehensively.
* The proposed techniques are integrated into an automated optimization framework -- _GACER_, which leverages a low-cost search method to identify the particular spatial and temporal deployment configuration for both offline and online multi-tenant deep learning scenarios.
_GACER_ could consistently provide high-utilization and high-throughput computing support to multi-tenant deep learning on GPUs. Compared to conventional computing framework without specific multi-tenant support (e.g., TVM), _GACER_ could consistently achieve almost \(\sim\)70% speeds up. And comparing to state-of-the-art multi-tenant optimization works, it could also achieve \(\sim\)30% acceleration with \(\sim\)40% resource utilization enhancement, and demonstrate outstanding capabilities for even more complex deployment scenarios.
## 2 Preliminary
### Multi-Tenant Deep Learning
This paper focuses on a generic multi-tenant deep learning deployment setting on GPUs [20], as shown in Fig. 11:
Footnote 1: Note that the techniques proposed in this paper are applicable to both the training and inference phases of multi-tenant deep learning. Hence, we will not make special distinctions in the following descriptions.
**Multi-Tenant Heterogeneity:** When multiple heterogeneous DNN models are deployed on a single GPU, and each is compiled into a data flow graph (DFG) that defines the computing sequence of a series of different operators by layers (e.g., Conv, ReLu, etc.). And each operator has a particular computational pattern and resource requirements, including computing resources measured by the occupancy of streaming multi-processors (SM), and memory resources measured by memory bandwidth utilization. Despite the particular operator design, the resource consumption of each operator is mainly determined by the batched job sizes.
**Multi-Tenant Deployment:** The DFG compilation is generally performed on the CPU side, and the DFG of each DNN tenant is wrapped into a processing thread for later GPU deployment. It's worth noting that the simultaneous multi-thread wrapping process has much less overhead than initializing operators one at a time. On the GPU side, each thread is assigned to a specific GPU computing stream [11] with a dedicated resource portion from the GPU's SM pool. As shown in Fig. 1, such a concurrent multi-tenant deployment significantly differs from traditional time-sliced deployment, where only one model runs at a time.
**Resource Sharing and Contention:** As multiple models are deployed concurrently and share the same GPU pool, resource sharing has become a major research focus. And various resource allocation and management schemes have been proposed to improve the complementary resource utilization across different model tenants, as will be reviewed in the next section. However, such resource-sharing mechanisms also lead to potential resource contention [18] among tenants when they compete for limited GPU resources, which can
Figure 1: Multi-Tenant Deep Learning with GPUs
result in certain performance issues such as latency increment and throughput reduction. And when we further take into account the heterogeneity between DNN tenants, multi-tenant computing support becomes even more challenging.
### Multi-Tenant Computing Support
Table 1 reviews state of the art for multi-tenant computing.
**Emerging GPU Techniques:** New GPU architecture features proposed by NVIDIA, such MIG [12] and MPS [10], have enabled GPUs to divide the SM pool into multiple concurrent portions for multi-tenant deployment. While these methods provide GPUs with spatial resource management capabilities, they often suffer from reconfiguration overhead and a lack of certain runtime flexibility.
Unlike MIG and MPS, which partition resources first and then deploy tenants, MS enables tenant-oriented dynamic resource allocation at runtime, facilitating post-compilation resource budget adjustment for multi-tenant models. Furthermore, MS supports frequent CPU-GPU synchronizations, which enable DFG reordering along stream issuing, thus providing the scheduling management capability from a temporal perspective. However, coordinating such multi-tenant GPU support is often overwhelming. Facing the multi-tenant complexity, MS usually simply adopted a greedy manner of runtime management, resulting in considerable resource contention and inappropriate scheduling cases [18].
**State-of-the-Art Optimization Works:** Based on these emerging techniques, many optimization works have been proposed. In the spatial domain, some studies have combined model-level workload/resource adaption with MIG or MPS to implement software-hardware co-optimization for enhancing multi-tenant resource sharing [17, 21]. In the temporal domain, very recent works have expanded the computing granularity of MS into the operator level. However, these works are still not comprehensive. Most of them either cannot handle multi-tenant scenarios or have overlooked the multi-tenant coordination overhead, leading to ineffective and unscalable runtime management [5, 18].
### Multi-Tenant Computing Granularity
In addition to reviewing the basic mechanisms, pros and cons of different techniques, Table 1 also provides a comparison of the optimization granularity. While these works have enabled multi-tenant deep learning on GPUs, most are still limited to a coarse granularity (esp. at the model level) and focus on a singular optimization domain (either spatial or temporal).
Fig 2 presents a brief comparison to demonstrate the impact of optimization granularity: (1) When multi-tenant DNNs are deployed with only model-level granularity, SMs are partitioned based on an average model resource requirement, as in the case of MPS and MIG. However, when considering individual operators by layer during runtime, the specific resource requirements may vary, resulting in resource underutilization with smaller operators or insufficient resources with larger ones, requiring additional GPU cycles. (2) As shown in Fig 2, a more fine-grained resource allocation at the operator level could overcome the fixed resource budgets and accommodate the varying resource requirements at individual model layers. However, due to the significant heterogeneity of multi-tenant models, significant resource contention can still occur. Although this could be addressed by the algorithm and compiling optimization (e.g., TVM [24]) or runtime scheduling (e.g., AutoMT [18]), additional efforts are still required to refine the dynamic resource allocations and related overhead.
\begin{table}
\begin{tabular}{c|c c c c} \hline \hline & **Techniques** & **Approaches** & **Granularity** & **Remaining Issues** \\ \hline \multirow{8}{*}{**Migelines**} & Multi-Instance GPU & • Physical GPU Instance Partition & • Model Level & • Fixed Instance Budget \\ & (MIG) [12] & • Instance-based Tenant Allocation & • Randomization Overhead & • Reconfiguration Overhead \\ \cline{2-5} & Multi-Process Service & • Virtualized GPU Resource Partition & • Resource Contention \\ & (MPS) [10] & • Dynamic Process Resource Budget & • Model Level & • Context Switch Overhead \\ \cline{2-5} & \multirow{4}{*}{Gslice[17], Salus[21]} & • Multi-Tenant with MPS & & \\ & & • Batching Configuration & • Model Level & • Coarse-Grained Optimization \\ & & • Overhead Optimization & & \\ \hline \multirow{8}{*}{**Migelines**} & Time-Sliced Sharing & • Runtime Singe-Tenant Switch & • Sub-Model/ & • Tennant Switch Overhead \\ & (EdgeBatch [22], Gpipe [23]) & • Post-Compilation Stream Resource Allocation & • Model Level & • Resource Contention \\ \cline{1-1} \cline{2-5} & Multi-Stream & • CPU-GPU Synchronization based Scheduling & • Model Level & • Greedy Runtime Management \\ \cline{1-1} \cline{2-5} & Operator Scheduling & • DFG-based Scheduling with MS & & \\ (IGS[5], AutoMT[18]) & • Resource Contention Optimization & • Operator Level & • Unregulated Resource Allocation \\ \cline{1-1} \cline{2-5} & & • Resource Contention Optimization & • Runtime Regulation Overhead \\ \hline \hline \end{tabular}
\end{table}
Table 1: Granularity and Performance of Recent Works on Multi-Tenant DL Computing Optimization
Figure 2: Comparison of Different Granularity
## 3. Design Analysis and Motivation
Therefore, this work focuses on advancing the manageability and efficiency of multi-tenant deep learning and at the fine-grained operator-level and expanding the spatio-temporal optimization space for comprehensive performance escalation. This section presents a detailed analysis of GPU resource utilization during multi-tenant DNN deployment and discusses the underlying design methodology of this work.
### Persistent Resource Utilization Issues
Fig. 3 highlights common multi-tenant computing issues that arise when deploying heterogeneous DNN models on a GPU with MS techniques and the ideal operator-level resource allocation. In this process, each operator is compiled with a defined resource budget and subsequently issued in a greedy order, as illustrated by the gray arrows in the DFG compiling figure. Such a subsequently issuing will allocate operators from different model tenants into individual CUDA streams to share the computing resource at each cycle.
Despite the utilization of the best available multi-tenant deployment settings above, resource underutilization issues still inevitable. In most deployment scenarios, even with operator-level resource allocation in place, the total resource requirements from concurrent tenants of a DFG stage will not exactly match the available GPU resource volume. As mentioned earlier, when there is a deficiency of any of the required resources (e.g., SM or memory) to deploy a model tenant's operator, the operator is moved to the next cycle. To illustrate this persistent issue, the lower-left figure displays an empirical example using the NVIDIA Nsight [25]. As shown there: In some GPU cycles, the concurrently deployed operators are unable to fully utilize the SM pool; and in some other cycles, concurrent operator deployment cannot be established for parallel processing, resulting in even worse resource underutilization.
### Design Motivation and Challenges
Here we refer to the unused portion of the GPU resources as "residue", and a GPU SM pool-oriented example is shown in the right part of Fig. 3, which is further analyzed from both spatial and temporal perspectives.
**Spatial Residue Optimization by Operator Resizing:** As aforementioned, a spatial optimization approach could be adjusting the deployed model tenant workload to accommodate resource availability with less contention [21]. And it will be even more effective when pushing the granularity from the model level to the operator level for operator resizing.
However, certain challenges arise: (1) Batching operation is always applied to an entire model in the current frameworks (e.g., PyTorch [26]). Since the total workload of individual operators is invariant, operator resizing can only follow the path of decomposing the workload into relatively small operator units. However, conventional operator decomposition remains at the model compilation stage and cannot satisfy the computational dynamics of multi-tenant scenarios, requiring a necessary framework modification. (2) Meanwhile, in multi-tenant scenarios, the optimization complexity also scales up to determine the appropriate operator for resizing, taking into account the runtime overhead. Thus, a comprehensive multi-tenant-specific optimization scheme is also required.
**Temporal Residue Optimization by Operator Reordering/Scheduling:** As shown in Fig. 3, the DFG reordering indicated by the green arrows could also exploit the underutilized GPU residual. In addition to conventional scheduling issues, such as layer dependency, new challenges are posed: (1) Multi-tenant scheduling involves multiple operators across various DFG settings, resulting in unprecedented dynamic complexity. Although the CPU-GPU synchronization provided by MS could achieve model reordering, specific library-level optimization is required to decompose the model DFG to the operator level for finer granularity. (2) Not coincidentally, significant DFG reconfiguration delay would be introduced, requiring an effective overhead regulation scheme.
**Spatial and Temporal Co-optimization** It is important to note that in a residue case, both spatial and temporal optimization methods can be adopted. Therefore, determining the trade-off between these methods is critical in selecting
Figure 3. GPU Resource Utilization Analysis from Spatial and Temporal Perspectives
the appropriate one. Furthermore, a series of optimization procedures through multi-tenant computation process are interdependent. In other words, the solution of one residue may affect the subsequent ones, making the previously determined locally optimal solution inefficient. Hence, this complexity renders spatial and temporal co-optimization as the most crucial aspect of this work.
## 4 Granularity-Aware
MULTI-Tenant Regulation
### Problem Formulation
**Resource and Tenant Formulation:** We abstract the total GPU SM resource pool \(S_{GPU}=100\%\). And we map the operator workload \(W(O^{B})\) to the SM occupancy, where \(O\) is the operator, \(B\) is the batch size of \(O\) We analyze DNN operators with different batch settings and formulate a lookup table for convenience, and give some examples of convolution and batchnorm operators in Fig. 4. The execution time \(T\) required by operators \(O\) also varies, and some operators need to span multiple time cycles for processing, so we also formulate the operator execution time.
**Multi-Tenant Runtime Deployment:** Multi-tenant DNN consists of several parallel tenants, and all tenants consist of \(N\) models: \(M_{1},M_{2},...,M_{n}\). And each model \(M\) can be represented by a DFG with a series of operators \(O\). So we have the model operator list \(M_{n}=[O_{n,1},O_{n,2},...,O_{n,i}]\).
The aforementioned formulation primarily illustrates the spatial mapping of resources and tenants, wherein a resource pool is utilized. This temporal deployment during runtime may be expressed for each cycle as follows:
\[S_{T_{0}}:[O_{1,1}],\;S_{T_{1}}:[O_{2,1},O_{3,1}],\] \[S_{T_{2}}:[O_{1,2},O_{1,3},O_{2,2},O_{3,2}],...,S_{T_{t}}:[O_{n, i},...] \tag{1}\] \[\text{where }S_{T_{0}},S_{T_{1}},...,S_{T_{t}}<=S_{GPU}.\]
In each time cycle, the aggregate SM occupancy \(S_{T}\) of operators must be maintained below the \(S_{GPU}\). This necessitates the deployment of certain operators across multiple time cycles, as exemplified by the notation \(S_{T_{0}\to T_{5}}:[O_{1,1}]\). This time cycle deployment essentially reflects the operator execution sequence \(S_{T_{0}}\to S_{T_{t}}\).
**Optimization Objective:** Upon the GPU resource and tenant formulation above, the residual resource in time cycle \(S_{T}\) mentioned in Section 3.3 could be formulated as follow:
\[R_{S_{T}}=S_{GPU}-S_{T}=S_{GPU}-\sum_{O\;in\;S_{T}}W(O_{n,i}^{B_{n,i}}). \tag{2}\]
And we subsequently sum the \(R_{S_{T}}\) across all cycles to compute the total residue \(R\).
\[R=\sum_{S_{T_{0}}\to S_{T_{t}}}(S_{GPU}-\sum_{O\;in\;S_{T}}W(O_{n,i}^{B_{n,i }})). \tag{3}\]
\(R\) has two variables: operator batch size \(B_{n,i}\) and operator execution sequence \(S_{T_{0}}\to S_{T_{t}}\), which are both in operator-level granularity. Therefore, we could optimize two sub-objective, finding the appropriate \(B\) and \(S_{T_{0}}\to S_{T_{t}}\) to minimize the \(R\). These could be translated into two sub-objective, finding the appropriate \(B\) and \(S_{T}\) to minimize the \(R\).
\[\tau\;(B_{n,i},\;S_{T_{0}}\to S_{T_{t}})\to Min\;R, \tag{4}\]
where \(\tau\) is an approximation approach.
Based on the two variables of Eq. 4, we need to extend the spatial and temporal granularity of the multi-tenant DNN to the operator level for solving \(R\). In the following context, we introduce the spatial and temporal regulation methods to adjust \(B_{n,i}\) and \(S_{T_{0}}\to S_{T_{t}}\) to fine granularity and present an optimal search strategy for \(\tau\) implementation.
### Spatial Granularity Regulation
**Regulation with Operator Resizing:** Instead of deploying all workloads of a certain operator to the GPU in the same time cycle, they can be resized into multiple smaller copies by decomposing the operator through batch size direction.
\[O_{n,i}^{B}\xrightarrow{\text{chunk}}O_{n,i}^{B^{1}},O_{n,i}^{B^{2}},...,O_{n,i}^{B^{j}},\;\text{ where }\sum_{1\text{ to }j}B^{j}=B. \tag{5}\]
The decomposed operators can be deployed to different time cycles, thus, we could only deploy part of the workload in a single time cycle. Therefore, the first objective could be translated into finding the batch size \(list_{B_{n,i}}=[B^{1},B^{2},...B^{j}]\) of each operator \(O_{n,i}\). By controlling the number \(j\), we can control the spatial granularity of each operator deployment.
**Regulation with Operator Decomposition** The current DNN model API structure employs a single batch size for all layers, resulting in a model-level granularity that fails to consider varying workloads in each layer. To address this issue, we used PyTorch's "torch.chunk()" and "torch.cat()" functions to rewrite the API model definition. By decomposing heavy workload operators into smaller ones and concatenating the resulting micro-batches, we can effectively control runtime resource consumption without sacrificing model accuracy. This approach increases spatial granularity and allows for finer-grained parallelism. Resizing the batch size in this manner enables us to adjust the workload at a more granular level, leading to optimized performance.
Figure 4: Resource Utilization and Time Profiling
**Resizing Regulation Overhead Analysis:** The resizing regulation needs to introduce additional decomposing and concatenation operations which also bring additional overhead. This makes the decomposed operator also have the trade-off between residue reduction and additional overhead. Therefore, the decomposed operator should be near the large residue in order to bring enough gain. However, not all operators in multi-tenant DL computing require decomposing along the batch direction. We design a mask list whose length is equal to the total number of operators in all DFGs, and each element in the mask is corresponding to an operator. If the \(mask(O_{n,i})\) is 0, that indicates \(O_{n,i}\) is no need to decompose, otherwise, a \(list_{B_{n,i}}\) is generated for \(O_{n,i}\) to represent how it was decomposed. During the optimization process, we will gradually find which operators need to be decomposed.
**Overall Spatial Regulation:** Based on the above overhead analysis, we calculate the biggest residue \(Max(R_{S_{T}})\) using Eq. 2 and decompose the operator with the largest size following this time cycle. After that, we decompose a batch that matches the residue size and update the mask list and \(list_{B}\) and update the decomposed operators to the DFG. And we check the dependence to make sure the decomposed operator could be deployed in this residue. However, due to the unequal length of some segments in the same cluster, operators in the tail of the longest segment have to execute individually and inevitably generate a larger residue. These residues do not need to be optimized, so we skip them.
Overall, the first sub-objective of spatial granularity regulation could be translated into finding the decomposition mask matrix and corresponding decomposition strategy \(list_{B}\).
### Temporal Granularity Regulation
In the temporal domain, we break through multiple DFGs into fine-grained operator segments, and therefore operator scheduling across multiple different models is implemented.
**Operator Level DFG Reordering:** In order to reorder the operator execution sequence \(S_{T_{b}}\to S_{T_{t}}\), we insert some pointers in DFG, and divide each DFG into several segments with different granularity (e.g. operator, stage, and the whole model), so that we could implement inter-model concurrent scheduling with different granularity. And then, we control which segments can be deployed simultaneously. By changing which segment the operator is located in, the order of execution of the operator is relatively changed. As shown in Fig 5, we use some demo models to show what is the pointer and how operator reordering could improve the parallelism performance, and each model is deployed in each stream. For simplicity, each sequence only consists of one type of the two operators \(C\) (e.g., Conv) and \(B\) (e.g., BatchNorm). As shown in Fig 5 (b), by inserting the pointer to reorder the execution sequence, the first operator \(C\) in stream 2 is able to perform concurrency with \(B\) in stream 1.
We formulate this process as follows, and taking stream 1 in Fig. 5 (c) as an example, the model \(M_{1}\) has 12 operators and is divided into three segments using two pointers. And then, the same index segments from different models are divided into the same cluster which can be deployed simultaneously, such as the cluster in Equation 6\(S_{T_{3}\to T_{B}}\). The operators in the same cluster can occupy multiple time cycles (assuming that all operators occupy only one cycle in this case).
\[\begin{split}& S_{T_{b}\to T_{2}}:[O_{1,1},O_{1,2}],[None],\\ & S_{T_{3}\to T_{3}}:[O_{1,3},...,O_{1,8}],[O_{2,1},...,O_{2,4}],\\ & S_{T_{b}\to T_{2}}:[O_{1,9},...,O_{1,12}],[O_{2,5},...,O_{2,8}]. \end{split} \tag{6}\]
Since the total time cycle of each DFG is not the same in the multi-tenant DL scenarios. Therefore, we do not need and cannot ensure that each segment is of equal length. We object to minimizing the overall time cycle occupied, which means that a smaller residue is generated.
Therefore, the operator execution sequence \(S_{T_{o}}\to S_{T_{t}}\) regulation could be translated to find the best segments and the optimal scheduling strategy. We use a synchronization pointer to detach each model's full sequence into several operator segments and establish a pointer matrix \(Matrix_{P}=[P_{1},P_{2},...P_{n}]\). Each model \(M\) has a list \(P\), and each \(P\) annotates the appropriate positions where we detach the DFG.
\[M_{1}:[O_{1,1},O_{1,2},...,O_{1,12}]+P_{1}:(2,8)=Seg(M_{1}). \tag{7}\]
Each number in \(P\) represents the position at which the pointer is inserted. Thus, the second sub-objective could be translated into finding the pointer matrix \(Matrix_{P}\) for each DFG. Each \(P\) has the same number of pointers.
**Runtime Synchronization Modification at Library Level:** The synchronization pointer is a GPU synchronization mechanism, that ensures that operators continue to be deployed only after previously issued operators have completed their
Figure 5: Reordering with Synchronization Pointers
Figure 6: CPU-GPU Synchronization Overhead
computation. Through these pointers, we could control which operators could be deployed simultaneously. However, the current DNN model structure makes the scheduler only take the whole model as a scheduling unit, which causes course-grained temporal granularity. This prevents us from playing the role of optimizing temporal granularity with synchronous pointers. Therefore, we also redesign the DNN API at the library level. Based on the PyTorch framework, we use the "model.named_modules()" method to get all the model layer objects and use "nn.Sequential" to refine all the objects into a list of DFG objects. Based on the above two operations, we are able to implement operator reordering and thus fine-grained scheduling of operators.
**Temporal Regulation Overhead Analysis:** However, the reordering method uses synchronization operation which is the runtime control, so this inevitably introduces scheduling overhead as shown in Fig. 6. The GPU waits until the CPU finishes synchronizing the pointer and then sends a new operator to GPU. This situation can lead to a lot of GPU wait time, resulting in additional residual resources. Adding a large number of pointers leads to frequent GPU waits, which can seriously slow down the overall efficiency of the GPU. So for different scenarios, we also need to trade off the gains and overheads that come with temporal granularity.
**Overall Temporal Regulation:** Based on the above overhead analysis, we readjust the computation of the residue so that the solution process can achieve granularity awareness. This waiting time is equivalent to introducing an additional residue of \(S_{GPU}*\) GPU synchronization wait time \(T_{SW}\). In the same computer system, this overhead is relatively stable and we can obtain roughly accurate values by profiling. Therefore, we can add the following term to Eq. 4 to adjust the objective function to be aware of the overhead caused by fine-grained scheduling.
\[R=\sum_{S_{B}\to S_{T}}(S_{GPU}-\sum_{O\ in\ S_{T}}W(O_{n,i}^{B_{n,i}}))+|P_{n }|*S_{GPU}*T_{SW}, \tag{8}\]
where \(|P_{n}|\) is the number of pointers. When \(|P_{n}|\) is larger, the antecedent term has a greater potential to obtain a smaller residue but will make the posterior term larger. This makes it profitable to fill only larger residues, which tend to occur in the first few time cycles of larger operators.
### Granularity-Aware Joint Optimization
**Search Framework for Joint Optimization:** Based on the formulation and regulation design, we propose our granularity-aware framework to minimize \(R\) by finding the optimal strategy from the search space of the mask, \(list_{B}\), and \(Matrix_{P}\).
We have three claims in our framework. (1) We do joint optimization of spatial and temporal granularity. And the adjustable factors in search space are coupled for the effect of residue, so finding the globally optimal solution is NP-hard. Therefore, we greedily alternate between spatial and temporal regulation until we reach the optimal concurrency strategy. (2) When doing joint optimization, we do not only consider the SM resource pool, but we can also extend this approach to other resources, such as GPU memory bandwidth. (3) Our framework also needs to be aware that fine-grained granularity optimization brings gain while also introducing overhead. we need to implement different granularity for different scenarios.
We propose a search approach that could consider all three claims and further propose a framework based on this search method as shown in Algorithm 1. We first initialize the elements in \(mask(O)\) and \(Matrix_{P}\) to \(0\) and calculate the residue using Eq. 8. For spatial granularity, we use the spatial regulation method in section 4.2. For temporal regulation, we implement a coordinate descent search algorithm, which takes the pointer number in \(Matrix_{P}\) as different coordinates. Then the optimal pointer number for each list \(P\) is searched alternately, during which other \(P\) lists are kept as the previous optimal one. The Spatial and Temporal search method is executed alternately. After a certain round of temporal execution, we switch to the spatial regulation search method in section 4.2 and update the DFG. And the decomposed operators are inserted between the pointers, without affecting the scheme of the existing \(Matrix_{P}\).
```
0:\(N\) DFGs, the initialized mask list, \(list_{B}\) and \(Matrix_{P}\), the rounds of search \(X\).
0: The optimal mask list, \(list_{B}\), and \(Matrix_{P}\)
1: Initialize a dictionary \(D\{R:Matrix_{P}\}\) to record residue \(R\) and the corresponding \(P\).
2:for rounds = \(x\) to \(X\)do
3:for model \(i=1\) to \(N\)do
4:for the \(j\) in \(P_{j}\)do
5: Calculate \(R\) using equation 8
6: Append \(R:Matrix_{P}\) to the records \(D\).
7: Update the \(P_{j}\) of \(Matrix_{P}\) with the smallest \(R\).
8: Sort the records \(D\) by the \(R\).
9:if the smallest \(R\) in \(D_{|P_{n}|}>R\) in \(D_{|P_{n}|-1}\)then
10:return\(Matrix_{P}\) with the smallest \(R\) in \(D_{|P_{n}|-1}\).
11: Add pointer in \(Matrix_{P}\)
12: Go back to Step 2
```
**Algorithm 1** Granularity-Aware Search
Our framework -- _GACER_, is a functional addition to the mainstream framework with good scalability and can be extended to a variety of applications at a low cost.
**Search Cost Analysis for Offline/Online Deployment** Since our automated optimization framework is a modeling-based search method, there is no need to profile the searched
strategy each time, so our automatic optimization framework is low-cost. In offline deployment, we can know all the multi-tenant deployment scenarios and can store the searched strategies in the device and use them directly when new requests appear. For online deployment, _GACER_ could yield near-optimal schedule solutions within a short time, and our search cost is acceptable for tasks that care about throughput and are not sensitive to real-time.
## 5. Performance Evaluation
In this section, we evaluate the proposed system performance, including end-to-end speedup and GPU utilization enhancement targeting multi-tenant batched-job tasks [27], in which each task has its own model batch size.
### Experimental Setting and Metric
**Model Selection:** We construct diverse multi-tenant DNN scenarios by leveraging three different types of applications collected from PyTorch: Vision models (including AlexNet (Alex), VGG16 (V16), ResNet18 (R18), ResNet34 (R34), ResNet50 (R50), ResNet101 (R101), MobileNetV3 (M3) and DenseNet (D121)), language model (LSTM [28]), and transformer-based recommendation model (BST [29]), which are commonly used for various applications such as photo auto-editing, image tagging, and video/voice processing. These models have distinctive depths and varying numbers of operators, resulting in unique computational and memory requirements for each model. Therefore, different model combinations based on the above models present varied resource utilization imbalances, thus requiring highly adaptive deployment and scheduling optimization strategies.
**Task and Benchmark:** We consider the multi-tenant execution of the aforementioned DNNs and create workloads with varied batch sizes for each model. For convolutional networks, we use an image scale of 224*224*3 with different batch size. For the language model, we use the text dataset ML2020spring which is used for emotion classification. And we use the Amazon dataset _Book_[30] as the benchmark dataset for the recommendation model. For generality, we also evaluate four types of NVIDIA GPUs from desktop-level to high-end ones: 1080Ti, P6000, and Titan V.
**Baseline Methods:** Several popular baseline resource allocation optimization and scheduling strategies are considered.
\(\bullet\) CuDNN-Seq [31]: The default strategy of PyTorch + CuDNN, which runs the models sequentially.
\(\bullet\) TVM-Seq [24]: A operator-level optimization method to search for the optimal kernel for each operator. However, it can only run these kernels sequentially.
\(\bullet\) Stream-Parallel [11]: The concurrent execution strategy from native GPU multi-stream support. It assigns models to different streams and leverages the default GPU scheduler to schedule the execution sequence.
\(\bullet\) MPS [10]: We distribute the resources to each model based on the models' FLOPS.
**Evaluation Metric:** In our experiments, we use the end-to-end execution latency of multi-tenant DNNs and monitored GPU resource utilization as the optimization objectives.
### Speed-Up Evaluation
We first compare the overall latency of the baselines and our methods _GACER_. To demonstrate each design component's effectiveness, we decompose and evaluate our method step by step, i.e., using spatial granularity regulation (_Spatial_), the method only using temporal granularity regulation (_Temporal_), and the combined optimization (_GACER_). The results are shown in Fig. 7. All latency is normalized by the CuDNN-Seq baseline to show the relative acceleration ratio. We use five multi-tenant DNN combination settings, which cover a wide range of concurrency scenarios. For example, ALEX+VGG+R18 is a relatively simple one (10 \(\sim\) 30 operators), and the number of operators of R101+D121+M3 can exceed 200. R34+LSTM+BST is a complex multi-tenant scenario, which includes a vision model, language model, and recommendation model with diverse resource requirements.
**Overall _GACER_ Speed-up:** Based on the results, it can be observed that our framework _GACER_ could consistently yield 1.37\(\times\)\(\sim\) 1.66\(\times\) speed-up compared to the sequential baselines across all five model combinations. The Stream-Parallel solution also yields a certain speed-up than CuDNN-Seq, but the acceleration ratio is much less usually 1.24\(\times\)\(\sim\) 1.51\(\times\). Also, the MPS acceleration effect is very unstable, specifically due to that fixed resource allocation cannot satisfy many particularly unbalanced model workload scenarios.
**Speed-up with Spatial Granularity Regulation:** The spatial granularity regulation is more advantageous for models with large operator workloads. In particular, when the workloads of multiple parallel models are large, such as R50+V16+M3 in Fig. 7, where both R50 and V16 both have large operator workloads, the decomposition of V16 can reduce more residue and obtain significant speedups. On the contrary, for the combination of R34+LSTM+BST, the latter two models have a low SM occupation, so the workload decomposition has less optimization space for the residue and almost no performance gain compared to Stream-Parallel.
**Speed-up with Temporal Granularity Regulation:** On the other hand, the temporal granularity regulation gives more significant speedup for model combinations with more layers. The most obvious scenario is R101+D121+M3, in which all three models have more layers and complex operators
in terms of timing and resource consumption. Therefore, dividing them into multiple operator clusters allows for better residue reduction within each concurrent cluster.
### GPU Utilization Evaluation
We further profiled and checked the GPU runtime statistics to analyze the overall GPU utilization within different scenarios. We use the achieved SM Occupancy from NVIDIA NSight Profiler as an indicator metric of GPU utilization information. Fig. 8 demonstrates the utilization statistics comparison between CuDNN-Seq, Stream-Parallel, and our method on the R101+D121+M3. As is observed, our method obtains about 60% utilization enhancement over the sequence method and almost 40% enhancement than Stream-Parallel, which is consistent with our speed-up performance. _GACER_ runs with a more even utilization and has less inefficient intervals due to the fact that our method fills the residue well.
### Framework Generality Analysis
We then evaluate the generality of our method with different GPU platforms. We perform operator profiling on different GPU platforms to obtain operator information lookup tables. Then we use our search framework to get the optimal regulation solution. We test five multi-tenant setups on different GPUs: the NVIDIA P6000 and the NVIDIA 1080Ti. The P6000 GPU is the last version before Titan-V and has a slightly lower peak computing performance (12.6 vs. 14.9 TFLOPS). The 1080Ti is a relatively early platform, with a calculated peak performance of only 10.4 TFLOPS. As the overall performance is shown in Table 2. C is CuDNN-Seq, and S is Stream-Parallel. Since these two platforms do not support MPS, it is not compared here. We set the batch size of each vision model to 8, the language model to 128, and the recommended model to 64, and we only test model inference here. Our scheduling framework _GACER_ also yields significant performance gain (1.38\(\times\)\(\sim\) 1.58\(\times\) acceleration on P6000 and 1.32\(\times\)\(\sim\) 1.70\(\times\) acceleration on 1080Ti) on the different GPU platforms.
### Framework Granularity Awareness
In this section, we conduct in-depth overhead profiling and analysis to demonstrate the full-spectrum system optimization trade-offs and how the _GACER_ framework decides coarse-to-fine temporal and spatial granularity to achieve optimal performance.
**Temporal Granularity Trade-off:** We compare the performance of multi-tenant DL computing under different scheduling granularity, including model-wise (Stream-Parallel), segment-wise (with a different number of pointers), and operator-wise. One interesting finding is that, as the scheduling granularity gets finer (from model to segment, and then operator), the latency performance tends to go through an improving and decreasing trend, forming a latency "sweet-zone" in the middle granularity as shown in Fig. 9.
We choose three scenarios of model combinations and execute them in different scheduling granularity. The "segment-2" means that we divide each model into 2 separate segments for scheduling. As shown in Fig. 9, although the "sweet-zone" always appears in the middle, the optimal granularity varies, which requires an adaptive granularity decision. Based on the results, we can find that complex model combination tends to benefit more from more fine-grained segments to get the best performance, e.g. R101+D121+M3. This is because they have more operators and thus the insertion of pointers usually has larger optimization space and freedom. In contrast, the second combination is relatively simple and achieves optimal performance with the optimization of a small number of pointers. And too much scheduling can lead to performance degradation due to synchronization overhead.
Figure 8. Analysis on GPU Utilization Enhancement
Figure 7. Runtime Performance of _GACER_ (with Titan V)
**Spatial Granularity Trade-off:** In this part, we analyze the parallel performance of some model combinations under different spatial granularity. For the baseline, we use two streams to parallel a VGG16 (V16) and ResNet18 (R18) model both with batch size 32. In order to improve the granularity of resource allocation, we decompose the batch of some operators in two models separately. We decompose all the convolution operators and the following Relu operators which are the main structure of these models The performance of different cases is shown in the Table. 3.
The spatial granularity also has a "sweet zone" phenomenon, and the optimal strategy is not the most fine-grained. This is because the decomposition and concatenation have the execution overhead and also increase more operators which could introduce more CPU operators issuing overhead. Another valuable phenomenon is the decomposition of the operator with a higher SM occupation can bring more benefits. This phenomenon is particularly evident in the decomposition of the VGG convolutional layers. Most of the layers of V16 are computationally intensive convolutional layers, and these convolutional layers tend to occupy a large amount of SM resources during computation. When computing the convolutional layers of V16, a large number of SMs are occupied and it is difficult for the operator from R18 to co-deploy, resulting in a large residue. The operator decomposition allows the convolutional layers in V16 to be more flexible to parallel with the operators in R18, thus achieving a huge performance improvement, such as the case in Table. 3
### Framework Overhead Analysis
In this section, we analyze the overhead of our search algorithm, our framework could usually yield near-optimal schedule solutions within a short search time. The framework's search running time overhead is demonstrated in Table 4. We profile the coordinate descent search with different search rounds from 100 to 10000. The results show that the running overhead of our framework stays in the range of several seconds to at most a few minutes. Also, since it is possible to pre-execute this automated search algorithm offline for a given multi-tenant scenario, we consider this offline tuning overhead to be very acceptable. Such a search overhead is also acceptable for some online tasks that are not very real-time, such as training.
for granularity optimization. Based on the optimization of granularity, we proposed _GACER_, an automated optimization framework that provides high-utilization, high-throughput, and low-latency multi-tenant deep learning computing support. This framework is a revolutionary approach to multi-tenant deployment and can provide a solid foundation for the future development of multi-tenant deep learning.
|
2305.05419 | Magneto-optical Properties of Reduced Titania Probed by First-principles
Calculations: Polarons | The magneto-optical properties of titanium dioxide systems are related to the
presence of impurity states in the band gap due to oxygen vacancies. To
understand about the interplay between localized electrons and structural
distortions at the vacancy sites and the magneto-optical properties, we employ
a self-interaction corrected density functional theory method to calculate bulk
and small nanoparticles of rutile, anatase, and brookite titania. Our
computations reveal bipolaron configurations associated to an oxygen vacancy
with optical transition levels in the band gap. The ground state for these
bipolarons is a spin-triplet state in bulk rutile TiO2 and also in the
nanoparticles independently of the crystal phase, a result which may support
the idea of oxygen vacancies as a source of magnetism in this material. The
ground state for bipolarons in bulk anatase TiO2 is however a spin-singlet
state, different from the spin-triplet configuration reported in a previous
work based on hybrid functionals. | C. Echeverria-Arrondo, H. Raebiger, J. Perez-Conde, C. Gomez-Polo, A. Ayuela | 2023-05-09T13:11:54Z | http://arxiv.org/abs/2305.05419v1 | # Magneto-optical Properties of Reduced Titania Probed by First-principles Calculations
###### Abstract
The magneto-optical properties of titanium dioxide systems are related to the presence of impurity states in the band gap due to oxygen vacancies. To understand about the interplay between localized electrons and structural distortions at the vacancy sites and the magneto-optical properties, we employ a self-interaction corrected density functional theory method to calculate bulk and small nanoparticles of rutile, anatase, and brookite titania. Our computations reveal bipolaron configurations associated to an oxygen vacancy with optical transition levels in the band gap. The ground state for these bipolarons is a spin-triplet state in bulk rutile TiO\({}_{2}\) and also in the nanoparticles independently of the crystal phase, a result which may support the idea of oxygen vacancies as a source of magnetism in this material. The ground state for bipolarons in bulk anatase TiO\({}_{2}\) is however a spin-singlet state, different from the spin-triplet configuration reported in a previous work based on hybrid functionals.
Introduction
Titanium dioxide is a wide band gap semiconductor metal oxide of great technological relevance for applications including perovskite solar cells[1; 2; 3], photocatalysis[4; 5], and spintronic devices[6; 7; 8]. Titanium dioxide doped with light elements such as N[9; 10; 11; 12] and C[13] and also with transition metal dopants shows ferromagnetism with Curie temperature above room temperature. Ferromagnetism is still observed in undoped crystals such as TiO\({}_{2}\) samples formed under strongly reducing synthesis conditions, wherein oxygen vacancies behave as intrinsic donors.[9; 14; 15; 16]
In this paper, to further understand the importance of oxygen vacancies themselves in the magneto-optical properties of TiO\({}_{2}\), we probe bulk and quasi-spherical nanocrystals of this material by quantum mechanical first-principles computations in the phases of rutile, anatase, and brookite. We show that the impurity electrons related to an oxygen vacancy are trapped by two of the three neighboring Ti\({}^{+4}\) cations which thereby are reduced into Ti\({}^{+3}\). These defect states appear in the band gap as polaron states which affect the magnetic and optical properties of reduced TiO\({}_{2}\) systems.
## II Method
The localized character of polarons, when treated with standard density functional theory (DFT) methods, is roughly described due to the self interaction error inherent to local and semilocal approximations. This error is expressed as a deviation from the correct Koopmans behavior of the highest electronic level; this level, in defective titania, corresponds to an impurity state[17]. To overcome the problem, we resorted to a self-interaction corrected DFT method [18; 19; 20; 21; 22; 23] known as NLEP, which is computationally less demanding than other methods such as those based on hybrid exchange correlation functionals[24] and Green's functions[25], and which corrects the band gap problem inherent to local and semi-local DFT approximations. In the NLEP formalism, non-local external potentials and \(U\) Hubbard terms are applied to Ti and O atomic orbitals. For computations we used the VASP code[26; 27; 28; 29], a plane-wave cutoff energy of 300 eV, an exchange correlation functional defined in the generalized gradient approximation of Perdew, Becke, and Ernzerhof [30], and Ti\({}_{\text{pv}}\) and O\({}_{\text{s}}\) pseudopotentials accounting for 10 and 6 electrons, respectively. The bulk first Brillouin zone was sampled with a shifted Monkhorst-Pack grid of \(4\times 4\times 4\)\(k\) points, and the quantum dots were placed in supercells with 10 A of vacuum space and calculated at the gamma point. The atomic structures were relaxed until the forces on the individual nuclei became
smaller than 0.02 eV/A.
## III Description of bulk and nanoparticle systems
We study rutile, anatase, and brookite crystals of bulk TiO\({}_{2}\) composed of 72, 108, and 96 atoms in supercells of \(2\times 2\times 3\), \(3\times 3\times 1\), and \(1\times 2\times 2\) unit cells, respectively. We use optimized lattice parameters which are close to the experimental ones [31; 32]; they are included in the SI. Because we have a single oxygen vacancy per supercell, the defect concentration amounts to \(\sim 2\times 10^{21}\) cm\({}^{-3}\), comparable to the one in highly reduced TiO\({}_{2}\)[33]. Furthermore, we calculate rutile, anatase, and brookite quasi-spherical nanocrystals of about 1.6 nm size and about 280 atoms, as shown in Fig. 1. To avoid the appearance of surface states in the band gap [34], pseudohydrogens of fractional
Figure 1: Atomistic description of bulk (left) and nanoparticles (right) of reduced TiO\({}_{2}\) in (a) rutile, (b) anatase, and (c) brookite phases. The numbers label the three neighboring Ti sites by the vacancy place, which is plotted as a yellow sphere.
charges[35] H\({}_{4/3}\) and H\({}_{2/3}\) are attached, respectively, to the edge titanium and oxygen atoms.
Figure 3: Densities of states (DOS) in bulk anatase TiO\({}_{2}\) with bipolarons in either (a) spin-triplet or (b) spin-singlet configuration. Insets show isosurfaces of constant spin density (0.0035 \(a_{0}^{-3}\)) with light (yellow) color when the spin is up and dark (blue) color when it is down. As before, the oxygen vacancy is represented as a yellow sphere.
Figure 2: Local magnetic moments at the three Ti sites surrounding the O vacancy in bulk rutile (R), anatase (A), and brookite (B). The given numbers indicate the positions of the Ti atoms as labeled in Fig. 1.
Results and Discussion
### Magnetic properties
First, we investigate oxygen defects, related polarons, and magnetic coupling properties in the bulk phases. The formation of an oxygen vacancy yields two impurity electrons which create the bipolaron states at the neighboring Ti sites in either spin-singlet or spin-triplet configuration, resulting in a total magnetic moment in the unit cell of zero and two Bohr magneton (\(\mu_{\rm B}\)), respectively. The polaronic nature of these defects emerge as large local magnetic moments and concomitant structural changes at the hosting Ti positions, as reported in Fig. 2. In the quantum dots, the local magnetic moments are closer to one Bohr magneton due to a loss in the number of lattice atoms around the vacancy.
In Fig. 3 we plot the densities of states for the anatase phase. It shows the presence of defect states and a 3.3 eV band gap close to the experimental one of 3.2 eV[36]. The densities of states in bulk rutile and brookite are supplied in the SI, with band gaps of 3.1 and 3.4 eV in consonance with the experimental gaps of 3.0 and 3.14 eV[36 ], respectively. In bulk anatase TiO\({}_{2}\), the spin-triplet state is formed by two electrons placed at bonding and antibonding states arising from the hybridization of two neighboring Ti-\(d\) orbitals (as labeled with numbers 1 and 2 in Fig. 1b). Moreover, the positions of the electronic energy levels in spin-singlet and spin-triplet states are consistent, on one hand, with the strong localized nature of polarons in bulk anatase TiO\({}_{2}\) and the consequent larger stability of the spin-singlet configuration, see Fig. 4, and, on the other hand, with comparable structural distortions around the vacancy for spin-singlet and triplet states. Previous calculations based on a hybrid density-functional method yielded, however, a spin-triplet ground state [24]. Analogous results on bulk rutile and brookite crystals can be found in the Supplementary Information. In Fig. 4 we report the energy differences between spin-singlet (S) and spin-triplet (T) configurations, \(\Delta=E^{S}-E^{T}\). The ground state for bipolarons in bulk rutile titania is a spin-triplet state. Furthermore, since \(\Delta\) exceeds the thermal energy at room temperature (\(k_{\rm B}T\simeq\)26 meV), polaron spins in bulk rutile TiO\({}_{2}\) may promote room-temperature ferromagnetism, in agreement with preliminary measurements[9]. For bulk anatase and brookite crystals, however, the ground state for bipolarons is a spin-singlet state. Hence, the ferromagnetic response observed in anatase TiO\({}_{2}\) would have another origin, possibly related with F\({}^{+}\) color centers characterized by a single electron at the vacancy site displaying a paramagnetic signal[37].
As for quantum dots, we investigate oxygen vacancies placed at central sites and below the
surface saturated by pseudohydrorogens (Fig. 1). The computed total energy decreases when the vacancy is moved from the center to the surface by 0.53, 0.59, and 0.87 eV in rutile, anatase, and brookite nanocrystals, respectively, in agreement with previous theoretical results which show an energy decrease of about 0.5 eV.[38] In addition, \(\Delta\) is almost null for vacancies by the nanoparticle surface. From Fig.4 we conclude, interestingly, that independently of the crystal phase the ground-state spin configuration for bipolarons in quantum dots is a spin-triplet state. As a consequence, oxygen vacancies in these small systems would produce magnetism, and this effect appears to be just induced by the crystal size.
### Optical properties
Second, we investigate the optical properties of reduced titania by addressing the transition energies of defect electrons to the conduction band. We focus on the transitions \(\rm V_{O}\to V_{O}^{+}\) and \(\rm V_{O}^{+}\to V_{O}^{++}\) (Fig. 5). In the Franck-Condon approximation, these optical levels are expressed as
\[\varepsilon_{\rm FC}(+1/0)=\rm E(V_{O}^{+}:V_{O})-\rm E(V_{O}) \tag{1}\]
and
\[\varepsilon_{\rm FC}(+2/+1)=\rm E(V_{O}^{++}:V_{O}^{+})-\rm E(V_{O}^{+}), \tag{2}\]
where \(E(\rm V_{O}^{+}\):\(\rm V_{O})\) is the total energy of a defective crystal with a neutral vacancy \(\rm V_{O}\), frozen
Figure 4: Energy difference \(\Delta=E^{S}-E^{T}\) between bipolaron spin-singlet (S) and spin-triplet (T) configurations for vacancies placed in the bulk (B, solid bars) and at a nanoparticle central site (NP, stripped bars).
atomic structure, and one electron in the conduction band, and \(E(\rm V_{O}^{++}\):\(\rm V_{O}^{+})\) is the total energy of a defective crystal with an excited vacancy \(\rm V_{O}^{+}\), frozen atomic structure, and two electrons in the conduction band, see Fig. 5. The thermodynamic transitions levels, however, involve fully relaxed structures and for this reason they are shallower than the optical ones; we express them as
\[\varepsilon_{\rm T}(+1/0)=\rm E(V_{O}^{+})-\rm E(V_{O}) \tag{3}\]
and
\[\varepsilon_{\rm T}(+2/+1)=\rm E(V_{O}^{++})-\rm E(V_{O}^{+}), \tag{4}\]
For bulk rutile and brookite, the optical levels lie significantly deeper in the band gap than the thermodynamic ones. However, for anatase, both optical and thermodynamic levels are close to each other due to a small excited-state relaxation of the lattice (Fig. 5). Interestingly, for bulk rutile, \(\varepsilon_{\rm FC}\)(+1/0)=0.78 eV and \(\varepsilon_{\rm FC}\)(+2/+1)=1.33 eV are in close agreement, respectively, with the 0.75 eV and 1.18 eV absorption peaks reported in the experiments.[40; 41; 42; 5; 4] We note that \(\varepsilon_{\rm FC}\)(+1/0)=0.78 eV is significantly closer to the experimental level than the previous value of 0.47 eV calculated with a hybrid density-functional method[24]. For bulk anatase, \(\varepsilon_{\rm FC}\)(+2/+1)=1.09 eV coincides with the 1.1 eV peak obtained from resonant photoemission and x-ray absorption spectroscopy[4; 5; 42]. As for the quantum dots, we study the transitions \(\rm V_{O}\rightarrow\rm V_{O}^{+}\) and calculate \(\varepsilon_{\rm FC}\)(+1/0), which is 1.76, 1.71, and 1.30 eV for rutile, anatase, and brookite phases, respectively.
Figure 5: Configuration coordinate diagram [39] illustrating the optical (\(\varepsilon_{\rm FC}\)) and thermodynamic (\(\varepsilon_{\rm T}\)) transitions of impurity electrons to the conduction band.
These values exceed the bulk ones because they are enhanced by the confinement of carriers within these quantum nanosystems.
## V Conclusions
We investigated the formation of polarons in reduced titania due to oxygen vacancies and their relevance to the magneto-optical properties such as the absorption spectrum. Based on a self-interaction corrected density functional theory method, we addressed both bulk and nanoparticles of TiO\({}_{2}\) in rutile, anatase, and brookite phases. Interestingly, the ground state for bipolarons in bulk rutile TiO\({}_{2}\) as well as in quantum dots is a spin-triplet state, what suggests that oxygen vacancies may yield a magnetic signal out of this oxide compound. Furthermore, we reported optical and thermodynamical transition levels from defect states in the band gap to the conduction band. For bulk rutile titania, the first optical transition level is at 0.78 eV, in good agreement with the 0.75 eV experimental value[40]. This result confirms the suitability of our theoretical approach for the study of optoelectronic properties in oxide compounds.
|
2305.14514 | Effects of wave propagation in canonical Poisson gauge theory under an
external magnetic field | The non-commutative electrodynamics based on the canonical Poisson gauge
theory is studied in this paper. For a pure spatial non-commutativity, we
investigate the plane wave solutions in the presence of a constant and uniform
magnetic background field for the classical electrodynamics in canonical
Poisson gauge theory. We obtain the properties of the medium ruled by the
permittivity and the permeability tensors in terms of the non-commutative
parameter, with the electrodynamics equations in the momentum space. Using the
plane wave solutions mentioned, the dispersion relations are modified by the
magnetic background, and the correspondent group velocity is affected by the
spatial non-commutative parameter. We construct the energy-momentum tensor and
discuss the conserved components of this tensor in the spatial non-commutative
case. The birefringence phenomenon is showed through the modified dispersion
relations, that depends directly on the non-commutative corrections and also on
the magnetic background field. Using the bound of the polarized vacuum with
laser (PVLAS) experiment for the vacuum magnetic birefringence, we estimate a
theoretical value for the spatial non-commutative parameter. | O. Abla, M. J. Neves | 2023-05-23T20:41:09Z | http://arxiv.org/abs/2305.14514v3 | # Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
###### Abstract
The noncommutative electrodynamics based on the canonical Poisson gauge theory is studied in this paper. We investigate the effects of the plane wave solutions in the noncommutative field equations when the theory is submitted to an external and uniform magnetic field. The energy-momentum tensor, symmetric and gauge invariant, is obtained from the noncommutative field equations. The plane wave solutions for the gauge potential yield the wave equations in the momentum space, using the linear approximation on the noncommutative parameter. Thereby, we obtain the dispersion relations of the gauge theory in the presence of an external and uniform magnetic field. The properties of the medium ruled by the permittivity and the permeability tensors are presented in terms of the noncommutative parameter, and of the magnetic field. We also show how the non-commutativity affects the energy density and the Poynting vector for the plane wave solution in a magnetic background field. From the dispersion relations, we discuss the birefringence phenomenon that depends directly on the noncommutative parameter. Using the bound of the PVLAS experiment for the vacuum magnetic birefringence, we estimate a theoretical value for the noncommutative parameter.
## I Introduction
The idea of a noncommutative space-time (NC) goes back to the early years of the foundation of quantum mechanics with the precision of a measure on quantum systems by Bronstein in the 30's Bronstein (1975). The precision is intimately related with the Compton wavelength for a test particle, in which the error on the quantum measure is minimized in higher energy levels. However, taking into account the gravitational effects, mass and energy are associated with the curvature of the space-time. In this case, the precision cannot exceed the Schwarzschild radius, since no information can be obtained from there. In other words, the space-time becomes non-local at the Planck scale, which suggests a minimal area so called Planck cell. This was the origin to study models with a NC space-time Bronstein (1976).
The usual way to introduce a NC field theory is inspired on the Snyder's formalism of quantizing the spacetime Snyder (1955). This approach considers that the order of the Planck cell is given by the NC parameter. The Weyl-Groenewold-Moyal formalism states that this is obtained by the quantization of the phase-space, and is order by order expanded on the so known star product Dirac (1957). However, the perturbation approach does not heal the divergences of the NC quantum field theory, and besides it, the emergence of the mixing between the ultraviolate and infrared (UV/IR) appears at the one loop order Nave and York (1979). The power series of the star product is not known until today in all orders, and also some problems were appointed, as the Lorentz symmetry violation.
After decades in the ostracism, the NC field theory was reconsidered when it emerged in quantum gravity and string theory Bronstein (1975)-Kosterov and York (1979), for example the Ramond-Neveu-Schwartz formalism for a constant and uniform magnetic background field (MBF), where the open string interactions in the correlation functions of the conformal field theory can be controlled by a star product. That suggests a NC gauge theory, such as a Yang-Mills coupled to scalar fields Bronstein (1975). For a non-constant MBF, the theory is more complicated, because the \(D\)-brane needs to live on a curved background, and the full machinery of the Kontsevich's quantization needs to be implemented Kontsevich (1999) (see also Bronstein and Nicolaev (2000) for an alternative approach to the construction of the star product).
Although the physics in the NC space-time was highly studied over the last twenty years Basso (2000)-Kosterov and York (1979), some aspects on the construction of NC gauge theories were not fully understood Kosterov and York (1979). Some attempts were developed, for example, the correspondence between NC gauge orbits induced by classical gauge orbits known as the Seiberg-Witten map Witten (1982); Witten (1982). In spite of the closed form be still unknown, most of the papers in the literature only consider the lowest orders on the deformation parameter. The \(L_{\infty}\)-bootstrap was proposed recently to deal with this problem revealing to be a powerful tool, even with the complicated calculations Witten (1983); Witten (1984). To understand the full picture of the NC gauge theories, a semi-classical formalism was developed on the last years, so known as the Poisson gauge theory (PGT). Although it is an approximation on the first order in \(\hbar\), the formalism is completely known order by order in the NC parameter Kosterov and York (1979)-Kosterov and York (1979). For a latest review on this approach, see the ref. Basso (2001).
The canonical formulation of the PGT is the main focus of this paper, in which the NC parameter does not
depend on the coordinates of the space-time. We study the electrodynamics defined on the canonical formulation of the PGT where the NC parameter has a key rule as the properties of this material medium and on the wave propagation. The conservation laws are obtained, as the components of the canonical energy-momentum tensor (EMT). Using the plane wave solutions superposed to a MBF, we investigate the influence of the NC parameter and of the magnetic field on the effects of the wave propagation. For example, we calculate the dispersion relations (DRs) and the group velocities of the plane wave (PW) in this medium. The permittivity and the permeability tensors give the characteristics of the material medium. Posteriorly, we study the energy density and the energy flux through the Poynting vector for three cases involving the MBF, the NC parameter, and the wave propagation direction. For end, we discuss the birefringence phenomenon associated with the DRs.
The plan of the paper is proposed as follow : In the section II, we review the structure of canonical PGT applied to the NC electrodynamics (ED). On section III is presented the conservation laws, defining a canonical NC EMT which is covariantly conserved, symmetric and gauge invariant. In section IV, we propose the PW solutions summed to a constant BF, and we discuss the properties of the NC ED equations, the DRs, and the group velocity in the presence of a an external magnetic field in a spatial NC. The section V is dedicated to the study of the energy density, and of the energy flux (Poynting vector) for the PW solution from the sec. IV. In the section VI, we show briefly the birefringence phenomenon with the contribution of the spatial NC, and using the bound of the PVLAS experiment for the vacuum magnetic birefringence, we estimate a theoretical value for the NC parameter [35]. In the last section (VII), we appoint some conclusions and perspectives for future projects.
We will use the indexes with latin letters to run from 0 to 3, besides the common use of \(i,j,k=\{1,2,3\}\), using the conventional signature of the Minkowski metric being \(\eta_{ab}=\mathrm{diag}(+1,-1,-1,-1)\), and the natural units \(c=\hbar=1\). Also we were always considering a closed system, which means that the Lagrangian is translational invariant, making the \(x^{a}\) dependence of \(\mathcal{L}\) due to the gauge field \(A_{a}\) and (at most) it first derivative, _i.e._, \(\mathcal{L}(A_{a},\partial_{b}A_{a})\).
## II The canonical Poisson gauge theory
The construction of Canonical PGT can be initiated with the discussion about the \(U(1)\) NC gauge theory, when one admits a flat space-time with a constant NC parameter. From the canonical Poisson gauge variation,
\[\delta_{f}A_{a}=\partial_{a}f+\{\,A_{a}\,,\,f\,\}=\{\,f\,,\,p_{a}-A_{a}\,\}\, \tag{1}\]
in which the bracket between two functions of \(x\), _i.e._, \(g(x)\) and \(h(x)\), on the canonical PGT is defined by
\[\{\,g\,,\,h\,\}:=\theta^{ab}\;\partial_{a}g\;\partial_{b}h\, \tag{2}\]
where \(\theta^{ab}=-\theta^{ba}\) is the antisymmetric NC parameter with area dimension. The commutative results are recovered in the limit \(\theta\to 0\). Since we consider a constant NC parameter, we can write it in terms of the components \(\theta^{ab}=(\,\theta^{0i}\,,\,\epsilon^{ijk}\,\vec{\theta}^{k}\,)\). Furthermore, spatial-time component can cause unitarity problems in the full NC theory [36; 37]. Although it is not a problem for PGT [25], we make the choice of a spatial NC, with \(\theta^{0i}=0\) in some examples on the next sections.
The commutator between two gauge variations yields the gauge algebra
\[[\delta_{f}\,,\delta_{g}]\ =\ \delta_{\{f\,,g\}}\,. \tag{3}\]
The covariant derivative operator on the canonical PGT may be defined as
\[\mathcal{D}_{a}\cdot\,:=\{\,\cdot\,,\,p_{a}-A_{a}\,\}\, \tag{4}\]
and the commutator of two covariant derivatives,
\[[\,\mathcal{D}_{a}\,,\,\mathcal{D}_{b}\,]\cdot\,=\{\,\mathcal{F}_{ab}\,,\, \cdot\,\}\, \tag{5}\]
where
\[\mathcal{F}_{ab}:=\{p_{a}-A_{a},p_{b}-A_{b}\}=F_{ab}+\{\,A_{a}\,,\,A_{b}\,\}\, \tag{6}\]
is the canonical Poisson field strength (CPFS), with \(F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}=(E_{i},-\epsilon_{ijk}B_{k})\), being the usual Maxwell's field strength.
The transformation property of the CPFS is
\[\delta_{f}\mathcal{F}_{ab}=\{\,\mathcal{F}_{ab}\,,\,f\,\}\, \tag{7}\]
meaning that it transforms covariantly under the gauge transformation \(\delta_{f}\). We then may construct the correspondent Lagrangian as usual,
\[\mathcal{L}=-\frac{1}{4}\,\mathcal{F}_{ab}\,\mathcal{F}^{ab}. \tag{8}\]
Note that the variation of the kinetic term \(\mathcal{F}_{ab}\,\mathcal{F}^{ab}\) satisfies the PGT, and the Lagrangian transforms covariantly,
\[\delta_{f}\left(\mathcal{F}_{ab}\mathcal{F}^{ab}\right)\ =\ \left\{\,\mathcal{F}_{ab}\,\mathcal{F}^{ab}\,,\,f\,\right\}. \tag{9}\]
In the presence of external source \(J^{a}=(\rho,\mathbf{J})\),
\[\mathcal{L}_{NED}=-\frac{1}{4}\,\mathcal{F}_{ab}\,\mathcal{F}^{ab}-J^{a}\,A_{ a}. \tag{10}\]
If we consider the \(4D\) functional action, it is invariant under the gauge transformations given by (1). The action principle applied to the Lagrangian (10) yields the NC field equation :
\[\mathcal{D}_{a}\mathcal{F}^{ab}=J^{b}\, \tag{11}\]
which recovers the Maxwell's equations with source \(J^{a}\) taking the commutative limit
\[\lim_{\theta\to 0}{\cal D}_{a}{\cal F}^{ab}=\partial_{a}F^{ab}=J^{b}\;. \tag{12}\]
Using the properties of the Poisson brackets, the NC Bianchi identity reads as
\[{\cal D}_{a}{\cal F}_{bc}+{\cal D}_{b}{\cal F}_{ca}+{\cal D}_{c}{\cal F}_{ab}=0\,, \tag{13}\]
which yields the Maxwell's equations without sources in the commutative limit
\[\lim_{\theta\to 0}{\cal D}_{a}{\cal F}_{bc}+{\cal D}_{b}{\cal F}_{ca}+{ \cal D}_{c}{\cal F}_{ab}\] \[=\partial_{a}F_{bc}+\partial_{b}F_{ca}+\partial_{c}F_{ab}=0\;. \tag{14}\]
Considering a spatial NC, the Poisson ED NC equations in the vector form are read :
\[{\cal D}\cdot{\bf E}+\widetilde{\mathbf{\theta}}\cdot[\,\nabla(\nabla \cdot{\bf A})\times\nabla V\,]\] \[+\widetilde{\theta}^{k}\,\epsilon^{ijk}\,\partial^{i}{\bf A}\cdot \partial^{j}(\nabla V)=\rho\;, \tag{15a}\] \[{\cal D}\cdot{\bf B}-\widetilde{\theta}^{k}\,\epsilon^{ijk}\, \nabla\cdot\left(\partial^{i}{\bf A}\times\partial^{j}{\bf A}\right)=0\;,\] (15b) \[{\cal D}\times{\bf E}+{\cal D}_{b}{\bf B}+\nabla\times(\widetilde {\theta}^{k}\,\epsilon^{ijk}\,\partial^{i}V\,\partial^{j}{\bf A})\] \[+\partial_{t}\left(\widetilde{\theta}^{k}\,\epsilon^{ijk}\, \partial_{i}{\bf A}\times\partial_{j}{\bf A}\right)={\bf 0}\;,\] (15c) \[{\cal D}\times{\bf B}+\partial_{c}\left(\widetilde{\theta}^{k}\, \epsilon^{ijk}\partial^{i}A^{c}\partial^{j}{\bf A}\right)={\bf J}+{\cal D}_{t} {\bf E}\;, \tag{15d}\]
where \({\cal D}\) and \({\cal D}_{t}\) means the spatial and time components of the operator \({\cal D}_{a}\), respectively, and we neglect the terms of higher orders in the NC parameter.
All the canonical PGT can be treated as the semi-classical limit of the full NC gauge theory (with constant deformation parameter). The Moyal product in the NC space-time between two functions of \(x^{a}\), say \(f=f(x)\) and \(g=g(x)\) is given by,
\[f(x)\star g(x)=\exp\left(\frac{i}{2}\,\theta^{ab}\,\partial_{a}^{x}\,\partial_ {b}^{y}\right)f(x)\,g(y)\bigg{|}_{y\to x}\;, \tag{16}\]
where the Moyal commutator is equal to the bracket on the canonical PGT in the first order of the NC parameter, _i.e._,
\[-i\,[\,f\,,\,g\,]_{\star}=-i\,(f\star g-g\star f)\approx\{\,f\,,\,g\,\}\;. \tag{17}\]
which means that the approach is completely equivalent, with field equations on the Moyal picture given by
\[{\cal D}_{a}{\cal F}_{M}^{ab}=\partial_{a}{\cal F}_{M}^{ab}-i\,\big{[}A_{a}\,,{\cal F}_{M}^{ab}\big{]}_{\star}=J^{b}\;, \tag{18}\]
and the NC tensor strength read as,
\[{\cal F}_{M}^{ab}=\partial^{a}A^{b}-\partial^{b}A^{a}-i\,\big{[}A^{a}\,,A^{b} \big{]}_{\star}\;. \tag{19}\]
The same results of the ref. [16] is then obtained on this approach.
## III The conservation laws and the canonical Poisson energy-momentum tensor
In this section, we investigate the conservation laws associated with the NC field equations (11) and (13). The components of the EMT yield the conserved physical quantities interpreted as the energy density, and the energy flux density of the NC field theory. Applying the operator \({\cal D}_{b}\) in both sides of (11), the classical source is covariantly conserved :
\[{\cal D}_{b}J^{b}=0\;, \tag{20}\]
since the covariant derivative commute, \([\,{\cal D}_{a}\,,\,{\cal D}_{b}\,]=0\), which means that \({\cal D}_{b}{\cal D}_{a}{\cal F}^{ab}=0\). In fact, if we define the bar current \(\bar{J}^{b}=J^{b}-\{\,A_{a}\,,\,{\cal F}^{ab}\}\), it is conserved \(\partial_{b}\bar{J}^{b}=0\), as is well known in Maxwell's ED.
The canonical Poisson EMT is so calculated combining the NC field equations. Contracting the Bianchi identity (13) with the CPFS, we obtain
\[{\cal D}_{a}\left({\cal F}^{ac}{\cal F}_{c}\,^{b}+\frac{1}{4}\eta^{ab}{\cal F} ^{cd}{\cal F}_{cd}\right)={\cal F}^{ab}J_{a}\;. \tag{21}\]
We use the properties of the canonical Poisson covariant derivative operator, and the source field equation (11) to obtain the equation [29]
\[{\cal D}_{a}{\cal T}^{ab}=J_{a}\,{\cal F}^{ab}\;. \tag{22}\]
In the absence of external sources (\(J^{a}=0\)), the quantity defined as canonical Poisson EMT is given by :
\[{\cal T}^{ab}={\cal F}^{ac}\,{\cal F}_{c}\,^{b}-\eta^{ab}\,{\cal L}_{ED}\;, \tag{23}\]
where \({\cal L}_{ED}\) is the Lagrangian (10), with \(J^{a}=0\). This tensor transforms covariantly under the gauge transformation \(\delta_{f}\), _i.e._,
\[\delta_{f}{\cal T}^{ab}=\{{\cal T}^{ab}\,,\,f\}\;, \tag{24}\]
is covariantly conserved in the absence of sources,
\[{\cal D}_{a}{\cal T}^{ab}=0\;. \tag{25}\]
and clearly symmetric. It can be written explicitly in terms of the gauge field in the linear approximation of the NC parameter,
\[{\cal T}^{ab}=F^{ac}\,F_{c}\,^{b}+\frac{1}{4}\eta^{ab}F_{cd}F^{cd}\] \[+\theta^{ef}\partial_{e}A_{c}\bigg{[}F^{ac}\,\partial_{f}A^{b}+F ^{bc}\partial_{f}A^{a}+\frac{1}{2}\,\eta^{ab}\,F^{cd}\partial_{f}A_{d}\bigg{]}\;. \tag{26}\]
The components \({\cal T}^{a0}=({\cal T}^{00},{\cal T}^{i0})\) yield the physical quantities that we will investigate in the section V, when it is submitted to PW solutions in a MBF. We define these components as \({\cal T}^{00}:=u\), that is the energy density, and \({\cal T}^{i0}:=S^{i},(i=1,2,3)\) are the components of the
Poynting vector for the canonical Poisson EMT, namely, given by
\[u = \frac{1}{2}\left(\mathbf{E}^{2}+\mathbf{B}^{2}\right)+\frac{3}{2}\, \theta^{ab}\,\mathbf{E}\cdot\partial_{a}\mathbf{A}\,\partial_{b}V \tag{27a}\] \[+\frac{1}{2}\,\theta^{ab}\,\mathbf{B}\cdot(\partial_{a}\mathbf{A} \times\partial_{b}\mathbf{A})\;,\] \[S^{i} = (\mathbf{E}\times\mathbf{B})^{i}-\theta^{ab}\,(\mathbf{E}\cdot \partial_{a}\mathbf{A})\,\partial_{b}A^{i}\] (27b) \[+\theta^{ab}(\partial_{a}\mathbf{A}\times\mathbf{B})^{i}\, \partial_{b}V\;.\]
These results show that the NC parameter corrects the energy density (even that it be a small contribution), and affects the propagation direction of the energy flux interpreted by the Poynting vector. The terms of \(\theta^{ab}\) in (27b) point on directions that, in general cases, are different of the usual direction of \(\mathbf{E}\times\mathbf{B}\). This fact will be explored in the section (V), in the case of the PW solutions.
## IV The dispersion relations and the group velocities under an external magnetic field
Using the definition of the canonical Poisson gauge covariant derivative (4) in the field equation (11), we can write this equation in terms of the gauge field \(A_{a}\). Thus, the free wave equation for the canonical PGT is
\[M_{ab}\,A^{b}=0\;, \tag{28}\]
where the wave operator has the following elements (in the linear \(\theta\)-approximation) after symmetrization,
\[M_{ab} = \left[\,\Box+\theta^{cd}\,(\partial_{e}\partial_{c}A^{e}\partial _{d}+2\partial_{c}A^{e}\partial_{e}\partial_{d})\,\right]\eta_{ab} \tag{29}\] \[-\partial_{a}\partial_{b}-\frac{1}{2}\,\theta^{cd}\,(\partial_{c} A_{b}\partial_{d}\partial_{a}+\partial_{c}A_{a}\partial_{d}\partial_{b})\;\;,\]
with \(\Box\) the D'Alembertian operator.
Therefore, the wave equation in the Lorenz gauge condition (\(\partial_{a}A^{a}=0\)) is
\[\Box A_{a}+\theta^{cd}\left(2\,\partial_{c}A_{b}\partial_{d}\partial^{b}A_{a} -\frac{1}{2}\,\partial_{c}A^{b}\partial_{d}\partial_{a}A_{b}\right)=0\;. \tag{30}\]
The idea here is to find the wave equation in the momentum space, when one substitutes the potential solution correspondingly to a single harmonic PW summed to an electromagnetic BF, namely,
\[A_{a}(x)=\tilde{A}_{a}\,\sin(k\cdot x)-\frac{1}{2}\,B_{ab}\,x^{b}\;, \tag{31}\]
where \(\tilde{A}_{a}\) is a constant and uniform amplitude of the potential, and the sinus is function of the scalar product \(k\cdot x=k_{a}x^{a}=\omega\,t-\mathbf{k}\cdot\mathbf{x}\), with \(k^{2}=\omega^{2}-\mathbf{k}^{2}\), and \(B^{ab}=\left(0,-\epsilon_{ijk}B_{0k}\right)\) sets a constant and uniform MBF. One may check that each solution of (31) satisfies the usual wave equation of the commutative case independently. Thereby, if one substitutes just the PW solution, the wave equation is \(M_{ab}A^{b}=k^{2}\,\tilde{A}_{a}=0\), which yields the usual photon DR \(\omega(\mathbf{k})=|\mathbf{k}|\). For \(A_{a}=-B_{ab}\,x^{b}/2\), with \(B_{ab}\) any uniform and constant BF, one may check that it is exactly the same of the Maxwell's ED. Using the solution (31) in the definition of the usual tensor, we write the electromagnetic field of the PW, \(F^{ab}(x)=\tilde{F}^{ab}\,\cos(k\cdot x)+B^{ab}\), where \(\tilde{F}^{ab}=k^{a}\,\tilde{A}^{b}-k^{b}\,\tilde{A}^{a}=(-\tilde{E}_{i},- \epsilon_{ijk}\tilde{B}_{k})\) are the electric and magnetic (constants and uniforms) amplitudes.
In the case of a purely magnetic background with a spatial NC, _i.e._, \(\theta^{0i}=0\), and \(\theta^{ij}=\epsilon^{ijk}\,\tilde{\theta}^{k}\), we substitute the solution (31) in the Eqs. (15a), (15b), (15c) and (15d) (with no sources), we obtain the relations
\[k_{i}\,\epsilon_{ij}\,\tilde{E}_{j}=-\frac{1}{2}\,(\mathbf{k} \times\widetilde{\mathbf{\theta}})\cdot(\mathbf{B}_{0}\times\mathbf{k})\,\tilde{V} \tan(k\cdot x)\;, \tag{32a}\] \[\mathbf{k}\cdot\tilde{\mathbf{B}}=\frac{1}{2}(\widetilde{\mathbf{ \theta}}\cdot\tilde{\mathbf{B}})(\mathbf{k}\cdot\mathbf{B}_{0})\] \[-\frac{1}{2}\,\tilde{\mathbf{A}}\cdot(\mathbf{k}\times\widetilde{ \mathbf{\theta}})(\mathbf{k}\cdot\mathbf{B}_{0})\tan(k\cdot x)\;,\] (32b) \[\tilde{\mathbf{B}}=\frac{1}{\omega}\,(\mathbf{k}\times\tilde{ \mathbf{E}})+\frac{1}{2\omega}\left[(\widetilde{\mathbf{\theta}}\times\tilde{ \mathbf{E}})(\mathbf{k}\cdot\mathbf{B}_{0})\right.\] \[\left.-(\mathbf{k}\times\tilde{\mathbf{E}})(\widetilde{\mathbf{ \theta}}\cdot\mathbf{B}_{0})\right]\left[1+\tan(k\cdot x)\right]\;,\] (32c) \[\tilde{E}_{i}=-\omega^{-1}\,\epsilon_{ijk}\,k_{l}\,(\mu^{-1})_{jl} \,\tilde{B}_{k}\;, \tag{32d}\]
where \(\epsilon_{ij}\) is the electric permittivity, and \((\mu^{-1})_{ij}\) is the inverse of the magnetic permeability tensor. Inverting the second tensor on (32d), we obtain
\[\epsilon_{ij} = \delta_{ij}+\frac{1}{2}\,B_{0i}\,\widetilde{\theta}_{j}\;, \tag{33a}\] \[\mu_{ij} = \delta_{ij}\left[1+\frac{1}{2}\,(\mathbf{B}_{0}\cdot\widetilde{\bm {\theta}})\right]-\frac{1}{2}\,\widetilde{\theta}_{i}\,B_{0j}\;. \tag{33b}\]
The eigenvalues of the permittivity tensor are \(\lambda_{1e}=\lambda_{2e}=1\) (two degenerate eigenvalues) and \(\lambda_{3e}=1+(\mathbf{B}_{0}\cdot\widetilde{\mathbf{\theta}})/2\). For the permeability tensor, we obtain the eigenvalues : \(\lambda_{1\mu}=1\) and \(\lambda_{2\mu}=\lambda_{3\mu}=1+(\mathbf{B}_{0}\cdot\widetilde{\mathbf{\theta}})/2\). The vacuum limit is so recovered when \(\widetilde{\mathbf{\theta}}\to 0\), or when \(\mathbf{B}_{0}\to 0\). Therefore, the NC space behaves like a material medium, in which the permittivity and permeability both are positive for leading order in \(\widetilde{\theta}\)-parameter.
Since \(\tilde{\mathbf{\theta}}\neq 0\), the properties of the wave propagation are such that the wave amplitudes are not, in general, perpendiculars to the propagation direction \(\mathbf{k}\). Furthermore, the relations (32a), (32b) and (32c) depend on the space-time coordinates through the tangent function. The time-average of these relations are read below
\[\langle\tilde{\mathbf{B}}\rangle=\frac{1}{\omega}\,(\mathbf{k} \times\tilde{\mathbf{E}})+\frac{1}{2\omega}\left[(\widetilde{\mathbf{\theta}}\times \tilde{\mathbf{E}})(\mathbf{k}\cdot\mathbf{B}_{0})\right.\] \[\left.-(\mathbf{k}\times\tilde{\mathbf{E}})(\widetilde{\mathbf{\theta}} \cdot\mathbf{B}_{0})\right]\;, \tag{34a}\] \[\langle\mathbf{k}\cdot\tilde{\mathbf{E}}\rangle=-\frac{1}{2}( \widetilde{\mathbf{\theta}}\cdot\tilde{\mathbf{E}})(\mathbf{k}\cdot\mathbf{B}_{0})\;,\] (34b) \[\langle\mathbf{k}\cdot\tilde{\mathbf{B}}\rangle=\frac{1}{2}( \widetilde{\mathbf{\theta}}\cdot\tilde{\mathbf{B}})(\mathbf{k}\cdot\mathbf{B}_{0})\;, \tag{34c}\]
where \(\langle{\bf k}\cdot\tilde{\bf E}\rangle=\langle{\bf k}\cdot\tilde{\bf B}\rangle=0\) in the situation in which the propagation \({\bf k}\) is perpendicular to the MBF.
One may observe also that, if we consider the particular situation of \(\widetilde{\boldsymbol{\theta}}=\ell^{2}\,\hat{\bf k}\), where \(\ell\) is a length scale, the NC contribution is null in the relation (34a), and if \({\bf k}\) is perpendicular to \({\bf B}_{0}\) (or the external \({\bf B}_{0}\) field is turned off), the divergence of the electric and magnetic amplitudes are equal to the commutative case.
In Maxwell's ED, the superposition (31) yields again the wave equation \(k^{2}A_{a}=0\). However, in the canonical PGT, an interference term emerges due to the NC contribution. Substituting the solution (31) in the wave equation (30), we obtain \(M_{ab}\tilde{A}^{b}=0\), where the wave matrix \(M_{ab}\) in the momentum space is
\[M_{ab}=\left(-k_{c}\,k^{c}+\theta^{cd}\,B_{cc}\,k^{e}\,k_{d}\right)\eta_{ab}- \frac{1}{4}\,\theta^{cd}\,B_{bc}\,k_{d}\,k_{a}\;. \tag{35}\]
The DRs are calculated by the non-trivial solution of \(M_{ab}\tilde{A}^{b}=0\), in which the determinant of \(M_{ab}\) is null, _i.e._, \(\det(M_{ab})=0\). Also in the case of a magnetic MBF in a spatial NC, _i.e._, the determinant in the linear \(\theta\)-approximation is,
\[\det(M_{ab})=(\omega^{2}-{\bf k}^{2})^{3}\left[\,{\bf k}^{2}-\omega^{2}+\frac{ 15}{4}\,({\bf B}_{0}\times{\bf k})\cdot(\widetilde{\boldsymbol{\theta}}\times {\bf k})\,\right]. \tag{36}\]
The solutions of (36) yield the usual photon DR, namely, \(\omega_{1}({\bf k})=|{\bf k}|\), and the DR modified by the spatial NC
\[\omega_{2}({\bf k}) = |{\bf k}|\,\sqrt{1+\frac{15}{4}\,({\bf B}_{0}\times\hat{\bf k}) \cdot(\widetilde{\boldsymbol{\theta}}\times\hat{\bf k})} \tag{37}\] \[\simeq |{\bf k}|\left[1+\frac{15}{8}\,({\bf B}_{0}\times\hat{\bf k}) \cdot(\widetilde{\boldsymbol{\theta}}\times\hat{\bf k})\right]\,,\]
where we have used a small \(\widetilde{\theta}\)-parameter.
The correspondent group velocity is read below :
\[{\bf V}_{g} = \frac{{\bf k}}{\omega}\,\frac{8\,({\bf k}^{2}-\omega^{2})+\frac{ 45}{2}\,({\bf B}_{0}\times{\bf k})\cdot(\widetilde{\boldsymbol{\theta}}\times {\bf k})}{8\,({\bf k}^{2}-\omega^{2})+\frac{45}{2}\,({\bf B}_{0}\times{\bf k })\cdot(\widetilde{\boldsymbol{\theta}}\times{\bf k})} \tag{38}\] \[+\frac{{\bf k}}{\omega}\,\frac{\frac{15}{2}({\bf k}^{2}-\omega^{2 })({\bf B}_{0}\cdot\widetilde{\boldsymbol{\theta}})}{8\,({\bf k}^{2}-\omega^{ 2})+\frac{45}{2}\,({\bf B}_{0}\times{\bf k})\cdot(\widetilde{\boldsymbol{ \theta}}\times{\bf k})}\] \[+\frac{15}{4}\,\frac{(\omega^{2}-{\bf k}^{2})[\,({\bf B}_{0} \cdot{\bf k})\,\widetilde{\boldsymbol{\theta}}+(\widetilde{\boldsymbol{ \theta}}\cdot{\bf k})\,{\bf B}_{0}\,]}{8\,({\bf k}^{2}-\omega^{2})+\frac{45}{2 }\,({\bf B}_{0}\times{\bf k})\cdot(\widetilde{\boldsymbol{\theta}}\times{\bf k })}\,.\]
Evaluating (38) in the DRs \(\omega_{1}=|{\bf k}|\) and \(\omega_{2}\) of (37), the first case is the same result from usual ED, \(\left.{\bf V}_{g}\right|_{\omega_{1}}=\hat{\bf k}\). The second DR (37) yields the group velocity,
\[{\bf V}_{g}\right|_{\omega_{2}} \simeq \hat{\bf k}\left[1-\frac{15}{4}({\bf B}_{0}\times{\bf k})\cdot( \widetilde{\boldsymbol{\theta}}\times{\bf k})\right] \tag{39}\] \[-\frac{15}{8}\,({\bf B}_{0}\cdot{\bf k})\,\widetilde{\boldsymbol {\theta}}-\frac{15}{8}\,(\widetilde{\boldsymbol{\theta}}\cdot{\bf k})\,{\bf B}_ {0}\,.\]
Notice that in both results, the usual photon DR and the group velocity are recovered when the MBF is null, _i.e._, \({\bf B}_{0}\to 0\). Both the results depend on the direction that the background vector \({\bf B}_{0}\) and the NC parameter \(\widetilde{\theta}\) do with the wave propagation direction \(\hat{\bf k}\). In this sense, the refractive index of this medium also changes with the direction of three vectors \(\widetilde{\boldsymbol{\theta}}\), \(\hat{\bf k}\) and \({\bf B}_{0}\). Furthermore, the group velocity (39) shows components in both directions \(\widetilde{\boldsymbol{\theta}}\) and \({\bf B}_{0}\). Thereby, the NC space is a medium that changes the propagation direction of the group velocity.
## V The energy density and the Poynting vector for plane wave solutions in an external magnetic field
The next step is the investigation of the plane wave solutions under an external magnetic field when applied to the components of the EMT. If one considers only the PW solutions for the gauge potential, _i.e._, \(A_{a}(x)=\tilde{A}_{a}\,\sin(k\cdot x)\), the canonical Poisson EMT is unchanged by the NC parameter, due to the contraction of the antisymmetric \(\theta\)-parameter with the product of two wave 4-vectors. Therefore, the analysis in the presence of an uniform and constant electromagnetic BF is needed to emerge the NC contribution. In a constant and uniform electromagnetic field, the gauge potential is written as \(A_{a}(x)=-B_{ab}\,x^{b}/2\), and one may check that the EMT now has corrections in the \(\theta\)-parameter in the presence of the BF. Thereby, we substitute the superposition (31) in the EMT to obtain the result
\[\mathcal{T}^{ab}|_{A}=\left(\,\tilde{F}^{ac}\,\tilde{F}_{c}^{\;\;b }+\eta^{ab}\,\frac{1}{4}\,\tilde{F}_{cd}\,\tilde{F}^{cd}\,\right)\cos^{2}(k \cdot x)\] \[+2\left(\tilde{F}^{ac}B_{c}^{\;\;b}+\frac{1}{4}\eta^{ab}\tilde{F }_{cd}B^{cd}\right)\cos(k\cdot x)\] \[+B^{ac}B_{c}^{\;\;b}+\frac{1}{4}\eta^{ab}B_{cd}B^{cd}\] \[-\,\theta^{ef}\Bigg{[}\left(k_{e}\tilde{A}_{c}\tilde{F}^{ac}B^{b \;}+k_{f}B_{ce}\tilde{A}^{a}\tilde{F}^{bc}\right.\] \[\left.-\eta^{ab}k_{e}\tilde{A}_{c}\tilde{F}^{cd}B_{df}\right)\cos^{ 2}(k\cdot x)\] \[-\frac{1}{4}\left[2k_{e}\left(B_{cf}\tilde{A}^{a}B^{bc}-\tilde{A} _{c}B^{ac}B^{b\;}_{f}\right)+B_{ce}\tilde{F}^{ac}B^{b\;}_{f}\right.\] \[\left.+\eta^{ab}\left(2k_{e}\tilde{A}_{c}B^{cd}+B_{ce}\tilde{F}^{ cd}\right)B_{df}\right]\cos(k\cdot x)\] \[-\frac{1}{8}B_{ce}\left(\,4B^{ac}B^{b\;}_{f}+\eta^{ab}\,B^{cd}\,B_ {df}\right)\Bigg{]}\;. \tag{40}\]
Considering the case of a purely MBF, \(B_{ab}=(0,-\epsilon_{ijk}\,B_{0k})\), the last terms in (40) does not contain any information on the propagation. Thereby, they are discarded from now on. Using the superposition (31), the energy density and the components of the Poynting
vector may be written, respectively, as
\[u|_{A}=\left[\frac{1}{2}\left(\tilde{\mathbf{E}}^{2}+\tilde{ \mathbf{B}}^{2}\right)+(\tilde{\mathbf{A}}\cdot\mathbf{B}_{0})(\tilde{B}\times \mathbf{k})\cdot\widetilde{\boldsymbol{\theta}}\right.\] \[\left.+(\mathbf{B}_{0}\cdot\tilde{\mathbf{B}})(\mathbf{k}\times \tilde{\mathbf{A}})\cdot\widetilde{\boldsymbol{\theta}}\,\right]\cos^{2}(k \cdot x)\] \[\left.+\frac{1}{4}\,\left[(\tilde{\mathbf{A}}\cdot\mathbf{B}_{0 })(\mathbf{B}_{0}\times\mathbf{k})\cdot\widetilde{\boldsymbol{\theta}}+ \mathbf{B}_{0}^{2}\,(\mathbf{k}\times\tilde{\mathbf{A}})\cdot\widetilde{ \boldsymbol{\theta}}\right.\right.\] \[\left.\left.-(\tilde{\mathbf{B}}\cdot\mathbf{B}_{0})( \widetilde{\boldsymbol{\theta}}\cdot\mathbf{B}_{0})+4\tilde{\mathbf{B}}\cdot \mathbf{B}_{0}\right]\cos(k\cdot x)\;,\right. \tag{41}\]
and
\[S^{i}|_{A}=(\tilde{\mathbf{E}}\times\tilde{\mathbf{B}})^{i}\, \cos^{2}(k\cdot x)+(\tilde{\mathbf{E}}\times\mathbf{B}_{0})^{i}\cos(k\cdot x)\] \[-\frac{1}{2}\,\left[k^{i}(\widetilde{\boldsymbol{\theta}}\cdot \mathbf{B}_{0})(\tilde{\mathbf{A}}\cdot\tilde{\mathbf{E}})-\widetilde{ \boldsymbol{\theta}}^{i}(\tilde{\mathbf{A}}\cdot\tilde{\mathbf{E}})(\mathbf{ k}\cdot\mathbf{B}_{0})\right.\] \[\left.+B_{0}^{i}\,\tilde{V}(\mathbf{k}\times\widetilde{ \boldsymbol{\theta}})\cdot\mathbf{B}_{0}-\tilde{V}\,\mathbf{B}_{0}^{2}\,( \mathbf{k}\times\widetilde{\boldsymbol{\theta}})^{i}\right.\] \[\left.+\tilde{A}^{i}\,(\mathbf{B}_{0}\times\tilde{\mathbf{E}}) \cdot(\mathbf{k}\times\widetilde{\boldsymbol{\theta}})\right]\cos^{2}(k\cdot x)\] \[-\frac{1}{2}\,\left[(\mathbf{B}_{0}\cdot\mathbf{k})(\widetilde{ \boldsymbol{\theta}}\times\mathbf{B}_{0})^{i}\,\tilde{V}-(\mathbf{B}_{0}\cdot \widetilde{\boldsymbol{\theta}})(\mathbf{k}\times\mathbf{B}_{0})^{i}\,\tilde{V}\right.\] \[\left.+\frac{1}{2}(\mathbf{B}_{0}\cdot\widetilde{\boldsymbol{ \theta}})(\tilde{\mathbf{E}}\times\mathbf{B}_{0})^{i}\right]\cos(k\cdot x)\;. \tag{42}\]
Taking the time average up to first order on the NC parameter, we obtain
\[\langle u\rangle|_{A} \approx \frac{1}{4}\,\tilde{\mathbf{E}}^{2}\left[1+\frac{\mathbf{k}^{2} }{\omega^{2}}-\frac{1}{\omega^{2}}(\mathbf{k}\times\mathbf{B}_{0})\cdot( \mathbf{k}\times\widetilde{\boldsymbol{\theta}})\right] \tag{43}\] \[+\frac{1}{4\omega}[\tilde{\mathbf{E}}\cdot(\mathbf{B}_{0}\times \mathbf{k})][\tilde{\mathbf{A}}\cdot(\widetilde{\boldsymbol{\theta}}\times \mathbf{k})]\] \[+\frac{\mathbf{k}^{2}}{2\omega}(\tilde{\mathbf{A}}\cdot\mathbf{B} _{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}})\;,\]
and
\[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2\omega}\,\mathbf{k}\,(1- \widetilde{\boldsymbol{\theta}}\cdot\mathbf{B}_{0})+\frac{\tilde{\mathbf{E}}^ {2}}{4\omega}\,(\mathbf{k}\cdot\mathbf{B}_{0})\,\widetilde{\boldsymbol{\theta}} \tag{44}\] \[-\frac{1}{4}\,\left[\,\left[\mathbf{k}\,(\widetilde{\boldsymbol{ \theta}}\cdot\mathbf{B}_{0})-\widetilde{\boldsymbol{\theta}}\,(\mathbf{k}\cdot \mathbf{B}_{0})\right](\tilde{\mathbf{A}}\cdot\tilde{\mathbf{E}})\right.\] \[\left.+\tilde{V}\mathbf{B}_{0}\,(\mathbf{k}\times\widetilde{ \boldsymbol{\theta}})\cdot\mathbf{B}_{0}-\tilde{V}\,\mathbf{B}_{0}^{2}\,( \mathbf{k}\times\widetilde{\boldsymbol{\theta}})\right.\] \[\left.+\tilde{\mathbf{A}}\,(\mathbf{B}_{0}\times\tilde{ \mathbf{E}})\cdot(\mathbf{k}\times\widetilde{\boldsymbol{\theta}})\,\right]\;.\]
We will analyze some cases of interest for the NC electromagnetic tensor when the vectors \(\mathbf{k}\), \(\widetilde{\boldsymbol{\theta}}\) and \(\mathbf{B}_{0}\) are parallel, or perpendiculars among themselves.
1. When \(\mathbf{k}\), \(\widetilde{\boldsymbol{\theta}}\) and \(\mathbf{B}_{0}\) are parallel, the energy density and the Poynting vector are given by \[\langle u\rangle|_{A}\approx\frac{1}{4}\,\tilde{\mathbf{E}}^{2}\left(1+\frac{ \mathbf{k}^{2}}{\omega^{2}}\right)+\frac{\mathbf{k}^{2}}{2\omega}(\tilde{ \mathbf{A}}\cdot\mathbf{B}_{0})(\tilde{\mathbf{E}}\cdot\widetilde{ \boldsymbol{\theta}})\;,\] (45) and \[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2\omega}\,(1-\widetilde{\theta}B_{0 })\mathbf{k}\,+\frac{\tilde{\mathbf{E}}^{2}}{4\omega}\,(kB_{0})\,\widetilde{ \boldsymbol{\theta}}\] (46) \[-\frac{1}{4}\,B_{0}(\tilde{\mathbf{A}}\cdot\tilde{\mathbf{E}}) \left(\widetilde{\theta}\mathbf{k}\,-k\widetilde{\boldsymbol{\theta}}\right)\;,\]
where we have used a small \(\widetilde{\theta}\)-parameter. It is interesting to note that in the case of \(\widetilde{\boldsymbol{\theta}}=\ell^{2}\,\hat{\mathbf{k}}\) mentioned previously, the contribution of \(\tilde{\mathbf{A}}\)-amplitude is null in the Poynting vector :
\[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2\omega}\,\mathbf{k}\left(1-\frac{ \ell^{2}B_{0}}{2}\right)\;, \tag{47}\]
and the energy density, when we rescale the terms, is positive definite for a small NC
\[\langle u\rangle|_{A} \approx \frac{1}{4}\,\tilde{\mathbf{E}}^{2}\left(1+\frac{\mathbf{k}^{2}}{ \omega^{2}}\right)+\frac{\mathbf{k}^{2}}{2\omega}(\tilde{\mathbf{A}}\cdot \mathbf{B}_{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}}). \tag{48}\]
We substitute now the DRs \(\omega_{1}(\mathbf{k})=|\mathbf{k}|\), and \(\omega_{2}(\mathbf{k})\) given by Eq. (37). The results are the same for both frequencies, since the contribution of \(\widetilde{\theta}\) is negligible for \(\omega_{2}(\mathbf{k})\), we have the expressions
\[\langle u\rangle|_{A} \approx \frac{1}{2}\,\tilde{\mathbf{E}}^{2}+\frac{k}{2}(\tilde{\mathbf{A} }\cdot\mathbf{B}_{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}}), \tag{49a}\] \[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2}\,\hat{\mathbf{k}}\left(\,1-\frac{ \ell^{2}B_{0}}{2}\right)\;. \tag{49b}\]
2. The case of \(\mathbf{k}\), \(\mathbf{B}_{0}\) and \(\widetilde{\boldsymbol{\theta}}\) perpendiculars among themselves yields the results \[\langle u\rangle|_{A} \approx \frac{1}{4}\,\tilde{\mathbf{E}}^{2}\left(1+\frac{\mathbf{k}^{2}}{ \omega^{2}}\right)+\frac{\mathbf{k}^{2}}{2\omega}(\tilde{\mathbf{A}}\cdot \mathbf{B}_{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}})\] (50) \[+\frac{1}{4\omega}[\tilde{\mathbf{E}}\cdot(\mathbf{B}_{0}\times \mathbf{k})][\tilde{\mathbf{A}}\cdot(\widetilde{\boldsymbol{\theta}}\times \mathbf{k})]\;,\]
and
\[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2\omega}\,\mathbf{k}\,(1- \widetilde{\boldsymbol{\theta}}\cdot\mathbf{B}_{0})+\frac{\tilde{\mathbf{E}}^{2}}{4 \omega}\,(\mathbf{k}\cdot\mathbf{B}_{0})\,\widetilde{\boldsymbol{\theta}}\] (51) \[-\frac{1}{4}\,\left[\,\left[\mathbf{k}\,(\widetilde{\boldsymbol{ \theta}}\cdot\mathbf{B}_{0})-\widetilde{\boldsymbol{\theta}}\left(\mathbf{k} \cdot\mathbf{B}_{0}\right)\right]\,(\tilde{\mathbf{A}}\cdot\tilde{\mathbf{E}})\right.\] \[\left.+\tilde{V}\mathbf{B}_{0}\,(\mathbf{k}\times\tilde{\boldsymbol{ \theta}})\cdot\mathbf{B}_{0}-\tilde{V}\,\mathbf{B}_{0}^{2}\,(\mathbf{k}\times \widetilde{\boldsymbol{\theta}})\right.\] \[\left.+\tilde{\
3. The third case of interest is when the MBF is parallel to the NC parameter, but it remains perpendicular to the wave propagation direction. Under these conditions, we obtain \[\langle u\rangle|_{A} \approx \frac{1}{4}\,\tilde{\mathbf{E}}^{2}\left[1+\frac{\mathbf{k}^{2}}{ \omega^{2}}\left(1-\tilde{\theta}B_{0}\right)\right]\] (54) \[+\frac{1}{4\omega}[\tilde{\mathbf{E}}\cdot(\mathbf{B}_{0}\times \mathbf{k})][\tilde{\mathbf{A}}\cdot(\widetilde{\boldsymbol{\theta}}\times \mathbf{k})]\] \[+\frac{\mathbf{k}^{2}}{2\omega}(\tilde{\mathbf{A}}\cdot\mathbf{B} _{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}})\;,\] and \[\langle\mathbf{S}\rangle|_{A} \approx \frac{\tilde{\mathbf{E}}^{2}}{2\omega}\left(1-\tilde{\theta}B_{0 }\right)\mathbf{k}+\frac{1}{4}B_{0}\left[\tilde{V}B_{0}\left(\mathbf{k}\times \widetilde{\boldsymbol{\theta}}\right)\right.\] (55) \[\left.-\tilde{\theta}\,\tilde{\mathbf{E}}\times(\mathbf{k}\times \tilde{\mathbf{A}})\right]\;.\] We observe that, in this case, the energy flow has a correction on the NC parameter, which can be controllable by the MBF. If the direction of \(\mathbf{B}_{0}\) is approximate to the wave propagation direction, when rescales the terms, the energy density (54) is positive definite for a small NC. Substituting the DRs \(\omega_{1}(\mathbf{k})\) and \(\omega_{2}(\mathbf{k})\), we will obtain different results for each frequency, since the NC contribution is not null for the second wave frequency. This fact will play a fundamental role on the next section, resulting on the birefringence phenomenon. Thus, the correspondent results are read as, \[\langle u\rangle|_{A,\omega_{1}} \approx \frac{1}{2}\,\tilde{\mathbf{E}}^{2}+\frac{k}{2}(\tilde{\mathbf{A} }\cdot\mathbf{B}_{0})(\tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}})\] (56a) \[+\frac{1}{4}[\tilde{\mathbf{E}}\cdot(\mathbf{B}_{0}\times\hat{ \mathbf{k}})][\tilde{\mathbf{A}}\cdot(\widetilde{\boldsymbol{\theta}}\times \mathbf{k})]\;,\] \[\langle\mathbf{S}\rangle|_{A,\omega_{1}} \approx \frac{\tilde{\mathbf{E}}^{2}}{2}\left(1-\tilde{\theta}B_{0} \right)\hat{\mathbf{k}}+\frac{1}{4}\,B_{0}\bigg{[}\tilde{V}B_{0}(\mathbf{k} \times\widetilde{\boldsymbol{\theta}})\] (56b) \[-\widetilde{\theta}\,\tilde{\mathbf{E}}\times(\mathbf{k}\times \tilde{\mathbf{A}})\bigg{]}\;,\] and \[\langle u\rangle|_{A,\omega_{2}} \approx \frac{1}{2}\,\tilde{\mathbf{E}}^{2}\left(1-\frac{23}{16}\widetilde {\theta}B_{0}\right)+\frac{k}{2}(\tilde{\mathbf{A}}\cdot\mathbf{B}_{0})( \tilde{\mathbf{E}}\cdot\widetilde{\boldsymbol{\theta}})\] (57a) \[+\frac{1}{4}[\tilde{\mathbf{E}}\cdot(\mathbf{B}_{0}\times\hat{ \mathbf{k}})][\tilde{\mathbf{A}}\cdot(\widetilde{\boldsymbol{\theta}}\times \mathbf{k})]\;,\] \[\langle\mathbf{S}\rangle|_{A,\omega_{2}} \approx \frac{\tilde{\mathbf{E}}^{2}}{2}\,\left(1-\frac{23}{8}\widetilde {\theta}B_{0}\right)\hat{\mathbf{k}}+\frac{1}{4}\,B_{0}\left[\tilde{V}B_{0}( \mathbf{k}\times\widetilde{\boldsymbol{\theta}})\right.\] (57b) \[\left.-\widetilde{\theta}\,\tilde{\mathbf{E}}\times(\mathbf{k} \times\tilde{\mathbf{A}})\right]\,.\]
## VI Non-commutativity as a material medium and birefringence phenomenon
In this section, we investigate the possible birefringence phenomenon that emerges from canonical PGT. We discuss the non-linear contribution of the NC space-time similar to the refs. [30]-[33]. The non-trivial solutions from the wave equation (35) are given by the null determinant of the matrix \(M_{ab}\), that leads to the DRs of the theory. The results depend on the direction of vectors \(\mathbf{k}\), \(\mathbf{B}_{0}\) and \(\widetilde{\boldsymbol{\theta}}\), in the case of purely spatial NC parameter. Notice that, if \(\mathbf{k}\) is parallel to \(\mathbf{B}_{0}\), the second DR is \(\omega_{2\parallel}(\mathbf{k})=|\mathbf{k}|\). The simplest case in which the \(\widetilde{\theta}\)-parameter has contribution is the third on the last section. Thus, the DR (37) is reduced to
\[\omega_{2\perp}(\mathbf{k})\simeq|\mathbf{k}|\left(\,1+\frac{15}{8}\,B_{0}\, \widetilde{\theta}\,\right)\;. \tag{58}\]
We define the correspondent refractive index as \(n_{\parallel}=|\mathbf{k}|/\omega_{2\parallel}=1\), and \(n_{\perp}=|\mathbf{k}|/\omega_{2\perp}\) in these situations. Thereby, the birefringence of \(\mathbf{B}_{0}\) in relation to \(\mathbf{k}\) is defined by the difference
\[\Delta n_{B_{0}}=n_{\parallel}-n_{\perp}\simeq\frac{15}{8}\,B_{0}\,\widetilde{ \theta}\;. \tag{59}\]
Using the bound from PVLAS experiment \(\Delta n_{B}/B_{ext}^{2}=(\,19\,\pm\,27)\times 10^{-24}\,\mathrm{T}^{-2}\) for an external magnetic field of \(B_{ext}=2.5\,\mathrm{T}\)[34], we obtain that
\[\sqrt{\widetilde{\theta}}\simeq 1.92\times(10\,\mathrm{TeV})^{-1}\;. \tag{60}\]
This result is consistent with the bounds on the \(\theta\)-parameter in the NC QED discussed in the ref. [35].
## VII Conclusions
We study the properties of the NC ED based on the canonical PGT approach. Since is known in the literature, the PGT is the semi-classical limit of the full NC gauge theory, where the NC parameter depends on the space-time coordinates. Thereby, the formulation of the canonical PGT defines a closed gauge algebra, whose the correspondent Lagrangian respects the gauge symmetry principle, and the dynamical equations lead to conservation laws in which physical quantities are affected by the NC space-time. Thus, the analysis of physical results, as the PW propagation, motivate us to understand the full NC picture. In this sense, the goal of this paper is the development of the field equations, and the conservation laws of the canonical PGT, where the NC parameter does not depend on the space-time coordinates. The symmetric and gauge-invariant EMT is so calculated in the linear approximation of the NC parameter. We obtain the conserved components, as the energy density, and the Poynting vector.
Posteriorly, we investigate the PW propagation in the presence of an external magnetic field (uniform and constant). Therefore, under the PW solution, we obtain some properties of the wave propagation, as the DRs, and the group velocity of the wave in this NC medium. The group velocity vector has contributions of the \(\widehat{\theta}\)-direction, and also of the magnetic background direction in the wave propagation. The NC space behaves like a material in which the permittivity and permeability tensors yield the medium characteristics that depend on the NC parameter and on the magnetic background. When the NC parameter is turned off, all these new effects go to zero, and the known results of the Maxwell's ED are recovered.
In the final part of the paper, we substitute the PW solution summed to the MBF in the conserved components of the EMT. We calculate the NC contribution for the energy density and the Poynting vector in terms of the three vectors : \(\mathbf{k}\) (wave propagation direction), \(\mathbf{B_{0}}\) (magnetic background field) and \(\widehat{\boldsymbol{\theta}}\) (vector for a spatial NC), that consequently, changes the direction of the energy flux density for the PW. For end, we examine briefly a birefringence phenomenon associated with the directions of \(\mathbf{k}\) and \(\mathbf{B}_{0}\). Using the known result from PVLAS experiment for birefringence [34], we obtain the estimative of \(\sqrt{\widehat{\theta}}\simeq 1.92\times(10\,\mathrm{TeV})^{-1}\) for the spatial NC parameter.
The perspective for future projects is to lead with the NC case where the parameter depends on the space-time coordinates, studying the fermionic case [38] and the spin interaction [39]. There are some examples already treated on PGT that can be consistent with the NC ED, like the \(\kappa\)-Minkowski space-time [40], which may be used for forthcoming investigations.
**Acknowledgments**: We are grateful to Vladislav Kupriyanov for early discussions on this subject and the valuable remarks. This study was partially financed by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES, Brazil) - Finance Code 001.
|
2308.12199 | Towards Real-Time Analysis of Broadcast Badminton Videos | Analysis of player movements is a crucial subset of sports analysis. Existing
player movement analysis methods use recorded videos after the match is over.
In this work, we propose an end-to-end framework for player movement analysis
for badminton matches on live broadcast match videos. We only use the visual
inputs from the match and, unlike other approaches which use multi-modal sensor
data, our approach uses only visual cues. We propose a method to calculate the
on-court distance covered by both the players from the video feed of a live
broadcast badminton match. To perform this analysis, we focus on the gameplay
by removing replays and other redundant parts of the broadcast match. We then
perform player tracking to identify and track the movements of both players in
each frame. Finally, we calculate the distance covered by each player and the
average speed with which they move on the court. We further show a heatmap of
the areas covered by the player on the court which is useful for analyzing the
gameplay of the player. Our proposed framework was successfully used to analyze
live broadcast matches in real-time during the Premier Badminton League 2019
(PBL 2019), with commentators and broadcasters appreciating the utility. | Nitin Nilesh, Tushar Sharma, Anurag Ghosh, C. V. Jawahar | 2023-08-23T15:38:26Z | http://arxiv.org/abs/2308.12199v1 | # Towards Real-Time Analysis of Broadcast
###### Abstract
Analysis of player movements is a crucial subset of sports analysis. Existing player movement analysis methods use recorded videos after the match is over. In this work, we propose an end-to-end framework for player movement analysis for badminton matches on live broadcast match videos. We only use the visual inputs from the match and, unlike other approaches which use multi-modal sensor data, our approach uses only visual cues. We propose a method to calculate the on-court distance covered by both the players from the video feed of a live broadcast badminton match. To perform this analysis, we focus on the gameplay by removing replays and other redundant parts of the broadcast match. We then perform player tracking to identify and track the movements of both players in each frame. Finally, we calculate the distance covered by each player and the average speed with which they move on the court. We further show a heatmap of the areas covered by the player on the court which is useful for analyzing the gameplay of the player.
Our proposed framework was successfully used to analyze live broadcast matches in real-time during the Premier Badminton League 2019 (PBL 2019), with commentators and broadcasters appreciating the utility.
## I Introduction
Sports analysis is one of the most challenging tasks in computer vision and several different approaches to sports analysis have been proposed. Sport analyses include match summarization [1], event prediction [2], ball tracking [3], structured analysis of the game [4] etc. This kind of analysis has proven to be useful in areas such as training and coaching,1 and has also been able to provide help to the referee during the game. Apart from the types of analysis mentioned above, research topics like analysis and prediction of how groups of players move [5], automatic identification of key stages [6] in a game for gathering statistics or automating the control of broadcast cameras [7] etc. have also been explored. Game statistics are important to both viewers as well as the players and their coaches. For viewers, having access to accurate statistics improves viewing experience and helps them understand the more technical aspects of the sport like complex strategies and playing styles. It also aids the players and their coaches to formulate better game strategies and improve performance. Analysing an opponent's gameplay also helps players to adapt their style of play to suit the particular opponent. In badminton, games statistics include number of wins, unforced errors, successful smashes, distance covered, speed of the serve and so on.
Footnote 1: [https://www.newscientist.com/ai-football](https://www.newscientist.com/ai-football)
While considerable research has been published regarding badminton analysis, the one common limitation of all existing methods is that they are applied post-factum on previously recorded match videos. This prevents the use of these methods to show statistics during a live broadcast match. The ability to compute these statistics in real-time from a live broadcast match not only makes the game more exciting for viewers but also allows the players to dynamically modify their strategies mid-game. However, broadcast videos are captured from multiple viewpoints for the best viewing experience and their 'unstructured' nature along with the rapid movement and complex human pose and motion makes real-time analysis a complex task. Moreover, broadcast delay for a typical badminton match is less than 15 seconds, usually 7 seconds
Fig. 1: An overview of the distance travelled by the player from the start of a set. Player in blue bounding box refers to the “top-player” and player in yellow bounding box refers to the “bottom-player”. Dot at the center-bottom of the player’s bounding box represents their current location. The “Rally Duration” is the time length of a particular rally. Four such rallies are depicted for reference. _(best viewed in color)_
2. Many of the existing analysis methods are constrained by these challenges. Our main focus in this work is to provide real-time analysis framework for the broadcast videos which is expected to generalize to almost all the badminton games captured from broadcast camera view.
Footnote 2: [https://en.wikipedia.org/wiki/Broadcast_delay](https://en.wikipedia.org/wiki/Broadcast_delay)
In this work, we introduce an approach which generates heatmaps of player movements, calculates distance run by players and the average speed of payer movements for the game of badminton (figure 1). We perform point segmentation, player detection and homography based techniques to compute the above defined metrics.
The major contribution of this paper is as follows:
1. We propose an end-to-end framework (figure 3) to analyze live broadcast match videos, the components of which can be trained using pre-recorded badminton matches. Unlike previous approaches, our method uses visual cues only and does not rely on any hardware setup.
2. Using recent advancements in object detection, we predict players' location, their heatmap across the court and distance covered by each player during gameplay. We do these analyses for both singles and doubles matches.
3. We tested our framework in a real-time scenario and analyzed live broadcast matches that were conducted as a part of the Premier Badminton League 2019.
4. We introduce two new datasets, one each for singles and doubles matches covering a wide range of broadcasting camera angles, court and lighting conditions, and featuring several different players. We also introduce a set of 25 rallies badminton match video of our own recording having the ground truth data for the distance covered by the players in each rally.
## II Related Work
**Sports analysis and Player Localization:** Traditionally, work in analyzing sports videos has focused on either tracking players [8] or balls [3] to analyze game formations or skill level of individual players. Among all sports, racket sports has received a lot of attention with applications in video summarization and highlight generation [9]. Yoshikawa et al. [10] implemented serve scene detection for badminton games using specialized overhead camera setup. Chu et al. [11] performed semi-automatic badminton video analysis by court and player detection, and clustering player strategy into offensive or defensive by classifying strokes. Mlakar et al. [12] performed shot classification while Chen and Wang [13] proposed a method based on 2-D seriate images to discover statistics of a badminton match.
Player detection and tracking methods for sports videos [14, 15] have been proposed in the past. Held et al. [16] also propose a method for handling occlusions between players. Wang et al. [17] used tracking data of players in the game of basketball to perform offensive playcall classification while Cervone et al. [18] did point-wise predictions and discussed defensive metrics. Unlike these approaches, our method uses only visual cues and does not rely on special camera setup or additional sensors. There has also been some work in analysing sports without reliance on anything other than the broadcast video. Sukhwani et al. [19] and Ghosh et al. [20] computed frame level annotations in tennis videos, however, where Sukhwani et al. used a dictionary learning method to co-cluster available textual descriptions, Ghosh et al. [20] used the scoreboard extraction approach to perform the index based analysis.
**Real-time Sports Analysis:** Zhong et. al. [21, 22] present a real-time framework for scene detection and structure analysis for tennis and baseball games using compressed-domain processing techniques. Further, they introduce another real-time framework to detect the syntactic structures that are at a level higher than shots for the game of tennis. Ekin et al. [23] propose a single generic real-time algorithm to detect playback events for multiple sports (football, tennis, basketball, and soccer). Their algorithm uses shot-based generic cinematic features to detect the events.
Our work is inspired by the work of [4] that proposes an end-to-end framework to automatically annotate badminton broadcast videos. Their work identifies various understandable metrics that can be computed for analyzing badminton matches which in turn help qualitative understanding of badminton games. We build on top of [4], and extend the work further by (a) covering wider match situations/tournaments/games (b) adapting the solution to real-time deployment and (c) a successful field trial in **PBL 2019** with commentators and broadcasters appreciating the utility.
## III Dataset
A badminton game is divided into sets which could be either two or three in number. Each set has certain number of points or rallies. We work on a collection of 20 badminton match videos taken from the official Badminton World Federation (BWF) channel on YouTube3. We focus on both "singles" and "doubles" (both men and women) matches played for two or three sets. These matches are typically around 60 to 90 minutes long depending upon the number of sets in the game. We also have our recorded badminton match video which consists of ground truth data for the distance covered by the players.
Footnote 3: [https://www.youtube.com/user/bwf](https://www.youtube.com/user/bwf)
In the work by Ghosh et al. [4], all the videos chosen to create the dataset were from the 2012 Summer Olympics. Due to this, all the match videos have the same court colors, illumination settings, and the same broadcasting camera angle. In order to introduce some variation, we create another dataset by taking videos from different tournaments held by the Badminton World Federation. For this, we select matches such that we try to cover all versatility of court structures, which includes different court colors, illumination settings, and all the variations in the broadcasting camera angle. All of these variations can be seen in fig 2. We choose the matches such that all opponents are unique, which ensures maximum variations in the dataset and also make sure that our analysis
model does not overfit on a particular player. Table I shows the comparison between our dataset and Ghosh et. al [4] dataset showing the differences between the matches.
To train and validate our approach, we took 10 matches each for both the singles and doubles matches, which contains 22 hours 16 minutes data, and annotate them for point segments and player bounding boxes. We split the ten matches into training set and testing set of 7 and 3 matches, respectively. We plan to release our dataset and annotations publicly post acceptance of the work.
### _Point Segmentation Annotation_
Broadcast badminton matches are divided into gameplays and replays. We define a "rally" frame as a frame where the actual gameplay is happening and a "non-rally" frame as a frame where the replay (highlights) has been shown, or players are resting, etc. (or in other words whether the gameplay is not happening). Generally, non-rally frames contain redundant information about the game and do not contribute much in the analysis part. Therefore, we first start classifying betweenrally and non-rally frames. We call this classification method as _Point Segmentation_. This allows us to keep all the rally frames, group them into rallies according to the points scored by the players and discard the non-rally frames. To make the dataset usable for point segmentation, we annotate every frame of a match as a rally or non-rally frame using the ELAN video annotation tool 4. We annotate 23156 frames in total, where 15147 were non-rally frames and 8009 were rally frames.
Footnote 4: [https://lta.mpi.nl/tools/lta-tools/clan/](https://lta.mpi.nl/tools/lta-tools/clan/)
### _Player Bounding Boxes Annotation_
We annotate the dataset to get the bounding box for each player using the LabelImg5 graphical image annotation tool. We define the player's position in the court with respect to the broadcast camera point of view where one player plays from the near side, and other player plays from the far side of the camera, and we define them as bottom and top player respectively (see figure 1 for more details). We manually annotate players' bounding boxes with two classes "PlayerTop" and "PlayerBottom" where "PlayerTop" corresponds to the player on the far side of the court and while "PlayerBottom" corresponds to the player on the near side of the court with respect to the viewpoint of the camera. We randomly pick 150 rally frames from each match and annotated them for player bounding boxes for both the players. Thus, we end up with 3000 (150 frames \(\times\) 2 players \(\times\) 10 matches) annotated frames in total. We use this annotation to learn player bounding boxes for the rest of the frames. We use the same annotation style for the doubles matches as well. As shown in figure 4, occasionally, players are occluded or blurred due to the fast-paced nature of the game. Such occlusions are even more prevalent in the doubles matches where there are four players on court as opposed to the two players in a singles match.
Footnote 5: [https://github.com/kcatalian/labellImg](https://github.com/kcatalian/labellImg)
### _Recorded Badminton Match Video_
Our analysis pipeline predicts distance covered by the players, and this forms the basis for other kinds of analysis (like average speed with which the players are moving). Therefore, it is imperative to rigorously evaluate the distance values predicted by our pipeline. Towards this end, we created a new dataset consisting of videos of amateur players playing badminton while wearing distance trackers. The distance trackers,
\begin{table}
\begin{tabular}{l|c c c} \hline \hline Component & \begin{tabular}{c} Ghosh et. al. [4] \\ (Singles) \\ \end{tabular} & \begin{tabular}{c} Ours \\ (Singles) \\ \end{tabular} &
\begin{tabular}{c} Ours \\ (Doubles) \\ \end{tabular} \\ \hline Matches & 10 & 10 & 10 \\ Players & 20 & 13 & 20 \\ Camera Positions & 1 & 4 & 3 \\ Tournaments & 1 & 6 & 5 \\ Illumination Settings & 1 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} TABLE I: A comparison of our datasets and the Badminton Olympic Dataset [4]. We show that our dataset is more diverse as we cover more tournaments, capturing multiple broadcast camera angles and a variety of illumination settings in the courts and background.
Fig. 2: Our dataset contains various broadcast camera positions, illuminations settings, different backgrounds and court colors, which helps generalize the framework for any kind of badminton match scenario. It should be observed that having audiences in the background may affect the player localization method if not generalized well.
which were accurate up to 1 meter, measured the distance covered by each player during each rally. We recorded 25 rallies and obtained the distance values for each player for all of them. All the videos were recorded at approximately the same broadcast angle as that of the singles and doubles datasets using a smartphone camera.
## IV Method
As shown in figure 3, our pipeline is largely based on the approach proposed by Ghosh et al [4]. It comprises of 4 steps: (1) Point Segmentation (2) Player Localization (3) Top View Space Conversion and (4) Analysis. We optimize each of these steps independently to make them more robust while simultaneously reducing latency. The improved pipeline can be used for real-time analysis of videos.
### _Point Segmentation_
The very first step of our pipeline is to keep the rally frames and discard the non-rally frames (which are redundant for our analysis) from the given match video. Rally frames are usually always shot from the broadcast camera view whereas the non-rally frames can include close-ups of players, additional graphics (score cards) and unusual camera angles. These differences help distinguish these frames. Unlike Ghosh et al [4] who use HOG + SVM, we use a convolutional neural network based classifier to classify rally and non-rally frames. We finetune the ResNet-18 [24] model (pre-trained on ImageNet [25] dataset) by changing the last layer of this network to label the frame as rally frame or non-rally frame. As shown in table II, our CNN based method is both more accurate and also exhibits lower latency compared to HOG + SVM.
At test time, we classify the frames retrieved from the match video and number the rally frames in a sequential manner. We assume that a rally is at least 3 seconds long and there is a minimum of 3 seconds gap between two rallies. This allows us to group frames of the match into individual rallies by separating sets of continuous rally frames with at least 3 seconds of non-rally frames in between them. Thus Point Segmentation gives us a set of rallies from the match. We number each rally sequentially as rally-1, rally-2 etc.
### _Player Localization_
Once we obtain individual rallies, we need to find player location in each frame of each rally. A YOLOv3 [26] network is finetuned for these two classes. In order to overcome the constraint of only having 3000 annotated samples for player localization, we use transfer learning to train the YOLOv3 network. We first pre-train the YOLOv3 network on PASCAL-VOC dataset [27] which has 20 classes including the "Person" class. Further, we finetune the YOLOv3 network on our own dataset to achieve better accuracy. We used the same method for both the single's and double's matches to get the location of the player. However, as can be seen in figure 4, player localization is a very difficult problem to solve in doubles matches.
Using a convolutional network for point segmentation and YOLOv3 for player localization helps us build a fast and robust pipeline that performs better than [4] both in terms of accuracy of predictions and latency. The improved accuracy as well as inference time make it possible to use our pipeline for a real-time implementation. Please refer to table II for more details.
### _Top View Space_
We know that the displacement of both the players would manifest differently in the camera coordinates as one player is near side of the camera and another one is on the far
Fig. 3: We propose a real-time end-to-end pipeline to analyze live broadcast match videos. Our pipeline consists of point segmentation, player localization, pictorial point summary and top-view space conversion to compute the distance and overall analysis.
side of the camera. In order to find out the distance or other kind of analysis, we need to switch the view of player positions from broadcast camera view to another view where the displacement of the player is not scaled by the near side or far side of the player position. We choose the top view of the badminton court where we map the player locations from broadcast camera view. So, using homography techniques, we map the 2-dimensional broadcast view camera coordinates to top view court coordinates. By doing this, we get an equivalent linear top view for the player locations. To perform homography calculation, we find the court lines from the video frame using Hough line finding method [28] and get the position of court corners. We compute a homography matrix between these four court corners and top view four court corners to get a mapping from camera coordinates to top view space coordinates. Finally, we map the player locations from broadcast camera view to top view. Please refer to figure 5 for output details.
### _Analysis_
**Players Heatmap:** The output of top-view space conversion gives us insight into the footwork of the players around the court. This top-view can also be seen as a heatmap of the players' locations. This heatmap can be useful for the players and their coaches to analyze the game.
**Player Distance Calculation:** To compute the distance covered by the player in a particular rally, we utilize our player tracks in top view space. Now the player locations are in 2-dimensional space and we take the "center bottom" point of the player bounding box as a proxy for the player's current location.
According to the standard badminton rules6, the length and width of the badminton court is 13.41 meters and 6.1 meters respectively. From this information, we know that from the top left corner of the court to the top right corner of the court, the distance of pixels is 6.1 meters and similarly, from the top left corner of the court to the bottom left corner of the court the distance is 13.41 meters. We use this information to calculate the distance of the player covered in meters. We calculate the euclidean distance between the locations of the player in successive frames for both players. Now we have the pixel-based distance between two successive frames. We use unitary method to scale this pixel-based distance into meters to get the actual distance values. We add all the distances (meters) successively for the whole rally to calculate the distance for each player in that particular rally. The distance calculation of
Fig. 4: Output of player tracking method. It should be noted that this method works well, even when there are levels of occlusion and irrespective of player poses.
Fig. 5: Conversion of player locations from camera coordinates to top view co-ordinates. The top-view space provides the player’s position and footwork information around the court for a particular gameplay. This view can also be seen as heatmap of the players for that particular rally. _(Best viewed in color)_
\begin{table}
\end{table} TABLE II: Comparison of methods between Ghosh et al. [4] and ours. We show that our method improves in terms of both latency and accuracy.
a rally for one player is formulated as:
\[\sum_{i=1}^{n-1}\sqrt{(x_{i+1}-x_{i})^{2}\times\frac{6.1}{D_{h}}\ +\ (y_{i+1}-y_{i})^{2}\times\frac{13.41}{D_{v}}}\]
where \(n\) is the number of frames in the rally, \(x_{i}\) and \(y_{i}\) are the co-ordinates of the player in the top-view space at \(i^{th}\) frame, \(D_{h}\) is the pixel-based distance from top-left corner to the top-right corner (horizontal distance) and \(D_{v}\) is the pixel-based distance from top-left corner to the bottom-left corner (vertical distance). For the game of double's, we have two players on each side of the court i.e. two "PlayerTop" (pt) and two "PlayerBottom" (pb). Therefore, for the first frame, we initialize \(pt_{1}\) and \(pt_{2}\) for the two top players and \(pb_{1}\) and \(pb_{2}\) for the two bottom players respectively. In subsequent frames, we use the player's previous position to track them individually. Again, we use the method same as singles to calculate the distance.
**Player Average Speed Calculation:** To calculate the average speed achieved by each of the players in a particular rally, we use the distance covered by the players. As we know the number of frames in a particular rally, we find out the total time elapsed (25 frames per second) and divide the player distance covered by the total time.
## V Evaluations
We evaluate two different aspects of our proposed pipeline: accuracy of predictions and latency.
### _Player Distance Calculation Evaluation_
To evaluate the accuracy of our proposed pipeline in terms of the distance covered by the players in a particular rally, we used our recorded badminton match video which has ground truth data of 25 rallies for the distance covered by the players. We use our proposed pipeline to obtain predicted distance values for each rally and compare them to the ground truth values. We used the same evaluation metric as [29], which uses paired t-tests to calculate differences between the ground truth distances and predicted distances. We report the mean, standard deviation, minimum, and maximum error between the ground truth and predicted values for both "Player Top" and "Player Bottom". We also report the t-stats value and degree of freedom of the paired t-test calculation between the ground truth and predicted values. Please refer to table III for the report.
### _Real-time Analysis Evaluation_
The proposed pipeline was successfully used in a real-world setting to analyze live broadcast matches in real-time during the Premier Badminton League 2019. As reproducing the exact same scenario is not feasible, we use a simulation of the real-time scenario to evaluate the performance (with respect to latency) of our model. Generally, as live matches are streamed, the camera recording the match captures 25 frames per second and thus the match video file size increases with time. In order to simulate the real-time scenario, we fed the pre-recorded match videos to our proposed pipeline at precisely 25 frames per second, sequentially. Thus our models did not have access to all the frames at the same time. We ran this simulation on 61 rallies (totaling up to 55 minutes) and timed our models. Figure 6 shows the results of this simulation.
## VI Experiments
We use the live feed of the broadcast video match from Premier Badminton League - 2019 (courtesy of Star Sports India7). We used frames from the live video in batches of 20 for our experiments. We attempt to extract the analysis from the data we got using the methods mentioned above. We have segmented the frames into either rally or non-rally frame using point segmentation method which in the end, gives us rally directories as soon as the rally is over in the live match, where each directory contains rally frames for that specific rally. Then we pass the rally frames to the player localization method to get the player bounding boxes. We perform homography to change the space from broadcast camera view to top-view. We perform each analysis on frame level for the accessed file.
\begin{table}
\begin{tabular}{c|c c c c c c} \hline \hline Player Position & Mean & Std & Min & Max & t-stats & df \\ \hline Player Top & 4.5 & 1.18 & 2 & 6 & 0.20 & 24 \\ Player Bottom & 4.73 & 1.76 & 1 & 3 & -1.46 & 24 \\ \hline \hline \end{tabular}
\end{table} TABLE III: We show the difference between the actual distance and the predicted distance covered by the players in our own recorded videos. It should be noted that maximum error we got is 6 meters and minimum error is 1 meter for player top and player bottom respectively.
Fig. 6: Comparison between time taken for gameplay and time taken for analysis in a simulated environment for a single pre-recorded game. The lower line denotes the time taken for gameplay and the upper line shows the time taken for analysis. We can see that the analysis curve closely follows the gameplay curve with a small time-lag. _(Best viewed in color)_
On an average, one really is \(10\) to \(12\) seconds long and it contains 250 to 300 rally frames. For a real time implementation, it is not feasible to process these many frames in one go. Hence, to solve this problem for this scenario, we process every \(3^{rd}\) really frame for player localization method. As there are 25 frames per second in the broadcast match video, we assume that there is no significant change in player location if we take every third rally frame. After getting the rally directories from the above point segmentation method we pass the rally directories to the player localization method to get the player location for each of the frames in a parallel manner. Using this rally-level parallelism for analysis helped us to achieve the goal in the real-time scenario.
## VII Field Trial
We now show the result of our analysis which is computed in real-time at Premier Badminton League - 2019 and was aired on "Hotstar", which is the official Star sports channel. We computed and showed the distances covered by the players in each set (typically a group of 15 - 30 rallies). Please refer to the figure 7 for output details.
## VIII Conclusion and Future Work
All existing work in badminton video analysis suffers from the drawback that it relies on recorded match videos and cannot be used to display the analysis or the computed match statistics in real time. In this work, we present an efficient and robust pipeline to analyze badminton matches (both singles and doubles) in real-time. We perform 3 different types of analysis: (1) heatmap generation (2) distance covered by players and (3) average speed of player movement. We optimize each component of our pipeline to ensure that it maintains high accuracy while also minimizing inference latency. We show the evaluations for the player distance calculation and its implementation in real-time scenario. We also empirically show that processing every third frame of the video is sufficient to perform the desired analysis. The real-time implementation of these kinds of analyses not only enhances the viewer experience but also helps players improve their game.
Another problem that hinders research in the area of sports analysis, and more specifically, badminton analysis, is the lack of labelled datasets. We introduce 3 different datasets to help combat this problem. Two of our datasets (singles and doubles) consist of Badminton World Federation match videos and feature professional badminton players and official courts and recording equipment. Both these datasets are annotated for Point Segmentation and Player Localization. The third dataset consists of amateur players recorded by using a smartphone camera from the approximately same broadcast angle as the other datasets. This dataset also consists of ground truth data for the distance covered by the players in each rally (which was obtained from the distance trackers worn by the players during gameplay). We hope that access to these datasets helps spur further research into real-time analysis of badminton videos.
The proposed pipeline was validated in a real-world setting when it was successfully used to analyze live broadcast matches in real-time during the Premier Badminton League 2019 (**PBL 2019**). In the future, we would like to extend this work to other types of badminton analysis like shot classification, shot recommendation etc. and build a pipeline that can do them in real-time. We are also optimistic about extending this pipeline to other racquet sports like tennis and squash.
|
2305.03068 | A generalized planar conchoid | This note presents the definition of a proposed generalization of the
conchoid at the plane. Known conchoids, such as the Nicomedes and the
Lima\c{c}on of Pascal are part of this set. Following the definition, one can
generate other conchoids. Examples are generated using of a computer code that
is available openly for download. In addition, two step-by-step examples are
described by detail, the first one which presents the results in calculation
tables. | Ludger O. Suarez-Burgoa | 2023-05-04T16:51:59Z | http://arxiv.org/abs/2305.03068v1 | # A generalized planar conchoid
###### Abstract
This note presents the definition of a proposed generalization of the conchoid at the plane. Known conchoids, such as the Nicomedes and the Limacon of Pascal are part of this set. Following the definition, one can generate other conchoids. Examples are generated using of a computer code that is available openly for download. In addition, two step-by-step examples are described by detail, the first one which presents the results in calculation tables.
**Keywords:** :conchoid, nicodemes conchoid, limacon of pascal
**Supplementary material:** :The computation code package, named genPlanarConchoid, is available at
[https://github.com/losuarezburgoa/genPlanarConchoid](https://github.com/losuarezburgoa/genPlanarConchoid).
Definition
Let be \(O\) a fixed point called a focus and let \(\mathcal{L}_{i}\) be a set of lines, where the lines are required to pass through \(O\) and intersect a curve \(\mathcal{C}\) at points \(P_{i}\). The geometric locus of points \(Q_{i}\) and \(Q^{\prime}_{i}\) on \(\mathcal{L}_{i}\) such that a variable offset Euclidean distance \(d_{i}=\overline{P_{i}Q_{i}}=\overline{P_{i}Q^{\prime}_{i}}\) (for \(d_{i}\in\mathbb{R}^{+}>0\)) responds to a function \(f(l_{i})\), which depends on the arc-length (\(l_{i}\)) measured from a starting point \(N\) and directly passes through any \(P_{i}\) and ending at point \(S\) in \(\mathcal{C}\), is defined here as a _generalized planar conchoid_ (GPC) and is denoted as \(\mathfrak{C}^{O}_{f(l)}(\mathcal{C})\). Because \(N\) and \(S\) are the starting and ending points of \(\mathcal{C}\), \(\mathcal{C}\) is finite and has a direction. The notation \(\mathfrak{C}^{O}_{f(l)}(\mathcal{C})_{N\to S}\) is read as:
a generalized planar conchoid at focus \(O\) with base curve \(\mathcal{C}\) from \(N\) to \(S\), and is based on function \(f(l)\).
Figure 1 shows the conceptual scheme of GPC with the names of each of the parts.
Point \(O\) is (as mentioned) the _focus_. The curve \(\mathcal{C}\) is the _base curve_. Lines \(\mathcal{L}_{i}\) are the rays that define two segments from point \(P_{i}\) to \(Q_{i}\) and from \(P_{i}\) to \(Q^{\prime}_{i}\), which are called the _interior branch_ (\(\overline{P_{i}Q_{i}}\)) and the _exterior branch_ (\(\overline{P_{i}Q^{\prime}_{i}}\)), respectively. The distance \(d_{i}\), which is equal to the interior and exterior branch distances, is called the _distance offset_. The distance from \(N\) to \(P_{i}\) through \(\mathcal{C}\) is the _arc length_ (\(l_{i}\)). The function \(f(l_{i})\) is called the _offset function_ or _distance function_. Point \(N\) is the starting point of the directed
curve \(\mathcal{C}\), upon which the _arc length_ is measured, and point \(S\) is the ending point of \(\mathcal{C}\).
The GPC \((\mathfrak{C}^{O}_{f(l)}(\mathcal{C})_{N\to S})\) is composed of the following mathematical objects.
1. The _focus_\(O\) is a point represented by a column vector with a size of 2 \(\times\) 1.
2. The _base curve_\(\mathcal{C}\) is represented by the following objects.
1. The function of the curve: \(c(x,y)\). 2. A starting point \(N\), where \(N\in c(x,y)\). 3. An ending point \(S\), where \(S\in c(x,y)\). 4. The function that defines the arc length of the base curve: \(l=g(x,y)\).
3. The function relating the distance \(d\) and the arc length: \(d=f(l)\).
In the next section, some examples that are used create a GPC that follows the defined rules are shown.
## 2 How to create a GPC
The creation of a GPC is straightforward when following the definition described in the previous section, but for a rapid implementation, a computation code in any programming language can be employed.
A particular difficulty may arise in the calculation of the _arc length_ of the curve \(\mathcal{C}\). Therefore, for each \(\mathcal{C}\) curve, a function for the arc-length estimation should be defined. For some particular planar curves, the arc length is not exactly defined, for example, when an ellipse is used as a base curve.
In this text, two Octave/MATLAB functions are implemented for the case when the base curves (\(\mathcal{C}\)) are a line and an arc circle; those functions are linegenconchoid and circarcgenconchoid, respectively. Both functions return a
data structure that is loaded into a plotting function (called plotgenconchoid) to obtain graphical representations.
With this implementation, some GPCs were created, as shown in Figure 2. In the first figure, the _Nicomedes conchoid_, which is a special case of these GPCs, is shown. In this case, the focus is at the origin; _i.e._\(O=[0,0]^{\intercal}\). It has a linear base curve parallel to the x-axis at \({\cal C}:y=1\) from \(N_{x}=-3\) to \(S_{x}=3\), and the arc-length function has a constant value; i.e., \(f(l)=2\). Other GPCs with the same linear base curves and foci at origin are shown in the figure. The arc length of the GPC shown in the second figure is described with a linear function \(f(l)=l\). By changing the arc-length function, one can obtain a different GPC, as shown in the third figure, where the arc length of the GPC is described with a sine function \(f(l)=\sin l\). Similarly, the fourth figure shows a GPC with an arc length described with a logarithmic function \(f(l)=\ln l\).
Other GPCs were created in Figure 3 by using a circular arc base curve. The first one plots the _Limacon of the Pascal conchoid_, which is also a special case of these GPCs. In this particular case, the GPC focus is as the origin of the coordinate system, and the base curve is a circular arc with a centre at \(c\) and a radius of \(r\); _i.e._\({\cal C}:c=[0,\frac{113}{100}]^{\intercal},r=\frac{80}{100}\). The base curve starts when \(\theta_{N}=0\) and ends when \(\theta_{S}=2\pi\). The arc length function is a constant value of \(f(l)=\frac{136}{100}\). The second GPC and the subsequent GPCs are generated with the same focus locations. The second GPC, in particular, has a circular arc base curve \({\cal C}:c=[0,\frac{7}{2}]^{\intercal},r=2\) from \(\theta_{N}=0\) to \(\theta_{S}=2\pi\) and an arc
length linear function of \(f(l)=l\). The third GPC is plotted considering a base curve with the properties \(\mathcal{C}:c=[0,\frac{7}{2}]^{\intercal},r=2\) and is generated from \(\theta_{N}=0\) to \(\theta_{S}=2\pi\), and the arc-length function is a trigonometric function; _i.e._\(f(l)=2\sin l\). Finally, the last GPC presented here is similar to the last three presented in the figure and is generated with a base curve of \(\mathcal{C}:c=[0,\frac{7}{2}]^{\intercal},r=2\) from \(\theta_{N}=0\) to \(\theta_{S}=2\pi\). Its arc length is bades on a natural logarithm function \(f(l)=\log l\).
The computer code scripts used to generate each of the GPCs, including those shown in Figure 2, are at file someLinearGenConchPlotsSCR, and the script for the GPC presented in Figure 3 is at file someCircarcGenConchPlotsSCR. For example, to generate the third GPC in Figure 2, the code described in 1 was used. Similarly, to generate the fourth GPC in Figure 3, the code described in 2 was used.
``` clearall focusColVec=zeros(2,1); lineFun=@(x)1; npts=180; abscissaIntval=4*[-1,1]; distFun=@(1)2*sin(1); gcLinPlotStruct=linegenconchoid(focusColVec,lineFun,abscissaIntval,... distFun,npts); figure() holdon plotgenconchoid(gcLinPlotStruct); holdoff axisequal
Figure 2: Generalized planar conchoid created when using a line segment as the base curve.
the thetaAngleIntval, distFun, npts); figure() holdon plotgenconchoid (gcCircarcPlotStruct); plot(circStruct.c(1), circStruct.c(2),'kx'); holdoff axis equal
Figure 3: Generalized planar conchoid created when using an arc circle as a base curve.
Descriptions of the code used in the above scripts are explained in the next section.
## 3 The computer code
The implementation is composed of three functions written in Octave, as can be deduced when reading the listings shown in the preceding paragraph. The linegenconchoid function generates a GPC with a base curve that is a _line_, while the cirarcgenconchoid function generates a GPC with a base curve that is a _circular arc_. Other functions can be created for different base curves. The line and circular-arc functions can be used as templates. Finally, the plotgenconchoid function can be called by any of the above (and newly created) functions to visualize the GPC plot.
The computer code can be downloaded from Suarez-Burgoa (2022). In the following subsections, the linear and circular arc functions are described through manual calculations.
To implement the code in MATLAB, only the end statements in every function should be changed. Octave uses particular end statements for each function; for example, the if statement ends with endif. In MATLAB, the ending word for this statement is simply end.
### The linegenconchoid function
To show how the linegenconchoid function creates a GPC data structure, we use an example. Our problem is to plot the following GPC:
\[\mathfrak{C}_{f(l)=l+\sin l}^{O=[2,1]^{\intercal}}\left(\mathcal{C}:l=\left[y(x )=\frac{3}{2}+\frac{1}{2}x\right]\right)_{x_{N}=-3\to x_{S}=0};\]
which is read as follows.
A generalized planar conchoid has a focus of
\[O=(2,1),\]
and _the base curve_ is a line given by the function
\[y(x)=\frac{3}{2}+\frac{1}{2}x\]
from point
\[N=(-3,y(-3))\]
to point
\[S=(0,y(0)).\]
The GPC is based the arc-length function
\[f(l)=l+\sin l.\]
For convenience, we approximate the GPC with \(m\) discrete points \((P_{i})\) generated in the base curve from point \(N\) to point \(S\); these extreme points are now represented by their corresponding vectors
\[N=\boldsymbol{n}=\begin{pmatrix}-3\\ 0\end{pmatrix}\]
and
\[S=\boldsymbol{s}=\begin{pmatrix}0\\ \frac{3}{2}\end{pmatrix}.\]
Those \(P_{i}\) points are expressed by vectors \(\boldsymbol{p}_{i}\)
\[\boldsymbol{p}_{i}=\boldsymbol{n}+(\boldsymbol{s}-\boldsymbol{n})k.\]
The scaling factor \(k\) varies from \(0\) to \(1\) and is divided into \(m\) parts
\[k=\frac{i}{m},\]
for \(i=\{0,2,\ldots,m-1\}\). The arc length \((l_{i})\) of each line from \(N\) to \(P_{i}\) is the norm of
\[\boldsymbol{l}_{i}=\boldsymbol{p}_{i}-\boldsymbol{n},\]
_id est_
\[l_{i}=|\boldsymbol{l}_{i}|=\sqrt{\boldsymbol{l}_{i}\cdot\boldsymbol{l}_{i}}.\]
Now, we need to calculate \(d_{i}=f(l_{i})\) to obtain the points that define the inner branch, that is, points \(\mathbf{q}_{i}\). Additionally, with this value, \(d_{i}\) can generate the points that define the outer branch, that is, points \(\mathbf{q}_{i}^{\prime}\). We must consider that \(f(l_{i})\) is given and that
\[d_{i}=l_{i}+\sin l_{i}\]
in this case. For the current example, Table 1 shows the values calculated for the following variables: \(\mathbf{p}_{i}\), \(\mathbf{l}_{i}\), \(|\mathbf{l}_{i}|\), and \(d_{i}\).
\begin{table}
\begin{tabular}{l c c c c c c} \hline \multirow{2}{*}{\(k\)} & \multicolumn{3}{c}{\(\mathbf{p}_{i}\)} & \multicolumn{3}{c}{\(\mathbf{l}_{i}\)} & \multirow{2}{*}{\(|\mathbf{l}_{i}|\)} & \multirow{2}{*}{\(d_{i}\)} \\ \cline{2-3} \cline{5-7} & \(x\) & \(y\) & \(x\) & & \(y\) & \(|\mathbf{l}_{i}|\) & \(d_{i}\) \\ \hline
[MISSING_PAGE_POST]
000 & 1.500 & 3.000 & 1.500 & 3.354 & 3.143 \\ \hline \end{tabular}
\end{table}
Table 1: Base curve, arc length and distance points.
The points \(\mathbf{q}_{i}\) and the locus of the inner branch of the GPC are obtained by a direction unit vector that joins the focus (point \(O\)) with point \(P_{i}\) and the distance \(d_{i}\) to the inside from \(O\). The unitary vector for each \(\mathbf{p}_{i}\) from \(O\) is
\[\mathbf{u}_{o} = \frac{\mathbf{p}_{i}-\mathbf{o}}{|\mathbf{p}_{i}-\mathbf{o}|}, \tag{1}\] \[= \frac{\mathbf{p}_{i}-\mathbf{o}}{\sqrt{(\mathbf{p}_{i}-\mathbf{o})\cdot(\mathbf{p}_{ i}-\mathbf{o})}}. \tag{2}\]
Table 3.1 shows the intermediate variable values needed to obtain the unitary vector \(\mathbf{u}_{o}\).
With the above calculated variables, we can calculate the points that define the locus of the inner branch of the GPC, which is obtained with the equation
\[\mathbf{q}_{i}=\mathbf{p}_{i}-d_{i}\mathbf{u}_{o}.\]
Similarly, for the case of the points \(\mathbf{q^{\prime}}_{i}\) that define the locus of the outer branch of the GPC, the equation is
\[\mathbf{q^{\prime}}_{i}=\mathbf{p}_{i}+d_{i}\mathbf{u}_{o}.\]
These calculated coordinates are shown in Table 3.1.
Finally, the GPC we are looking for is approximated by the set of discrete points \(\mathbf{q}_{i}\) and \(\mathbf{q^{\prime}}_{i}\), i.e.,
\[\mathfrak{C}=\{\mathbf{q}_{i},\mathbf{q^{\prime}}_{i}\}.\]
The GPC that is calculated the step-by-step using intermediate variables to finally obtain the coordinates is shown in Table 3.1.
### The circarcgenconchoid function
Similar to the preceding case, to show how the circarcgenconchoid function creates the GPC data structure, we make use of another example that is translated for solving the problem of plotting the GPC; it is written as
\[\mathfrak{C}^{O=[0,0]^{\intercal}}_{f(l)=l+\frac{1}{l}}(\mathcal{C}:c=[5,10]^{ \intercal},r=6)_{\theta_{N}=0\rightarrow\theta_{S}=\frac{9}{8}\pi}\]
\begin{table}
\begin{tabular}{l l l l l l} \hline \multirow{2}{*}{\(k\)} & \multicolumn{2}{c}{\((\boldsymbol{p}_{i}-\boldsymbol{o})\)} & \multicolumn{2}{c}{\(\boldsymbol{u}_{o}\)} \\ \cline{2-7} & \(x\) & \(y\) & \(|\boldsymbol{p}_{i}-\boldsymbol{o}|\) & \(x\) & \(y\) \\ \hline
[MISSING_PAGE_POST]
000 & 1.500 & 1.500 & 0.000 & 1.000 \\ \hline \end{tabular}
\end{table}
Table 2: Vector \((\boldsymbol{p}_{i}-\boldsymbol{o})\) and its unit vector \(\boldsymbol{u}_{o}\).
which is read as follows.
A generalized planar conchoid has a focus of
\[O=(0,0),\]
and a _base curve_ that is a circular arc with a centre \(C\) and radius \(r\), which are equal to
\[C=(5,10)\]
\[r=6,\]
\begin{table}
\begin{tabular}{l c c c c} \hline \multirow{2}{*}{\(k\)} & \multicolumn{2}{c}{\(\boldsymbol{q}_{i}\)} & \multicolumn{2}{c}{\(\boldsymbol{q}^{\prime}_{i}\)} \\ \cline{2-5} & \(x\) & \(y\) & \(x\) & \(y\) \\ \hline
[MISSING_PAGE_POST]
000 & 4.643 \\ \hline \end{tabular}
\end{table}
Table 3: Points on the inner and outer branches.
that starts from an angle of
\[\theta_{N}=0\]
and ends at an angle of
\[\theta_{S}=\frac{9}{8}\pi\]
is based on the arc-length function
\[f(l)=l+\frac{1}{l}.\]
Here, we show how to approximate this GPC with \(m=180\)\(P_{i}\) points generated on the base curve from \(N\) to \(S\).
The circle to which the circular arc belongs is defined by its centre (now as a vector \(\mathbf{c}\)) and radius \(r\). As defined by a polar equation, the circle's function is
\[\begin{pmatrix}x\\ y\end{pmatrix}=\mathbf{c}+r\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}.\]
Then, points \(N\) and \(S\), which are represented by vectors in this case, are
\[N=\mathbf{n} = \begin{pmatrix}5\\ 10\end{pmatrix}+6\begin{pmatrix}\cos 0\\ \sin 0\end{pmatrix} \tag{3}\] \[= \begin{pmatrix}11\\ 10\end{pmatrix} \tag{4}\]
\[S=\mathbf{n} = \begin{pmatrix}5\\ 10\end{pmatrix}+6\begin{pmatrix}\cos\left(\frac{9}{8}\pi\right)\\ \sin\left(\frac{9}{8}\pi\right)\end{pmatrix}\] \[= \begin{pmatrix}5-3\sqrt{2+\sqrt{2}}\\ 10-3\sqrt{2-\sqrt{2}}\end{pmatrix}\] \[\approx \begin{pmatrix}-0.543\\ 7.704\end{pmatrix}. \tag{7}\]
The arc length between these two points is
\[L=r(\theta_{S}-\theta_{N}).\]
Every point at the base curve \(\mathbf{p}_{i}\) is distributed on the circle between \(N\) and \(S\); then,
\[\mathbf{p}_{i}=\mathbf{c}+r\begin{pmatrix}\cos\theta_{i}\\ \sin\theta_{i}\end{pmatrix};\]
where
\[\theta_{i}=\theta_{N}+\frac{L}{r}k\]
and the scaling factor \(k\) varies from \(0\) to \(1\) and is divided into \(m\) parts; _i.e._
\[k=\frac{i}{m},\]
for \(i=\{0,2,\ldots,m-1\}\). The arc lengths from \(N\) to every \(\boldsymbol{p}_{i}\) are
\[l_{i}=L\;k.\]
The following examples are similar to those described in the preceding subsection when using the linear base curve; because now, we have an expression for \(l_{i}\), which can be passed through the function \(f(l_{i})\) in this example.
\[f(l)=l+\frac{1}{l};\]
Therefore,
\[d_{i}=l_{i}+\frac{1}{l_{i}}.\]
This finally gives that
\[\boldsymbol{q}_{i}=\boldsymbol{p}_{i}-d_{i}\boldsymbol{u}_{o}.\]
and
\[\boldsymbol{q^{\prime}}_{i}=\boldsymbol{p}_{i}+d_{i}\boldsymbol{u}_{o}.\]
The same expression is used for \(\boldsymbol{u}_{o}\); _i.e._
\[\boldsymbol{u}_{o}=\frac{\boldsymbol{p}_{i}-\boldsymbol{o}}{\sqrt{( \boldsymbol{p}_{i}-\boldsymbol{o})\cdot(\boldsymbol{p}_{i}-\boldsymbol{o})}}.\]
The GPC we are looking for is approximated by the sets of discrete points \(\boldsymbol{q}_{i}\) and \(\boldsymbol{q^{\prime}}_{i}\)
\[\mathfrak{C}=\{\boldsymbol{q}_{i},\boldsymbol{q^{\prime}}_{i}\}.\]
Indeed, this GPC is the same as the GPC plotted in Figure 1. This time, it was necessary to further discretize the curve to find a good approximation (180 points were used). For that reason, a step-by-step calculation table is not presented, but the script shown in 3 can be used to visualize this result.
```
clearall
##comment focusColVec=zeros(2,1); thetaAngleIntval=[0*pi,9/8*pi]; npts=18*10; circStruct=struct("c",[5;10], "r",6); distFun=@(1)1./1+1; gcCircarcPlotStruct=circarcgenconchoid(focusColVec,circStruct,... thetaAngleIntval,distFun,npts); figure() holdon plotgenconchoid(gcCircarcPlotStruct); plot(circStruct.c(1),circStruct.c(2), 'kx'); holdoff axisequal
```
Listing 3: Code to generate Figure 1b
## 4 Final remark
The user can create many GPCs depending on the different focus places, base curves, arc-length functions and intervals. |
2306.05149 | Engineering flat bands in twisted-bilayer graphene away from the magic
angle with chiral optical cavities | Twisted bilayer graphene (TBG) is a recently discovered two-dimensional
superlattice structure which exhibits strongly-correlated quantum many-body
physics, including strange metallic behavior and unconventional
superconductivity. Most of TBG exotic properties are connected to the emergence
of a pair of isolated and topological flat electronic bands at the so-called
magic angle, $\theta \approx 1.05^{\circ}$, which are nevertheless very
fragile. In this work, we show that, by employing chiral optical cavities, the
topological flat bands can be stabilized away from the magic angle in an
interval of approximately $0.8^{\circ}<\theta<1.3^{\circ}$. As highlighted by a
simplified theoretical model, time reversal symmetry breaking (TRSB), induced
by the chiral nature of the cavity, plays a fundamental role in flattening the
isolated bands and gapping out the rest of the spectrum. Additionally, TRSB
suppresses the Berry curvature and induces a topological phase transition, with
a gap closing at the $\Gamma$ point, towards a band structure with two isolated
flat bands with Chern number equal to $0$. The efficiency of the cavity is
discussed as a function of the twisting angle, the light-matter coupling and
the optical cavity characteristic frequency. Our results demonstrate the
possibility of engineering flat bands in TBG using optical devices, extending
the onset of strongly-correlated topological electronic phases in moir\'e
superlattices to a wider range in the twisting angle. | Cunyuan Jiang, Matteo Baggioli, Qing-Dong Jiang | 2023-06-08T12:17:44Z | http://arxiv.org/abs/2306.05149v2 | # Engineering flat bands in twisted-bilayer graphene
###### Abstract
Twisted bilayer graphene (TBG) is a recently discovered two-dimensional superlattice structure which exhibits strongly-correlated quantum many-body physics, including strange metallic behavior and unconventional superconductivity. Most of TBG exotic properties are connected to the emergence of a pair of isolated and topological flat electronic bands at the so-called magic angle, \(\theta\approx 1.05^{\circ}\), which are nevertheless very fragile. In this work, we show that, by employing chiral optical cavities, the topological flat bands can be stabilized away from the magic angle in an interval of approximately \(0.8^{\circ}<\theta<1.3^{\circ}\). As highlighted by a simplified theoretical model, time reversal symmetry breaking, induced by the chiral nature of the cavity, plays a fundamental role in flattening the isolated bands and gapping out the rest of the spectrum. The efficiency of the cavity is discussed as a function of the twisting angle, the light-matter coupling and the optical cavity characteristic frequency. Our results demonstrate the possibility of engineering flat bands in TBG using optical devices, extending the onset of strongly-correlated topological electronic phases in Moire superlattices to a wider range in the twisting angle.
_Introduction -_ Controlling and engineering quantum phases of matter is a central task in condensed matter physics. Inspired by the original discovery of single-layer graphene [1], two-dimensional (2D) materials have emerged as a versatile platform to realize strongly-correlated physics in quantum many body systems [2]. Recently, unconventional superconductivity was discovered in twisted bilayer graphene (TBG), a two dimensional superlattice where one layer of graphene is stacked on top of another at a special magic twisting angle, _i.e._, \(\theta\approx 1.05^{\circ}\)[3; 4; 5; 6]. G galvanized by this breakthrough, several other stacked two-dimensional systems that host exotic superconductivity, such as twisted multilayer graphene, have been revealed [7; 8; 9; 10; 11]. While the underneath physical mechanism of superconductivity in twisted 2D systems is still under debate [12; 13; 14; 15; 16; 17; 18; 19], it is clear that the isolated electronic flat band appearing at the magical angle plays an essential role. Besides superconductivity, flat bands are also indispensable for the emergence of strongly-correlated insulating states and the strange-metal phase near the superconducting dome in the phase diagram of TBG, which closely mimics that of cuprate superconductors [20; 21; 22; 23; 24; 25; 26; 27; 28].
However, despite being a promising platform for studying strongly correlated physics, the unavoidable and uncontrollable non-uniformity of the twist angle across the sample, and the consequent difficulty in keeping the twist angle at its magic value, prevented a wide realization of these phenomena [29; 30]. More precisely, because the magical-angle configuration is unstable, a little offset (around \(\pm 0.1^{\circ}\)) of the twisting angle easily destroys all the emergent exotic properties of TBG. In this regard, one of the most important challenges in the field is therefore to achieve superconductivity at non-magic values of the twisting angle. To achieve this final goal, it is desirable to realize a primary step, namely to create and stabilize electronic flat bands in a wider range of the twisting angle [31; 32; 33; 34; 35].
In this Letter, we propose a new method to engineer stable flat bands at non-magic angles by embedding twisted-bilayer graphene in a vacuum chiral cavity (see top panel in Figure 1 for a cartoon of the setup). Using vacuum cavities to control materials and molecules has emerged as a fruitful playground connecting quantum optics to condensed matter and chemistry [36; 37; 38; 39]. Vacuum cavity engineering is superior to the Floquet method, where external electromagnetic radiation drives the system out of equilibrium, since this second route inevitably heats up the system, destroying quantum coherence and inducing transient phenomena away from thermal equilibrium states. In the past, the usage of vacuum cavities has been proposed to design material conductivity [40; 41], unconventional superconductivity [42; 43; 44; 45], topological properties [46; 47; 48], and even chemical reactivity [49; 50]. Some of these proposals have been already successfully realized experimentally.
A fundamental property of vacuum chiral cavities is that time-reversal symmetry is broken without the need of an external driving. Time-reversal symmetry breaking is essential since, as we will see, quantum fluctuations alone can not significantly influence the electronic bands in TBG. In single-layer graphene, a band gap can be induced by quantum fluctuations in a chiral cavity as well [47]. However, the effect is too small to be directly
observed due to the large bandwidth. As we will demonstrate, the situation is different in TBG near the magic angle, where the small bandwidth enables time-reversal symmetry-broken quantum fluctuations to play a significant role.
In recent years, a number of works have realized the vital impact of symmetry breaking on quantum-fluctuations-related phenomena, such as anomalous Casimir effects [51; 52], topological gap generation [46; 47], and selection of chiral molecules in chemical reactions [53; 54]. A recent work by one of us [55] highlighted the combined power of symmetry breaking and quantum fluctuations, proving that symmetry breaking effects can be transmitted from a material to its vicinity by vacuum quantum fluctuations. In this scenario, the vacuum in proximity of a material with broken symmetries is referred to as its _Quantum Atmosphere_.
In this Letter, we investigate the band renormalization of TBG in the time-reversal symmetry broken quantum atmosphere provided by a chiral cavity. We start from a faithful tight-binding model of TBG and calculate the one-loop self-energy induced by the light-matter coupling. The bottom panel of Figure 1 displays the specific Feynman diagram considered. We find that, for experimentally realizable values of the light-matter coupling and cavity frequency, the topological flat bands in TBG can be stabilized away from the magic angle in an interval of approximately \(0.8^{\circ}<\theta<1.3^{\circ}\). Our derivation and calculations can be directly generalized to other twisted 2D systems.
_Setup and methods_ - To set the stage, we model the Hamiltonian of the combined system, TBG and cavity, as follows:
\[\hat{H}=\hat{H}_{\text{TBG}}(\mathbf{q}-e\hat{\mathbf{A}})+\hbar\omega_{c}\hat{a} ^{\dagger}\hat{a}, \tag{1}\]
where \(H_{\text{TBG}}(\mathbf{q})\) represents the TBG Hamiltonian in reciprocal space, and \(\omega_{c}\) is the cavity photonic mode frequency. TBG and cavity photonic modes are coupled through Peierls substitution \(\mathbf{q}\mapsto\mathbf{q}-e\hat{\mathbf{A}}\), where \(\hat{\mathbf{A}}\) can be expressed in terms of photonic creation and annihilation operators, _i.e._, \(\hat{\mathbf{A}}=A_{0}\left(\mathbf{\varepsilon}^{*}\hat{a}^{\dagger}+\mathbf{ \varepsilon}\hat{a}\right)\). Here, \(\mathbf{\varepsilon}\) is the polarization tensor of the cavity photonic modes and \(A_{0}=\sqrt{\frac{\hbar}{2\epsilon_{0}V\omega_{c}}}\) is the mode amplitude in terms of the cavity volume \(V\). We focus on chiral cavities where the photonic polarization is given by \(\mathbf{\varepsilon}=\frac{1}{\sqrt{2}}\left(\mathbf{e_{x}}+i\mathbf{e_{y}}\right)\), with \(\mathbf{e_{x(y)}}\) the unit vector in the x(y)-direction. Our setup can be straightforwardly generalized to the multi-mode case.
To be concrete, let us consider the effective tight-binding Hamiltonian \(H_{\text{TBG}}(\mathbf{q})\)[56]:
\[\begin{pmatrix}H_{1}(\mathbf{q})&T_{\mathbf{q_{k}}}&T_{\mathbf{q_{tr}}}&T_{ \mathbf{q_{il}}}&\cdots\\ T_{\mathbf{q_{k}}}^{\dagger}&H_{2}(\mathbf{q}-\mathbf{q_{b}})&0&0&\cdots\\ T_{\mathbf{q_{tr}}}^{\dagger}&0&H_{2}(\mathbf{q}-\mathbf{q_{tr}})&0&\cdots\\ T_{\mathbf{q_{tl}}}^{\dagger}&0&0&H_{2}(\mathbf{q}-\mathbf{q_{tl}})&\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix} \tag{2}\]
where \(\mathbf{q}\) is the wave-vector, and \(H_{1,2}(\mathbf{q})\) indicate the Hamiltonian of the top/bottom layer respectively. Moreover, we have defined:
\[\mathbf{q}_{b}=\frac{1}{3}(\mathbf{b}_{1}^{m}-\mathbf{b}_{2}^{m}),\quad\mathbf{q}_{tr}=\frac{1}{3}(\mathbf{b}_{1}^{m}+2\mathbf{b}_{2}^{m}),\] \[\mathbf{q}_{tl}=\frac{1}{3}(-2\mathbf{b}_{1}^{m}-\mathbf{b}_{2}^{ m}), \tag{3}\]
where \(\mathbf{b}_{1}^{m}\) and \(\mathbf{b}_{2}^{m}\) are Moire reciprocal vectors. The twisting angle \(\theta\) is hidden in these vectors; see Fig.6.11 in Ref.[56] for more details. The hopping matrix elements are given by,
\[T_{\mathbf{q}_{b}}=t\begin{pmatrix}1&1\\ 1&1\end{pmatrix},\quad T_{\mathbf{q}_{tr}}=t\begin{pmatrix}e^{i\phi}&1\\ e^{-i\phi}&e^{i\phi}\end{pmatrix},\] \[T_{\mathbf{q}_{tl}}=t\begin{pmatrix}e^{-i\phi}&1\\ e^{i\phi}&e^{-i\phi}\end{pmatrix}, \tag{4}\]
where \(\phi=2\pi/3\) and \(t=0.11\) is the hopping parameter. For more details about the TBG Hamiltonian we refer to the Supplementary Information (SI). Once the effective Hamiltonian is known, the bare electron propagator can be obtained using [47]
\[G_{0}^{-1}(\epsilon,\mathbf{q})=\left[\left(\epsilon+i0^{+}\right)\mathbf{I}-H _{\text{TBG}}(\mathbf{q})\right]^{-1}. \tag{5}\]
Figure 1: **Top panel.** A cartoon of the optical setup considered in this work. Two skewed sheets of graphene are stacked on top of each other with a twisting angle \(\theta\) creating a characteristic Moiré pattern. They are then put inside a chiral optical cavity with a light-matter coupling \(g\) and a characteristic frequency \(\omega_{c}\). **Bottom panel.** The dominant Feynman diagram describing light-matter interaction in the optical chiral cavity and giving rise to the self-energy \(\Sigma_{0}(\epsilon,\mathbf{q})\).
The full electron propagator, taking into account the interaction with the vacuum cavity, can then be derived from the Dyson equation
\[G^{-1}(\epsilon,\mathbf{q})=G_{0}^{-1}(\epsilon,\mathbf{q})-\Sigma_{0}(\epsilon, \mathbf{q}), \tag{6}\]
where \(\Sigma_{0}\) is the self-energy (see bottom panel of Fig.1) given by
\[\Sigma_{0}(\epsilon,\mathbf{q})=-\frac{g^{2}}{\beta}\sum_{m=1}^{\infty}G_{0}( \epsilon+i\omega_{m},\mathbf{q})D_{0}(\omega_{m}). \tag{7}\]
Here, \(\omega_{m}=2\pi mk_{B}T\) is the \(m\)th Matsubara frequency and \(g=v_{F}eA_{0}\) denotes the electron-photon coupling strength with \(v_{F}\) the Fermi velocity of monolayer graphene and \(e\) the electromagnetic coupling. Finally, \(D_{0}(\omega_{m})\) is the photon propagator given by
\[D_{0}(\omega_{m})=\begin{pmatrix}\dfrac{-1}{i\omega_{m}+\omega_{c}}&0&\cdots \\ 0&\dfrac{1}{i\omega_{m}-\omega_{c}}&\cdots\\ \vdots&\vdots&\ddots\end{pmatrix}, \tag{8}\]
with \(\omega_{c}\) the cavity frequency. It should be noticed here that all quantities \(G_{0}\), \(G\), \(D_{0}\), \(\Sigma_{0}\) are matrices with the same dimension of the effective TBG Hamiltonian. With the full propagator in Eq.(6), the spectral function, giving the renormalized electronic band structure, can be calculated from
\[A(\epsilon,\mathbf{q})=-\frac{1}{\pi}\mathrm{Im}\mathrm{Tr}G(\epsilon, \mathbf{q}). \tag{9}\]
More details about the TBG Hamiltonian, the structure in reciprocal space and the numerical methods employed can be found in the SI. In the rest of this manuscript, we will be mainly interested in studying the electronic spectral function \(A(\epsilon,\mathbf{q})\) as a function of the twisting angle \(\theta\), the light-matter coupling \(g\) and the cavity frequency \(\omega_{c}\). For convenience, we define the dimensionless coupling \(\tilde{g}\equiv g/k_{B}T\).
_Electronic spectrum_ - It is well known that TBG exhibits a pair of topological flat bands at the magic angle, \(\theta\approx 1.05^{\circ}\)[57; 58; 59], which play a key role for the underlying strongly-correlated physics. Away from that magic value, the flat bands, and most of the exotic and interesting related properties of TBG, disappear. In panels (a) and (c) of Fig.2, we show two examples of the electronic spectrum of TBG at respectively \(\theta=0.8^{\circ}\) and \(\theta=1.5^{\circ}\). As already mentioned, no isolated flat bands are present anymore in the electronic spectrum just moving of \(\pm 0.3^{\circ}\) from the magic angle. In other words, the flat bands are very fragile and sensitive to the twisting angle. In panels (b) and (d) of Fig.2, we show the same electronic spectra in presence of the chiral optical cavity, with a coupling \(\tilde{g}=2\) and a characteristic frequency \(\omega_{c}=0.3\)eV. A pair of nearly-flat bands re-appear away from the magic angle thanks to the coupling to the chiral cavity. Importantly, the two bands are not anymore degenerate as their energy is shifted from the Fermi energy and grows with the light-matter coupling \(\tilde{g}\). This is a direct consequence of the breaking of time reversal symmetry induced by the chiral cavity. At the same time, the other bands, similarly to the case of the Dirac cone in monolayer graphene (see SI), are also gapped away as a consequence of the same symmetry breaking pattern.
_Theoretical model_ - The emergence of the two isolated quasi-flat bands and their energy splitting, shown in Fig.2, are intimately connected to the chiral nature of the optical cavity, which plays a fundamental role in this regard. To explicitly prove this statement we construct a simplified analytical model which, as we will see, possesses all the minimal ingredients to describe our setup. In order to model the effects of the chiral cavity on TBG, we consider the following deformed hamiltonian
\[H_{\mathrm{TBG+\tau}}(\mathbf{q})=H_{\mathrm{TBG}}(\mathbf{q})+\tau\begin{pmatrix} \sigma_{z}&0&0&0&\cdots\\ 0&\sigma_{z}&0&0&\cdots\\ 0&0&\sigma_{z}&0&\cdots\\ 0&0&0&\sigma_{z}\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}, \tag{10}\]
where the coupling \(\tau\) parameterizes the breaking of time-reversal symmetry on top of the original TBG Hamiltonian in Eq.(2). This is unlikely the most general defor
mation which breaks time-reversal symmetry, but it will be sufficient to qualitatively reproduce the numerical results displayed in Fig.2 and identify the main underlying physical principle behind them. By diagonalizing the above Hamiltonian \(H_{\rm TBG+\tau}({\bf q})\), the band structure with broken time reversal symmetry can be obtained. The results are shown in Fig.3 for different strength of the time-reversal symmetry breaking \(\tau\). As clearly demonstrated, the effects of \(\tau\) is to gap away the higher energy bands and create a pair of isolated quasi-flat bands with nondegenerate energy. At least at a qualitative level, these results are in perfect agreement with the more realistic scenario of TBG in a chiral cavity shown in Fig.2, where the light-matter coupling \(\tilde{g}\) plays an analogous role of the phenomenological parameter \(\tau\) in Eq.(10). This simplified but tractable analytical model highlights the fundamental role of time reversal symmetry breaking, induced by the chiral cavity, in stabilizing the flat band of TBG.
_Phase diagram and quasi-flat bands_ - In order to explore the effects of the chiral cavity in more detail, we have performed an extensive analysis of the band structure for different values of \(\theta\) and \(\tilde{g}\) covering a wide range of the phase diagram around the magic-angle value. To give a quantitative estimate of the flatness of the bands, we define the energy bandwidth parameter
\[\Delta\epsilon(g,\theta)\equiv\epsilon_{b}^{\rm max}(g,\theta)-\epsilon_{b}^{ \rm min}(g,\theta) \tag{11}\]
which quantifies the variation of the energy along the isolated nearly-flat bands. As a reference, a completely flat band would correspond to \(\Delta\epsilon(g,\theta)=0\). Our results are shown in Fig.4 in the interval \(0.8^{\circ}<\theta<1.5^{\circ}\) and \(0<\tilde{g}<2.5\) for a fixed and reasonable value of the cavity characteristic frequency \(\omega_{c}=0.3\)eV. By increasing the value of the dimensionless coupling \(\tilde{g}\), the energy bandwidth becomes smaller and therefore the isolated bands can be flattened away from the magic angle. As expected from simple intuition, the isolated bands can be flattened more efficiently for angles which are closer to the magic value. Interestingly, we find that it is easier to flatten the bands for angles smaller than the magic one compared to angles larger than the latter.
In general, we observe that "quasi-flat" bands, with a variation of the energy within \(0.01\)eV, can be easily tuned using reasonable values of the light-matter coupling \(\tilde{g}\sim\mathcal{O}(1)\) for angles of \(\pm 0.3^{\circ}\) away from the magic value \(\approx 1.05^{\circ}\).
_Conclusions_ - In this Letter, we have revealed the possibility of extending the onset of topological flat-bands in twisted bilayer graphene away from the magic angle by using optical chiral cavities. We have demonstrated that the effects of light-matter coupling can stabilize and flatten the topological flat bands for a large range of the twisting angle without the need of fine-tuning. Using physical values for the optical cavity frequency and the strength of the light-matter coupling, we have estimated that quasi-flat bands can be achieved at least in an interval of \(0.8^{\circ}<\theta<1.3^{\circ}\). From a theoretical point of view, taking advantage of a simplified analytical model, we have identified the breaking of time-reversal symmetry as the fundamental ingredient behind the achieved flattening.
One immediate future task is to verify whether all the interesting strongly-correlated physics related to the topological flat bands survive in presence of the chiral
Figure 3: The band structure of TBG at \(\theta=1.5^{\circ}\) obtained from the deformed Hamiltonian in Eq.(10). Panels **(a)**, **(b)**, **(c)**, **(d)** respectively correspond to an increasing value for the time reversal breaking parameter \(\tau=0,0.1,0.2,0.3\).
Figure 4: The value of the energy bandwidth \(\Delta\epsilon(\tilde{g},\theta)\) for different \(\tilde{g}\) and \(\theta\). Darker color corresponds to a flatter band. The black solid lines indicates a few constant \(\Delta\epsilon\) values as reference. The optical cavity frequency is set to \(\omega_{c}=0.3\)eV.
cavity or how that is modified. For example, it would be interesting to understand further the effects of time-reversal symmetry breaking on the emergent exotic superconductivity of TBG. A second and more pressing point is to verify our theoretical predictions within an experimental setup. Following our preliminary estimates, we conclude that the results shown in this Letter might already be within experimental reach.
In general, we expect the combination of twistronics and photonics engineering to become a powerful platform to study strongly-correlated electronic systems and topological matter beyond the case of twisted bilayer graphene.
_Acknowledgments_ - We would like to thank Dario Rosa for collaboration at an early stage of this project. CJ and MB acknowledge the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01). MB acknowledges the support of the sponsorship from the Yangyang Development Fund. Q.-D. Jiang was sponsored by Pujiang Talent Program 21PJ1405400, Jiaoda 2030 program WH510363001-1, and the Innovation Program for Quajtum Science and Technology Grant No.2021ZD0301900.
|
2310.18198 | On the Fidelity Distribution of Link-level Entanglements under
Purification | Quantum entanglement is the key to quantum communications over considerable
distances. The first step for entanglement distribution among quantum
communication nodes is to generate link-level Einstein-Podolsky-Rosen (EPR)
pairs between adjacent communication nodes. EPR pairs may be continuously
generated and stored in a few quantum memories to be ready for utilization by
quantum applications. A major challenge is that qubits suffer from unavoidable
noise due to their interaction with the environment, which is called
decoherence. This decoherence results in the known exponential decay model of
the fidelity of the qubits with time, thus, limiting the lifetime of a qubit in
a quantum memory and the performance of quantum applications.
In this paper, we evaluate the fidelity of the stored EPR pairs under two
opposite dynamical and probabilistic phenomena, first, the aforementioned
decoherence and second purification, i.e. an operation to improve the fidelity
of an EPR pair at the expense of sacrificing another EPR pair. Instead of
applying the purification as soon as two EPR pairs are generated, we introduce
a Purification scheme Beyond the Generation time (PBG) of two EPR pairs. We
analytically show the probability distribution of the fidelity of stored
link-level EPR pairs in a system with two quantum memories at each node
allowing a maximum of two stored EPR pairs. In addition, we apply a PBG scheme
that purifies the two stored EPR pairs upon the generation of an additional
one. We finally provide numerical evaluations of the analytical approach and
show the fidelity-rate trade-off of the considered purification scheme. | Karim Elsayed, Wasiur R. KhudaBukhsh, Amr Rizk | 2023-10-27T15:16:19Z | http://arxiv.org/abs/2310.18198v1 | # On the Fidelity Distribution
###### Abstract
Quantum entanglement is the key to quantum communications over considerable distances. The first step for entanglement distribution among quantum communication nodes is to generate link-level Einstein-Podolsky-Rosen (EPR) pairs between adjacent communication nodes. EPR pairs may be continuously generated and stored in a few quantum memories to be ready for utilization by quantum applications. A major challenge is that qubits suffer from unavoidable noise due to their interaction with the environment, which is called decoherence. This decoherence results in the known exponential decay model of the fidelity of the qubits with time, thus, limiting the lifetime of a qubit in a quantum memory and the performance of quantum applications.
In this paper, we evaluate the fidelity of the stored EPR pairs under two opposite dynamical and probabilistic phenomena, first, the aforementioned decoherence and second purification, i.e. an operation to improve the fidelity of an EPR pair at the expense of sacrificing another EPR pair. Instead of applying the purification as soon as two EPR pairs are generated, we introduce a Purification scheme Beyond the Generation time (PBG) of two EPR pairs. We analytically show the probability distribution of the fidelity of stored link-level EPR pairs in a system with two quantum memories at each node allowing a maximum of two stored EPR pairs. In addition, we apply a PBG scheme that purifies the two stored EPR pairs upon the generation of an additional one. We finally provide numerical evaluations of the analytical approach and show the fidelity-rate trade-off of the considered purification scheme.
## 1 Introduction
Quantum entanglement lies at the core of the quantum Internet which enables quantum applications including quantum communications [1, 2], quantum key distribution [3, 4] and distributed quantum computation [5]. A major challenge is that qubits suffer from unavoidable decoherence, which results in a rapid decay in the quality of the entangled Einstein-Podolsky-Rosen (EPR) qubit pair with time [6, 7]. A corresponding quality metric, also denoted fidelity, measures the closeness between the noisy EPR pairs and the original (desired) one. In the phase damping decoherence model, the fidelity decays exponentially with time [8].
A canonical model for quantum networks with quantum memories or queues assumes that EPR pairs are continuously generated and stored to be ready to respond to transmission requests of qubits resulting in a high-capacity network [9]. The quantum network must guarantee a sufficient fidelity for the desired application and due to the probabilistic nature of quantum operations the higher the fidelity, the better the quality attained by the application. To this end, the goal of network nodes is to generate high-fidelity entanglements, ensure the validity of the stored ones and apply purification to them.
The generation of high-fidelity entanglements through purification involves consuming a smaller or equal fidelity EPR pair to improve the fidelity of another pair. In [10, 11, 12, 13] different recurrence purification schemes are proposed using multiple purification rounds to generate one very high-fidelity EPR pair. Specifically, we start from the purification scheme in [11] as a baseline to compute the fidelity distribution. Also note that some works consider entanglement cut-off times, i.e., a deadline after which the fidelity is assumed below a required threshold, to ensure a minimum validity of the stored EPR pairs. For example, the work in [14] assumes the cut-off times are probabilistic and modeled by an exponential distribution based on which the EPR pairs stored in the quantum queue are dropped.
In this paper, we address the gap in the literature on the derivation of the fidelity steady-state distribution of stored EPR pairs under purification. Purification is usually treated as a mechanism to initially generate high fidelity EPR pairs [10, 11, 12, 13]. Its application beyond the initial generation on the stored EPR pairs is rarely considered. The purification of the stored EPR pairs has the potential to improve the fidelity at the expense of reducing the average number of EPR pairs in the system, which inherently leads to a rate-fidelity trade-off. We denote the purification scheme applied beyond the generation of an EPR pair as PBG.
In this paper, we derive the steady-state probability distribution of the fidelity of the link-level entanglement in a system with a few quantum memories in isolation of any request process. In addition, we apply a PBG scheme that distillates the stored EPR pairs before storing a newly generated pair when the quantum memory is full. To the best of our knowledge, this is the first work that evaluates the fidelity distribution of the stored EPR pairs in the quantum memories and the effect of the PBG schemes.
The remainder of the paper is structured as follows: We first describe the model and problem statement in Sect. 2. In Sect. 3, we derive the steady-state fidelity distribution of the stored EPR pairs. We numerically evaluate the proposed approach in Sect. 4 and summarize the related work in Sect. 5 before concluding the paper and discussing open problems in Sect. 6.
## 2 Model and Problem statement
We model the entanglement generation as Bernoulli trials with success probability \(p_{g}\) within a time slot \(\triangle t\) similar to [15, 16]. One rationale that the entanglement generation is probabilistic is that the optical fiber is assumed to absorb the transmitted qubit from one node to the other with probability \(1-p_{g}=1-\mathrm{e}^{-\eta l}\), where \(l\) is the fiber length between the communication nodes and \(\eta\) is the attenuation coefficient [17]. This is associated with the link-level entanglement generation schemes that require qubit transmission through a fiber of length \(l\) as discussed in [1, 10]. The scheme that we consider in this paper involves, first, the preparation of an EPR pair at one node before sending half of it, i.e., one of the two entangled qubits, to the other node, hence, \(l\) denotes the link length. In addition, each entanglement generation attempt takes ideally \(\triangle t=l/c\) duration, where \(c\) is the speed of light. Following the formulation from [10] the fidelity of an EPR pair at time \(t_{0}\) decays with time due to decoherence as
\[F(t)=\frac{1}{2}\left(1+(2F(t_{0})-1)\,\mathrm{e}^{-(t-t_{0})/t_{c}}\right), \tag{1}\]
where \(1/t_{c}\) is the decoherence rate and \(F(t_{0})\) is the fidelity of the EPR pair at time \(t_{0}\). In this work, we assume a perfect EPR generation. Note that this assumption does not affect our analytical approach to obtain the fidelity distribution.
We assume a PBG scheme to maintain high fidelity. This entails attempting to purify the two stored EPR pairs at the moment of a successful generation of an additional one. Instead of dropping the lowest fidelity EPR pair to be replaced by the freshly generated one, we use it to purify the other stored EPR pair.
Figure 1: A sample path realization of the fidelity of two EPR pairs stored in the size-two memory system with fidelities \(F_{1}(t)\) and \(F_{2}(t)\) such that \(F_{2}(t)\) always represents the lower fidelity EPR pair. Entanglement inter-generation times are given by the random sequence \(\{\tau_{i}\}\). Purification takes place upon a new entanglement generation _to a full system_ where the fidelity improves in case of purification success while the two stored pairs are lost in case of purification failure.
Specifically, we consider the purification scheme in [11], where the fidelity of the purified EPR pair becomes
\[F_{p}(F_{1},F_{2})=\frac{F_{1}F_{2}}{F_{1}F_{2}+(1-F_{1})(1-F_{2})}, \tag{2}\]
with a purification success probability \(p_{s}\) given by
\[p_{s}(F_{1},F_{2})=F_{1}F_{2}+(1-F_{1})(1-F_{2}). \tag{3}\]
Here, \(F_{1}\) and \(F_{2}\) denote the fidelity of the first and the second pair, respectively.
In Fig. 1, we illustrate the model of the purification protocol using a sample path realization of the fidelity of the EPR pairs over time. We assume the system contains one EPR pair at time \(t=0\) and its fidelity decays with time due to decoherence as in (1). As per the Bernoulli assumption on the generation from above the inter-generation times \(\{\tau_{i}\}_{i}\) come from a geometric distribution denoting the time between two successful EPR pair generations. When an EPR pair is generated and the quantum memories are full, purification takes place between the two stored EPR pairs. The figure shows the improved fidelity obtained from purification as well as the random event of purification failure leading to losing the two stored EPR pairs.
Next, we calculate the steady-state distribution of the fidelity of the EPR pairs in the system with a few quantum memories. The hardness of the problem originates from the hardness of tracking the fidelity due to its dependence on the purification outcome which in turn recursively depends on the fidelity at the previous purification attempts.
## 3 Approach
Motivated by the Bernoulli modeling of the EPR generation in Sect. 2, our key idea for calculating the fidelity distribution is to track the fidelity decay at each time slot by discretizing the fidelity proportional to the time slots. This allows modeling the fidelity using a discrete time Markov chain (DTMC). We divide the fidelity range into \(N+1\) discrete levels proportional to its decay ranging from the lowest fidelity value \(F_{\epsilon}\) to the initial fidelity of the generated EPR pair \(F_{0}\) as
\[F(\triangle n)=\frac{1}{2}\left(1+(2F_{0}-1)\mathrm{e}^{-\alpha\triangle n} \right), \tag{4}\]
where \(\triangle n\in\{0,1,...,N\}\)_is the time duration elapsed since the entanglement generation_, which we denote _the age_ (given in discrete time) and \(\alpha:=\triangle t/t_{c}\) denotes the decoherence coefficient in one time slot. We do not consider the fidelity beyond the lowest value \(F_{\epsilon}\). Since the age \(\triangle n\) uniquely defines the fidelity level, we model the fidelity level as a result of a successful purification of two EPR pairs by an EPR pair with equal
Figure 2: A sample path realization showing the application of PBG on the two stored EPR pairs with ages \(\triangle n_{1}\) (shown) and \(\triangle n_{2}\) (not shown), respectively, obtains a purified EPR pair with a fidelity value equivalent to an EPR generated at a later time point with shortened age \(\triangle n_{p}<\triangle n_{1}\). The time point \(t_{g}^{\prime}\) is the hypothetical generation time of the EPR pair with an equivalent fidelity to the purified EPR pair at \(t_{p}\).
or smaller age \(\triangle n_{p}\) according to
\[\triangle n_{p}(\triangle n_{1},\triangle n_{2})=\] \[\max\left(\left\lceil\frac{-1}{\alpha}{\rm ln}\left(\frac{2F_{p}( \triangle n_{1},\triangle n_{2})-1}{2F_{0}-1}\right)\right\rceil,0\right), \tag{5}\]
where \(F_{p}(\triangle n_{1},\triangle n_{2})\) is the fidelity after purification of the two EPR pairs from (2) and \(\triangle n_{1}\) and \(\triangle n_{2}\) are the ages corresponding to the fidelities of the stored EPR pairs \(F_{1}(n)\) and \(F_{2}(n)\), respectively. Here \(F_{i}(n)\) is the fidelity at slot \(n\) on the discrete time lattice. Since \(F_{p}(\triangle n_{1},\triangle n_{2})\) may not correspond to one of the discrete fidelity levels, we use \(\lceil.\rceil\) to map the _purification age_ to the next larger integer to lower bound the purified fidelity. In case \(F_{p}>F_{0}\), which may occur for small initial EPR fidelity, the maximum operation in (5) maintains \(\triangle n_{p}\geq 0\) corresponding to the highest fidelity \(F_{0}\).
Note that the reduced age due to purification does not reflect the actual time the EPR pair spent in the memory. In our model, the purified pair obtains a fidelity value from (2) with success probability (3) that is equivalent to the fidelity of an EPR with a later generation time. Hence, as shown in 2 the age \(\triangle n\) is shortened accordingly through the purification operation.
Similarly, we calculate the maximum age \(N\) that achieves the lowest fidelity threshold according to \(F(N)=F_{\epsilon}\) as
\[N=\left\lceil\frac{-1}{\alpha}{\rm ln}\left(\frac{2F_{\epsilon}-1}{2F_{0}-1} \right)\right\rceil. \tag{6}\]
Note that the fidelity \(F_{1}(n)\) always represents the larger fidelity EPR pair out of the two stored ones when the memories are full and is exactly calculated using (4). Hence, right after a fidelity jump in Fig. 1, \(F_{2}(n)\) represents the older EPR pair and the fidelity of the only EPR pair in the system when it is not full (cf. the figure). The value of \(F_{2}(n)\) is quantized according to (5) during purification.
### DTMC model of the age of the stored EPR pairs
We model the fidelity of the EPR pairs stored in the system by a DTMC with states \((\triangle n_{1},\triangle n_{2})\sim(F_{1}(n),F_{2}(n))\) representing their age such that \(\triangle n_{2}\) always represents the oldest (smallest fidelity) EPR
Figure 3: DTMC modelling of the fidelity of the EPR pairs stored in the system. A state \((x,y)\) corresponds to the age of the stored EPR pairs \(x,y\), respectively where \(x\leq y\) denotes the ordering of the pairs. Recall that the discretization in (4) provides a one to one mapping of the age to the discrete fidelity levels. The description of the transitions represented by the dotted arrows is in the text below.
pair in the system. We assume that the system has initially one EPR pair with perfect fidelity, thus the initial system state is \((-\infty,0)\) at time \(n=0\), where \(-\infty\) stands for the non-existing second EPR pair. We illustrate the system DTMC in Fig. 3, where we denote the state transitions to be either _forward_ or _backward_. The forward transitions represent the time evolution before attempting purification, i.e., the age progression of EPR pairs. We summarize the forward transitions as
\[(i,j)\rightarrow(\min\{i+1,N\},\min\left\{j+1,N\right\})\,\text{ w.p. }1-p_{g},\] \[(-\infty,j)\rightarrow(0,\min\left\{j+1,N\right\})\,\text{ w.p. }p_{g}. \tag{7}\]
The backward transitions are a result of a purification attempt as in
\[\text{Success: }(i,j)\rightarrow \left(0,\triangle n_{p}(i,j)\right)\text{ w.p. }p_{g}\ p_{s}(i,j),\] \[\text{Fail: }(i,j)\rightarrow \left(-\infty,0\right)\text{ w.p. }p_{g}(1-p_{s}(i,j)),\] \[\forall i\in\left\{0,1,...,N\right\},\ j\geq i, \tag{8}\]
where \(\triangle n_{p}(i,j)\) and \(p_{s}(i,j)\) are the age of the successfully purified EPR pair (5) and the probability of purification success (3) at state \((i,j)\), respectively. The purification attempt occurs upon entanglement generation subject to a full quantum memory, thus it only appears when \(i\neq-\infty\). In case of a purification failure, the two stored EPR pairs are lost and only the newly generated EPR pair remains, thus the system state resets to \((-\infty,0)\).
The backward transition probabilities are state-dependent since the success probability depends on the fidelity levels (3), i.e., the age and \(\alpha\). We describe this dependence in Fig. 3 by the dotted arrow representing the existence of a state-dependent transition from each state within a block to a corresponding state in the destination block. We define a _block_ in Fig. 3 to comprise the states within a horizontal row which represents the states \(S_{m}=\{(m,j)\}\ \forall j\geq m,j\in\{0,1,..,N\}\). Note that not only do the transition probabilities vary in the case of successful purification but also the destination state \((0,\triangle n_{p}(i,j))\). As illustrated the destination state is a function of the current state as well as \(\alpha\). Note that a careful choice of \(\alpha\), i.e., the time discretization with respect to the decoherence rate is crucial for the design of the DTMC.
We represent the transition matrix of this Markov chain in terms of sub-matrices describing the transitions between blocks of the DTMC as depicted in Fig. 3 with the states ordered as
\[\mathbf{Q}=\begin{bmatrix}\mathbf{0}_{N+1,1}\mathbf{X}_{0}&\mathbf{X}_{- \infty}&\mathbf{0}_{N+1,N}&\ldots&\ldots&0\\ \boldsymbol{f}_{0}\ \mathbf{0}_{N+1,N}&\mathbf{D}_{0}&\mathbf{X}_{0}&\ddots&\ddots& \vdots\\ \boldsymbol{f}_{1}\ \mathbf{0}_{N,N}&\mathbf{D}_{1}&\mathbf{0}_{N,N}&\mathbf{X}_{1}& \ddots&\vdots\\ \vdots&\vdots&\vdots&\ddots&\ddots&0\\ \vdots&\vdots&\vdots&\ddots&\mathbf{0}_{2,2}&\mathbf{X}_{N-1}\\ \boldsymbol{f}_{N}\ \mathbf{0}_{1,N}&\mathbf{D}_{N}&\mathbf{0}_{1,N}&\ldots& \mathbf{0}_{1,2}&1-p_{g}\end{bmatrix}. \tag{9}\]
The forward transition implies the transition from one block to the next one, thus resulting in the sparse matrix structure, where \(\mathbf{X}_{m}\) is an \(N-m+1\times N-m\) matrix, with \(0\leq m\leq N-1\), representing the forward transitions in (7). We express this matrix as
\[\mathbf{X}_{m}=(1-p_{g})\begin{bmatrix}\mathbf{I}_{N-m}\\ \mathbf{0}_{1,N-m-1}&1\end{bmatrix},0\leq m\leq N-1, \tag{10}\]
while the \(N+1\times N+1\) matrix \(\mathbf{X}_{-\infty}\) represents the forward transitions due to the successful generation of an EPR pair when only one EPR pair is stored, which we express as
\[\mathbf{X}_{-\infty}=\left[\mathbf{0}_{N+1,1}|1-\mathbf{X}_{0}\right], \tag{11}\]
where \(\left[.|.\right]\) represents the column-wise concatenation operation. The probabilities of entanglement generation resulting in a failed purification attempt, thus the backward transitions in (8), are represented by the \(N-m+1\times 1\) vectors
\[\boldsymbol{f}_{m}:=p_{g}\left(\mathbf{1}-\left[p_{s}(m,m),p_{s}(m,m+1),....,p_ {s}(m,N)\right]^{T}\right), \tag{12}\]
with \(p_{s}(i,j)\) being the probability of purification success at state \((i,j)\) known from (3). Additionally, \(\mathbf{D}_{m}\) includes the backward transitions due to a successful purification expressed in (8). We express the elements of the matrix representing the transition from state \((m,j)\) to state \((0,k)\) by
\[\mathbf{D}_{m}[(m,j),(0,k)]=p_{g}p_{s}(m,j)\mathds{1}_{k=\triangle n_{p}(m,j)}, \ 0\leq m\leq N, \tag{13}\]
where \(\mathds{1}\) is the indicator function.
### Obtaining the fidelity distribution from the DTMC
The classical steady-state solution to the DTMC to obtain the steady-state probability vector \(\boldsymbol{p}\) involves solving the linear system of equations \(\boldsymbol{p}^{T}\mathbf{Q}=\boldsymbol{p}^{T}\) with the normalization condition \(\boldsymbol{p}^{T}\boldsymbol{e}_{n_{s}}=1\), where \(\mathbf{Q}\) is the transition matrix, \(\boldsymbol{e}_{n_{s}}\) is an all-one column vector of length \(n_{s}\) while \(n_{s}\) being the number of states. Since the number of equations in the linear system grows quadratically as \(O(N^{2})\), we make use of the problem structure and derive next a reduced problem that requires solving only \(N+1\) equations. We denote the probability of a state \((i,j)\) as \(p_{i,j}\) and the column probability vector of the block states \(S_{i}\) as \(\boldsymbol{p}_{i}\). Moreover, we denote the part of the transition matrix representing the transitions from all the states to the states \(S_{i}\), i.e., a block column in \(\mathbf{Q}\), by \(\mathbf{Q}_{i}\). For example \(\mathbf{Q}_{-\infty}\) and \(\mathbf{Q}_{0}\) represent the first and the second block column in \(\mathbf{Q}\) as given in (9). Using the steady-state description from above and (9), we express \(\boldsymbol{p}_{i}\) in terms of \(\mathbf{Q}_{i}\) as
\[\boldsymbol{p}^{T}\mathbf{Q}_{i}=\boldsymbol{p}_{i}^{T}. \tag{14}\]
The key idea to reducing the system of equations to \(N+1\) is by relating the steady-state probabilities of all the states in terms of \(\boldsymbol{p}_{0}\) using the structure of the DTMC and the transition matrix (9). The structure of the DTMC implies that the states \(S_{0}\) link all the states together. First, the states \(S_{i},\ i>0\) recursively originate from the forward transitions of \(S_{0}\) as given by the corresponding block columns \(\mathbf{Q}_{\mathbf{i}}\). Equipped with this idea, we can recursively derive \(\boldsymbol{p}_{i}:i>0\) in terms of \(\boldsymbol{p}_{0}\) using (9) and (14), i.e., the recursive structure starts from the third block column in (9). This recursive structure leads to
\[\boldsymbol{p}_{i}^{T}=\boldsymbol{p}_{i-1}^{T}\mathbf{X}_{i-1}=\boldsymbol{ p}_{0}^{T}\prod_{m=0}^{i-1}\mathbf{X}_{m},\ 0<i<N. \tag{15}\]
Similarly, we derive \(p_{N}\) as
\[p_{N}=\boldsymbol{p}_{N-1}^{T}\mathbf{X}_{\mathbf{N-1}}+(1-p_{g})p_{N}=\frac{ 1}{p_{g}}\boldsymbol{p}_{N-1}^{T}\mathbf{X}_{\mathbf{N-1}}.\]
Note that \(p_{N}\) represents only one state, i.e., \(p_{N,N}\). We further derive \(p_{N}\) using the expression of \(\boldsymbol{p}_{N-1}\) in terms of \(\boldsymbol{p}_{0}\) from (15) as
\[p_{N}=\frac{1}{p_{g}}\boldsymbol{p}_{0}^{T}\prod_{m=0}^{N-1}X_{m}. \tag{16}\]
Now, the state \((-\infty,0)\) is the destination of the states \(S_{i},\ i\geq 0\) as a result of the backward transitions capturing the failed purification attempt which is represented by the first column in \(\mathbf{Q}\). Therefore, using (14), we derive \(p_{-\infty,0}\) in terms of \(\boldsymbol{p}_{i},\ i\geq 0\) as
\[p_{-\infty,0}=\sum_{m=0}^{N}\boldsymbol{p}_{m}^{T}\boldsymbol{f}_{m}= \boldsymbol{p}_{0}^{T}\boldsymbol{f}_{0}+\boldsymbol{p}_{N}^{T}\boldsymbol{f} _{N}+\sum_{m=1}^{N-1}\boldsymbol{p}_{m}^{T}\boldsymbol{f}_{m}.\]
Consequently, using the expressions in (15) and (16) we obtain
\[p_{-\infty,0}= \boldsymbol{p}_{0}^{T}\left[\boldsymbol{f}_{0}+\frac{1}{p_{g}} \prod_{m=0}^{N-1}\mathbf{X}_{m}\boldsymbol{f}_{N}+\sum_{m=1}^{N-1}\prod_{n=0}^{ m-1}\mathbf{X}_{n}\boldsymbol{f}_{m}\right],\] \[:= \boldsymbol{p}_{0}^{T}\boldsymbol{\Phi}. \tag{17}\]
Next, the states \(S_{-\infty}\) are recursively related by the forward transitions according to \(\mathbf{Q}_{-\infty}\) as
\[p_{-\infty,j} =(1-p_{g})p_{-\infty,j-1}=(1-p_{g})^{j}p_{-\infty,0},\ 0<j<N,\] \[p_{-\infty,N} =\frac{1-p_{g}}{p_{g}}p_{-\infty,N-1}=\frac{(1-p_{g})^{N}}{p_{g}}p _{-\infty,0}.\]
Let \(\boldsymbol{\rho}=\left[1,(1-p_{g}),\ldots,(1-p_{g})^{N-1},(1-p_{g})^{N}/p_{g} \right]^{T}\), we rewrite \(\boldsymbol{p}_{-\infty}\) in vector form in terms of \(\boldsymbol{p}_{0}\) as
\[\boldsymbol{p}_{-\infty}^{T}=p_{-\infty,0}\ \boldsymbol{\rho}^{T}=\boldsymbol{p}_{0}^{T} \boldsymbol{\Phi}\boldsymbol{\rho}^{T}. \tag{18}\]
Finally, \(S_{0}\) is the destination of all the states according to \(\mathbf{Q}_{0}\), i.e., from \(S_{-\infty}\) according to the forward transitions in (7) and from all the other states according to the backward transitions due to successful purification in (8). Therefore, we describe this relation using (14) as
\[\boldsymbol{p}_{0}^{T}=\boldsymbol{p}_{-\infty}^{T}\mathbf{X}_{-\infty}+\sum _{m=0}^{N}\boldsymbol{p}_{m}^{T}\mathbf{D}_{m}. \tag{19}\]
As a result, the linear system of equation to be solved is reduced to
\[\boldsymbol{p}_{0}^{T}\boldsymbol{\Psi} =\boldsymbol{0}_{N+1,1},\] \[\boldsymbol{p}_{0}^{T}\boldsymbol{\beta} =1, \tag{20}\]
where we derive \(\boldsymbol{\Psi}\) using (15), (16) and (18) in (19) as
\[\boldsymbol{\Psi}:=\mathbf{I}_{N}-\boldsymbol{\Phi}\boldsymbol{\rho}^{T} \mathbf{X}_{-\infty}-\mathbf{D}_{0}-\sum_{m=1}^{N-1}\prod_{n=0}^{m-1}\mathbf{ X}_{n}\mathbf{D}_{m}-\frac{1}{p_{g}}\prod_{m=0}^{N-1}\mathbf{X}_{m}\boldsymbol{D}_{N}, \tag{21}\]
in addition to \(\boldsymbol{\beta}\) using (15), (16) and (18) in the normalization equation as
\[\boldsymbol{\beta}:=\boldsymbol{\Phi}\boldsymbol{\rho}^{T}\boldsymbol{e}_{N+1 }+\boldsymbol{e}_{N+1}+\sum_{m=1}^{N-1}\prod_{n=0}^{m-1}\mathbf{X}_{n} \boldsymbol{e}_{N-m+1}+\frac{1}{p_{g}}\prod_{m=0}^{N-1}\mathbf{X}_{m}. \tag{22}\]
We rewrite (20) in a short form as
\[\boldsymbol{p}_{0}^{T}\left[\boldsymbol{\Psi}|\boldsymbol{\beta}\right]=[ \boldsymbol{0}_{1,N+1}|1]\,. \tag{23}\]
using the column-wise concatenation operation \([.|.]\).
The linear system of equations in (23) is of rank \(N+1\), where its solution yields the value of \(\boldsymbol{p}_{0}\). In addition, we obtain the other steady-state probabilities by substituting \(\boldsymbol{p}_{0}\) in (15), (16) and (18).
Figure 4: The simulation of a link with \(l=15\)km validates the analytical distribution of the fidelity of the older EPR pair as calculated from the DTMC model.
## 4 Numerical Validation
In this section, we validate our DTMC analytical approach with simulations and show the trade-off between the steady-state average fidelity of the stored EPR pairs defined as \(\bar{F}_{i}:=\underset{n\rightarrow\infty}{\lim}\mathrm{E}[F_{i}(n)]\) and their average number for an increasing link length ranging between 5km and 30km. We set the attenuation \(\eta=0.15\) dB/km and the decoherence time \(t_{c}=1\) ms similar to [18, 19]. We assume a perfect generation of EPR pairs and use a fidelity threshold \(F_{\epsilon}=0.55\).
In Fig. 4, we validate the steady-state analytical cumulative mass function (CMF) of the older EPR pair with the result from the simulation for \(l=15\)km.
We illustrate in Fig. 5 the rate-fidelity trade-off achieved by applying _purification beyond generation_ to the stored EPR pairs in our system with two quantum memories. Intuitively, while purification improves the average steady-state fidelity of the two stored EPR pairs as shown in Fig. 4(a), it results in a reduction in the average number of the EPR pairs as shown in Fig. 4(b) since we sacrifice one EPR pair for successful purification and both in case of failure. Note that \(\bar{F}_{1}\) represents the average fidelity of the higher fidelity EPR pair when it exists, i.e., when the quantum memories are full.
## 5 Related Work
Link-level entanglement is the first step towards long distant quantum communication. The authors of [9] propose a physical and link layer protocol to provide a robust link-level entanglement generation between quantum communication nodes. Specifically, the proposed protocol organizes the link-level entanglement generation requests to ensure the fidelity desired by the applications at the expense of the increased generation time. Nitrogen vacancy (NV) centers in diamond platform [20] is one way to generate desired fidelity EPR pairs, where higher fidelity EPR pairs require longer generation times. A different method relies on recurrence purification algorithms, which use two EPR pairs per round to obtain a higher fidelity one. The work in [11] proposes an approach that purifies two EPR pairs using polarization mode dispersion and derives an expression of the improved fidelity as well as the probability of purification success. Several other works such as [10, 13] provide quantum operation-based procedures for the purification of two EPR pairs.
Starting from the Lindblad formalization of the qubit interaction with the environment, i.e., decoherence, as time first order differential equation [21], the time dynamics of the fidelity can be analytically expressed for different phase damping models [6]. Using this concept, the works in [10] express the exponentially decaying fidelity over time of the EPR pairs. Hence, quantum communication nodes need to address the effect of the
Figure 5: The trade-off between the average number of stored EPR pairs and their fidelity by applying purification on the stored qubits for an increasing link distance: (a) The analytical average fidelity of the two EPR pairs \(\bar{F}_{1}\) and \(\bar{F}_{2}\) in case of applying purification is larger. (b) The average number of the stored EPR pairs is, however, smaller in the case of applying purification.
decoherence on the stored link-level EPR pairs by estimating their fidelity to ensure meeting the desired application requirements. For that reason, the works in [14, 19] drop qubits from the memory after specific cut-off times to ensure a minimum fidelity requirement. Specifically, the authors in [14] probabilistically model the cut-off times by an exponential distribution. On the other hand, the work in [16] models a quantum queue without dropping qubits and derives an expression on the average queuing delay, thus it can estimate the average decoherence a qubit suffers in the queue. Overall, these works differ from this paper in the sense that we target the derivation of the _steady-state distribution_ of the fidelity of EPR pairs on one link given a continuous purification after generation protocol.
## 6 Discussion & Open problems
In this paper, we used a DTMC to model the fidelity of the EPR pairs for a quantum communication link in a few (two) quantum memory system. We used this model to calculate the steady-state distribution of the fidelity of the EPR pairs. The model shows the improvement of the fidelity in terms of its distribution of the existing EPR pairs by applying a purification beyond generation protocol at the expense of a decrease in the average number of ready EPR pairs in the system.
Extending the model to more than two quantum memories or a quantum memory queue is open for future work as well as incorporating a request process that consumes the EPR pairs as required by the desired application. Moreover, having more than a few EPR pairs stored in the queue raises a question about the appropriate purification beyond generation protocol and when it should be applied. Further, the problem of calculating the distribution of the continuous fidelity is open and is considered much more complex due to the stochastic behavior of the entanglement generation and purification as well as the dependence between the fidelity at the purification points resulting in random recursive equations.
|
2306.12435 | Modeling T1 Resting-State MRI Variants Using Convolutional Neural
Networks in Diagnosis of OCD | Obsessive-compulsive disorder (OCD) presents itself as a highly debilitating
disorder. The disorder has common associations with the prefrontal cortex and
the glutamate receptor known as Metabotropic Glutamate Receptor 5 (mGluR5).
This receptor has been observed to demonstrate higher levels of signaling from
positron emission tomography scans measured by its distribution volume ratios
in mice. Despite this evidence, studies are unable to fully verify the
involvement of mGluR5 as more empirical data is needed. Computational modeling
methods were used as a means of validation for previous hypotheses involving
mGluR5. The inadequacies in relation to the causal factor of OCD were answered
by utilizing T1 resting-state magnetic resonance imaging (TRS-MRI) scans of
patients suffering from schizophrenia, major depressive disorder, and
obsessive-compulsive disorder. Because comorbid cases often occur within these
disorders, cross-comparative abilities become necessary to find distinctive
characteristics. Two-dimensional convolutional neural networks alongside
ResNet50 and MobileNet models were constructed and evaluated for efficiency.
Activation heatmaps of TRS-MRI scans were outputted, allowing for
transcriptomics analysis. Though, a lack of ability to predict OCD cases
prevented gene expression analysis. Across all models, there was an 88.75%
validation accuracy for MDD, and 82.08% validation accuracy for SZD under the
framework of ResNet50 as well as novel computation. OCD yielded an accuracy
rate of around 54.4%. These results provided further evidence for the p-factor
theory regarding mental disorders. Future work involves the application of
alternate transfer learning networks than those used in this paper to bolster
accuracy rates. | Tarun Eswar | 2023-06-15T12:02:57Z | http://arxiv.org/abs/2306.12435v2 | # Modeling T1 Resting-State MRI Variants Using Convolutional Neural Networks in Diagnosis of OCD
###### Abstract
Obessive-compulsive disorder (OCD) presents itself as a highly debilitating disorder. The disorder has common associations with the prefrontal cortex and the glutamate receptor known as Metabotropic Glutamate Receptor 5 (mGluR5). This receptor has been observed to demonstrate higher levels of signaling from positron emission tomography scans measured by its distribution volume ratios in mice. Despite this evidence, studies are unable to fully verify the involvement of mGluR5 as more empirical data is needed. Computational modeling methods were used as a means of validation for previous hypotheses involving mGluR5. The inadequacies in relation to the causal factor of OCD were answered by utilizing T1 resting-state magnetic resonance imaging (TRS-MRI) scans of patients suffering from schizophrenia, major depressive disorder, and obsessive-compulsive disorder. Because comorbid cases often occur within these disorders, cross-comparative abilities become necessary to find distinctive characteristics. Two-dimensional convolutional neural networks alongside ResNet50 and MobileNet models were constructed and evaluated for efficiency. Activation heatmaps of TRS-MRI scans were outputted, allowing for transcriptomics analysis. Though, a lack of ability to predict OCD cases prevented gene expression analysis. Across all models, there was an 88.75% validation accuracy for MDD, and 82.08% validation accuracy for SZD under the framework of ResNet50 as well as novel computation. OCD yielded an accuracy rate of around 54.4%. These results provided further evidence for the _p-factor_ theory regarding mental disorders. Future work involves the application of alternate transfer learning networks than those used in this paper to bolster accuracy rates.
Obessive-compulsive disorder magnetic resonance imaging major depressive disorder schizophrenia
## 1 Introduction
Over recent decades, obsessive-compulsive disorder (OCD) has been ranked as one of the ten most disabling disorders [1]. A patient suffering from OCD will often experience a variety of symptoms that fall into two main categories: obsessions and compulsions. Obsessions refer to being overly focused on a specific issue, involving overthinking in the form of impulsions. Furthermore, compulsions reflect specific actions to counteract the obsessive symptoms. These habits can include checking and mental compulsions, though the list of specific compulsions varies by case [2]. The symptoms themselves seem relatively benign; however, a concern arises in relation to how much time the disorder occupies in a sufferer's daily life. The fear associated with failing to fulfill an impulse is the major component in the factors that push a diagnosed patient to follow their obsession(s). As a result, a sufferer of OCD typically spends an hour or more each day fixated on these debilitating symptoms [3]. Those facing extreme cases of OCD can often endure increased disruptions to daily life, including the inability to participate at places of work or in school [4].
### Current Solutions
Though OCD has been established as a severely debilitating condition, treatments and knowledge of the disorder are still developing. Current treatments involve utilizing Selective Serotonin Reuptake Inhibitors (SSRIs) as prior research hypothesized serotonin to be a target for effective treatment of the disorder. However, when tested, 40 to 60 percent of patients noticed zero to partial improvements in their symptoms [5]. Low success rates with this specific class of drug demonstrate a lack of understanding of targeting the source of OCD; however, the issue is further convoluted by SSRIs remaining the most common choice for medicating OCD patients [6]. Rather than focusing on serotonin-based solutions, glutamate-based treatments have risen as a novel approach to understanding the causal factors involved. For instance, these treatments have recently been used to target NMDA receptors for OCD patients [7]. However, findings based on glutamate could be difficult to generalize as the substance is abundant in the brain and underpins various aspects of learning and memory; therefore, unintended consequences could abound as it is not clear that the substance could be targeted specifically for OCD with the exclusion of its other functions. Our work is important in isolating a more narrowed aspect of glutamate for the causation of OCD.
### Glutamate
Glutamate is an excitatory neurotransmitter, with the function of stimulating nerve cells that send a chemical message between different nerve cells. Glutamate is made from glial cells in the brain and is recycled as the older glutamate is simply refreshed with new glutamate naturally. Beyond serving the different trigger actions, glutamate also helps to process gamma-aminobutyric acid, which is another neurotransmitter to calm the brain. In the body, glutamate serves to enhance learning and memory, energy sources for brain cells, chemical messengers, sleep-wake cycles, and pain signaling [8]. Therefore, in the scope of OCD, where obsessive behaviors--such as constant checking--are prevalent, the involvement of glutamate becomes a potential target for therapy. At Ruhr University in Germany, researchers were able to determine that excessive glutamate led to a higher cerebrospinal fluid level in OCD patients compared to non-OCD patients [9]. High levels of glutamate were also observed in OCD patients based on a magnetic resonance spectroscopy scan at Wayne State University [9]. To confirm correlation, gene expression data in varying regions of the brain can confirm the up-regulation of metabotropic glutamate receptors (GRM), validating the involvement of glutamate.
### T1 resting-state MRI
In the case of GRM, T1 resting-state MRI scans can be utilized for analysis. These scans can identify structural regions of the brain, unnoticeable to the human eye, that deviate across disorders that may correlate with mGlu5. Gene expression analysis based on these scans then provides an outlet to map MRI scans to data points that undergo analysis. However, if no significant features are noted from these MRI scans for OCD, the results may instead indicate support for the _p-factor_ theory, which states that all disorders are on a continuum rather than discrete [10].
### Engineering Statement
Determining the root cause of obsessive-compulsive disorder is highly difficult. At a high level, it is often difficult to differentiate obsessive-compulsive disorder from major depressive disorder and schizophrenia. As a result, the overall aim of this project was to design models for each disorder, develop activation heatmaps, and extract regions of interest. These models serve as a stepping stone in reaching the significance of GRM in OCD patients. To supplement these models, gene expression analysis was conducted afterward to determine the involvement of mGlu5, encoded by GRM, in OCD patients.
### Engineering Objective
OCD diagnosis is currently understudied and misunderstood within the field of neuroscience. Based on these observations, a few main objectives were enacted:
* Obj. 1a: Construct individual CNNs with guidance from pre-trained networks for OCD, MDD, and Schizophrenia respectively, with accuracy rates of at least 80%.
* Obj. 1b: Develop activation heatmaps, demonstrating regions of interest unique to each disorder.
* Obj. 1c: Perform gene expression analysis on T1 resting-state MRIs with transcriptomics.
* Obj. 1d: Provide an online web application to allow patients to receive data from the model's details in Obj. 1a, as well as to feed more data into the models.
Research on obsessive-compulsive disorder regarding root causes is still misunderstood because of the privatization of datasets and knowledge. As a result, Obj. 1d provides a method to aid future research in being able to expand upon past knowledge in a more accessible and reasonable manner. Obj. 1d will also allow patients to receive helpful metrics free of charge.
## 2 Methodology
### Role of Student vs. Mentor
Over 6 months, I conducted work within the general area of machine learning. For this project, I take accountability for the work done with modeling and results. I received guidance from my advisors in developing my ideas and how to probe further into findings. I also received assistance in developing a mastery of machine learning-based technologies from my mentors.
### Equipment and Materials
To achieve the objectives outlined, a plethora of resources were utilized. Models were constructed on Python 3.10.0 with TensorFlow Keras. Within these models, numerous technologies were required for development: SimpleITK, Pandas, Matplotlib, NumPy, MedPy, Skimage, seaborn, ResNet50, MobileNet, Scikit-learn, NPM, node.js, AWS, and Imaging-transcriptomics. Furthermore, in terms of hardware, this project was conducted using a 2022 Apple MacBook Pro (M2 processor, 8GB ram). Additionally, parts of the models were constructed on a 2022 Apple MacBook Mini (M2 processor, 8GB ram).
### Datasets
To construct models for the disorders in question, T1 resting-state MRI is required in plentiful amounts for each. Data acquired for MDD was sourced from Bezmaternykh D.D et al. 2021. This database contains 72 patients with T1 resting-state MRI scans. The repetition time for these scans was 2.5 seconds with a 90-degree flip angle. T1 resting-state MRI scans for schizophrenia were acquired from Poldrack, R. et al. 2021. The dataset from Poldrack, R. et al. provided MRI scans for several disorders, though only schizophrenia and control patients were used. OCD T1 fractional anisotropy scans were sourced from Kim, Seung-Goo, et al., 2015. Each of these datasets was preprocessed with NumPy resizing techniques to fit the data to size requirements. Afterward, scikit-learn distributed the datasets into the configuration of 70% train, and 30% test [11]. Missing anatomy sub-folders were excluded before model creation. Incorrectly sized images or mistimed scans were also subject to removal. AWS and npm were utilized for large file downloads from the databases.
### Novel 2D CNNs
Novel 2D convolutional neural networks were established for each disorder to provide proof of concept for further testing. Each novel CNN was composed of a sequential base, containing pooling, batch, and dropout layers to condense T1 MRI slices into a 1x1 matrix within the sigmoid layer. The model was compiled with an Adam optimizer to reduce computation time [12]. Furthermore, this optimization allowed for an easier load during this proof of concept. Each model compiled over 25 to 100 epochs with a batch size of 32, amounting to roughly 11.7 hours of runtime per trial.
### Optimized Neural Networks
Pre-trained frameworks were used post-confirmation of functioning novel models. Slices were re-scaled by a scale factor of 0.874 per ResNet50 requirements. During pre-processing, slices with differentiating time stamps due to issues within the scan were excluded. Furthermore, T1 MRI scans noted as corrupted or missing by the primary author of each dataset were also excluded. During the usage of ResNet50, the ram requirement for 23.2 billion parameters surpassed the number of resources available with the given hardware. As a result, the central 40 scans per disorder model were utilized to allow for model compilation. As per ResNet requirements, the coordinate arrays
Figure 1: TRS-MRI Sample: An example of a 2D resting-state MRI scan acquired from the UCLA Consortium and pre-processed with NumPy.
from STIK were stacked by 3 and fitted to the respective shape per disorder (X, Y, 3). The default weight of ImageNet from ResNet50 was in use for modeling. The model ran with a base learning rate of 0.001 throughout 30 epochs. The first 10 epochs were run without interference from ResNet weights whereas pre-trained models and the activation of ImageNet executed from epoch 10 onwards. A similar approach was adopted to make use of MobileNet.
### Activation Heatmaps
Activation heatmaps of the neural network models were created following model development. To construct heatmaps, testing data was classified based on the prediction attribute of the model. Afterward, the layer at index -1 was extracted to obtain weights of size 2048. Based on the usage of ResNet, the final conv layer, conv5block3out, was extracted at sizes 7, 7, and 1. This matrix was then resized to a T1 resting-state size of 189 X 189 X 2048 to provide an overlay of the heatmap onto the original MRI. A Cmap of "jet" was incorporated at an alpha level of 0.5 to highlight regions of importance per each disorder.
### Statistical Tests
To analyze the performance of the models, the F1 score, and confusion matrix were generated. Additionally, the Matthews Correlation Coefficient score was analyzed. These metrics were selected based on their specificity to binary classification models [13]. Classifying the results of a model using traditional statistics was averted because of variations in data sets and methodology. The F1 score was calculated as follows:
\[F1=\frac{2\cdot\text{precision}\cdot\text{recall}}{\text{precision}+\text{ recall}}\]
## 3 Results
### Dataset Creation
In the initial stages of this project, large datasets from various sources were compiled before analysis. In total, over 150 sub-sessions of T1 resting-state scans were developed through the course of this project. In light of limited data accessibility, acquiring data of this amount presents a means for future use in finding MRI-related datasets.
### Novel 2D CNNs Predictions
Through the course of model development, the novel 2D CNN was utilized as a means of providing reliable evidence of functionality before moving forward. In the case of MDD, the model accuracy approached 99.99% as, the validation accuracy fluctuated through 100 epochs as demonstrated by Figure 2. These results are then shown by a t-SNE plot in Figure 3 --a visualization technique that demonstrates the low-dimensional representation of high-dimensional of predictions. These plots will reveal the underlying patterns, clusters, or similarities among the MRI scans. Furthermore, the statistical analysis technique of a confusion matrix was utilized for the novel models as shown in Figure 4. This measure allows the generalization of the overall effectiveness of the model at a glance. Precision, recall, F1, and detailed analysis were constructed for definite pre-trained models. These results enabled the usage of pre-trained neural networks. Furthermore, the novel 2D CNN constructed for schizophrenia also yielded promising results with a model accuracy of 99.77% and a validation accuracy of 82.08%. The final accuracy fell to around 79% but the model restores best weights on run. Figure 4 demonstrates the confusion matrix for schizophrenia. Based on the results provided by ResNet50, a novel model was not constructed for OCD.
Figure 4: MDD Confusion Matrix: Confusion matrix statistical analysis for MDD model performance and accuracy.
Figure 3: t-SNE Visualization: A low-dimensional representation of high-dimension, complex MDD MRI slices.
Figure 2: MDD Validation: Validation and model accuracy metrics over the interval of 100 epochs for MDD.
### Optimized Neural Networks Predictions
After the creation of novel networks, ResNet50 and MobileNet were investigated to achieve higher rates of validation accuracy. To visualize the variance created by the usage of a pre-trained network, a green fine-tuning line is incorporated at epoch 10. For MDD, training approached the limit of 1.00 while the validation accuracy stabilized at 88.75%. Cross entropy readings fluctuated but reached a natural deviation after epoch 15. Further insight into the overall performance of the model is provided by the calculated statistical scores in Table 1. These scores were all produced for the pre-trained models to protect against potential overfitting in the novel 2D CNNs.
Figure 6 demonstrates the change in accuracy and cross-entropy over time, while Figure 7 is the corresponding matrix results. Similar results were produced regarding the Schizophrenia ResNet50 model. Patient 126 was excluded from the sample due to difference in numpy dimensions. Afterward, the remaining scans were cross-validated for shape, as shown in Figure 10. The model yielded a validation accuracy of 80.5% and a model accuracy that approached 99.9%. Before fine-tuning, the model averages a validation accuracy of around 70%. After weight and pre-train activation, the model climbs in terms of model accuracy, though the validation accuracy remains stable. The final classifications of patients can be seen in the confusion matrix of Figure 9. The t-SNE plot in Figure 8 demonstrates the model performance with mixed clusters. The resulting statistical measures are shown in Table 2.
### Activation Heatmap Findings
After confirming relatively fair accuracy within the models constructed, activation heatmaps were constructed with both patients diagnosed with a disease and healthy patients. Scaled convolution layers were placed over the original image to extract the regions of interest. Cmap "jet" outputted the heatmaps as zones. Red and orange zones are representative of areas in which the neural network detected differences in structural patterns. Heatmaps were not outputted for OCD. Figure 12 depicts the heatmap configuration for MDD while Figure 13 demonstrates the configuration for schizophrenia. Partial scans from MDD and SZD are displayed; all heatmaps can be found in appendices B and C.
## 4 Discussion
Determining the root cause of obsessive-compulsive disorder is highly difficult. As a result, the overall aim of this project was to design models for each disorder, develop activation heatmaps, and extract regions of interest. From these models, heatmaps became an additional objective to provide an overview of regions of interest for the 2D CNN. Furthermore, these models were intended to be used in gene expression analysis to identify significant gene encodings that demonstrate higher levels of influence in OCD as compared to other disorders A two-proportion z-test was utilized throughout all model cases due to the
Figure 11: (left) ResNet50 validation and cross entropy with respect to change in epoch for OCD (right) ResNet50 validation and cross entropy with respect to change in epoch for OCD.
Figure 12: MDD Activation Heatmaps: The figures above demonstrate the activation heatmaps cast onto the 72 patients within the MDD model.
Figure 10: Verifying SZN TRS-MRI Data: Remaining T1 resting-state MRI scans after removal of sub-126. Shape validation was conducted on these scans for dimensions (151, 40, 199).
Figure 13: Schizophrenia Activation Heatmaps: The figures above demonstrate the activation heatmaps for the schizophrenia model. Regions of red are areas of interest in diagnosis.
Figure 9: ResNet50 SZD Confusion Matrix: Confusion matrix generation for performance of the ResNet50 schizophrenia model.
nature of the data: varying percentages requiring a metric to be standardized and compared.
Through conducting this study, respective models for MDD, SZD, and OCD have been successful in construction. Each demonstrates an important and different aspect in terms of OCD. The MDD model provided a validation accuracy of 88.75%. Within the field of computational neuroscience, this accuracy serves to provide more reliable results as compared to other models. When compared to other models, our MDD classification network produced a P-Value of 0.04363, significant at an alpha level of 0.05. Achieving this level of significance allows for the model to be presented as a viable classification method in the future. The t-SNE plot from Figure 3 also depicts separation in clusters of healthy vs. MDD patients. Even within the healthy MRI scans, there are some distinct clusters, which may be an area of future research. However, the primary clusters show that there is a clear factor the model is able to identify when classifying patients. This factor is found when analyzing the heatmaps which demonstrated implications in the corpus callosum. This finding is consistent with past literature where implications in the corpus callosum were found [14]. By demonstrating further support for the involvement of the corpus callosum, future research can have greater assurance of results based on our work.
The schizophrenia model provided similarly assuring results as with MDD. To test the concept of computational modeling, a novel network was built as a primary means. Though we theorized that shifting to ResNet50 to bolster accuracy with ImageNet, the novel network provided a greater level of accuracy at 82.08% in comparison to the optimized network. This novel network was not optimized per layer, meaning that future work has the potential to bolster accuracy rates with a different framework for transfer learning. ResNet50's _include-top_ parameter was declared as _false_ meaning that it effectively acted as a transfer learning model; however, in the future VGG16 may be explored as a different means of transfer learning. The t-SNE plot from Figure 8 demonstrates a high level of overlap between the healthy and SZD patients. This overlap may suggest why the model validation accuracy stalled at around 80%. At this range, due to overlapping clusters of features, improvement may have been different. Though for the separations identified, the GradCam heatmaps demonstrate implications within the right frontal lobe. Determining the importance of the right frontal lobe, or more broadly the frontal lobe, aligns with previous works within the field of neuroscience that also find particular importance in the right frontal lobe [15]. These results can narrow down future research regarding schizophrenia--a contribution that resides outside of the topic of focus.
Unlike MDD and SZD, the OCD novel and optimized networks failed to predict healthy patients from OCD patients. When first observing the results, this failure appeared to be the cause of TRS-MRI distortions during the scans from the dataset; however, further analysis demonstrated that the framework was able to sufficiently read the coordinate points of scans. Furthermore, the authors of the data with FSL [16]. As a result, OCD is evidenced to possess no unique characteristics that define it differently from a healthy brain, leading to the inability to predict a diagnosis. As demonstrated by the confusion matrix in Figure 11 (left), the model outputted that all scan slices were healthy patients. If no defining characteristic can be found by the classification network, it must default to healthy as per the logistics of binary classification. These findings are consistent with a meta-analysis conducted by the ADAA [17]. The authors found that across 100 studies collected, they were unable to find a consistency of OCD function implications that could be a cause of a cognitive aspect. In addition, they note that due to these inconsistencies in findings across studies, pushing toward adequate treatments in the future becomes increasingly difficult. Our model summarizes these descriptions by demonstrating that for multiple reasons, the model fails to provide a viable method to predict future cases due to comorbidity and a lack of differentiability in scans. Therefore, the objective for OCD TRS-MRI slices was not satisfied, but a deeper understanding of the field and its reflection within the model was understood.
Though SZD and MDD neural networks point to specific regions of scan slices, the overall findings of this project further the _p-factor theory_. This theory alludes to the fact that all psychopathological disorders can be generalized under one umbrella. Authors of this paper further contribute to the concept of being inability to utilize TRS-MRIs instating that groups attempting to neglect the _p-factor_ and find regions of interest tend to create contradictory results to preexisting studies [10]. Understanding disorders with lacking knowledge will require utilization of the _p-factor_. OCD lacks fundamental understanding; though, the use of the _p-factor_ presents a viable solution to treat OCD as well as other under-studied disorders in the future.
### Limitations
Throughout conducting this project, the major limitation faced was computing power. Due to the 8GB RAM limit which resulted in computer crashes, only certain subsets of data were able to be processed. Surpassing this barrier will allow for the usage of deep transfer learning in the form of a support vector machine in the future. Data acquisition arose as another limitation and restricted the project to publicly available datasets. For instance, this study intended to utilize PET scan data in the initial stages but switched to TRS-MRI due to data availability.
### Future Research
Future research may involve the optimization of the models constructed in this paper. Though the accuracy rates were promising, model validation can be presented with deep learning methods. Furthermore, in terms of SZD and MDD, rather than being stratified into two separate
disorders, future work may focus on condensing these models under one network that can differentiate the disorders across multiple classes. The creation of this combined model has the potential to provide more supplementary evidence to the _p-factor_. Providing further evidence for the _p-factor_ is essential to develop more effective and widely used treatments in the future.
## 5 Conclusion
Our engineering project served to identify the differentiation factor between OCD and disorders comorbid with OCD. As a result, SZD and MDD were selected for cross-comparative analysis. From there, the project served to generate heatmaps as well as a gene expression analysis model to determine the significance of gene encodings across the disorders. These objectives were set on the basis that mGlu8 was implicated in those suffering from OCD. To conduct the study, this study we focused on using convolutional neural networks to classify disorders with the means of novel, ResNet5, and MobileNet models. Afterward, heatmaps were generated following ResNet50 guidelines and with GradCam. However, before approaching the gene expression analysis stage, the TRS-MRI scans were unable to predict cases of OCD based on slices. As a result, we interpreted the model as unable to predict OCD cases because of multiple regions were connected to OCD as a neurological disorder. Though diverting from the original objective of finding the differentiation factor for OCD, the lack of prediction still points to a fundamental piece of motivation for this project--a lacking form of treatment for OCD. Lacking predictions, though, still provided useful insight as they pointed towards a theory known as the _p- factor_[10]. This theory states that within the field of neuroscience, researchers are unable to find distinctive characteristics respective to individual disorders because these disorders are not discrete and share many overlapping regions of implication. Rather, they Therefore, these disorders fall onto a continuum that must be treated in an empirical manner rather than case-by-case. As a result, it becomes clear that searching for universal characteristics that can facilitate diagnosis is paramount. Though a singular factor was unable to be uncovered through this project, further support for the _p-factor_ was found, providing further evidence that disorders are on a continuum rather than discrete. These findings guide future research and provide a feasible theory and mode for finding a treatment for OCD.
## Acknowledgments
I would like to give special thanks to Dr. Kevin Crowthers and Mr. Nicholas Medeiros for their support throughout the journey of this research project. Additionally, I am grateful for the help provided by Varshini Ramanathan of Tufts University as well as Revathi Ravi of Massachusetts General Hospital for their guidance in topic-specific knowledge as well as with data needs. Finally, I am thankful for the guidance provided by Joseph Yu in aiding my understanding of machine learning concepts for this project.
|
2301.01388 | The cosmic web of X-ray active galactic nuclei seen through the eROSITA
Final Equatorial Depth Survey (eFEDS) | Which galaxies in the general population turn into active galactic nuclei
(AGNs) is a keystone of galaxy formation and evolution. Thanks to SRG/eROSITA's
contiguous 140 square degree pilot survey field, we constructed a large,
complete, and unbiased soft X-ray flux-limited ($F_X>6.5\times 10^{-15}$ erg
s$^{-1}$ cm$^{-2}$) AGN sample at low redshift, $0.05<z<0.55$. Two summary
statistics, the clustering using spectra from SDSS-V and galaxy-galaxy lensing
with imaging from HSC, are measured and interpreted with halo occupation
distribution and abundance matching models. Both models successfully account
for the observations. We obtain an exceptionally complete view of the AGN halo
occupation distribution. The population of AGNs is broadly distributed among
halos with a mean mass of $3.9 _{- 2.4 }^{+ 2.0 }\times10^{12}M_\odot$. This
corresponds to a large-scale halo bias of $b(z=0.34)= 0.99 ^{+0.08}_{-0.10}$.
The central occupation has a large transition parameter,
$\sigma_{\log_{10}(M)}=1.28\pm0.2$. The satellite occupation distribution is
characterized by a shallow slope, $\alpha_{{\rm sat}}=0.73\pm0.38$. We find
that AGNs in satellites are rare, with $f_{{\rm sat}}<20\%$. Most soft
X-ray-selected AGNs are hosted by central galaxies in their dark matter halo. A
weak correlation between soft X-ray luminosity and large-scale halo bias is
confirmed (3.3$\sigma$). We discuss the implications of environmental-dependent
AGN triggering. This study paves the way toward fully charting, in the coming
decade, the coevolution of X-ray AGNs, their host galaxies, and dark matter
halos by combining eROSITA with SDSS-V, 4MOST, DESI, LSST, and \textit{Euclid}
data. | Johan Comparat, Wentao Luo, Andrea Merloni, Surhud More, Mara Salvato, Mirko Krumpe, Takamitsu Miyaji, William Brandt, Antonis Georgakakis, Masayuki Akiyama, Johannes Buchner, Tom Dwelly, Toshihiro Kawaguchi, Teng Liu, Tohru Nagao, Kirpal Nandra, John Silverman, Yoshiki Toba, Scott F. Anderson, Juna Kollmeier | 2023-01-03T23:11:28Z | http://arxiv.org/abs/2301.01388v2 | # The cosmic web of X-ray active galactic nuclei
###### Abstract
Which galaxies in the general population turn into active galactic nuclei (AGN) is a keystone of galaxy formation and evolution. Thanks to SRG/eROSITA's contiguous 140 square degrees pilot survey field, we constructed a large, complete, and unbiased soft X-ray flux-limited AGN sample at low redshift \(0.05<z<0.55\). Two summary statistics, the clustering using spectra from SDSS-V and galaxy-galaxy lensing with imaging from HSC, are measured and interpreted with halo occupation distribution and abundance matching models. Both models successfully account for the observations. We obtain an exceptional complete view of the AGN halo occupation distribution. The population of AGN is broadly distributed among halos with a mean mass of \(3.9^{+2.0}_{-2.4}\times 10^{12}\mathrm{M}_{\odot}\). This corresponds to a large-scale halo bias of \(b(z=0.34)=0.99^{+0.08}_{-0.10}\). The central occupation has a large transition parameter \(\sigma_{\mathrm{high(M)}}=1.28\pm 0.2\). The satellite occupation distribution is characterized by a shallow slope \(\alpha_{\mathrm{at}}=0.73\pm 0.38\). We find that AGNs in satellites are rare, with \(f_{\mathrm{at}}<20\%\). Most soft X-ray-selected AGNs are hosted by central galaxies in their dark matter halo. A weak correlation between soft X-ray luminosity and large-scale halo bias is confirmed (\(3.3\sigma\)). We discuss the implications of environmental-dependent AGN triggering. This study paves the way towards fully charting, in the coming decade, the co-evolution of X-ray AGN, their host galaxies, and dark matter haloes by combining eROSITA with SDSS-V, 4MOST, DESI, LSST, and Euclid.
## 1 Introduction
Active galactic nuclei (AGN) are a keystone in galaxy evolution. How they are triggered and are fueled are essential ques
tions. Answering them will deepen our understanding of the co-evolution between galaxies, the gas surrounding them, and their central supermassive black holes (SMBH; see reviews from Padovani et al.2017; Eckert et al.2021). This article focuses on the large-scale environment of X-ray-selected AGN, namely the population of dark matter haloes hosting them. X-ray selection provides AGN samples with higher completeness and purity than selections at different wavelengths (Hickox et al.2009). As devised in simulations, this population is diverse (Georgakakis et al.2018; Comparat et al.2019). To infer the population of dark matter haloes hosting a sample of galaxies, the best technique to date consists of interpreting the complementary signals from clustering and weak gravitational lensing (see for example Comparat et al.2013; More et al.2015; Coupon et al.2015; Favole et al.2016; Zhang et al.2021).
Previous studies of the clustering of X-ray-selected AGN were limited by the total number of X-ray AGN or the small survey area. They typically measured the large-scale halo bias of AGN selected in different fashions (Gilli et al.2009; Cappelluti et al.2010; Starikova et al.2011; Koutoulidis et al.2013, 2018; Leauthaud et al.2015; Viitanen et al.2019; Allevato et al.2019). The auto-correlation of X-ray-selected AGN was studied locally (\(z\sim 0.045\)) with 199 AGN in the Swift-BAT all-sky survey (Cappelluti et al.2010). They found these bright low redshift AGN to be hosted on average by dark matter haloes of mass \(1.6-2.5\times 10^{13}h^{-1}\mathrm{M}_{\odot}\) corresponding to a large-scale halo bias of \(1.2\pm 0.1\). At higher redshift (\(z\sim 1\)) with deep pencil beam surveys (COSMOS observed with XMM and Chandra, Bootes, and Chandra compilations) and larger numbers of AGN (ranging from 500 to 1,500), Gilli et al. (2009); Starikova et al. (2011); Koutoulidis et al. (2013); Viitanen et al. (2019); Allevato et al. (2019) inferred a large-scale halo bias of \(\sim 2\pm 0.2\) corresponding to halo masses \(4-9\times 10^{12}h^{-1}\mathrm{M}_{\odot}\). In these studies, further splitting of the samples as a function of AGN type, luminosity, and host-galaxy properties, is not very conclusive due to small statistics. There are hints of correlation with X-ray luminosity and an indication of a low satellite fraction. The study of the angular auto-correlation of photometrically selected AGN, so with much larger samples, led to similar large-scale halo bias and typical dark matter halo masses (Myers et al.2007; Donoso et al.2014; Koutoulidis et al.2018). Finally, Leauthaud et al. (2015) studied the galaxy-galaxy lensing signal around 382 X-ray-selected AGNs in the COSMOS field. They find that AGN host occupation is no different from that of galaxies. They explain the issue of quoting a mean for the halo mass when, instead, complete halo occupation distributions should be discussed; see also Georgakakis et al. (2018) for an extended discussion. Also, after controlling for stellar mass, Yang et al. (2018) found no clear dependence between the environment and the sample-averaged SMBH accretion rate or the AGN fraction, which indicates that environment-related physical mechanisms might not significantly affect SMBH growth.
To circumvent the low signal-to-noise ratio in the auto-correlation functions, the cross-correlation with a controlled galaxy population has been recently fruitful. Such studies relate AGN populations to their host dark matter haloes (Krumpe et al.2010, 2012, 2015, 2018; Mendez et al.2016; Mountrichas et al.2019; Zhang et al.2021). They cross-correlated a similar number of X-ray-selected AGN (between 300 and 1500) with spectroscopic galaxy surveys: 2MASS, SDSS, VIPERS, COSMOS (Skrutskie et al.2006; York et al.2000; Guzzo et al.2014; Scoville et al.2007). They obtain similar large-scale halo bias values as the auto-correlation studies and investigate the correlation with host-galaxy properties hinting at possible correlations with stellar mass. This powerful technique works only with access to a well-studied galaxy sample (Zehavi et al.2011; Marulli et al.2013). The limited signal-to-noise impedes establishing a clear definitive picture of how X-ray AGNs populate the cosmic web.
With the advent of eROSITA (Predehl et al.2021), the number density of X-ray AGN increased to more than a hundred per square degree in the eROSITA Final Equatorial Depth Survey (eFEDS, 140 deg\({}^{2}\), \(\sim\)1,400 ks Brunner et al.2022; Salvato et al.2022). Accurate redshifts are required for precise clustering and lensing analysis. The dedicated spectroscopic observations of the X-ray sources detected in eFEDS (SDSS-IV, SDSS-V Abdurrouf et al.2022; Kollmeier et al.2017, Merloni et al. in preparation) enabled the accurate measurement of redshifts for about eleven thousand X-ray point sources in eFEDS (i.e., for \(\sim 50\%\) of the sources). This number of X-ray AGN with spectra is already comparable to its predecessor follow-up of ROSAT point sources (Comparat et al.2020).
Outstanding weak-lensing data products are now available over wide areas thanks to the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) (Aihara et al.2019). They measured accurate galaxy shapes for more than 20 source galaxies per square arc minute over vast areas (1,400 deg\({}^{2}\)), which almost completely cover the eFEDS field (Mandelbaum et al.2018).
With these two outstanding observational advances, we measure the auto-correlation function and the galaxy-galaxy lensing signal of X-ray-selected AGN to study their underlying dark matter halo distribution. We detail, in Sect. 2, the construction of the X-ray AGN sample, and the weak-lensing data products used. We describe the method to measure the clustering and galaxy-galaxy lensing in Sect. 3. The halo occupation distribution and sub-halo abundance matching models used are detailed in Sect. 4. Results are discussed in Sect. 5, 6. Throughout, we assume a Flat LCDM cosmology with \(H_{0}=67.74\) km s\({}^{-1}\) Mpc\({}^{-1}\) and \(\Omega_{m}(z=0)\)=0.3089 (Planck Collaboration et al.2020). The uncertainties are \(1\sigma\) unless stated otherwise. Magnitudes are in the AB system (Oke and Gunn, 1983). Throughout the article, we use AGN to designate X-ray-selected AGN.
## 2 Data
In this section, we describe the X-ray observations in Sect. 2.1 and the weak-lensing data products in Sect. 2.2.
### eRosita eFEDS
We use the public Early Data Release eROSITA point source catalog of the eFEDS Performance Verification survey (Brunner et al., 2022), The catalog contains 20,191 primary sources, over 140 deg\({}^{2}\), detected with a likelihood greater than 8 (ERO_DET_LIKE \(>8\)) and with reliable counterpart (\(\mathrm{CTP_{quality}}\geq 2\)) determined as described in Salvato et al. (2022). Simulations are only at X-ray wavelength and not in the optical, so the impact of the determination of the counterpart is studied empirically in Sect. 4 of Salvato et al. (2022). The trade-off between purity and completeness study shows that counterparts with a threshold of \(\mathrm{CTP_{quality}}\geq 2\) (p_any \(>0.035\)) have a purity and completeness both equal to 95%. We follow the recommendation of Salvato et al. (2022) and use the set of reliable counterparts above threshold with \(\mathrm{CTP_{quality}}\geq 2\). There are 1,219 extra sources \(\mathrm{ERO\_DET\_LIKE}>8\) & \(\mathrm{CTP_{quality}}<2\) that are then discarded. These sources are on the faint end of the X-ray flux distribution, inducing a 5% incompleteness at the faint end.
Given the large distribution of fluxes (luminosities) considered here, we neglect this bias.
2,160 sources are classified as stars either via astrometry, spectroscopy, X-ray, and opt/IR colors or via a dedicated analysis as described in Schneider et al. (2022) and removed from the rest of the study.
As shown by simulations (Liu et al., 2022; Seppi et al., 2022), faint clusters are contaminants of the point source catalog. In eFEDS, 129 clusters are present in the point source catalog (Bulbul et al., 2022). They are identified in Salvato et al. (2022) with the flag CLUSTER_CLASS \(\geq 3\) and are masked.
After these cuts on the eFEDS point source catalog, we obtain 17,902 AGN candidates over 140 deg\({}^{2}\) (density of 127.9 per square degree). Fig. 1 illustrates the light cone considered in this analysis.
#### 2.1.1 Masks
We must propagate the masks applied to the source catalog to the random catalog to estimate clustering. The random catalog is a set of un-clustered data points that cover the same sky area as the observations, see description in Sect. 2.1.4. As the masking radius for each detected source, we use its radius of maximum signal-to-noise augmented by 40 percent. This radius is determined while extracting the X-ray spectrum of each source (Liu et al., 2022), see Appendix A.1 for complete details.
The edges of the survey have a lower exposure time. We find that trimming the survey edges by requiring a minimum exposure time of 830 seconds minimizes the KS test values (between random and data vectors) with a minimal area loss, see Sec SS 2.1.4. After applying the minimum exposure time cut, we are left with 16,308 AGN candidates over 128 deg\({}^{2}\), resulting in a density of \(\sim\)127.4 deg\({}^{-2}\).
#### 2.1.2 Photometric redshifts
Photometric redshift estimation for galaxies hosting active galactic nuclei is complex (e.g. Salvato et al., 2018). In the eROSITA/eFEDs case, Salvato et al. (2022) measured photometric redshifts to have \(\sigma_{NMAD}=1.48\times{\rm median}\left(\frac{\left[v_{\rm true}-v_{\rm global }\right]}{1+v_{\rm true}}\right)\sim 0.05\) and a fraction of outliers, with \(\frac{\left[v_{\rm true}-v_{\rm global}\right]}{1+v_{\rm true}}>0.15\), of the order of 20%. At the bright end (r\(<\)21.5), we find that \(\sigma_{NMAD}\) decreases to \(\sim 0.03\) while the outlier fraction remains the same, 20%.
With the help of the simulation from Comparat et al. (2019), we find that the measured clustering using photometric redshift with such dispersion and fraction of outliers would result in losing between one-third and one-half of the amplitude of the clustering signal. So we will not use the photometric redshift to measure clustering statistics; instead, we focus on the sub-sample of 10,680 AGNs with spectroscopic observations, see the following Sect. 2.1.3.
#### 2.1.3 Spectroscopic redshifts
The eFEDS field was observed with the SDSS infrastructure (Gunn et al., 2006; Smee et al., 2013) in March-April 2020 with both BOSS spectrographs (1000 fibers per plate, SDSS-IV, Blanton et al., 2017) and March-April 2021 with a single BOSS spectrograph (500 fibers per plate, SDSS-V, Kollmeier et al., 2017, Merloni et al. in preparation). A total of 31 plates were observed, see Section 'SPIDERS' of Abdurro' et al. (2022) and the spectra are part of the SDSS DR18 (SDSS collaboration in preparation). The total area covered by SDSS-IV and V spectroscopic observations is 133.77 deg\({}^{2}\) (95% of the eFEDS area). The obtained spectroscopic redshift completeness depends on (_i_) the position in the sky; (_ii_) the optical magnitude of the source. We consider the z-band AB magnitude measured as in the legacy survey DR8 (Dey et al., 2019) and based on observations made with DECam(Flaugher et al., 2015). Although photometric redshifts are not accurate enough for clustering studies, they are of sufficient quality to compare the distribution of magnitudes and fluxes in broad redshift bins. Overall, we find that at a z-band magnitude of 21.25 (19.0), the completeness is 50% (90%). We find that, up to redshift \(\sim\)0.55, the spectroscopic sample is a fair sub-sample (as a function of optical magnitude and X-ray flux) of the entire population. SDSS-V observations being limited to z-band magnitudes brighter than 21.5, beyond a redshift of 0.55, we are missing a significant fraction of the AGN that are optically faint X-ray-selected AGN, see Fig. 2.
We estimate the spectroscopic completeness in \(\sim\)3.5 deg\({}^{2}\) equal area pixels (half the size of an SDSS plate \(\sim\)7 deg\({}^{2}\)). The minimum (maximum) completeness measured in a pixel is 13% (69%). The relative variations of the spectroscopic redshift distribution as a function of completeness are within the expected
Figure 1: Slice of the light cone sampled by the X-ray-selected eFEDS AGN sample in the redshift range \(0.05<z<0.55\) (blue crosses). The surrounding large-scale structure is sampled by GAMA galaxies (grey) and GAMA galaxy groups (purple) (Driver et al., 2022) as well as by eROSITA eFEDS clusters (red) (Liu et al., 2022).
fluctuations for pixels with completeness levels above 40%. So, we discard areas with completeness lower than 40%. It removes about 20 deg\({}^{2}\) of the area located at the edge of the eFEDs field (most of it overlaps with the low-exposure regions).
To summarize, to measure the clustering, we create an AGN sample covering redshifts between 0.05 and 0.55, where the spectroscopic sample is not biased compared to the parent photo-z sample. We obtain a sample of 1,992 AGN with spectroscopic redshift covering 122.3 deg\({}^{2}\). Figure 1 illustrates how the AGN considered in this analysis sample the large-scale structure observed with galaxies and groups from the GAMA survey and eROSITA eFEDS clusters. Table 1 summarizes the main properties of the considered sample. The mean redshift of the sample is 0.34, with a standard deviation of 0.13. The sample's mean X-ray luminosity in the soft 0.5-2 keV band is 42.91. The distribution around the mean is broad and has a standard deviation of 0.65.
We verified that the edges of the selection do not impact the clustering and lensing summary statistics: by moving the redshift cut from 0.05 to 0.1 and from 0.55 to 0.5 and by adding a minimum luminosity threshold of 41.5. Further splitting the sample in soft X-ray luminosity-limited samples (or following a visual inspection of the optical spectra) significantly decreases the signal-to-noise in the measurements, and HOD model parameters become unconstrained. Larger numbers of AGN with spectroscopic redshifts are required to investigate trends with parameters defining the sample.
Among the 1992 AGN studied here, 1648 (82.7%) have their spectroscopic redshift coming from SDSS observations. 270 (13.5%) come from GAMA observations (Liske et al., 2015). Then in smaller numbers, spectroscopic redshifts originate from: 46 from WiggleZ (Drinkwater et al., 2018), 8 from LAMOST DR5 v3 (Luo et al., 2015), 7 from 2SLAQ (Cannon et al., 2006), 5 from 2MASS (Skrutskie et al., 2006), 3 from HCSC (Oguri et al., 2018), 2 from 6dFGS (Jones et al., 2009), 1 from HYPERLEDA (Paturel et al., 2003), 1 from (Veron-Cetty and Veron, 2010), and 1 from RCSEDv2 (Chilingarian et al., 2021).
#### 2.1.4 Random catalogue
To measure clustering, one compares the set of observed points to a group of points with no clustering but all other aspects equal (window function, redshift distribution). In this section, we explain how the set of random points is constructed.
We draw a set of random points with a large uniform density of \(\sim\)81,000 deg\({}^{-2}\) on the sky (about 11.5 million points on eFEDS). We first trim it to follow precisely the edges of the survey. We then follow the methodology of Georgakakis et al. (2008) to downsample the uniform random catalog with the sensitivity map, see details in Appendix A.2. Heuristically, this step applies the X-ray flux limit and its variations across the field to the set of random points. The total number of random points remaining after downsampling, masking (see the previous section), and trimming (low exposure time region) is 3,713,726. The density of random points, \(\sim 30,000\) deg\({}^{-2}\), is more than 200 times larger than that of the data points (127.4 deg\({}^{-2}\)), which is largely sufficient.
We downsample the random catalog to follow the spectroscopic redshift completeness map and its dependency on R.A. and Dec. We cut the areas where spectroscopic completeness is lower than 40 %. As the relative variation in the redshift distribution is independent of the completeness (see the previous section), we shuffle the set of observed redshifts and assign them to the random points, regardless of the completeness level.
### HSC-SSP weak-lensing data
We use the HSC S19A weak-lensing products based on the HSC-SSP accumulated \(i\)-band imaging data from 2014 to 2019.
\begin{table}
\begin{tabular}{c|c c c c} \hline \hline property & min & mean & max & std \\ \hline redshift & 0.05 & 0.34 & 0.55 & 0.13 \\ soft L\({}_{X}\) & 40.49 & 42.91 & 45.12 & 0.65 \\ g\({}_{AB}\) & 15.16 & 20.1 & 23.13 & 1.41 \\ r\({}_{AB}\) & 14.26 & 19.23 & 22.39 & 1.35 \\ z\({}_{AB}\) & 13.62 & 18.57 & 21.82 & 1.32 \\ \hline \end{tabular}
\end{table}
Table 1: Properties of the sample. Minimum, mean, maximum, and standard deviation of the redshift, soft band X-ray luminosity, and g, r, z magnitudes from the legacy survey (Dey et al., 2019).
Figure 2: Spectroscopic completeness as a function of r-band magnitude vs. redshift (top) and soft X-ray luminosity (0.5-2 keV) vs. redshift (bottom). The completeness coverage is homogeneous below the redshift of 0.6. At higher redshift, completeness at the faint end impacts the sample significantly.
The original HSC-SSP S19A wide-layer data covers about 512 deg\({}^{2}\), and it reduces to 433.48deg\({}^{2}\) after the full-color full-depth (FCFD) selection. With the \(i\)-band magnitude cut of 24.5, the observed number density reaches up to 22.9 arcmin\({}^{-2}\). The deep imaging data enable a comprehensive redshift coverage ranging from 0 to 3, and the calibrated bias residual shows no dependence on the redshift.
There are several major updates in the shape catalog version from hscPipe4 to hscPipe7 (used here). These improvements include PSF modeling, image co-addition, bright star masking, and background subtraction.
Due to the bad weather, volcanic activity, and other telescope downtimes, the observation time has significantly decreased. To compensate for the loss, besides the 30 nights additional observation time, the survey strategy has been modified by _(i)_ reducing to 80% the time in the Deep/Ultra Deep fields, _(ii)_ lowering the seeing conditions in \(i\)-band, _(iii)_ changing the dither pattern from 6 to 5 in the \(i\), \(z\) and \(y\) bands. The latter results in a 0.1 magnitude difference in the \(i\)-band depth. Altogether, the expected coverage should still reach up to 1200 deg\({}^{2}\). We detail several essential aspects of the S19A imaging data below.
#### 2.2.1 Photometry
The most important updates to the pipeline and data processing are _(i)_ improved sky background subtraction; _(ii)_ improved image co-addition warping kernel from Lanczos3 to Lanczos5; _(iii)_ the addition of two new filters \(r2\) and \(i2\) that substitute the original \(r\) and \(i\)- band filters; _(iv)_ improved Point Spread Function (PSF) based on PSFEx; _(v)_ new masks around bright stars composed of ghosts, blooming, halo. About the sky subtraction algorithm, we apply the newest version based on Aihara et al. (2022) rather than the global-global algorithm Aihara et al. (2019). The latter leads to about 10% loss of extended source near the shape catalog cut at \(i\)-band magnitude equals 24.5.
We use the Forward Global Calibration Method (FGCM; Burke et al., 2018) that was firstly developed for the Dark Energy Survey (DES; Sevilla-Noarbe et al., 2021) and now has been merged into the LSST/HSC pipeline. It starts with modeling the instrumental throughput measurements such as mirrors, filters, and detectors. Then, the atmospheric model (MODTRAN4 Berk et al., 1999) carries out detailed modeling of atmospheric throughput as a function of zenith distance at the location of the SUBARU telescope on Mauna Kea.
The performance of the photometry is tested in two ways, an internal test by comparing the PSF magnitude to the Kron magnitude for a bright star sample (\(i\)\(<\)21.5) in the Wide XMM-LSS field. The standard deviation of the difference (PsfMag-KronMag) achieved is better than 1%, independently of the filters and fields. One exception is the \(y\)-band, which shows a slightly larger scatter of 1.5%. In addition, the difference between the CModel magnitude and PSF magnitude is below 0.2%. For the external test, PanSTARR1 stars brighter than \(r\)-band 20 mag are used (PS1; Chambers et al., 2016). The scatter level is also at about the 1% level indicating good photometric performance. Observations are mixing the \(i\) (\(r\)) and the \(i2\) (\(r2\)) filters. It results in a small offset in the \(i\) (\(r\)) photometry in the Deep+UltraDeep fields and small regions of the Wide fields. Good photometric performance is a pre-condition for the redshift calibration, described below.
#### 2.2.2 Photometric redshifts for sources
The overlapping photometric and public spectroscopic surveys with HSC-SSP provide a wealth of data for photometric redshift calibration. These data sets include zCOSMOS DR3 (Lilly et al., 2009), UDSz (Bradshaw et al., 2013; McLure et al., 2013), 3D-HST (Skelton et al., 2014; Momcheva et al., 2016), FMOS-COSMOS (Silverman et al., 2015) and so forth. There are about 170k spectroscopic redshifts (spec-\(z\)) and 37k g/prism-\(z\) with high-quality.
To cover the wide range of photo-\(z\) methods (Tanaka et al., 2018; Nishizawa et al., 2020) used different techniques on the HSC-SSP data: template-fitting, empirical-fitting, and machine learning. These include the Mizuki template-fitting method Tanaka (2015), MLK Self-Organise Map (SOM; More et al. in prep), NNPZ Nearest Neighbors P(z) (Cunha et al., 2009), FRANKEN-Z Flexible Regression over Associated Neighbors with Kernel density estimation for Redshifts, DEMP Direct Empirical Photometric code ( Hsieh & Yee, 2014) and Ephor Extended Photometric Redshift (EPHOR). There is a newly developed machine learning photometric method for HSC-SSP Y3 shape catalog, i.e., \(\alpha\)NHz (A. J. Nishizawa et al. in preparation).
The metrics to quantify the performance of each method are the bias defined as \(\Delta z=(z_{phot}-z_{ref})/(1+z_{ref})\), the dispersion \(\sigma_{phot}=1.48\times MAD(\delta z)\) (MAD is the median absolute deviation), the outlier rate \(f_{outlier}=N(\Delta z)>0.15/N_{total}\) and the loss function \(L(\Delta z)=1-1/(1+(\Delta z/\gamma)^{2})\) with \(\gamma=0.15\). The photometric redshift used in this work is based on dNNz method, which achieves an accuracy with a \(\Delta z=10^{-4}\) bias, a dispersion \(\sigma_{phot}=3\%\), and an outlier rate smaller than \(f_{outlier}\leq 10\%\).
#### 2.2.3 Shape catalog
The HSC galaxy sample is selected following a series of basic flag cuts, such as i_detect_isprimary, i_extendedness_value and i_sdsscentroid_flag. The detailed descriptions are listed in Table 2 of Li et al. (2022). The shapes of galaxies are measured with re-Gaussianization method (Hirata & Seljak, 2003) (reGauss which has been merged to GalSim Rowe et al. (2015)), the PSF effects are corrected during the measurement process. HSC covers six discrete fields named after overlapping regions from previous surveys: XMM, HECTOMAP, WIDE12H, GAMA09H, GAMA15H, and VVDS. The eFEDS region overlaps with GAMA09H. Note that the HSC region GAMA09H covers a larger area than the original GAMA09H field and completely encompasses the eFEDS field.
The final shape catalog contains the two components of the ellipticity:
\[(\mathrm{e_{1},e_{2}})=\frac{1-(\mathrm{b/a})^{2}}{1+(\mathrm{b/a})^{2}}(\mathrm{ cos2}\phi,\mathrm{sin2}\phi), \tag{1}\]
where \(b/a\) is the ratio between the minor axis and major axis, and \(\phi\) is the position angle of the major axis with respect to the sky coordinates. The shear distortion, \(\gamma_{i}\), is then related to the \(e_{i}\) (i=1,2) such that
\[\gamma_{i}=\frac{1}{2\mathcal{R}}\langle\mathrm{e_{i}}\rangle(\mathrm{i}=1,2), \tag{2}\]
where \(\mathcal{R}\) is the response of the galaxy ellipticity to a small distortion defined in Kaiser et al. (1995); Bernstein & Jarvis (2002). The response is calculated from the calibrated parameters \(e_{rms}\)
and \(\sigma_{e}\) based on simulations (Mandelbaum et al., 2018; Li et al., 2022) as follows
\[\mathcal{R}=1-\frac{\sum_{\rm i}{\rm w_{i}}{\rm e}^{2}_{\rm rms,\rm i}}{\sum_{ \rm i}{\rm w_{i}}}. \tag{3}\]
The weighting term \({\rm w_{i}}\) in Eq.3 is composed of the per-component error from simulation due to photon noise \(\sigma_{e;\rm i}\) and the rms of galaxy shape distribution \(e_{rms;\rm i}\), \({\rm w_{i}}=1/(\sigma_{e;\rm i}^{2}+{\rm e}^{2}_{\rm rms;\rm i})\).
The reGauss algorithm suffers from several estimation biases, e.g., model bias, noise bias, and selection bias which can be classified into multiplicative bias \({\rm m_{i}}\) and additive bias \({\rm c_{i}}\) (i=1,2), so that
\[\gamma_{\rm i}=(1+{\rm m_{i}})\gamma_{\rm i}^{\rm true}+{\rm c_{i}}. \tag{4}\]
The final shear estimator is obtained with Eq. 5. It does not incorporate the geometry factor \(\Sigma_{crit}\) described in Sec.3.2.
\[\langle\gamma_{\rm i}\rangle=\frac{\sum_{\rm i}w_{\rm i}{\rm e}_{\rm ij}}{2 \mathcal{R}(1+\langle{\rm m_{i}}\rangle)\sum_{\rm i}w_{\rm j}}-\frac{\langle{ \rm c_{i}}\rangle}{1+\langle{\rm m}\rangle}. \tag{5}\]
Both multiplicative and additive biases are calibrated based on the simulations mentioned above. The two biases are then assigned to each galaxy as a function of SNR and resolution \(R_{2}\). Additionally, there is selection bias as well as weight bias. The overall bias is quantified as the residuals for both multiplicative \(\delta m\) and additive bias \(\delta a\), both of which can reach below 1% level for HSC-SSP Y3 shape catalog Li et al. (2022).
## 3 Summary statistics
Galaxy clustering and gravitational lensing probe the galaxy and matter over-density field's auto and cross-correlations as a function of scale via a biasing function (Tegmark and Peebles, 1998; Tegmark and Bromley, 1999; Dekel and Lahav, 1999). These measurements are well suited to constrain the biasing function, also named more generically the galaxy-halo connection (Sheth and Tormen, 1999; Wechsler and Tinker, 2018). With the data described above, we compute two summary statistics: the AGN-AGN auto-correlation (clustering, Sect. 3.1) and galaxy-galaxy lensing with the AGN population being the lenses (Sect. 3.2).
### Clustering measurement
We use the Landy and Szalay (1993) estimator to measure the projected two-point correlation function, labeled \(w_{p}(r_{p})\) (for detailed definition, see e.g., Davis and Peebles, 1983). To count pairs and integrate along the line of sight, we use the corffunc software (Sinha and Garrison, 2020). For the integration, we use \(\pi_{max}=40\)\(h^{-1}\)Mpc. We carried out measurements with shorter and longer \(\pi_{max}\) and found that with 40, we would obtain the largest signal-to-noise in the clustering measurement.
We randomly down-sample the catalog of random points for the clustering measurement to have twenty times the number of AGN. To have consistent 3D positions between the optical spectra and the X-ray sources, we compute the clustering using the position on the sky of the optical counterparts (Salvato et al., 2022); we do not use the positions of the X-ray sources. The projected correlation function obtained is shown in Fig. 3 (black error bars). The clustering measurement's uncertainty is estimated using the diagonal component of the covariance matrix obtained with 18 eFEDS simulated catalogs (Liu et al., 2022). These simulated eFEDS observations are based on the empirical models of the X-ray cosmic web from Comparat et al. (2019, 2020). The yellow shaded area in Fig. 3 shows the prediction from the 18 mocks. We find that the forecast is faithful to the observations.
Following Driver and Robotham (2010), we estimate the cosmic variance in this field to be 1%, which we add as a constant systematic uncertainty at all scales to the clustering measurement. Note that it is small compared to statistical uncertainties. Using the eFEDS simulations, we find that clustering summary statistics are significantly biased low for separations \(r_{p}>\)40 \(h^{-1}\)Mpc. This is due to the finite volume observed. So, we exclude from the fitting procedure clustering measurements with a separation larger than 40 \(h^{-1}\)Mpc. The total signal-to-noise in the clustering measurement is 17.7, split into 11 radial bins. We sample the separation range with five bins per decade evenly log-spaced (0.2 dex steps) between 0.25 (10\({}^{-0.6}\)) and 39.8 (10\({}^{0.6}\)) \(h^{-1}\)Mpc.
The fiber collision radius in SDSS is 62 arc seconds (\(\sim\)0.25\(h^{-1}\)Mpc at the mean redshift of the sample). The eROSITA PSF is 30 arc seconds, so X-ray-selected AGN pairs with a separation smaller than one arc minute are hardly detected. Moreover, since the AGN sample considered here is sparse (120 deg\({}^{-2}\sim\)0.03 arc minutes\({}^{-2}\)) and spread over a long line of sight, AGN close pairs are small in numbers. Using the mock catalogs limited to redshifts \(0.05<z<0.55\), we estimate the number of expected pairs with an angular separation smaller than 62 arc seconds to be typically a handful: less than 10. Only half have physical separation smaller than 40 \(h^{-1}\)Mpc. So, the number of missed pairs due to fiber collisions is negligible. So in our case, we consider that fiber collisions are not an issue, and we define our lowest separation bin at 0.25 \(h^{-1}\)Mpc.
### Galaxy-galaxy lensing measurement
The galaxy-galaxy lensing measurement is a cross-correlation between positions of foreground lenses (AGN in our case) and shapes of background galaxies acting as sources (HSC galaxies), see reviews from Bartelmann and Schneider (2001); Refregier (2003). This measurement directly traces the galaxy halo
Figure 3: Projected clustering measurement of the X-ray flux-limited eFEDS AGN sample in the redshift range \(0.05<z<0.55\) (black). The prediction from the 18 eFEDS simulations appears in yellow. The best-fit model (jointly with weak-lensing observations) is in red.
connection (e.g. Mandelbaum et al., 2005; Seljak et al., 2005). Numerous studies have used galaxy-galaxy lensing (sometimes combined with galaxy clustering) to trace the galaxy-halo connection in general (Leauthaud et al., 2011; Coupon et al., 2015; Zu & Mandelbaum, 2015; Dvornik et al., 2018; Zacharegkas et al., 2022).
We combine the X-ray point sources from the eFEDS region and the HSC shape catalog to compute the galaxy-galaxy lensing using each source galaxy and its probability distribution function as a function of redshift (\(p(z)\)). The physical interpretation of the galaxy-galaxy lensing signal is the difference between the average density inside a certain projected radius \(R\) and the average density at that same radius, so an excess surface density (ESD, \(\Delta\Sigma\)), that is
\[\Delta\Sigma(\mathrm{R})=\bar{\Sigma}(\leq\mathrm{R})-\Sigma(\mathrm{R}). \tag{6}\]
We follow the measurement procedure described in Miyatake et al. (2019); Luo et al. (2022), that
\[\Delta\Sigma(\mathrm{R})=\frac{1}{2\mathcal{R}(\mathrm{R})}\frac{\sum_{1}^{ \mathrm{N_{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ }}}}}}}}}}}}}}}}{\mathrm{N_{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ }}}}}}}}}}}}}}} {\mathrm{N_{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ { \mathrm{ \mathrm{ \mathrm{ { \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ { \mathrm{ { \mathrm{ { \mathrm{ \mathrm{ { \mathrm{ { \mathrm{ \mathrm{ \mathrm{ { \mathrm{ \mathrm{ { \mathrm{ \mathrm{ \mathrm{ }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\ \ \ \ \ \}\)\}\)\}\\\\\\\\\\\}}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\}
AGN. In this study, since the correlation function measurements do not depend on the normalization of the occupation distribution, we arbitrarily set \(f_{A}\) to 1.
To avoid sampling un-physical values of \(M_{\rm sat}\), the parameter passed to the fitting routine is \(M_{\rm sat}-M_{\rm min}\) with boundaries specified in Table 2.
To fit for the \(\Delta\Sigma\) measurement at small separations and benefit from the signal present, we need to add the prediction for a point-like mass term that represents the baryonic lensing mass of the AGN host galaxies. It adds the parameter \(M_{12}^{*}\) as follows:
\[\Delta\Sigma^{*}(r)=\frac{10^{M_{12}^{*}+12}}{\pi r^{2}}. \tag{11}\]
The posterior of this parameter represents the mean baryonic lensing mass of the galaxies hosting AGN. This mass is related to stellar mass (inferred with stellar population synthesis models) but will also encompass gas in and around the galaxy. This baryonic lensing mass can be considered the upper limit of the mean stellar mass of galaxies hosting AGN.
In total, we fit for five parameters on the two measurements \(\Delta\Sigma\) (\(w_{p}(r_{p})\)) which have S/N=46 (17.7) in 15 (11) radial bins. The parameters are sampled with a flat prior (in linear space) within broad boundaries as specified in Table 2.
### Sub halo abundance matching models
The Comparat et al. (2019, 2020) empirical AGN model statistically links the dark matter haloes to the probability of hosting an AGN and its spectral energy distribution. By construction, it follows the X-ray luminosity function from Aird et al. (2015). Importantly for interpretation, the assignment is done regardless of the environment in which haloes live. The model has two parameters: the fraction of AGN in satellites sub haloes and the scatter in the abundance matching relation between stellar mass and hard X-ray luminosity.
We show the direct \(w_{r}(r_{p})\) prediction from the mock catalogues of Liu et al. (2022) in Fig. 3. It is consistent with observations. It was obtained with \(f_{\rm sat}=10\%\) (fraction of AGN being satellites) and \(\sigma=1\) (scatter in the abundance matching procedure between hard X-ray luminosity and stellar mass). These parameters were chosen by hand by Comparat et al. (2019, 2020). At that time (before this study), such parameters resulted in reasonable predictions.
Creating a complete SHAM-base mock catalog is long (order of a few CPU hours) and thus impractical for fitting purposes. Furthermore, with current light cones constructed with replications, predicting the galaxy-galaxy lensing signal is tedious as the dark matter particles are not kept. So, instead of predicting summary statistics as measured as a function of the SHAM parameters, we directly predict the halo occupation distribution curves as a function of \(f_{\rm sat}\) and \(\sigma\). Thus, we sample a small and finite number of (\(f_{\rm sat}\), \(\sigma\)) combinations and create individual mock catalogs to predict the HOD curves.
## 5 Results
We discuss here the results of the fitting procedure and the comparison between models. In Sec. 5.1 (5.2), we discuss the results obtained with the HOD (SHAM) model. For the first time, we measure with relatively small uncertainties the complete halo occupation distribution of a low redshift flux-limited X-ray-selected sample of AGN. We obtain a global view of the distribution of haloes hosting X-ray AGN; see Fig. 6.
### HOD results
We fit the parameters of the HOD model with a nested sampling method ultranest(Buchner, 2021). The resulting parameters are given in Table 2. The constraints on the HOD parameters (when fitting individually each summary statistic or both jointly) are shown in Fig. 5. It illustrates the complementary nature of the two measurements. The comparison between the joint best-fit model and the clustering measurements and lensing measurements are shown in Figs. 3, 4. The models are sensible. They account for the observations.
The five parameters are meaningful, although not precisely constrained by the joint fit of both summary statistics. For the central haloes, \(M_{\rm min}\) takes a median posterior value of 13.06\(\pm\)0.44, the width of the error function is found at \(\sigma_{\log_{fit}M}=1.3\pm 0.2\). There is a low 1\(\sigma\)-level tension between the constraints on these parameters obtained by each summary statistic; see Fig. 5. Due to the higher signal-to-noise on the lensing statistic, the combined best-fit values are closer to the individual best-fit value on the lensing statistic. For the satellites, the slope is best-fit at \(\alpha_{\rm sat}=\)0.73\(\pm\)0.38 and the transition occurs in haloes 10 to 100 times more massive than the typical halo : \(M_{\rm sat}-M_{\rm min}=1.46\)\(\pm\) 0.52. Both summary statistics point to these parameter values (Fig. 5). The typical baryonic lensing mass of galaxies hosting
\begin{table}
\begin{tabular}{c c c} \hline \hline parameters & min & max & 0.05 \(<z<\) 0.55 \\ \hline \(M_{\rm min}\) & 8.0 & 15.0 & 13.06 \(\pm\) 0.44 \\ \(\sigma_{\log_{10}M}\) & 0.05 & 1.5 & 1.28 \(\pm\) 0.2 \\ \(\alpha_{\rm sat}\) & 0.1 & 1.5 & 0.73 \(\pm\) 0.38 \\ \(M_{\rm sat}-M_{\rm min}\) & -3.0 & 2.45 & 1.46 \(\pm\) 0.52 \\ \(M_{12}^{*}\) & -4.0 & 0.1 & -0.96 \(\pm\) 0.45 \\ evidence (logZ) & \multicolumn{3}{c}{-39.88 \(\pm\) 0.14} \\ & \multicolumn{3}{c}{ deduced parameters} \\ & \multicolumn{3}{c}{\(b(z=\bar{z})=0.991^{+0.078}_{-0.096}\)} \\ & \multicolumn{3}{c}{\(b(z=0.1)=0.915^{+0.065}_{-0.08}\)} \\ & \multicolumn{3}{c}{\(f_{\rm sat}<20.6\%\)} \\ \hline \multicolumn{3}{c}{4-parameter fit, \(\sigma_{\log_{10}M}=1.3\)} \\ \hline \(M_{\rm min}\) & 13.09 \(\pm\) 0.19 \\ \(\alpha_{\rm sat}\) & 0.75 \(\pm\) 0.39 \\ \(M_{\rm sat}-M_{\rm min}\) & 1.56 \(\pm\) 0.46 \\ \(M_{12}^{*}\) & -0.97 \(\pm\) 0.46 \\ evidence (logZ) & \multicolumn{3}{c}{-38.6 \(\pm\) 0.12} \\ & \multicolumn{3}{c}{ deduced parameters} \\ & \multicolumn{3}{c}{\(b(z=\bar{z})=1.001^{+0.075}_{-0.094}\)} \\ & \multicolumn{3}{c}{\(b(z=0.1)=0.918^{+0.065}_{-0.076}\)} \\ & \multicolumn{3}{c}{\(f_{\rm sat}<16.8\%\)} \\ \hline \multicolumn{3}{c}{4-parameter fit, \(\sigma_{\log_{10}M}=1.0\)} \\ \hline \(M_{\rm min}\) & 12.47 \(\pm\) 0.26 \\ \(\alpha_{\rm sat}\) & 0.75 \(\pm\) 0.39 \\ \(M_{\rm sat}-M_{\rm min}\) & 1.84 \(\pm\) 0.53 \\ \(M_{12}^{*}\) & -1.0 \(\pm\) 0.5 \\ \(M_{\rm 12}\) & -40.34 \(\pm\) 0.11 \\ \multicolumn{3}{c}{ deduced parameters} \\ & \multicolumn{3}{c}{\(b(z=\bar{z})=0.996^{+0.067}_{-0.092}\)} \\ & \multicolumn{3}{c}{\(b(z=0.1)=0.919^{+0.059}_{-0.081}\)} \\ & \multicolumn{3}{c}{\(f_{\rm sat}<66.4\%\)} \\ \hline \end{tabular}
\end{table}
Table 2: HOD parameters obtained (median of the posterior) by jointly fitting the auto-correlation function and the galaxy-galaxy lensing of the X-ray flux-limited AGN sample. The uncertainties quoted are 1\(\sigma\) (15.9–84.1 percentiles). Priors are flat in linear space.
these AGN is \(M_{12}^{*}=\) -0.96 \(\pm\) 0.45. It sets an upper limit to the mean stellar mass of galaxies hosting AGN of \(\sim 10^{11}\)M\({}_{\odot}\). The 1\(\sigma\) boundaries encompass 3.9 \(\times 10^{10}\) and 3.1 \(\times\) 10\({}^{11}\)M\({}_{\odot}\), which is in fair agreement with expectations from AGN host stellar mass function (Bongiorno et al., 2016; Yang et al., 2018). This parameter is not degenerate with others and is constrained only by the lensing measurements at small separations.
We note a degeneracy between \(M_{\rm min}\) and \(\sigma_{\log_{10}M}\), which we investigate. We decrease the number of parameters by fixing \(\sigma_{\log_{10}M}\) to 1.3 (similar to the best-fit value) and 1 (to force a less broad halo distribution, a sharper transition). We obtain a set of best-fit parameters (see Table 2) that are compatible with the 5-parameter fit. When fixing \(\sigma_{\log_{10}M}\) to 1.3, we obtain \(M_{\rm min}13.09\pm 0.19\), similar value to the 5-parameter fit but with half the uncertainty. Results for other parameters remain unchanged. When fixing \(\sigma_{\log_{10}M}\) to 1, \(M_{\rm min}\) is logically forced to lower values to \(\sim 12.5\) to be able to fit the overall signal. The halo occupation distribution posterior is close to that of the five parameter fit, see Fig. 6 black and yellow/orange contours.
We obtain a global view of the distribution of haloes hosting X-ray AGN; see Fig. 6. The 4-parameter best-fit model is within the 1\(\sigma\) uncertainty of the 5-parameter best-fit model. Due to the degeneracy between \(M_{\rm min}\) and \(\sigma_{\log_{10}M}\), the 4-parameter fit HOD with \(\sigma_{\log_{10}M}=1\) is skewed towards lower masses compared to the 5-parameter fit. With the HOD model, we derive the average halo mass hosting central (central or satellite) AGN is 3.93\({}^{+2.03}_{-2.4}\times 10^{12}\)M\({}_{\odot}\) (4.95\({}^{+2.63}_{-1.99}\times 10^{12}\)M\({}_{\odot}\)). These values are comparable to the findings of Rodriguez-Torres et al. (2017). We measure that
Figure 5: Constraints were obtained on the HOD parameters when fitting only the clustering measurement (yellow), the lensing measurement (purple), or both jointly (blue). Contours show 1 and 2 \(\sigma\) constraints. Most of the constraining power comes from galaxy-galaxy lensing.
the distribution of halo masses is broad. We thus confirm that quoting a typical halo mass will be extremely sensitive to the definition of what 'typical' means; see discussion in Leauthaud et al. (2015).
The direct HOD predictions from mock catalogs from Leauthaud et al. (2015); Georgakakis et al. (2018); Comparat et al. (2019) are shown on the right panel of Fig. 6. They are within the fitted contours obtained. The mocks from Leauthaud et al. (2015); Georgakakis et al. (2018) have a lower \(\sigma_{\log_{10}M}\) value (sharper transition) and are thus more in line with the 4-parameter fit. The mock from Comparat et al. (2019) has a higher \(\sigma_{\log_{10}M}\) value and is comparable to the 5-parameter fit (see more discussion in the SHAM section below). The normalization of \(\langle N(M)\rangle\) can be added as a parameter and possibly constrained by jointly fitting the clustering and lensing summary statistics with the stellar mass (or luminosity) function of galaxies hosting X-ray AGN to have a handle on the fraction of galaxies hosting an AGN. Though measuring reliable host-galaxy stellar masses in the case of type 1 AGN is complex (Ciesla et al., 2015; Zou et al., 2022), and is left for future studies.
The large-scale halo bias inferred is given in Table 2 and shown in Figs. 7. At the mean redshift (0.34), it takes a value of \(b(z=0.34)=0.99^{+0.08}_{-0.10}\), which extrapolated to redshift \(z=0.1\) becomes \(b(z=0.1)=0.92^{+0.07}_{-0.08}\). The deduced large-scale halo bias is the same if we fit the HOD model with 4 or 5 parameters, see Table 2. Krumpe et al. (2015) measured the bias of X-ray-selected AGN in a similar redshift range but for intrinsically more luminous AGN. With this analysis, we add a new measurement of the bias at lower soft X-ray luminosity: \(8.1\times 10^{42}\) erg s\({}^{-1}\). We confirm the weak positive correlation between bias and soft X-ray luminosity found by Krumpe et al. (2015), see Fig. 7 bottom panel. We fit a linear relationship between the quantities and obtain \(b=(0.48\pm 0.14)L_{X}+(-19.68\pm 6.17)\). The slope value obtained is \(3.3\sigma\) (0.48/0.14=3.3) away from 0.
Other X-ray-selected AGN clustering studies were either at lower redshift (Cappelluti et al., 2010) or higher redshift (Gilli et al., 2009; Starikova et al., 2011; Koutoulidis et al., 2013; Viitanen et al., 2019; Allevato et al., 2019) and always covered higher luminosities. This new study is complementary to them.
### Halo abundance matching results
The model has two parameters: the fraction of satellite AGNs and the scatter in the relation between stellar mass and X-ray luminosity. The predicted curves extend to halo masses of \(10^{11.5}\)M\({}_{\odot}\) (and not lower) due to the resolution of the simulation used. Both impact the shape and amplitude of the clustering signal and the HOD. Figure 8 shows the predicted HOD curves for a subset of the parameter space explored. In the top left panel, the satellite fraction is fixed to 10%, and the \(\sigma\) parameter varies. The lower the \(\sigma\), the sharper the transition is. The \(\sigma\) parameter from SHAM is related to the \(\sigma_{\log_{10}M}\) parameter from the HOD model. Mock catalogs with higher \(\sigma\) have a distribution of dark matter haloes more extended towards lower masses. In the top right panel, \(\sigma\) is fixed to 0.8, and the \(f_{\rm sat}\) varies. The higher the \(f_{\rm sat}\), the steeper the slope of the satellite occupation curve (the larger the \(\alpha\) parameter). The \(f_{\rm sat}\) parameter is related to both the \(\alpha_{\rm sat}\) and the \(M_{\rm st}\) HOD parameters.
We compute a distance, denoted \(d\), between each predicted HOD curve (\(N^{SHAM}(M)\)) and the 50th precentle of the inferred HOD model as follows
\[d=\Sigma_{M=11.5}^{M=15.5}\frac{\left[N^{SHAM}(M,f_{\rm sat},\sigma)-N_{HOD}^ {50\%}(M)\right]^{2}}{\left[N_{HOD}^{84.19}(M)-N_{HOD}^{15.9\%}(M)\right]^{ 2}}. \tag{12}\]
Figure 8 (bottom panels) shows the distances as a function of \(\sigma\) and \(f_{\rm sat}\). We find mock catalogs constructed with parameters satisfying \(\sigma<2-f_{\rm sat}/10\) predict halo occupation distributions well
Figure 6: Inferred halo occupation distribution (solid) split into central (dashes) and satellite (dots) for the 4 and 5 parameter HOD fits (yellow/orange and black). The 4-parameter best-fit model is within the 1\(\sigma\) uncertainty of the 5-parameter best-fit model. Due to the degeneracy between \(M_{\rm halo}\) and \(\sigma_{\log_{10}M}\), the 4-parameter fit HOD with \(\sigma_{\log_{10}M}=1\) is skewed towards lower masses compared to the 5-parameter fit. The direct predictions from mock catalogs from Leauthaud et al. (2015); Georgakakis et al. (2018); Comparat et al. (2019) are shown on the right panel. They are within the fitted contours obtained. The mocks from Leauthaud et al. (2015); Georgakakis et al. (2018) have a lower \(\sigma_{\log_{10}M}\) value (sharper transition) and are thus more in line with the 4-parameter fit. The mock from Comparat et al. (2019) has a higher \(\sigma_{\log_{10}M}\) value and is comparable to the 5-parameter fit.
within the contours of the 5-parameter best-fit HOD inferred from the observations (Fig. 8 bottom left panel). Parameter combinations such as \(\sigma>2-f_{\rm sat}/10\) are less preferred (top right corner of the bottom left panel). \(d\) is minimized for \(\sigma=0.8\) and \(f_{\rm sat}=4\%\), see the star in the Figure. The six smallest distances are pointed out with empty circles. When comparing to the 4-parameter best-fit HOD contours (Fig. 8 bottom right panel), parameters within \(\sigma<2.4-f_{\rm sat}/10\) and \(\sigma>0.5\) are acceptable. Here we find that solutions with low \(\sigma\) are less preferred. In that case, \(d\) is minimized for \(\sigma=1.2\), and \(f_{\rm sat}=4\), see the star in the Figure. The six smallest distances are pointed out with empty circles. In both cases (comparing either to the 4-parameter of the 5-parameter HOD fit), the best solutions point towards low \(f_{\rm sat}\) values.
Mocks with both high \(\sigma\) and high \(f_{\rm sat}\) are far from the observations and are ruled out.
## 6 Summary and discussion
This article provides a complete picture of how soft X-ray AGNs populate the cosmic web (Fig. 1). This achievement is possible thanks to two factors: _(i)_ the combination of the eROSITA eFEDS X-ray survey with its dedicated SDSS spectroscopic follow-up and with the HSC S19A lensing products _(ii)_ the complementary nature of the two summary statistics fitted (Figs. 3, 4). We obtain meaningful HOD constraints for an X-ray-selected AGN sample (Fig. 5, 6). We interpret the summary statistics with state-of-the-art HOD and SHAM models (Sect. 5). We set upon firm footage the fact that the mass distribution of haloes hosting X-ray-selected AGN is broad, as hinted by previous studies. Both models point to a shallower satellite slope than for galaxy surveys meaning that the satellite fraction for X-ray-selected AGN is low, similar to the findings of Miyaji et al. (2011). Interestingly, we find a relatively large \(\sigma_{\log_{10}M}\) that is likely related to the width of the specific accretion rate distribution. Contrasting our results with those of Krumpe et al. (2015), the large-scale halo bias of X-ray-selected AGN appears to correlate (3.3\(\sigma\) significance) with soft band X-ray luminosity (Fig. 7). We compare the results with predictions from SHAM models and can rule out a portion of the parameter space (Fig. 8).
### On the \(\sigma\) and \(\sigma_{\log_{10}M}\) parameters
The \(\sigma\) parameter in the SHAM model is the scatter in the abundance matching relation between the stellar mass of the galaxy hosting the AGN and the AGN hard X-ray luminosity (2-10 keV). The probability distribution function of specific accretion rate resulting from a broad range of galaxy stellar masses (hosting AGN) is close to a power-law (with slope -1) in the range \(31.5<\log_{10}(\lambda_{SAR})<33.5\) (see Fig. 5 of Comparat et al. (2019)). The distribution on obtained deviates from the power law at high accretion rates. Indeed the scatter induces an exponential cut-off. The distribution at the faint end of the function would require higher-resolution simulations to be populated. The HOD results obtained here are compatible with SHAM models if \(\sigma\in[0.8,1.2]\) and incompatible for low values of \(\sigma<0.5\) (for the four parameters HOD fit) or high values (for the 5-parameter HOD fit). Indeed low values of \(\sigma\) induce a steeper probability distribution function of specific accretion rate, which are excluded by observations (Georgakakis et al., 2017). In the opposite regime, large values of \(\sigma\) induce a shallow (tending to become flat) probability distribution function of specific accretion rate when considering the entire population. In a sense, the \(\sigma\) SHAM parameter is related to how broad the distribution of specific accretion rate and its slope is.
The \(\sigma_{\log_{10}M}\) parameter characterizes how broad the host halo distribution is. It is related to the diversity of host galaxies and their stellar mass via the stellar-to-halo mass relation (Moster et al., 2013; Behroozi et al., 2013). The relatively high \(\sigma_{\log_{10}M}\sim 1.3\) parameter obtained indicates that the host halo mass distribution and, thus, the host stellar mass distribution are both broad. This is consistent with studies of the AGN host-galaxy stellar mass function (Bongiorno et al., 2016).
So, it seems that both models point to the same general interpretation: the distribution of host-galaxy stellar mass and that of specific accretion rate are 'broad', which strengthens direct observations of these distributions (Bongiorno et al., 2016; Georgakakis et al., 2017), even if these might be subject to systematic effects in the measurement of the stellar mass of type 1 AGN (Ciesla et al., 2015).
Figure 7: Inferred large-scale halo bias as a function of redshift (top panel) and luminosity (bottom panel) compared to (Krumpe et al., 2015) (red-shift range 0.16-0.36). We confirm the trend with soft X-ray luminosity and obtain a best-fit of \(y=(0.482\pm 0.143)x+(-19.684\pm 6.173)\) between the soft X-ray luminosity and the large-scale halo bias.
With the innermost lensing measurements, we measure the baryonic lensing mass for this sample (\(M_{12}^{*}\)) to be within \(4\times 10^{10}\) and \(3\times 10^{11}\)M\({}_{\odot}\). From the mock catalog, we predict a broad stellar mass distribution of AGN host galaxies with a median \(4\times 10^{10}\)M\({}_{\odot}\) and a large standard deviation of 0.6 dex. So, the SHAM predicted stellar mass being smaller than the HOD inferred baryonic lensing mass; we find that interpretations from the two models are consistent.
### On the satellite occupation
As suggested in Leauthaud et al. (2015), the combination of clustering and lensing best constraints satellite occupation statistics. Compared to previous studies, we take here a significant step forward. Indeed satellite fractions inferred from clustering studies are limited by the precision of redshift in the presence of broad line AGNs, and for example Shen et al. (2013) or Rodriguez-Torres et al. (2017) could not constrain it. Lensing studies were limited by small numbers of X-ray-selected AGN (Leauthaud et al., 2015) and showed large uncertainties in the satellite occupation statistics. By combining eFEDS with HSC, we find a preference for low satellite fractions (HOD upper limit is \(f_{\rm sat}<20\%\) and SHAM best fits are with \(f_{\rm sat}<12\%\)). The HOD result shows a preference for a shallow satellite slope (\(\sim 0.75\)) that is smaller than measured for galaxy samples (\(\alpha\sim 1-1.1\) and \(f_{\rm sat}\) of 40% for galaxies with a stellar mass of \(3\times 10^{10}\)M\({}_{\odot}\)) (Zehavi et al., 2011; Zu & Mandelbaum, 2015).
The low satellite fraction could, in part, be due to the soft X-ray selection of the AGN. Indeed, satellite AGN could be obscured and only detectable in hard X-ray or the infrared (Kocevski et al., 2015; Krumpe et al., 2018). Krumpe et al. (2018)
Figure 8: SHAM predictions plotted with the fitted HOD model 1 \(\sigma\) contour of the 4 and 5 parameters HOD fit results. On the top left panel, the satellite fraction is fixed to 10%, and the \(\sigma\) parameter is varied. On the top right panel, \(\sigma\) is fixed to 0.8, and the \(f_{\rm sat}\) is varied. On the bottom left (right) panel is shown, as a function of \(\sigma\) and \(f_{\rm sat}\), the distance between the HOD predicted by SHAM models and the 5-parameter (4-parameter) HOD inferred from the observations. The black dashed line on the bottom left panel corresponds to \(\sigma=2-f_{\rm sat}/10\). When comparing to the 5-parameter HOD fit, the bottom left half, below the \(\sigma<2-f_{\rm sat}/10\) line of the parameter space, is preferred. Compared to the 4-parameter HOD fit (bottom right panel), the bottom left half is also preferred. The dashed line represents \(\sigma=2.4-f_{\rm sat}/10\). The star identifies the lowest distance model. Empty circles identify the six lowest distance models.
compared the cross-correlation functions (CCF) of _Swift_ BAT AGNs with 2MASS redshift survey galaxies and their HODs. Since _Swift_ BAT AGN sample is hard (14-195 keV) X-ray-selected, it contains a larger fraction of type 2 obscured AGN than eROSITA-based samples. They found clear suppression of the 1-halo term in type 1 AGN CCF compared to type 2. The HOD analysis shows \(\alpha\sim 1\) for the type 2 AGN HOD while that of the type 1 AGN was \(\alpha\lesssim 0.6\). Powell et al. (2018) obtained similar results. A possible scenario causing the low \(\alpha\) is the suppression of sub-halo mergers in high velocity encounters in high mass halos (Altamirano-Devora et al., 2016; Oogi et al., 2020). An alternative interpretation of the apparent shallow slope is that the satellite HOD slope is not shallow, but the satellite distribution profile within the dark matter halo does not follow the mass density profile assumed in the HOD modeling. If the satellite distribution is suppressed towards the outer part of the halo, the ordinary HOD modeling would result in low fitted \(\alpha\). Indeed, it would appear as if the satellites were suppressed in high-mass halos with large virial radii. However, one should be cautious as such interpretations are still a debate.
### Triggering mechanism for soft X-ray AGN in the cosmic web
The general SHAM scheme applied to populate mock catalogs with AGN accounts for the observations satisfactorily. One important assumption made in the SHAM model is that the assignment of an AGN to a galaxy is independent of the environment: it ignores the properties of the neighboring haloes. It implies that, to the first order, the larger scale environment, beyond the galaxy host halo, is not the primary driver to turn on the AGN. Instead, the local environment (within the virial radius) i.e., the circumgalactic medium, the interstellar medium, and the stellar populations, are likely more decisive parameters. It agrees with the findings of Yang et al. (2018); Allevato et al. (2019); Siudek et al. (2023). It emphasizes internal processes and their role as AGN triggers, for example, disc instabilities (e.g. Bournaud et al., 2011) or the presence of bars (e.g. Ohta et al., 2007).
The fact that the satellite slope is shallower than that of galaxies with equivalent stellar mass means that the in-fall of a satellite on a larger structure makes it less likely to host an AGN, even more, when structures are larger. It likely illustrates that the gas stripping from satellite galaxies in deep potential wells suppresses AGN. It is compatible with the environment quenching mechanism described by Peng et al. (2010, 2012).
### Outlook
The eROSITA eFEDs observations constitute about 1% of the full eROSITA All-Sky Survey (eRASS). This study paves the way towards charting the co-evolution of X-ray AGN and their host galaxies and dark matter haloes.
In the coming decade, by combining eROSITA with SDSS-V, 4MOST, and DESI spectroscopic redshifts (Kollmeier et al., 2017; Merloni et al., 2019; DESI Collaboration et al., 2016) and with LSST and Euclid lensing products, one will be able to carry out the similar analysis over a larger area and on an extended redshift range, up to z=1. Between eFEDS (120 deg\({}^{2}\), \(z<0.55\)) and future analysis (13,000 deg\({}^{2}\), \(z<1\)), the comoving volume will increase by a factor 450. HOD parameters should be inferred to the percent level. We will accurately measure the halo occupation distributions as a function of host-galaxy properties and AGN properties towards characterizing possible correlations between HOD parameters and host-galaxy, AGN, and environmental properties. With that, one should unravel the role of AGN in shaping the galaxy population and its hot circumgalactic medium (Hopkins et al., 2006; Comparat et al., 2022).
Complementary to HOD analysis is the direct or partial correlations with host galaxy properties; see reviews from Brandt and Yang (2021); Brandt and Alexander (2015). In recent years, spectroscopic energy distribution fitting has dramatically improved in retrieving unbiased galaxy stellar parameters of galaxies hosting AGN (e.g., Mountrichas et al., 2021; Yang et al., 2022, Buchner et al. in prep). The upcoming Rubin observatory LSST survey1(Ivezic et al., 2019) will provide deep multi-band imaging to be used to determine host galaxy properties. In addition, the future Euclid2 imaging space mission (Laureijs et al., 2011) will enable accurate morphological measurements of AGN hosts on a significant fraction of the extra-galactic sky. Together these will allow charting of the physics of the connection between AGN, host galaxy morphology, and stellar properties (e.g. Yang et al., 2019; Ni et al., 2019) and give further insight on the ecology of the cosmic web of X-ray AGN.
Footnote 1: [https://www.lsst.org](https://www.lsst.org)
Footnote 2: [https://www.cosmos.esa.int/web/euclid](https://www.cosmos.esa.int/web/euclid)
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Universe (Kavli PMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (UST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for Vera C. Rubin Observatory. We thank the Rubin Observatory for making their code available as free software at [http://pipelines.lsst.io/](http://pipelines.lsst.io/). This paper is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by the Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics (CICA), NAOJ. We are honored and grateful for the opportunity of observing the Universe from Muankae, which has the cultural, historical and natural significance in Hawaii. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Hasing-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss5.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including the Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, the Gohtan Papitrou Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique federale de Lausanne (EPFL), Leibniz-Institut fur Astrophysik Potsdam (AIP), Max-Planck-Institut fur Extromotrie (MPIA Heidelberg), Max-Planck-Institut fur Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autonoma de Mexico, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian (CFA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofisica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/ University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut fur Astrophysik Potsdam (AIP), Max-Planck-Institut fur Astronomie (MPIA Heidelberg), Max-Planck-Institut fur Astrophysik (MPA Garching), Max-Planck-Institut fur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatorio Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autonoma de Mexico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
## Appendix A X-ray data analysis
### X-ray mask
We use the region files created by eSASS/srctool to create X-ray masks for point sources (PS) and extended sources (EXT) (Liu et al., 2022, 2020). Each source has its signal-to-noise ratio measured as a function of radius (circular apertures). An optimal radius for source extraction is found by maximizing the signal-to-noise ratio given the local background surface brightness. It is clipped to a minimum radius of \(10^{\prime\prime}\)(MINIMUM_SOURCE_RADIUS parameter) and a maximum radius of the 99% energy enclosed fraction radius of the point spread function. We use this maximum signal-to-noise radius as a starting point to determine the area to be masked around sources.
We measure the cross-correlation as a function of scale between events (0.2-2.3 keV) and sources in the catalog. We measure it for bins of the number of counts measured per source in the detection band. The cross-correlation becomes constant above a particular angular scale, which corresponds to a conservative masking radius of a source (with a given number of counts), i.e., its average imprint on the sky, see Fig. 10 top panels. For each cross-correlation curve, we measure the radius at which its value is between 1.25 and two times that of the constant values measured at large separation. This brackets the masking radius within the black vertical error bars shown in Fig. 10 bottom panels. We find that this cross-correlation masking radius for point sources is, on average, 40 percent (20 for extended sources) larger than the eSASS/srctool radius of maximum signal-to-noise, see Fig. 10 bottom panels. The sterol mask is likely not conservative enough for our purpose. For instance, the detection of a point source just beyond the eSASS/srctool masking radius of another point source will be subject to biases due to the residual events measured via the cross-correlation. Though, if we followed the average masking radius suggested by the cross-correlation (Fig. 10 bottom panels), the large scatter in the relation between the maximum signal-to-noise radius and the total number of counts would be missed. So to have a conservative mask that closely follows the data, we multiply the masking radii from eSASS/srctool by a factor of 1.4 (this is more conservative than required for the extended sources, but it simplifies the procedure). In that way, the masking radius will reach, on average, the line obtained from the cross-correlation. Doing so ensures no remaining correlation between the set of events outside the mask and the source catalog. We are conservatively masking both point sources and extended sources individually.
After applying the mask, we are left with 17,523 AGN candidates. We are using the fraction of random points (see SS 2.1.4) that fall in the masks, we estimate the area of the observed X-ray sky effectively occupied by sources. In all, sources occupy 9.805 deg\({}^{2}\) out of 141.97 deg\({}^{2}\). AGN occupy 6.914 deg\({}^{2}\), stars 0.988 deg\({}^{2}\), and extended sources 2.057 deg\({}^{2}\).
### Random catalogue
We use the sensitivity map produced by the eROSITA pipeline Brunner et al. (eSASS, apetool 2022) with a parameter \(P_{thres}=e^{-8}=0.00033\) (the Poisson probability threshold below which an excess of counts is considered a source), corresponding to a detection likelihood of 8. It is a pixelated fits image with size \([0,9000)\times[0,18000)\) containing the sensitivity limit (in counts). Each random point falls in a pixel of this map, and we attach the corresponding count limit \(C_{X}^{\mathrm{lin}}\) to the random number. We draw a large set of redshift and X-ray fluxes (\(f_{x}\), \(z\)) from the AGN X-ray luminosity function projection to assign to each random point Aird et al. (2015); Comparat et al. (2019). It is sampled down to \(2\times 10^{-15}\)erg cm\({}^{-2}\) s\({}^{-1}\), a flux value at which the area curve is smaller than 0.5 deg\({}^{2}\). We convert the flux into an expected number of counts
\[CT^{expected}=f_{x}\times ECF\times EEF\times t_{exp}+CT^{background}. \tag{10}\]
where the energy conversion factor is \(ECF=1.164\times 10^{12}\). The encircled energy fraction is set to \(EEF=0.65\). The exposure time, \(t_{exp}\), is obtained with the exposure map. \(CT^{background}\) is obtained from the background map. We draw a random Poisson variable \(R_{v}\) for each \(CT^{expected}\). If this value exceeds the count limit, \(R_{v}>C_{X}^{lin}\), the point is accepted in the random sample. We remove the shallower areas at the edge of the field through a minimum exposure time threshold to minimize the maximum offset between the normalized cumulative distribution of the data sample and the random sample. We find that an 830 seconds threshold minimizes the KS-test values at 0.19% (0.81%) for R.A. (Dec.). It removes \(\sim 10\) deg\({}^{2}\). It is sufficiently accurate to estimate clustering on the photometric sample. After masking extended sources and stars and trimming the low exposure time region, the total number of random points remaining is 3,713,726. |
2310.19292 | Fusing Temporal Graphs into Transformers for Time-Sensitive Question
Answering | Answering time-sensitive questions from long documents requires temporal
reasoning over the times in questions and documents. An important open question
is whether large language models can perform such reasoning solely using a
provided text document, or whether they can benefit from additional temporal
information extracted using other systems. We address this research question by
applying existing temporal information extraction systems to construct temporal
graphs of events, times, and temporal relations in questions and documents. We
then investigate different approaches for fusing these graphs into Transformer
models. Experimental results show that our proposed approach for fusing
temporal graphs into input text substantially enhances the temporal reasoning
capabilities of Transformer models with or without fine-tuning. Additionally,
our proposed method outperforms various graph convolution-based approaches and
establishes a new state-of-the-art performance on SituatedQA and three splits
of TimeQA. | Xin Su, Phillip Howard, Nagib Hakim, Steven Bethard | 2023-10-30T06:12:50Z | http://arxiv.org/abs/2310.19292v1 | # Fusing Temporal Graphs into Transformers for Time-Sensitive Question Answering
###### Abstract
Answering time-sensitive questions from long documents requires temporal reasoning over the times in questions and documents. An important open question is whether large language models can perform such reasoning solely using a provided text document, or whether they can benefit from additional temporal information extracted using other systems. We address this research question by applying existing temporal information extraction systems to construct temporal graphs of events, times, and temporal relations in questions and documents. We then investigate different approaches for fusing these graphs into Transformer models. Experimental results show that our proposed approach for fusing temporal graphs into input text substantially enhances the temporal reasoning capabilities of Transformer models with or without fine-tuning. Additionally, our proposed method outperforms various graph convolution-based approaches and establishes a new state-of-the-art performance on SituatedQA and three splits of TimeQA.
## 1 Introduction
Long-document time-sensitive question answering [1] requires temporal reasoning over the events and times in a question and an accompanying long context document. Answering such questions is a challenging task in natural language processing (NLP) as models must comprehend and interpret the temporal scope of the question as well as the associated temporal information dispersed throughout the long document. For example, consider the time-sensitive questions about George Washington's position provided in Figure 1. The relevant events and temporal information regarding George Washington's position are scattered across many different sentences in the context document. Since there is no one single text segment containing the answer, the model must integrate and reason over events and times throughout the context document. Additionally, this example illustrates how changing the time expression in the question may also result in a change in the answer: in this case, replacing _between 1776 - 1780_ with _from 1790 to 1797_ changes the answer from _Commander-in-Chief_ to _Presidency_ and _Chancellor_.
Though not designed directly for question answering, there is a substantial amount of research on temporal information extraction [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Such models can help reveal the structure of the timeline underlying a document. However, there is little existing research on combining such information extraction systems with question answering Transformer models [11] to effectively reason over temporal information in long documents.
In this work, we utilize existing temporal information extraction systems to construct temporal
Figure 1: Time-sensitive question-answer pairs with a context document. The times and answers are in red and blue, respectively.
graphs and investigate different fusion methods to inject them into Transformer models.
We evaluate the effectiveness of each temporal graph fusion approach on long-document time-sensitive question answering datasets. Our contributions are as follows:
1. We introduce a simple but novel approach to fuse temporal graphs into the input text of question answering Transformer models.
2. We compare our method with prior approaches such as fusion via graph convolutions, and show that our input fusion method outperforms these alternative approaches.
3. We demonstrate that our input fusion approach can be used seamlessly with large language models in an in-context learning setting.
4. We perform a detailed error analysis, revealing the efficacy of our method in fixing temporal reasoning errors in Transformer models.
## 2 Related Work
### Extracting Temporal Graphs
Research on extracting temporal graphs from text can be grouped into the extraction of event and time graphs (Chambers et al., 2014; Ning et al., 2018), contextualized event graphs (Madaan and Yang, 2021), and temporal dependency trees and graphs (Zhang and Xue, 2018, 2019; Yao et al., 2020; Ross et al., 2020; Choubey and Huang, 2022; Mathur et al., 2022). Additionally, some prior work has focused on the problem of extracting temporal relations between times and events (Ning et al., 2018, 2019; Vashishtha et al., 2019; Han et al., 2019; Ballesteros et al., 2020; Zhang et al., 2022). The outputs of these temporal relation extraction systems are often used to construct temporal graphs. In this work, we use CAEVO (Chambers et al., 2014) and SUTime (Chang and Manning, 2012) to construct our temporal graphs because they are publicly available (unlike more recent models such as the temporal dependency graph parser proposed by Mathur et al. (2022)) and can easily scale to large amounts of long documents without requiring additional training.
### Temporal Question Answering
Jia et al. (2018) decomposes questions and applies temporal constraints to allow general question-answering systems to answer knowledge-base-temporal questions. Saxena et al. (2021), Jia et al. (2021), Mavromatis et al. (2021), and Sharma et al. (2023) use time-aware embeddings to reason over temporal knowledge graphs. Similar to our work, Huang et al. (2022) and Shang et al. (2021) answer temporal questions on text, but they focus on temporal event ordering questions over short texts rather than time-sensitive questions over long documents. Li et al. (2023) focus on exploring large language models for information extraction in structured temporal contexts. They represent the extracted time-sensitive information in code and then execute Python scripts to derive the answers. In contrast, we concentrate on temporal reasoning in the reading comprehension setting, using unstructured long documents to deduce answers. This poses more challenges in information extraction and involves more complex reasoning, which motivates our integration of existing temporal information extraction systems with transformer-based language models. The most similar work to ours is Mathur et al. (2022), which extracts temporal dependency graphs and merges them with Transformer models using learnable attention mask weights. We compare directly to this approach, and also explore both graph convolutions and input modifications as alternatives to fusing temporal graphs into Transformer models.
### Fusing Graphs into Transformer Models
The most common approaches for fusing graphs into Transformer models are graph neural networks (GNN) and self-attention. In the GNN-based approach, a GNN is used to encode and learn graph representations which are then fused into the Transformer model (Yang et al., 2019; Feng et al., 2020; Yasunaga et al., 2021; Zhang et al., 2022). In the self-attention approach, the relations in the graphs are converted into token-to-token relations and are then fused into the self-attention mechanism. For example, Wang et al. (2020) uses relative position encoding (Shaw et al., 2018) to encode a database schema graph into the BERT representation. Similarly, Bai et al. (2021) utilize attention masks to fuse syntax trees into Transformer models. We explore GNN-based fusion of temporal graphs into question answering models, comparing this approach to the attention-based approach of Mathur et al. (2022), as well as our simpler approach which fuses the temporal graph directly into the Transformer model's input.
## 3 Method
Our approach applies temporal information extraction systems to construct temporal graphs and then fuses the graphs into pre-trained Transformer models. We consider two fusion methods:
1. Explicit edge representation fusion (ERR): a simple but novel approach to fuse the graphs into the input text.
2. Graph neural network fusion: a GNN is used to fuse the graphs into the token embeddings or the last hidden layer representations (i.e., contextualized embeddings) of the Transformer model.
The overall approach is illustrated in Figure 2.
### Graph Construction
Given a time-sensitive question and a corresponding context document, we construct a directed temporal graph where events and time expressions are nodes and temporal relations are edges of the type BEFORE, AFTER, INCLUDES, INCLUD BY, SIMULTANEOUS, or OVERLAP.
We extract the single timestamp included in each question, which is either explicitly provided by the dataset (as in SituatedQA [22]) or alternatively is extracted via simple regular expressions (the regular expressions we use achieve 100% extraction accuracy on TimeQA [3]). We add a single question-time node to the graph for this time.
For the document, we apply CAEVO1 to identify the events, time expressions, and the temporal relations between them. CAEVO follows the standard convention in temporal information extraction that events are only simple actions (typically verbs) with linking of these actions to subjects, objects, and other arguments left to dependency parsers [23, 24, 25, 26]. We add document-event and document-time nodes for each identified event and time, respectively, and add edges between nodes for each identified temporal relation.
Footnote 1: [https://github.com/nchambers/caevo](https://github.com/nchambers/caevo)
To link the question-time node to the document-time nodes, we use SUTime2 to normalize time expressions to time intervals, and deterministically compute temporal relations between question time and document times as edges. For example, given a document-time node "the year 2022" and a question-time node "from 1789 to 1797" from the question "What was George Washington's position from 1789 to 1797?", the times will be normalized to [2022-01-01, 2022-12-31] and [1789-01-01, 1797-12-31] respectively, and the temporal relation between them can then be computed as AFTER.
Footnote 2: [https://nlp.stanford.edu/software/sutime.html](https://nlp.stanford.edu/software/sutime.html)
To link the question-time node to document events, for each document-event node, we calculate the shortest path in the temporal graph between it and the question-time node and recursively apply standard transitivity rules (see Appendix A.1) along the path to infer the temporal relation. For example, given a path A is BEFORE B and B INCLUDES C, we can infer the relation between A and C is BEFORE. An example of a constructed temporal graph for Q1 in Figure 1 is illustrated in Figure 3.
### Graph Fusion
For both fusion methods, we concatenate the question and corresponding context document as an input sequence to the Transformer model. For example, given the question and document from Figure 1 Q1, the input is:
Figure 3: Temporal graph example.
Figure 2: Overview of our approach. The ERR method allows for optional fine-tuning of the Transformer model, whereas for the GNN method requires fine-tuning.
question: What was George Washington's position between 1776 - 1780? context:...Congress created the Continental Army on June 14, 1775..._
#### 3.2.1 Explicit Edge Representation
In the ERR method, we mark a temporal graph's nodes and edges in the input sequence, using <question time> and </question time> to mark the question-time node and relation markers such as <before> and </before> to mark the nodes in the context document and their relations to the question time. Thus, the ERR input for the above example is:
_question: What was George Washington's position <question time> between 1776-1780</question time>? context:...Congress <before>created</before> the Continental Army on <before>June 14, 1775</before>..._
This approach aims to make the model learn to attend to parts of the input sequence that may contain answer information. For instance, the model may learn that information related to the answer may be found near markers such as <overlap>, <includes>, <included by>, and <simultaneous>. Additionally, the model may learn that answer-related information may exist between <before> and <after>, even if the answer does not have any nearby temporal information.
#### 3.2.2 GNN-based Fusion
In GNN-based fusion, we add <e> and </e> markers around each node, and apply a relational graph convolution (RelGraphConv; Schlichtkrull et al., 2018) over the marked nodes. RelGraphConv is a variant of graph convolution (GCN; Kipf and Welling, 2017) that can learn different transformations for different relation types. We employ the RelGraphConv to encode a temporal graph and update the Transformer encoder's token embedding layer or last hidden layer representations (i.e., contextualized embeddings). We utilize the RelGraphConv in its original form without any modifications.
Formally, given a temporal graph \(G=(V,E)\), we use representations of the <e> markers from the Transformer model's token embedding layer or the last hidden layer as initial node embeddings. The output of layer \(l+1\) for node \(i\in V\) is:
\[h_{i}^{l+1}=\sigma(\sum_{r\in R}\sum_{j\in\mathcal{N}^{r}(i)}\frac{1}{c_{i,r} }W_{r}^{(l)}h_{j}^{(l)}+W_{0}^{(0)}h_{i}^{(l)})\]
where \(\mathcal{N}^{r}(i)\) denotes all neighbor nodes that have relation \(r\) with node \(i\), \(\frac{1}{c_{i,r}}\) is a normalization constant that can be learned or manually specified, \(\sigma\) is the activation function, \(W_{0}\) is the self-loop weight, and \(W_{r}\) is the learnable weights. We refer readers to Schlichtkrull et al. (2018) for more details.
## 4 Experiments
### Datasets
We evaluate on two time-sensitive question answering datasets: Time-Sensitive Question Answering (TimeQA; Chen et al., 2021) and SituatedQA (Zhang and Choi, 2021). We briefly describe these two datasets below and provide statistics on each dataset in Table 9 of Appendix A.6.
TimeQAThe TimeQA dataset is comprised of time-sensitive questions about time-evolving facts paired with long Wikipedia pages as context. The dataset has two non-overlapping splits generated by diverse templates which are referred to as TimeQA Easy and TimeQA Hard, with each split containing 20k question-answer pairs regarding 5.5K time-evolving facts and 70 relations. TimeQA Easy contains questions which typically include time expressions that are explicitly mentioned in the context document. In contrast, TimeQA Hard has time expressions that are not explicitly specified and therefore require more advanced temporal reasoning. For example, both questions in Figure 1 are hard questions, but if we replace the time expressions in the questions with _In 1788_, they will become easy questions. In addition, smaller Human-paraphrased Easy and Hard splits are also provided.
by FiD Izacard and Grave (2021)3 for each question as context documents.
Footnote 3: [https://github.com/facebookresearch/FiD](https://github.com/facebookresearch/FiD)
Footnote 4: wenhuchen/Time-Sensitive-QA
Evaluation MetricsWe use the official evaluation methods and metrics provided in the code release of the datasets. For TimeQA, we report the exact match and F1 scores. For SituatedQA, we report the exact match score.
### Baselines
We compare our models with the previous state-of-the-art on the TimeQA and SituatedQA, which we describe in this section and summarize in Table 1.
For TimeQA, we compare against:
**FiD & BigBird**: Chen et al. (2021) adapt two long-document question answering models, BigBird Zaheer et al. (2020) and FiD Izacard and Grave (2021), to the TimeQA dataset. Before fine-tuning the models on TimeQA, they also fine-tune on Natural Questions Kwiatkowski et al. (2019) and TriviaQA Joshi et al. (2017), finding that training on Natural Questions results in the best performance. The best model from Chen et al. (2021) on TimeQA Easy and Hard is FiD, while BigBird performs best on the Human-paraphrased TimeQA Easy and Hard splits.
**Replicated FiD**: We also report our replication of Chen et al. (2021)'s FiD model using the code provided by their GitHub repository4, but even with extensive hyperparameter tuning, we were unable to reproduce their reported performance of the FiD model on TimeQA Easy5.
Footnote 5: Other researchers have reported the same issue on GitHub
**DocTime**: DocTime Mathur et al. (2022) first uses CAEVO to identify events and time expressions in context documents. Then, a custom-trained temporal dependency parser is applied to parse temporal dependency graphs. Finally, the parsed temporal dependency graphs are fused into the attention mechanism of the FiD model, which is fine-tuned on Natural Questions.
**BigBird + MTL & Longformer + MTL**: Chen et al. (2023) inject temporal information into long text question answering systems using a multi-task learning (MTL) approach. They train BigBird and Longformer models on time-sensitive question answering tasks, along with three auxiliary temporal-awareness tasks. They explore first fine-tuning on Natural Questions and TriviaQA datasets, finding that TriviaQA results in the best performance. BigBird + MTL is their best model on TimeQA Easy, while Longformer + MTL performs best on TimeQA Hard.
For SituatedQA, we compare against:
**DPR + Query Modified**: Zhang and Choi (2021) adapt a retrieval-based model, DPR Karpukhin et al. (2020), for SituatedQA. They first fine-tune the retriever and reader of DPR on Natural Questions and then further fine-tune on SituatedQA.
**TSM**: Cole et al. (2023) uses SuTime to identify and mask time expressions throughout all Wikipedia documents. They then fine-tune T5-1.1-XXL Raffel et al. (2020) to predict the masked time expressions. Finally, they fine-tune the resulting T5-1.1-XXL model on SituatedQA.
### Implementation Details
We use LongT5-base Guo et al. (2022), a transformer sequence-to-sequence model, as our base model. Our experiments demonstrate that LongT5 outperforms the FiD model Izacard and Grave (2020) commonly used on long-document question answering tasks. To fairly compare with previous work Chen et al. (2021); Mathur et al. (2022), we pre-train the LongT5 model on Natural Questions Kwiatkowski et al. (2019) and then fine-tune it
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Method & Model Size & External Tool (Public Avail.) & Add. Data & Knowl. Fusion \\ \hline BigBird Chen et al. (2021) & 128M & - & NaturalQ & - \\ FiD \& Replicated FiD Chen et al. (2021) & 220M & - & NaturalQ & - \\ DocTime Mathur et al. (2022) & 220M & CAEVO (\(\checkmark\)), TDGP (\(\mathbf{\mathcal{X}}\)) & NaturalQ & Attention \\ BigBird + MTL Chen et al. (2023) & 128M & TT (\(\checkmark\)), TR (\(\mathbf{\mathcal{X}}\)) & TriviaQA & Multi-Task \\ Longformer + MTL Chen et al. (2023) & 435M & TT (\(\checkmark\)), TR (\(\mathbf{\mathcal{X}}\)) & TriviaQA & Multi-Task \\ DPR + Query Modified Zhang and Choi (2021) & 110 M & - & - & Question Text \\ TSM Cole et al. (2023) & 11B & SUTime (\(\checkmark\)) & Entire Wiki & Pre-Training \\ LongT5\_ERR (Ours) & 250M & SUTime (\(\checkmark\)) & NaturalQ & Input Text \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of methods, base models’ size, external tools, additional data, and temporal knowledge fusion methods. NaturalQ: Natural Questions, TDGP: Temporal Dependency Graph Parser, TT: BERT-based Temporal Tagger, TR: timestamps retriever.
on TimeQA and SituatedQA, respectively. Appendix A.5 provides other implementation details such as hyperparameters, graph statistics, software versions, and external tool performance.
We perform model selection before evaluating on the test sets, exploring different graph subsets with both the ERR and GNN based fusion approaches introduced in Section 3.2. Table 6 in appendix A.2 shows that the best ERR method uses a document-time-to-question-time (DT2QT) subgraph and the best GNN method uses the full temporal graph by fusing it into the token embedding layer representations of the Transformer model. We hereafter refer to the LongT5 model fused with a DT2QT graph using the ERR method as LongT5\({}_{\text{ERR}}\), and the LongT5 model fused with a full temporal graph using the GNN method as LongT5\({}_{\text{GNN}}\).
### Main Results
We summarize the performance of baseline models and those trained with our graph fusion methods in Table 2.
Which baseline models perform best?On TimeQA, our LongT5 model without temporal graph fusion performs better than or equivalent to all other baseline models across every split and metric except for the Easy split. The best-performing model reported on TimeQA Easy is DocTime. On SituatedQA, LongT5 with no fusion performs as well as the best-reported results on this dataset.
Which graph fusion methods perform best?Using LongT5, we consider both of our ERR and GNN fusion methods described in Section 3.2. On TimeQA, the LongT5\({}_{\text{GNN}}\) model fails to outperform LongT5 without fusion, while the LongT5\({}_{\text{ERR}}\) model improves over LongT5 on every split and dataset, exhibiting particularly large gains on the Hard splits. On Situated QA, both LongT5\({}_{\text{ERR}}\) and LongT5\({}_{\text{GNN}}\) models improve over the no-fusion LongT5 baseline, with ERR again providing the best performance. The somewhat inconsistent performance of the GNN fusion method across datasets (beneficial on SituatedQA while detrimental on TimeQA) suggests the need for a different GNN design for TimeQA, which we leave to future work.
To explore the differences between LongT5\({}_{\text{ERR}}\) and LongT5\({}_{\text{GNN}}\) models, we analyze 20 randomly sampled examples from TimeQA Hard where LongT5\({}_{\text{ERR}}\) is correct but LongT5\({}_{\text{GNN}}\) is incorrect. From our manual analysis of these 20 examples, all 20 examples share the same pattern: LongT5\({}_{\text{GNN}}\) fails to capture explicit temporal expressions in the context and relate them to the question's timeline, which is crucial for deducing the right answer. This suggests that directly embedding pre-computed temporal relations between time nodes into the input is more efficient than implicitly doing so through the GNN, allowing the model to utilize them more easily. Table 14 of appendix A.11 shows three of the analyzed examples.
Does our approach outperform prior work?On TimeQA, the LongT5\({}_{\text{ERR}}\) model achieves a new state-of-the-art on three of the four splits, with TimeQA Easy being the exception. On SituatedQA, the LongT5\({}_{\text{ERR}}\) model achieves a new state-of-the-art, outperforming the best reported results on this dataset. Our model excels on datasets that require
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline & \multicolumn{2}{c}{TimeQA Easy} & \multicolumn{2}{c}{TimeQA Hard} & \multicolumn{2}{c}{TimeQA\({}_{\text{HP}}\) Easy} & \multicolumn{2}{c}{TimeQA\({}_{\text{HP}}\) Hard} & \multicolumn{2}{c}{SituatedQA} \\ \cline{2-11} & EM & F1 & EM & F1 & EM & F1 & EM & F1 & EM \\ \hline BigBird (Chen et al., 2021) & 51.2 & 60.3 & 42.4 & 50.9 & 47.6 & 53.7 & 38.8 & 45.9 & - \\ FiD (Chen et al., 2021) & 60.5 & 67.9 & 46.8 & 54.6 & - & - & - & - & - \\ DocTime (Mathur et al., 2022) & **62.4** & **69.6** & 48.2 & 56.4 & - & - & - & - & - \\ BigBird + TML (Chen et al., 2023b) & 55.4 & 63.8 & 45.2 & 55.3 & - & - & - & - & - \\ Longformer + TML (Chen et al., 2023b) & 52.4 & 62.6 & 48.7 & 58.5 & - & - & - & - & - \\ DPR + Query Modified (Zhang and Choi, 2021) & - & - & - & - & - & - & - & - & 23.0 \\ TSM (Cole et al., 2023) & - & - & - & - & - & - & - & - & - & 24.6 \\ \hline Replicated FiD & 55.3 & 65.1 & 45.2 & 54.5 & 52.7 & 59.7 & 39.7 & 47.0 & 23.7 \\ LongT5 & 55.4 & 64.3 & 49.9 & 58.4 & 53.7 & 61.6 & 45.3 & 52.3 & 24.7 \\ LongT5\({}_{\text{GNN}}\) & 54.2 & 63.4 & 49.0 & 57.6 & 46.5 & 55.3 & 34.3 & 40.2 & 27.3 \\ LongT5\({}_{\text{ERR}}\) & 56.9 & 66.2 & **54.0** & **62.0** & **54.9** & **62.9** & **49.3** & **56.1** & **29.0** \\ \hline Replicated FiD (re-annotated) & 64.0 & 68.3 & & & & & & & \\ LongT5\({}_{\text{ERR}}\) (re-annotated) & **68.8** & **70.4** & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance on the test sets. TimeQA\({}_{\text{HP}}\) denotes the human-paraphrased splits of TimeQA. The highest-performing model is bolded. Confidence intervals for our results are provided in table 7 in appendix A.3.
more temporal reasoning skills, like TimeQA Hard (where our model achieves a 5.8-point higher exact match score than DocTime) and SituatedQA (where our model achieves a 4.3-point higher exact match score than TSM).
Our approach offers further advantages over alternative models due to its simplicity, as summarized in Table 1. The best prior work on TimeQA Easy, DocTime, requires training a temporal dependency parser on additional data, using CAEVO, and modifying the Transformer model's attention mechanism. The best prior work on SituatedQA, TSM, requires an 11-billion parameter T5 model which is pre-trained on the entirety of Wikipedia. In contrast, our approach only uses SUTime to construct a graph, requires only minor adjustments to the Transformer model's input, and outperforms prior work using a model with only 250 million parameters.
Why does our model not achieve state-of-art performance on TimeQA Easy as it does on other splits and datasets?On TimeQA Easy, there is a performance gap between our LongT5\({}_{\text{ERR}}\) model and DocTime. Because the DocTime model has not been released we cannot directly compare with its predicted results. Instead, we randomly select 50 errors from our LongT5\({}_{\text{ERR}}\)'s output on the TimeQA Easy development set for error analysis. Table 3 shows that most of the errors are false negatives, where the model's predicted answers are typically co-references to the correct answers (as in Table 3 example 1) or additional correct answers that are applicable in the given context but are not included in the gold annotations (as in Table 3 example 2). The remaining errors are primarily related to semantic understanding, including the inability to accurately identify answer entities (e.g. identifying Greek Prime Minister George Papandreou as an employer in Table 3 example 3), the inability to interpret negation (e.g. in Table 3 example 4, where "rejected" implies that Veloso did not join Benfica), and the inability to reason about time with numerical calculations (e.g. "a year later" in Table 3 example 5 implies 1846). Addressing the semantic understanding errors may require incorporating additional entities and their types into the graphs, as well as better processing of negation information and relative times.
To better understand the extent of false negatives in TimeQA Easy, we re-annotated the 392 test examples where the predictions of the replicated FiD model and our LongT5\({}_{\text{ERR}}\) model are partially correct (i.e., \(\text{EM}=0\) and \(\text{F1}>0\)). We then incorporated additional coreferent mentions into the gold label set for these examples. For instance, if the original gold answer was "University of California," we added its coreferent mention "University of California, Los Angeles" to the gold answers. We then evaluate both the replicated FiD (the best-performing model we can reproduce) and our LongT5\({}_{\text{ERR}}\) model on the re-annotated TimeQA Easy split. The last two rows of Table 2 show that while the exact match score for FiD increases by 8.7, the exact match score for our LongT5\({}_{\text{ERR}}\)
\begin{table}
\begin{tabular}{l c c l l l l} \hline \hline
**Type** & **Freq** & **\#** & **Question** & **Relevant Context** & **Prediction** & **Answer** \\ \hline \multirow{4}{*}{False Negative} & \multirow{4}{*}{40} & 1 & Which team did the player & After the Olympics \(\ldots\)Rivaldo moved to & Deorviucs \(\ldots\)Rivaldo moved to & Deorviucs \(\ldots\)Rivaldo belong to to 1996 to 1997? & Deorviucs \(\ldots\)In La Liga. & Deorviucs \(\ldots\)In La Liga. & Deorviucs \(\ldots\)In La Corvina. \\ \cline{3-6} & & 2 & What was the position & Albert Reynolds was an Irish Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fianaa Fiana Fiana Fiana Fiana Fianaa Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fianaa Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fianaa Fiana Fianaa Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fiana Fianaa Fiana Fiana Fiana Fianaa Fiana Fiana Fianaa Fiana Fiana Fianaa Fianaa Fiana Fiana Fiana Fianaa Fiana Fiana Fianaa Fianaa Fianaa Fianaa Fianaa Fiana Fianaa Fianaa Fianaa Fianaa Fianaa Fianaa Fianaa Fianaaa Fianaa Fianaa Fianaa Fianaaa Fianaa Fianaaa Fianaaaa Fiana
model increases by 11.9. This suggests that our model may be incurring greater penalties for finding valid coreferent answers than baseline methods.
Does our ERR method benefit large language models using in-context learning?We have focused so far on temporal graph fusion when fine-tuning models, but large language models such as ChatGPT [14] and LLaMA [15] can achieve impressive performance without additional fine-tuning via in-context learning. Therefore, we tested the performance of ChatGPT (gpt-3.5-turbo) both with and without ERR for fusing the question-time-to-document-time graph. Following previous work [10, 11, 13] and considering the cost of ChatGPT's commercial API, we randomly sample 500 examples from TimeQA Easy and TimeQA Hard for evaluation. The prompt format for ChatGPT remains the same as the input format described in Section 3.2.1, except that we concatenate in-context learning few-shot exemplars and task instructions before the input. We evaluate ChatGPT with and without graph fusion using an 8-shot setting. Examples of prompts are provided in Table 11 of Appendix A.8. Table 4 shows that our ERR graph fusion method improves the performance of ChatGPT on TimeQA Easy and particularly on TimeQA Hard. We note that this improvement is possible because our method can easily integrate with state-of-the-art large language models, as our approach to temporal graph fusion modifies only the input sequence. Prior work which relies on modifying attention mechanisms or adding graph neural network layers is incompatible with this in-context learning setting.
## 5 Analysis
In this section, we analyze our LongT5\({}_{\text{ERR}}\) model on the TimeQA development set.
How do predictions differ compared to FiD?We compare the predictions of LongT5\({}_{\text{ERR}}\) to the replicated FiD model in Table 5. While LongT5\({}_{\text{ERR}}\) and FiD both correct about the same number of each other's errors on TimeQA Easy (269 vs. 243), LongT5\({}_{\text{ERR}}\) corrects many more of FiD's errors than the reverse on TimeQA Hard (494 vs. 260). To further analyze these cases, we sampled 10 errors from the set where LongT5\({}_{\text{ERR}}\) was correct while FiD was incorrect as well as the set where the FiD was correct while LongT5\({}_{\text{ERR}}\) was incorrect. We did this across both TimeQA Easy and TimeQA Hard, totaling 40 examples. Among the 20 examples in which LongT5\({}_{\text{ERR}}\) was correct and FiD was incorrect, 17 have node markers near the answers in the ERR input sequence, and the most frequent ones are <included by> and <overlap>. The remaining 3 examples have unanswerable questions. In the examples in which FiD was correct while LongT5\({}_{\text{ERR}}\) was incorrect, we observe that 13 examples are additional correct answers (i.e., false negatives), while the other 7 examples are semantic understanding errors similar to those discussed previously. These results suggest that our ERR graph fusion approach is providing the model with useful targets for attention which allow it to produce more correct answers.
How does the length of the document affect performance?We compare the performance of LongT5\({}_{\text{ERR}}\) to the replicated FiD model on various document lengths, as depicted in Figure 4.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline Model & \multicolumn{2}{c}{TimeQA Easy} & \multicolumn{2}{c}{TimeQA Hard} \\ \cline{2-5} & EM & F1 & EM & F1 \\ \hline ChatGPT & 41.6 & 56.6 & 25.0 & 32.0 \\ w/ ERR + DT2QT Subgraph & 43.2 & 57.2 & 30.4 & 38.5 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance of ChatGPT on 500 sampled test examples, with and without our ERR method.
Figure 4: Comparison of the performance on different lengths of context documents.
\begin{table}
\begin{tabular}{c c c c} \hline \hline LongT5\({}_{\text{ERR}}\) & FiD & Time QA Easy & TimeQA Hard \\ \hline Correct & Correct & 1427 & 1113 \\ Correct & Wrong & 269 & 494 \\ Wrong & Correct & 243 & 260 \\ Wrong & Wrong & 1082 & 1220 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of the predictions of LongT5\({}_{\text{ERR}}\) vs. the predictions of FiD.
LongT\({}_{\text{ERR}}\) performs less competitively than FiD on the Easy split for longer documents. This could be attributed to a high frequency of false negatives in LongT\({}_{\text{ERR}}\), as discussed previously. Additionally, it could be that LongT\({}_{\text{ERR}}\) is less efficient at string matching on longer documents than FiD. Most of the question times in the Easy split are explicitly mentioned in the context document, which can be solved via string matching rather than temporal reasoning. However, our LongT\({}_{\text{ERR}}\) model shows a substantial improvement on TimeQA Hard, outperforming the FiD model across most lengths.
## 6 Conclusion
In this paper, we compared different methods for fusing temporal graphs into Transformer models for time-sensitive question answering. We found that our ERR method, which fuses the temporal graph into the Transformer model's input, outperforms GNN-based fusion as well as attention-based fusion models. We also showed that, unlike prior work on temporal graph fusion, our approach is compatible with in-context learning and yields improvements when applied to large language models such as ChatGPT. Our work establishes a promising research direction on fusing structured graphs with the inputs of Transformer models. In future work, we intend to use better-performing information extraction systems to construct our temporal graphs, enhance our approach by adding entities and entity type information to the graphs, and extend our method to spatial reasoning.
## Limitations
We use CAEVO and SUTime to construct temporal graphs because of their scalability and availability. Using more accurate neural network-based temporal information extraction tools may provide better temporal graphs but may require domain adaptation and retraining.
While we did not find graph convolutions to yield successful fusion on TimeQA, we mainly explored variations of such methods proposed by prior work. We also did not explore self-attention based fusion methods, as preliminary experiments with those methods yielded no gains, and DocTime provided an instance of such methods for comparison. But there may exist other variations of graph convolution and self-attention based fusion methods beyond those used in prior work that would make such methods more competitive with our input-based approach to fusion.
We also did not deeply explore the places where graph neural networks failed. For example, the graph convolution over the final layer contextualized embeddings using the full temporal graph yielded performance much lower than all other graph convolution variants we explored. We limited our exploration of failures to the more successful explicit edge representation models.
## Ethics Statement
Wikipedia contains certain biases (Falenska and Cetinoglu, 2021), and we use data from Wikipedia to train our model, thus we are potentially introducing similar biases into our models.
|
2304.12258 | Nonrandom behavior in the Projection of Random bipartite networks | There are two main categories of networks that are investigated in the
complexity physics community: monopartite and bipartite networks. In this
letter, we report a general finding between these two classes. If a random
bipartite network is projected into a monopartite network, under quite general
conditions, we obtain a non-random monopartite network with special features.
We believe this finding is very general and has important real-world
implications. | Izat B. Baybusinov, Enrico Maria Fenoaltea, Yi-Cheng Zhang | 2023-04-24T16:51:22Z | http://arxiv.org/abs/2304.12258v1 | # Non random behavior in the projection of random bipartite networks
###### Abstract
There are two main categories of networks that are investigated in the complexity physics community: monopartite and bipartite networks. In this letter, we report a general finding between these two classes. If a random bipartite network is projected into a monopartite network, under quite general conditions, we obtain a non-random monopartite network with special features. We believe this finding is very general and has important real-world implications.
pacs: 03.65.-a, 03.65.-b, 03.65.-b In the last three decades, networks have been a major research subject[1; 2; 3]. Among them, bipartite networks are one particular class [4; 5; 6; 7]. We can present a bipartite network as people connected to events. For example, an agent can participate in a few among many events and if everyone randomly chooses a few events independently, we obtain a typical random bipartite graph. Traditional networks, instead, belong to the so-called class of monopartite networks. We can present a random monopartite network as formed by people only. If each agent chooses randomly to connect with a few other agents, then we obtain a random monopartite network with nodes solely represented by people. This is the well-known Erdos-Renyi network [8].
Networks of the latter class can be constructed from those of the former class by projection [9; 10]: A new monopartite network can be constructed where two individuals are connected if, in the bipartite counterpart, they have at least one common event. In the literature, the people events network is called affiliation network [11; 12], and the projection is studied in many different systems. For instance, scientists connected by collaborating on the same project [13; 14], movie actors connected by appearing in the same movie [15; 16], and so forth. In addition, the projected network is the basis of recommender systems [17; 18].
In this letter, we study the properties of the projected network in the general case of a random bipartite network. Similar studies [19; 20] have examined the projected network by randomizing a configuration model [21] with a given degree sequence. Here we show that in the most simple approximation, i.e., a random bipartite graph, the projected network has interesting features.
Keeping with the people and events metaphor for bipartite networks, we construct a random network consisting of \(K\) people and \(N\) events and assume that each person connects to any of the events with probability \(\beta\in[0,1]\). In a large sample size, the decision to participate in an event can be assumed to be random. The elements of the adjacency matrix \(\mathbf{n}\) of the bipartite network is written as
\[n_{i\alpha}=\begin{cases}1\text{ with probability }\beta\\ 0\text{ with probability }1-\beta,\end{cases} \tag{1}\]
where individuals are labeled by \(i=1,..,K\) and events by \(\alpha=1,..,N\). From this adjacency matrix, we can compute all the network observables. For our purposes, it is useful to work with the degree distribution, i.e., the probability \(B_{N}(m)\) that an individual is connected to \(m\) events out of \(N\). Since the network is random, it follows a binomial distribution:
\[B_{N}(m)=\binom{N}{m}\beta^{m}(1-\beta)^{N-m}. \tag{2}\]
Now, we focus on the network of people resulting from the projection of the bipartite network. Since two agents are connected when they share at least one event, the elements of the adjacency matrix \(\mathbf{A}\) in the projected monopartite network can be written as:
\[A_{ij}=\theta\left(\sum_{\alpha=1}^{N}n_{i\alpha}n_{j\alpha}\right), \tag{3}\]
where \(\theta(\cdot)\) is the Heaviside function, and \(i,j=1,..,K\).
We want to study the properties of \(\mathbf{A}\) averaged over all the realizations of the bipartite network. However, writing the probability \(\mathbf{A}\) involves \(K^{2}\) conditions on the adjacency matrix. Therefore, we compute a simpler quantity, that is, the probability \(p\) that an element \(A_{ij}\) is \(1\). For \(N\) events this is the extremal distribution of sampling at least \(1\) element out of \(N\), and correlations due to the transition from a bipartite to a monopartite structure are averaged out. We have
\[p=\left\langle\theta\left(\sum_{\alpha}n_{i\alpha}n_{j\alpha}\right)\right\rangle \!\!=1-(1-\beta^{2})^{N} \tag{4}\]
By symmetry, this is the same for any \(i\) and \(j\), so the link probability gives no information about the bipartite structure in the monopartite version, and one might think that the network is random if we neglect correlations. In the following, we compute \(P(d)\) of the degree distribution of the projected network. The degree \(d_{i}\) of the individual \(i\) in the monopartite network is given by \(d_{i}=\sum_{j}A_{ij}\). If an individual with degree \(d\) participates in \(m\) events, there must be \(d\) individuals out of \(K-1\) (we do not allow self-link) meeting her at one of those
events. Summing over \(m\) we have
\[P(d)=\sum_{m=0}^{N}B_{N}(m)\binom{K-1}{d}[1-B_{m}(0)]^{d}B_{m}(0)^{K-1-d}. \tag{5}\]
When \(N\rightarrow\infty\) and \(\beta\) is fixed, the sum in Eq.5 is dominated by the maximum of \(B_{N}(m)\), i.e., when \(m=\beta N\). In this limit, \(P(d)\) is a binomial distribution with the probability parameter given by \(1-e^{-\beta^{2}N}\approx p\). Thus, one retrieves the Erdos-Reny network with connection probability \(p\).
In general, one can calculate the moment-generating function to show that \(P(d)\) is binomial in the large \(N\) limit:
\[\begin{split}\left\langle e^{\lambda d}\right\rangle=e^{\lambda (K-1)}\sum_{l=0}^{K-1}&\Big{[}(\beta(1-\beta)^{l}+(1-\beta))^{N }\\ &\times\binom{K-1}{l}(e^{-\lambda}-1)^{l}\Big{]},\end{split} \tag{6}\]
and at large \(N\) we have \(\left[\beta(1-\beta)^{l}+(1-\beta)\right]^{N}\approx e^{-l\beta^{2}N}\). So we obtain
\[\left\langle e^{\lambda d_{i}}\right\rangle\approx e^{\lambda(K-1)}\left[1-e ^{-\beta^{2}N}(1-e^{-\lambda})\right]^{K-1}, \tag{7}\]
which correspond to the moment-generating function of a binomial distribution with probability parameter \(1-e^{-\beta^{2}N}\). Hence, with many events, the projected network and a random monopartite network cannot be distinguished when measuring their degree distributions. Note that the above argument fails if \(K\) scales with \(N\).
Besides the degree distribution, we also consider a higher-order measure, namely, the clustering coefficient \(C\) of the projected network [16]:
\[C=3\frac{\left\langle triplets\right\rangle}{\left\langle open\ triplets\right\rangle}=\frac{\sum_{i,j,k}(A_{ij}A_{jk}A_{ki})}{ \sum_{i}(\left\langle\sum_{j}A_{ij}\right\rangle(\sum_{j}A_{ij}-1))} \tag{8}\]
This quantity measures how connected a node's neighbors are to one another. Note that, in the case of a random network generated with probability \(p\), the clustering coefficient is \(p\)[22]. To compute the numerator of Eq.8 in our projected network, we must consider the probability that three agents \(i,j,k\) are connected. There are two contributions to this probability. The first is the probability \(P_{2}\) that in the bipartite network \(i,j,k\) participate in the same event. In this case, we have \(P_{2}=1-(1-\beta^{3})^{N}\). The second is the probability \(P_{3}\) that the three individuals participate pairwise in three different events. This is computed as follows:
\[P_{3}=\sum_{m,n,l}B_{N}(m)B_{m}(n)\left[B_{n}(0)B_{m-n}(l)\right](1-B_{N-m}(0)) \tag{9}\]
The first factor in this sum is the probability that the first individual, say \(i\), participates in \(m\) events out of \(N\); the second and third factors are the conditions that \(j\) participates in \(n\) of these \(m\) events and that \(k\) participates in \(l\) of the remaining \(m-n\) events, respectively. In this way, individuals \(j\) and \(k\) both meet \(i\), but they do not meet each other in any of the \(m\) events chosen by \(i\). The last factor is the condition that \(j\) and \(k\) meet in the other \(N-m\) events. The denominator of Eq.8 is straightforwardly written as \(K\langle d_{i}(d_{i}-1)\rangle\). After some algebra, the clustering coefficient can be written as:
\[C=3-\frac{2+(1+2\beta)^{N}(1-\beta)^{2N}-3(1-\beta^{2})^{N}}{1-2(1-\beta^{2})^ {N}+(1-2\beta^{2}+\beta^{3})^{N}}. \tag{10}\]
Fig.2 shows that our analytical result is consistent with numerical simulations. The clustering is \(1\) at \(\beta\approx 1/N\), where the network is separated into disconnected communities composed of individuals that share all the events they participate in (i.e., in a community two people participate in the same events and do not participate in all the others). Increasing \(\beta\), spurious links start to appear between communities, reducing the clustering until \(\beta\sim N^{-1/2}\) (see Fig.3), where links between two randomly sampled individuals are highly frequent.From this point on, the increase in link density generates a higher clustering coefficient. Actually, the clustering reaches its minimum value when the network transits from a phase of isolated communities to one with a unique connected component. To evaluate this minimum we compute \(C\) for small \(\beta\):
\[C=\frac{1+N^{2}\beta^{3}}{N\beta+1}+o(\beta^{3}). \tag{11}\]
The minimum is obtained by solving the following equation:
\[2\beta^{3}N^{3}+3\beta^{2}N^{2}-N=0. \tag{12}\]
Figure 1: Difference between a projected bipartite network and a random network (i.e., the projected network without considering correlations) with \(4\) events and \(3\) agents. In the bipartite configuration, there is a unique list of events (in green). Meanwhile, in the random network case, there is an independent list of events (in red) for each possible link, i.e., for each pair of individuals.
For large \(N\), the solution \(\beta_{c}\) scales as \(N^{-2/3}\). Instead, the clustering evaluated at \(\beta_{c}\) decreases quite slowly with \(N\):
\[C(\beta_{c})\approx N^{-1/3}. \tag{13}\]
Note that, when looking at the clustering coefficient, our projected network is fundamentally different from a random network, as shown in Fig.2. Unlike the case of the degree distribution, this is also true for large \(N\). This is a consequence of the correlations arising from dimensionality reduction. Therefore in the projection of a random network, the degree distribution is not a sufficient measure to extract all the information from a network.
The presence of correlations can be explained by a geometrical argument. For any individual \(i\), let us define \(\mathbf{v}_{i}=\{n_{i\alpha}\}_{\alpha=1,..,N}\) as the vector whose elements are \(1\) if \(i\) participates in event \(\alpha\), and zero otherwise. We can interpret this vector also as a vertex of an \(N\)-dimensional hypercube. Thus, each vector (or individual) is a randomly drawn vertex on the hypercube with an average distance from the origin equal to \(\beta N\). On this hypercube two individuals are connected whenever their inner product is positive, that is, when \(\mathbf{v}_{i}\cdot\mathbf{v}_{j}>0\). From this perspective, correlations in the projected network naturally appear as dimensional constraints on the hypercube.
Now, it is interesting to study the properties of our system in function of \(N\) as the scaling gives different behaviors. For example, in [4], Newman describes a similar model where he fixes the average degree \(z=O(1)\) in the bipartite network so that \(\beta=z/N\). Since \(\beta\) scale as \(1/N\), for large \(N\), the resulting monopartite network of people is clustered into different communities, as we have shown above. Thus, with the Newman approach, the effects arising at higher values of \(\beta\) cannot be observed (see Fig.2), and the network always remains fragmented.
Remarkably, the number of events \(N\) can be used as a scaling tool of the network: studying the system with few events (for example, by aggregating similar events or by limiting the data set) lead to a richer scenario where links between different communities appear. One can argue that interesting social phenomena can be detected by observing the system at a fixed scale (in this case \(N\)).
In the real world there is a cognitive constraint in the number of connections each individual can handle [23]. To take this into account, we can fix the average degree in the projected network to be of the order of \(1\):
\[\langle d_{i}\rangle=Kp=K[1-(1-\beta^{2})^{N}]\sim O(1), \tag{14}\]
for any \(i\). In this way, we obtain a new scaling:
\[1-(1-\beta^{2})^{N}\sim 1/K\quad\rightarrow\quad\beta\sim\sqrt{\frac{1}{NK}} \tag{15}\]
So the probability to participate in an event depends on both \(N\) and \(K\). If \(K\approx\sqrt{N}\) the population is clustered in different communities. Instead, if \(K<\sqrt{N}\), the system is more cohesive. This mechanism can help to understand and manage social phenomena such as the observed social fragmentation and polarization [24; 25].
To generalize these results from a statistical mechanics perspective, one can introduce a Hamiltonian for the matrix \(\mathbf{A}^{\prime}\):
\[H[\mathbf{A}^{\prime}]=\sum_{i=1}^{K}\sum_{j=1}^{K}A^{\prime}_{ij}(1-\mathbf{ v}_{i}\cdot\mathbf{v}_{j}). \tag{16}\]
Figure 2: Clustering coefficient vs \(\beta\). The clustering of our projected network is compared with that of a random network with connection probability \(p\) and that of the Newman model in [4].
Figure 3: Phase diagram of the clustering coefficient of Eq.10 in the \([\beta,N]\) parameter space. When \(N\) increases, the width of the minimum valley increases since the left border scales as \(1/N\) and the right one as \(1/\sqrt{N}\).
This describes a model where individuals have a nonzero probability (defined by the temperature) of being connected in the projected network even though they have no common events in the bipartite network. The minimum of the Hamiltonian is when \(\mathbf{A}^{\prime}\) is identically the adjacency matrix in Eq.3. So, by studying this Hamiltonian in the \(0\) temperature limit, one can study the properties of \(\mathbf{A}\). Computing the partition function \(Z\), we find
\[Z=\prod_{ij}\sum_{A^{\prime}_{ij}=0}^{1}e^{-\beta H[\mathbf{A}^{\prime}]}=\prod _{ij}\left(1+e^{-\beta(1-\mathbf{v}_{i}\cdot\mathbf{v}_{j})}\right), \tag{17}\]
and the link probability is:
\[n_{ij}:=\sum_{A_{ij}=0}^{1}A_{ij}P(\mathbf{A}_{ij})=\frac{1}{e^{\beta(1- \mathbf{v}_{i}\cdot\mathbf{v}_{j})}+1}, \tag{18}\]
which is a Fermi distribution with chemical potential \(\mu_{ij}=\mathbf{v}_{i}\cdot\mathbf{v}_{j}\). Note that these chemical potentials are not independent, so when averaging, correlations appear when computing quantities involving three or more individuals.
To summarize, in this letter we have examined the projection of a random bipartite network of people connected to events. Averaging over all the realizations, it is possible to extract information about the bipartite structure only with measures involving three or more agents. Indeed, lower-order quantities (e.g., the degree distribution) are not distinguishable from those of a random network if the number of events is large enough. This is evident by noting that the bipartite structure mapped in a hypercube imposes geometrical constraints. We have shown analytically how these constraints affect the properties of the projected network. In particular, we have investigated the scaling properties of the clustering coefficient in the monopartite network showing, for example, that when there are many events the system is fragmented into smaller communities.
In conclusion, our findings offer a new perspective to understanding the deep differences between monopartite and bipartite networks. Moreover, in most cases, we only see the monopartite network of connected individuals, and the reasons behind the observed social structure are hidden. These are too complex to trace, but our findings can provide insights into people's diversification and relationships. Our work can offer a way to infer the underlying mechanisms (such as the co-participation in different events) giving rise to the apparent person-to-person relationship network.
|
2308.03080 | Peakless Motzkin paths of bounded height | There was recent interest in Motzkin paths without peaks (peak: up-step
followed immediately by down-step); additional results
about this interesting family is worked out. The new results are the
enumeration of such paths that live in a strip $[0..\ell]$, and as
consequence the asymptotics of the average height, which is given by $2\cdot
5^{-1/4}\sqrt{\pi n}$. Methods include the kernel method
and singularity analysis of generating functions. | Helmut Prodinger | 2023-08-06T10:19:00Z | http://arxiv.org/abs/2308.03080v1 | # Peakless Motzkin paths of bounded height
###### Abstract.
There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a strip \([0..\ell]\), and as consequence the asymptotics of the average height, which is given by \(2\cdot 5^{-1/4}\sqrt{\pi n}\). Methods include the kernel method and singularity analysis of generating functions.
Key words and phrases:Motzkin paths, peakless, height, generating functions, asymptotics 2010 Mathematics Subject Classification: 05A15
## 1. Introduction
Motzkin paths are cousins of the more famous Dyck paths. They appear first in [7]. In the encyclopedia [12] they are sequence A001006, with many references given. They consist of up-steps \(U=(1,1)\), down-steps \(D=(1,-1)\) and horizontal (flat) steps \(F=(1,0)\). Slightly different notations are also in use. They start at the origin and must never go below the \(x\)-axis. Usually one requires the path to end on the \(x\)-axis as well, but occasionally one uses the term _Motzkin path_ also for paths that end on a different level. Figure 1 shows all Motzkin paths of 4 steps (=length 4).
An important concept is the _height_ of a path. It is the maximal \(y\)-coordinate when scanning the path (from left to right, say). For the paths in Figure 1, the heights are (in this order) \(0,1,1,1,1,1,1,2\). The average height of all Motzkin paths of length \(n\) was computed in an early paper of the present writer [9].
Recently, I learned from the paper [2] that there is interest in _peakless Motzkin paths_. A peak in a Motzkin path is a sequence of an up-step followed immediately by a down-step. Figure 2 indicates all peaks in the list of Motzkin paths of length 4. The enumerating sequence is A004148 in [12], where one can find several references; one recent paper about the subject is [3]. A general discussion about forbidden patterns, concentrating on analytic aspects, is in [1].
The paths without peaks are called peakless, and there are four of them, as shown in Figure 3.
Figure 1. All 9 Motzkin of 4 steps (length 4).
## 2. A warmup: Enumeration of peakless Motzkin paths via the kernel method
We will use the following generating functions: \([z^{n}]f_{i}(z)\) is the number of peakless paths ending at state \(i\) in the top layer (Figure 4), \([z^{n}]g_{i}(z)\) is the number of peakless paths ending at state \(i\) in the bottom layer; for convenience, we mostly write \(f_{i}\) and \(g_{i}\). The following recursions can be read off the automaton, by considering the last step separately.
\[f_{0} =1+zf_{0}+zf_{1}+zg_{0},\] \[f_{i} =zf_{i}+zf_{i+1}+zg_{i},\quad i\geq 1,\] \[g_{0} =0,\] \[g_{i+1} =zf_{i}+zg_{i},\quad i\geq 0.\]
To solve this system, one introduces double generating functions
\[F(u,z)=\sum_{i\geq 0}u^{i}f_{i}(z)\quad\text{and}\quad G(u,z)=\sum_{i\geq 0}u^{i} g_{i}(z).\]
Again, for convenience, we mostly write \(F(u)\) and \(G(u)\). By summing the recursions, we find
\[F(u) =1+zF(u)+\frac{z}{u}(F(u)-F(0))+zG(u),\] \[G(u) =zuF(u)+zuG(u)=\frac{zuF(u)}{1-zu}.\]
Figure 4. Graph (automaton) to recognize peakless Motzkin paths. Starting at the origin and ending at nodes labelled \(0\) corresponds to Motzkin paths, and ending at a node labelled \(k\) to a path that ends at level \(k\).
Figure 3. Peakless Motzkin paths of length \(4\).
Figure 2. Motzkin paths with peaks indicated.
Eliminating one function, we are left to solve (note that \(F(0)=f_{0}\))
\[F(u)=1+zF(u)+\frac{z}{u}(F(u)-F(0))+\frac{z^{2}uF(u)}{1-zu}.\]
Rewriting the functional equation, we find
\[F(u)=\frac{(-u+zF(0))(1-zu)}{zu^{2}+(z-z^{2}-1)u+z}=\frac{(-u+zF(0))(1-zu)}{z(u- s_{1})(u-s_{2})}\]
with
\[s_{1}=\frac{1-z+z^{2}+\sqrt{(1+z+z^{2})(1-3z+z^{2})}}{2z} \tag{1}\]
and
\[s_{2}=\frac{1-z+z^{2}-\sqrt{(1+z+z^{2})(1-3z+z^{2})}}{2z}. \tag{2}\]
Note that \(s_{1}s_{2}=1\). Plugging \(u=0\) into that equation does not help, but one of the factors from the denominator can be cancelled. This is (a simple instance of) the kernel method. Some twenty years ago, I collected various related examples [10], and many more just recently [11].
Since \(s_{2}=z+z^{2}+\cdots\), the factor \((u-s_{2})\) must cancel, since \(1/(u-s_{2})\) would not have a power series expansion around \(u\), \(z\) close to zero. Performing the cancellation, we get
\[F(u)=\frac{zs_{2}-1+zu-z^{2}F(0)}{z(u-s_{1})}\quad\text{and}\quad F(0)=\frac{ zs_{2}-1-z^{2}F(0)}{-zs_{1}}.\]
Solving,
\[F(0)=f_{0} =\frac{s_{2}}{z}=\frac{1-z+z^{2}-\sqrt{(1+z+z^{2})(1-3z+z^{2})}}{ 2z^{2}}\] \[=1+z+z^{2}+2z^{3}+4z^{4}+8z^{5}+17z^{6}+\cdots\]
We are mostly interested in
\[H(u):=F(u)+G(u)=\frac{-u+zF(0)}{zu+z-z^{2}u-u+zu^{2}},\]
although \(F(u)\) and \(G(u)\) could be computed separately as well. The coefficients of \([u^{k}]H(u)\) are just enumerating all peakless Motzkin paths ending on level \(k\). Hence
\[H(u)=\frac{-u+zF(0)}{zu+z-z^{2}u-u+zu^{2}}=\frac{-1}{z(u-s_{1})}.\]
Expanding and noting that \(s_{1}=1/s_{2}\), we find
\[[u^{k}]H(u)=\frac{s_{2}^{k+1}}{z}.\]
This could be seen as well by a canonical decomposition of a peakless Motzkin path according to the last return to the \(x\)-axis.
The denominator of \(H(u)\) has a special significance; if we write \(h_{k}=[u^{k}]H(u)\), the recursion for these quantities can be read off from the denominator:
\[zh_{k}+(z-z^{2}-1)h_{k-1}+zh_{k-2}=0, \tag{3}\]
this can be checked directly as well by inserting \(h_{k}=s_{2}^{k+1}/z\) and simplifying.
The sequence enumerating peakless Motzkin paths is A004148 in [12]. If we call them \(m(n)=[z^{n}]s_{2}/z\), then the software \(\mathsf{Gfun}\), implemented in Maple, produces the recursion
\[nm(n)-(2n+3)m(n+1)-(n+3)m(n+2)-(2n+9)m(n+3)+(n+6)m(n+4)=0,\]
with initial values \(m(0)=1\), \(m(1)=1\), \(m(2)=1\), \(m(3)=2\).
A second order linear recursion with constant coefficients is driven by the characteristic equation and its two roots. Not surprisingly, they are \(s_{1}\) and \(s_{2}\).
For completeness, we mention the asymptotics of the coefficients \(m(n)\) of
\[\frac{s_{2}}{z}=\frac{1-z+z^{2}-\sqrt{(1+z+z^{2})(1-3z+z^{2})}}{2z^{2}}.\]
This is a standard application of _singularity analysis of generating functions_, as described in [5]. First, we must consider the closest singularity to the origin. The candidates are the solutions of \((1+z+z^{2})(1-3z+z^{2})=0\). There are two complex solutions of absolute value \(1\), which are irrelevant, and then \(\frac{3\pm\sqrt{5}}{2}\). The relevant value is \(\varrho=\frac{3-\sqrt{5}}{2}\); note that \(1/\varrho=\frac{3+\sqrt{5}}{2}\), which is the square of the golden ratio \(\phi=\frac{1+\sqrt{5}}{2}\) from the Fibonacci fame.
The local expansion around \(z\sim\varrho\) looks like
\[\frac{s_{2}}{z}\sim\frac{1}{\varrho}-\frac{5^{1/4}}{\varrho}\sqrt{1-\frac{z}{ \varrho}},\]
and following the principles of singularity analysis we might translate this to the coefficients:
\[[z^{n}]\frac{s_{2}}{z}\sim\frac{5^{1/4}\varrho^{-n-1}}{2\sqrt{\pi}n^{3/2}}.\]
## 3. Peakless Motzkin paths of bounded height
We fix a parameter \(\ell\geq 0\) and postulate that \([u^{j}]H(u)=0\) for \(j>\ell\). This means that states \(\ell+1,\ell+2,\dots\) (on both layers) can never be reached. The recursion, compare (3)
\[zh_{k+1}+(z-z^{2}-1)h_{k}+zh_{k-1}=0\]
is then best written as a matrix equation:
\[\begin{pmatrix}z-z^{2}-1&z&0&0&\dots&&\\ z&z-z^{2}-1&z&0&0&\dots\\ 0&z&z-z^{2}-1&z&0&\dots\\ \vdots&\vdots&\vdots&\ddots&\ddots&&\\ &&&&z&z-z^{2}-1\end{pmatrix}\begin{pmatrix}h_{0}\\ h_{1}\\ h_{2}\\ \vdots\\ h_{\ell}\end{pmatrix}=\begin{pmatrix}-1\\ 0\\ 0\\ \vdots\\ 0\end{pmatrix}\]
Let \(\mathscr{D}_{\ell}\) be the determinant of the \((\ell+1)\times(\ell+1)\) matrix. \(\mathscr{D}_{0}=z-z^{2}-1\), \(\mathscr{D}_{1}=(z-z^{2}-1)^{2}-z^{2}=(1-z)^{2}(1+z^{2})\). It is a bit easier to work with \(\mathscr{D}_{-1}=1\) instead of \(\mathscr{D}_{1}\). The solution is, by standard methods,
\[\mathscr{D}_{\ell}=-\frac{1}{W}(-zs_{1})^{\ell+2}+\frac{1}{W}(-zs_{2})^{\ell+ 2},\]
where we use the abbreviation \(W=\sqrt{(1+z+z^{2})(1-3z+z^{2})}\).
The quantity \(h_{0}\), describing _all_ paths, restricted as described, can, by Cramer's rule, be written as
\[h_{0}=\frac{-\mathscr{D}_{\ell-1}}{\mathscr{D}_{\ell}} =-\frac{-\frac{1}{W}(-zs_{1})^{\ell+1}+\frac{1}{W}(-zs_{2})^{\ell+ 1}}{-\frac{1}{W}(-zs_{1})^{\ell+2}+\frac{1}{W}(-zs_{2})^{\ell+2}}\] \[=\frac{1}{z}\frac{-(-s_{1})^{\ell+1}+(-s_{2})^{\ell+1}}{(-s_{1}) ^{\ell+2}-(-s_{2})^{\ell+2}}=\frac{1}{z}\frac{s_{1}^{\ell+1}-s_{2}^{\ell+1}}{ s_{1}^{\ell+2}-s_{2}^{\ell+2}}.\]
In the limit \(\ell\to\infty\), \(h_{0}=\frac{1}{zs_{1}}=\frac{s_{2}}{z}\). This is, as we have seen already, the generating function of peakless Motzkin paths without boundary. The other functions \(h_{i}\) could be computed by Cramer's rule as well, but we concentrate only on the paths that return to the origin and are bounded by \(\ell\). They live in the strip \([0..\ell]\). At this stage, we drop the '\(0\)' from the notation and make the '\(\ell\)' explicit by writing \(A_{n,\ell}\). We summarize the results:
**Theorem 1**.: _The number of peakless Motzkin paths of length \(n\) (returning to the \(x\)-axis), bounded by \(\ell\), is given by_
\[A_{n,\ell}=[z^{n}]\frac{1}{z}\frac{s_{1}^{\ell+1}-s_{2}^{\ell+1}}{s_{1}^{\ell +2}-s_{2}^{\ell+2}},\]
_with the functions \(s_{1}\) and \(s_{2}\) given in (1) and (2). The formula is only correct for \(\ell\geq 1\); if \(\ell=0\), there is only one such path of length of \(n\), namely consisting of flat steps only; the generating function must then be replaced by \(\frac{1}{1-z}\)._
We will use the notations (\(\ell\geq 1\))
\[A_{\ell}(z)=\frac{1}{z}\frac{s_{1}^{\ell+1}-s_{2}^{\ell+1}}{s_{1}^{\ell+2}-s_ {2}^{\ell+2}}\quad\text{and}\quad A_{\infty}(z)=\frac{s_{2}}{z}.\]
As can be checked directly (best with a computer), there is the recursion of the continued fraction type
\[A_{\ell}=\frac{1}{1-z+z^{2}-z^{2}A_{\ell-1}},\]
and thus
\[A_{1}=\frac{1}{1-z+z^{2}-\frac{z^{2}}{1-z+z^{2}}},\quad A_{2}=\frac{1}{1-z+z^ {2}-\frac{z^{2}}{1-z+z^{2}-\frac{z^{2}}{1-z+z^{2}}}}\]
and so on. As discussed, the formula for \(A_{0}=\frac{1}{1-z}\) is slightly different.
Continued fraction expansions are very common in the context of generating functions of lattice paths of bounded height; the first example is (perhaps) [4].
The total generating function \(A_{\infty}\) has a pretty expansion as well, viz.
\[A_{\infty}=1+\cfrac{z}{1-\cfrac{z}{1-\cfrac{z}{1-\cfrac{z^{3}}{1-\cfrac{z}{1- \cfrac{z}{1-\cfrac{z^{3}}{1-\ddots}}}}}}}\]
this can be checked directly. I am wondering whether this might have an easy combinatorial interpretation.
Now we consider the _average height_ of peakless Motzkin paths of length \(n\), assuming that all of them are equally likely. As always, when enumerating the average height, the relevant formula is
\[\frac{[z^{n}]\sum_{\ell\geq 0}\big{(}A_{\infty}(z)-A_{\ell}(z)\big{)}}{[z^{n}]A _{\infty}(z)}.\]
The difference \(A_{\infty}(z)-A_{\ell}(z)\) enumerates paths of height \(>\ell\), or
\[\frac{s_{2}}{z}-\frac{1}{z}\frac{s_{1}^{\ell+1}-s_{2}^{\ell+2}}{s_{1}^{\ell+2 }-s_{2}^{\ell+2}}=\frac{1}{z}\frac{(1-s_{2}^{2})s_{2}^{\ell+1}}{s_{1}^{\ell+2} -s_{2}^{\ell+2}}=\frac{W}{z^{2}}\frac{s_{2}^{\ell+2}}{s_{1}^{\ell+2}-s_{2}^{ \ell+2}}\]
This must be expanded around \(z=\varrho\); we decrease \(\ell\) by one, since then we enumerate paths of height \(\geq\ell\). Since \(s_{2}\sim 1\), we find the approximation
\[\frac{W}{\varrho^{2}}\frac{1}{s_{1}^{\ell+1}-1}=\frac{W}{\varrho^{2}}\frac{s_{ 2}^{\ell+1}}{1-s_{2}^{\ell+1}}=\frac{W}{\varrho^{2}}\sum_{k\geq 1}s_{2}^{( \ell+1)k}\]
A local expansion yields
\[\frac{W}{\varrho^{2}}\sim\frac{2\cdot 5^{1/4}}{\varrho}\Big{(}1-\frac{z}{ \varrho}\Big{)}^{1/2}.\]
From our earlier computation,
\[s_{2}\sim 1-5^{1/4}\Big{(}1-\frac{z}{\varrho}\Big{)}^{1/2}.\]
From [6] we conclude that
\[\sum_{k\geq 1}\frac{s_{2}^{k}}{1-s_{2}^{k}}\sim-\frac{\log(1-s_{2})}{1-s_{2}},\]
as \(s_{2}\to 1\). Our paper [6] has many more technical details about a similar scenario. Putting both expansions together (we don't care about a missing term in the sum as we only want to work out the leading term),
\[\frac{W}{\varrho^{2}}\sum_{k,\ell\geq 1}s_{2}^{(\ell+1)k} \sim\frac{2\cdot 5^{1/4}}{\varrho}\Big{(}1-\frac{z}{\varrho}\Big{)}^{1 /2}\cdot\frac{-\log(1-s_{2})}{1-s_{2}}\] \[\sim\frac{2\cdot 5^{1/4}}{\varrho}\Big{(}1-\frac{z}{\varrho} \Big{)}^{1/2}\cdot\frac{-\log\Big{(}5^{1/4}\big{(}1-\frac{z}{\varrho}\big{)}^ {1/2}\Big{)}}{5^{1/4}\big{(}1-\frac{z}{\varrho}\big{)}^{1/2}}\] \[\sim-\frac{2}{\varrho}\log\Big{(}5^{1/4}\big{(}1-\frac{z}{\varrho }\big{)}^{1/2}\Big{)}\sim-\frac{2}{\varrho}\log\Big{(}1-\frac{z}{\varrho} \Big{)}^{1/2}\sim-\frac{1}{\varrho}\log\Big{(}1-\frac{z}{\varrho}\Big{)}.\]
By singularity analysis (transfer theorem) we find that
\[[z^{n}]\frac{W}{\varrho^{2}}\sum_{k,\ell\geq 1}s_{2}^{(\ell+1)k}\sim-[z^{n}] \frac{1}{\varrho}\log\Big{(}1-\frac{z}{\varrho}\Big{)}\sim\frac{\varrho^{-n- 1}}{n}.\]
As discussed before, the total number of peakless Motzkin paths of length \(n\) is asymptotic to
\[[z^{n}]\frac{s_{2}}{z}\sim\frac{5^{1/4}\varrho^{-n-1}}{2\sqrt{\pi}n^{3/2}}.\]
and for the average height we have to consider the quotient of the last two expressions, which is
\[\frac{\varrho^{-n-1}}{n}\frac{2\sqrt{\pi}n^{3/2}}{5^{1/4}\varrho^{-n-1}}=\frac {2\sqrt{\pi n}}{5^{1/4}}.\]
The numerical constant \(2/5^{1/4}=1.337480610\). This can be compared with the average height of _all_ Motzkin paths of length \(n\), which is asymptotic to \(\sqrt{\frac{\pi n}{3}}\), see [8]; \(3^{-1/2}=0.5773502693\).
|
2305.12171 | Diffusion Co-Policy for Synergistic Human-Robot Collaborative Tasks | Modeling multimodal human behavior has been a key barrier to increasing the
level of interaction between human and robot, particularly for collaborative
tasks. Our key insight is that an effective, learned robot policy used for
human-robot collaborative tasks must be able to express a high degree of
multimodality, predict actions in a temporally consistent manner, and recognize
a wide range of frequencies of human actions in order to seamlessly integrate
with a human in the control loop. We present Diffusion Co-policy, a method for
planning sequences of actions that synergize well with humans during test time.
The co-policy predicts joint human-robot action sequences via a
Transformer-based diffusion model, which is trained on a dataset of
collaborative human-human demonstrations, and directly executes the robot
actions in a receding horizon control framework. We demonstrate in both
simulation and real environments that the method outperforms other state-of-art
learning methods on the task of human-robot table-carrying with a human in the
loop. Moreover, we qualitatively highlight compelling robot behaviors that
demonstrate evidence of true human-robot collaboration, including mutual
adaptation, shared task understanding, leadership switching, and low levels of
wasteful interaction forces arising from dissent. | Eley Ng, Ziang Liu, Monroe Kennedy III | 2023-05-20T11:21:34Z | http://arxiv.org/abs/2305.12171v4 | # Diffusion Co-Policy for Synergistic Human-Robot Collaborative Tasks
###### Abstract
Modeling multimodal human behavior accurately has been a key barrier to increasing the level of interaction between human and robot, particularly for collaborative tasks. Our key insight is that the predictive accuracy of human behaviors on physical tasks is bottlenecked by the model for methods involving human behavior prediction. We present a method for training denoising diffusion probabilistic models on a dataset of collaborative human-human demonstrations and conditioning on past human partner actions to plan sequences of robot actions that synergize well with humans during test time. We demonstrate the method outperforms other state-of-art learning methods on human-robot table-carrying, a continuous state-action task, in both simulation and real settings with a human in the loop. Moreover, we qualitatively highlight compelling robot behaviors that arise during evaluations that demonstrate evidence of true human-robot collaboration, including mutual adaptation, shared task understanding, leadership switching, learned partner behaviors, and low levels of wasteful interaction forces arising from dissent.
## I Introduction
Multimodal behavior poses a key barrier to achieving effective human-robot coordination in collaborative tasks. In collaborative tasks, decentralized agents execute joint actions, which are defined in cognitive neuroscience as "any form of social interaction whereby two or more individuals coordinate their actions in space and time to bring about a change in the environment" [1]. For such scenarios, the ability to anticipate and predict partners' actions is crucial, as it can significantly enhance the capacity to plan synergistic actions that contribute to the team's success with an understanding of the task (i.e. how the team's actions affects the dynamics). Collaborative table carrying, the epitome of such tasks, demands on-the-fly mutual adaptation, and precise timing of movements without verbal communication. This paper introduces a new approach to coordinating and executing behaviors seamlessly, demonstrated through human-robot collaborative carrying.
Given the recent successes of denoising diffusion probabilistic models (DDPM) in learning multimodal behaviors for single agents [2, 3], we propose leveraging diffusion models for human-robot collaboration, and demonstrate that the gains in model expressivity and quality enabled by DDPM are highly suited for physical tasks that involve human and robot executing tightly-coupled actions. We train a robot co-policy using a Transformer-based diffusion model that conditions on past observations and human actions to generate sequences of future joint human and robot actions, as learned from human-human collaborative demonstrations, and show the effectiveness of the diffusion-based co-policy in both simulation and real robot experiments by highlighting compelling collaborative behaviors exhibited by the robot and human.
**Contributions**: In this paper, our primary contribution is the application of diffusion models for learning robot collaborative policies that can synergize with a real human partner. Our approach utilizes a transformer-based diffusion model to sample future joint actions based on past observations, including past joint actions. By modeling joint behavior as a series of individual actions taken by each agent, our method can generate plausible trajectories of joint actions in real-time, facilitating smooth, coordinated actions that can be executed without further planning or use of hand-engineered collaborative reward functions. We show that this method outperforms state-of-the-art approaches to human-robot coordination in both simulated and real-world experiments. Specifically, diffusion co-polic
Fig. 1: Using a diffusion co-policy for human-in-the-loop table-carrying, wherein the objective is to reach the goal (flags) while avoiding the obstacles (red squares). (_Left_) A sequence of past observations of the state and human actions from current time step \(t\) is passed as conditioning input to a Transformer-based diffusion co-policy (_center_). To generate a sequence of length of joint action predictions from \(\tau=[t,...,t+T_{a}]\), the model samples a sequence of Gaussian noise, \(a^{H}_{\tau,k}\) and \(a^{R}_{\tau,k}\), and iteratively refines the sequence for \(k\) steps into a sequence of joint action predictions, \(a^{H}_{\tau,0}\) and \(a^{R}_{\tau,0}\). (_Right_) The robot executes the robot action simultaneously with a live human, continuously predicting and executing until the goal is reached or collision occurs.
rates and lower wasteful interaction forces than previous state-of-art imitation learning methods used for human-robot interaction and collaboration. We evaluate the diffusion co-policy on both simulated and real robots with a human partner in the loop. By leveraging the generative capabilities of diffusion models, we have presented a significant step towards enabling effective human-robot collaboration on continuous state, joint-action tasks that require rapid mutual co-adaptation, behavior multimodality, shared task understanding and implicit constraints from human interaction, and coordinated movement.
## II Related Work
**Imitation Learning**: A multitude of work has been done in improving the quality of policies trained with offline datasets, particularly to account for multimodality. In particular, behavioral cloning has made significant progress due to modeling advances (e.g. LSTM-GMM [4], Transformers [5], Diffusion [3]). While these methods demonstrate the generative power of diffusion models, their effectiveness in scenarios that involve collaboration remains to be seen.
**Multiagent Reinforcement Learning**: Multiagent RL (MARL) is a widely studied problem, with most recent approaches leveraging RL algorithms developed for Markovian and stationary environments, and leveraging action information of other agents for the centralized critic [6, 7, 8]. Few methods [9] incorporate human behaviors, which are notably different than behaviors trained in RL algorithms [10]. Our approach is non-Markovian and non-stationary; we model the human non-stationary agent as a part of the environment. We take this approach for tasks in which predicting your partner's actions becomes critical to the task's success. In particular, just knowing the state of the table and agents is not sufficient information on the next table state since the table is over-actuated by two agents. Moreover, it is a non-trivial problem to design reward functions that are effective for cooperative MARL.
**Human intent modelling for HRI**: Prior work in human-robot interaction (HRI) have generally fallen along a spectrum defined by the degree to which the human is modelled, as well as how significantly such modelling can affect the human-robot interaction. Theory of Mind (ToM) methods, which ascribe mental states to the human with whom the robot interacts, generally involve learning human reward functions [11, 12], learning user type [13, 14], or latent strategies [9]. These approaches are predicated on the hypothesis that human movement follows a goal-directed policy, but do not account for humans having multimodal behaviors during interactions, or are limited to modeling policies across interactions. Rather than attributing behavior observations to mental states or strategies, black box methods leverage data to capture transitions in human actions and observations, thereby learning human behaviors with which to train robot policies. Recent advances in human-robot interaction suggest that a promising approach for incorporating the challenges of multimodal behavior is to leverage the expressiveness of generative models trained on offline demonstration datasets to learn human behavior, and leverage it for planning (e.g. Gaussian Mixture Models [15], conditional variational autoencoders [16], variational recurrent neural networks [17]) or learning a joint action policy (e.g. Co-GAIL [18]). In this work, we leverage diffusion models to learn a co-policy, wherein we predict actions of both agents involved in the task, and condition the predictions on past partner actions as well as observations.
## III Diffusion Co-Policy Formulation
This section describes the formulation of the collaborative robot policy as a Denoising Diffusion Probabilistic Model (DDPM). First, we describe the collaborative task and motivate the use of diffusion models, leading to the formulation of a co-policy which incorporates human action and scene conditioning for significantly improved human-robot coordination on continuous state-action tasks.
### _Problem Setting_
Consider a human-robot system wherein the dynamics of the world state \(s_{t}\in\mathcal{S}\subset\mathbb{R}^{n}\), are
\[s_{t+1}=f(s_{t},\textbf{a}_{t}) \tag{1}\]
and \(\textbf{a}_{t}=(a_{t}^{H},a_{t}^{R})\) denotes the joint action human \(H\) and robot \(R\) at time step \(t\). Provided with a control sequence of joint actions, a rollout starting from initial state \(s_{0}\) results in trajectory \(\tau=(s_{0},\textbf{a}_{0},...s_{T},\textbf{a}_{T})\).
Typically, the goal is to find a sequence of actions \(\textbf{a}_{0:T}^{*}\) that maximizes the sum of rewards \(\sum_{t=0}^{T}r(s_{t},\textbf{a}_{t})\) via trajectory optimization or reinforcement learning methods.
An instance of such a system is the collaborative carrying task [17], a human and a robot both carry a table at opposite ends, moving it from a start pose to a goal location while avoiding obstacles. Each agent in the collaborative carrying task applies a force at the opposing ends of the table, causing an acceleration and moment on the table. We assume that each agent has full observability of the map.
### _Approach_
While we could formulate the diffusion model for planning with RL using classifier-guided sampling [2, 19], doing so would require hand-designing a collaborative reward function. However, manually designing reward functions are tricky and prone to over-specification, and more so for multi-agent collaborative scenarios, where multimodal behaviors arise from preference, or other factors like diversity and even inconsistency. Despite promising directions in increasing the nuance of learned reward functions, such as learning multimodal rewards from demonstrations [20], querying methods would not be viable in interactive, long-horizon tasks like collaborative carrying, as specifying the query itself would be non-trivial, especially if deployed while executing the task.
Our **key insight** is that the predictive accuracy of human behaviors on physical tasks is bottlenecked by the model for methods involving human behavior learning. Given recent empirical breakthroughs in generative quality of diffusion models [21, 22], we propose leveraging diffusion models to
enable coordination on long-horizon, continuous state-action, human-robot collaborative tasks in various novel settings during test time. Rather than optimizing over a multimodal reward function for multimodal behaviors, we rely on the dataset of human-human collaborative demonstrations to capture and generalize interaction dynamics.
### _Denoising Diffusion Probabilistic Models_
DDPMs [23, 24, 21] are generative models that approach sample generation with an iterative denoising process modeled by Langevin dynamics. Data generation via diffusion works by denoising (reversing) a forward diffusion process that iteratively adds noise to data until it resembles as standard Gaussian. Specifically, samples from a standard Gaussian prior, \(p(x_{K})=\mathcal{N}(0,I)\) pass through \(K\) iterations of noise reduction based on a fixed iteration-dependent variance schedule (parameterized by \(\sigma_{k}\), \(\alpha_{k}\) and \(\gamma_{k}\)), producing \(K\) intermediate latent variables, \(x_{k-1},...,x_{0}\), where \(x_{0}\) is the noiseless output. The Gaussian noise predicted is parameterized by a network, \(\epsilon_{\theta}(x_{k},k)\). Thus, to sample \(x_{k-1}\sim p(x_{k-1}|x_{k})\), we compute:
\[x_{k-1}=\alpha_{k}\Big{(}x_{k}-\gamma_{k}\epsilon_{\theta}(x_{k},k)\Big{)}+ \sigma_{k}z \tag{2}\]
where \(z\sim\mathcal{N}(0,I)\).
_Clarification of notation:_ In this work, we use two different time steps in subscript: \(k\) to denote the diffusion timestep, and \(t\) to denote the prediction time step, i.e. \(s_{t,k}\) refers to the \(t^{th}\) state in the \(k^{th}\) diffusion step. Subscripts of noiseless quantities are omitted, e.g. \(s_{t}\). Subscripts of constants parameterized only by \(k\) do not have a time-indexed subscript, e.g. \(\epsilon_{k}\).
### _Diffusion Co-Policy for Coordinating with Humans_
We consider the task of modeling a robot co-policy as learning a probabilistic model for the robot that infers future _sequences of joint human-robot_ actions, conditioned on past states, map information, and past human actions. Motivated by the collaborative carrying task, we modify the DDPM in two ways: 1. Conditioning on past human actions, which allows the robot to derive an understanding of human strategy from past human action trajectories to aid future team predictions; and 2. High-level goal conditioning, wherein the robot can condition its predictions on where the carried table should land.
Therefore, we seek to learn \(p(\mathbf{a}_{t}|s^{\prime}_{t},a^{H}_{t-1})\), where \(s^{\prime}_{t}\) is an augmented state vector with a vector of fixed size with the locations of obstacles, as well as the initial pose and goal location. The fixed size of the obstacle representation is limited to the maximum number of obstacles in the maps (3 in this work). _Since our focus is not on perception_, this is a sufficient representation for learning map obstacles locations for robot navigation, but future work will focus on improving the observation model.
The conditional distribution \(p(\mathbf{a}_{t}|s^{\prime}_{t},a^{H}_{t-1})\) can be captured by modifying Eq. 2:
\[\mathbf{a}_{t,k-1}=\alpha_{k}\Big{(}\mathbf{a}_{t,k}-\gamma_{k}\epsilon_{ \theta}(s^{\prime}_{t},a^{H}_{t-1},\mathbf{a}_{t,0}+\epsilon_{k},k)\Big{)}+ \sigma_{k}z \tag{3}\]
### _Network Architecture_
We adopt the Transformer-Based Diffusion architecture from Chi, et. al. [3], which uses the minGPT [25] transformer decoder model for action prediction, and modify the inputs as follows. Noise-injected joint actions from the human-human dataset, \(\mathbf{a}_{t,k}\), are tokenized and passed to the transformer decoder, which uses a sinusoidal embedding to encode the \(k^{th}\) diffusion step inputs as well as \(k\), which is prepended as first token. Positional embedding is applied to conditional inputs, \(s^{\prime}_{t}\) and \(a^{H}_{t-1}\), which are converted to a sequence before passed to the transformer decoder. The decoder then predicts the noise corresponding to each input in the time dimension for the \(k^{th}\) iteration. A causal attention mask constrains the attention of each action to itself and prior actions. Since _past human actions_ are treated as observations, each joint action embedding can attend to all past human actions. The predicted joint action sequence is constructed only after the predicted noise is propagated through the noise scheduler following the reverse diffusion process.
As expected [3], the 1D temporal CNN diffusion model does not work as well as the Transformer-based model for this task due to oversmoothing the action space. For the task we focus on, the actions are inertial forces and torques. Depending on the user and playing style, people tend to apply short impulses due to damping and oscillations when pivoting around obstacles or correcting speed, hence making the Transformer-based architecture useful for the table-carrying task.
### _Training_
To train the diffusion model, a value of \(k\) is randomly sampled, which is then used to sample noise \(\epsilon_{k}\) with variance \(\sigma_{k}\). The sampled noise is then added to input data, which, in this work, is a sequence consisting of \(T_{o}\) steps of augmented states \(s^{\prime}\) and past human actions \(a^{H}\), after which the noise model \(\epsilon_{\theta}(x_{k},k)\) predicts the added noise. Therefore, the loss for the noise model is
\[\mathcal{L}=\|\epsilon_{k}-\epsilon_{\theta}(s^{\prime}_{t},a^{H}_{t-1}, \mathbf{a}_{t,0}+\epsilon_{k},k))\|^{2}_{2} \tag{4}\]
## IV Evaluation
A key advantage of the Diffusion Co-policy is its generative quality; however, the model's effectiveness on human-robot collaborative tasks remains unexplored. The collaborative carrying task itself poses several interesting questions for evaluation: 1) Can it learn to effectively condition on obstacles? 2) Can it learn to effectively condition on its partner's behaviors and shared task representation? 3) Can it mutually adapt with real humans in test time? To attempt to address these questions, we compare our diffusion co-policy against several state-of-art imitation learning methods on a suite of evaluations described in the following sections.
We select three learning-based methods for comparison:
* **BC-LSTM-GMM [4]**: Multiple works in human-robot interaction have leveraged this method or variants of it, notably due to the performance increase with the GMM output layer; thus, we select this as a baseline and
adapt the implementation from [4]. Originally, we had added human action conditioning as well; however, this resulted in the learned policy performing much more poorly.
* **VRNN Planner [17]**: This is our only baseline with a sampling-based planning approach geared for this task. It predicts team waypoints learned from demonstrations autoregressively for receding horizon planning, and does not condition on human actions.
* **Co-GAIL [18]**: Similar to our approach, Co-GAIL learns collaborative behaviors from demonstrations. However, Co-GAIL explicitly disentangles latent behaviors with a human-action recognition model, and trains a co-policy using Generative Adversarial Neural Networks (GANs).
Since all baselines were not originally trained with map information (obstacles and goal locations, as well as initial table state), we trained all baselines with the same observation model we used for the diffusion co-policies, i.e. with the augmented state vector, \(s^{\prime}_{t}\). We also compare two variants of the Diffusion Co-Policy: one with past human action conditioning (**CoDP-H**), and the other without (**CoDP**).
### _Experimental Setup_
We trained the diffusion co-policy, CoDP, and variant (CoDP-H) to output joint actions at 10Hz, which were then executed on a simulator running at 30Hz by applying a zero-order hold of 3 time steps for each planned. All other baselines were trained to produce outputs at 30Hz. We conduct four simulation experiments and a real robot experiment for the collaborative carrying task to address the questions posed in the prior section:
* **Co-planning (_Sim. only_)**: To test each method's ability to complete the task and learn a representation of the map without the potential added benefit of human co-piloting corrections, we varied out-of-distribution obstacle locations while keeping the same distribution of initial and goal states from the training data. We executed \(T_{a}=8\) actions sampled with 100 denoising steps before replanning, which takes roughly 0.3 sec on a NVIDIA 3090Ti GPU.
* **Human-in-the-loop evaluation (_Sim._)**: In this evaluation, we have the robot policies complete the task with a real human-in-the-loop, in various out-of-training-distribution settings. The human teleoperates the orange circle agent in the simulation using joystick control. The sampling scheme for the human-in-the-loop simulation evaluation is different than the co-planning setting due to having a human in the loop. To account for visual latency and reaction time, we execute \(T_{a}=1\) sampled actions with 34 inference steps to allow for planning time of roughly 0.1 sec, and zero-order hold each planned action for 3 time steps before executing the next planned action, resulting in low visual latency in the simulator.
* **Human-in-the-loop evaluation (_Real robot_)**: In the real robot experiments, a human teleoperates an Interbotix Locobot with a pin connection (the table) to another Locobot operating with the policy or planner. We use the same sampling scheme from the human-in-the-loop evaluation.
### _Dataset and Map Details_
#### Iv-B1 Training data
The training dataset consists of 376 human-human demonstrations (179,993 environment interactions) on the collaborative carrying task on a total of 36 possible configurations. Due to the added complexity of obstacle
Fig. 3: **Qualitative Observations: (a) _Mutual adaptation and shared task understanding_ between human (orange circle) and CoDP-H agent (blue triangle). (b) _Leadership switching_ with CoDP-H. (c) _Without conditioning on past human behavior_, robot behavior is notably affected: the human becomes de-facto leader as the robot displays passive behavior. (d) _With conditioning_ on its partner’s past actions, the robot actively takes the lead, and displays interesting leadership switching behaviors via pivoting.
Fig. 2: Co-planning evaluation training (yellow box) and test (green box) maps. Start poses and goal positions (green “X”s) are shown, along with obstacle layouts (red boxes).
representation learning, cost of demonstration collection, and requirement of hand-engineered obstacle placement to allow for multimodal behaviors, we used up to a maximum of three obstacles in each training demonstration, with each initial, goal, and obstacle location illustrated in Fig. 1(a). Note that while many offline dataset learning methods [26] augment data with trained RL policies, planners paired with PID controllers, etc., we recognize that such augmented data could skew our dataset distribution, particularly if these methods are do not contain demonstrations of multimodal, sub-optimal, and inconsistent, yet "human-like" behaviors. For example, RRT planners do not exhibit the same behaviors (e.g. rotations, distance from obstacles) that human demonstrators do on the collaborative carrying task [17].
#### Iv-B2 Test maps
For the co-planning evaluation, we evaluated on all possible combinations of unseen map settings outlined in Fig. 1(b). For the human-in-the-loop simulation evaluation, we sampled from a subset of unseen maps, goals, in addition to different initial orientations. For reference, \(\pi\) is the initial table orientation depicted in Fig. 2, and we included four total initial orientations in our sampling, i.e. \([0,\frac{\pi}{2},\pi,\frac{3\pi}{2}]\). We then evaluated on the same sampled subset per method. On the real robot evaluation, we used a one obstacle unseen map with two initial orientations: robot facing the obstacle first, and human facing the obstacle first.
### _Simulation Results: Key Findings_
#### Iv-C1 Co-planning
We subjected the co-policies to the collaborative carrying task on maps seen in training, as well as maps unseen in training. **CoDP-H** consistently outperforms other baselines on all maps tested (Table I). Considering the few maps and obstacle configurations used during training, methods leveraging diffusion models exhibited unexpectedly high performance on maps with novel obstacle locations, suggesting that the diffusion co-policy was able to learn obstacle representations efficiently.
#### Iv-C2 Human-in-the-loop Trials
CoDP-H outperforms other baselines on the human-in-the-loop evaluation on task success rate (Table II). Fig. 5 shows heat maps of the state visitation frequencies for all of the human-in-the-loop simulation evaluations for each policy/planner, posed against state visitation frequencies and obstacle layouts present in the human-human demonstration dataset. Note in the test maps that obstacles were placed in locations where the table frequently visited in the training data. This suggests that CoDP-H is able to learn a shared task representation and perform well on unseen maps with a real human partner over multimodal behaviors.
Table II also demonstrates the planning time disadvantage of diffusion-based methods; yet, despite planning less frequently and requiring interpolation methods, diffusion-based methods achieve higher success rate on the task.
of the task and its partner's intentions. Fig. (d)d shows that the CoDP-H robot is capable of proactive behavior since it has learned to associate past partner actions with past observations. Without this past action conditioning, the robot acts passively, leaving the human to lead (Fig. (c)c).
#### Iii-B4 Low interaction forces
Interaction forces are forces that do not contribute to motion; if the magnitude of the interaction forces between forces applied by two different agents on the carried load are non-zero, then this implies that the carried load is being stretched (positive) or compressed (negative) by the two agents [27]. Interaction forces are considered "wasteful" since they do not lead to motion. Moreover, on the table-carrying task, examining the interaction forces on the table can lend insight into periods of collaboration, disagreement, and/or other decision points in the trajectory. Fig. 6 shows that the interaction forces between the diffusion co-policies and human are lower in magnitude over the course of the example trajectory with a human in the loop in simulation. To get a better understanding of the interaction forces exhibited during human-in-the-loop evaluations over the _maps_, we plot a heat map of the magnitudes of normalized interaction forces over all trajectories for each method in Fig. 7. Interaction forces less than 0.25 are generally negligible, and all human-human demonstrations displayed a negligible frequency of non-zero interaction forces. Magnitudes of normalized interaction forces between 0.25 - 0.75 may indicate a decision point or dissent; interaction forces above 0.75 is a strong indicator of dissent or human taking corrective action. Across all methods, it is clear that CoDP-H displays less dissent over most states, including those surrounding obstacles. Interaction forces generally were higher across all methods near the lower boundary of the map.
both settings, while BC-LSTM-GMM is biased towards one modality.
Table III shows that while BC-LSTM-GMM appears a strong baseline in the robot "leading" case, it performs poorly in the human "leading" case. BC-LSTM-GMM prefers a route below the obstacle, as seen in Fig. 4; if the human partner happens to adapt to the robot or pick the same route, this tends to lead to success. However, unlike CoDP-H, it is unable to adapt to move above the obstacle when necessary to achieve task success.
## V Conclusion
In this work, we explore using action predictions from diffusion models directly to plan collaborative actions that synergize well with real humans in the loop and demonstrate shared task understanding. Through a suite of evaluations on both simulation and real settings, with and without a human-in-the-loop, we show that a co-policy developed with a Transformer-based diffusion model conditioning on past human actions can not only plan multimodal action sequences with real humans-in-the-loop to achieve high success rates, but also qualitatively display compelling collaborative behaviors in novel, out-of-training-distribution settings, including mutual adaptation, shared task understanding, learning partner behaviors, and leadership switching.
Our study has several limitations. On the implementation side, the time required for diffusion planning is significantly longer than other model types. However, ongoing efforts to enhance the sampling process may increase the speed and quality of this framework as model development advances [22]. Moreover, the real robot experiments had several limitations, not least including physical capabilities of the robots, physical space constraints, and variance among human subjects. On the model side, the current method does not have a semantic latent space for generating behavior. To generate specific future behaviors that influence team interactions, we may need explicit guidance. Janner, et. al. has shown that sampling with diffusion for planned behaviors can be guided by learned cost functions [2]. To avoid hand-designed rewards, future work could separately train learned multimodal human reward functions to guide the sampling process accordingly. Despite these limitations, the diffusion co-policy has demonstrated the significance of the model bottleneck for quality action prediction that truly
Fig. 6: Visualization of interaction forces over rollouts on a sample map configuration played between each policy/planner (blue triangle) with a human (orange circle) in the loop in simulation, with the goal (yellow region) and obstacles (red squares) shown. Closer to zero is better, as demonstrated in the human-human exemplar. The diffusion policies show overall less (in magnitude) interaction forces over the course of the trajectory. Co-GAIL in this case is constantly checked by the human, who exerts an opposing force to prevent the table from hitting obstacles. BC-LSTM-GMM crashes when the human lets it continue moving without interference since it cannot reorient itself to pass between the two obstacles. VRNN shows relatively higher levels of the table compressing and stretching than any of the diffusion policies.
Fig. 7: From left to right: interaction forces from the human-human demonstration dataset, as well as each method evaluated with a human-in-the-loop during simulation evaluations on novel, unseen maps. For each trajectory of each method, we binned the interaction force _magnitudes_ (see color-map bar for bins) at each location in the discretized map, then _normalized_ the magnitudes across _all_ trajectories. Each map shows one to three obstacles that were placed in the the potential obstacle locations shown as red squares for each map. CODP-H clearly shows lower magnitudes of interaction forces across all human-in-the-loop simulation evaluations.
leverages information on the task and human behaviors, and enables work on robots that can act in a tightly coupled, interdependent manner with humans on collaborative tasks like table-carrying.
|
2303.00823 | Automated control and optimisation of laser driven ion acceleration | The interaction of relativistically intense lasers with opaque targets
represents a highly non-linear, multi-dimensional parameter space. This limits
the utility of sequential 1D scanning of experimental parameters for the
optimisation of secondary radiation, although to-date this has been the
accepted methodology due to low data acquisition rates. High repetition-rate
(HRR) lasers augmented by machine learning present a valuable opportunity for
efficient source optimisation. Here, an automated, HRR-compatible system
produced high fidelity parameter scans, revealing the influence of laser
intensity on target pre-heating and proton generation. A closed-loop Bayesian
optimisation of maximum proton energy, through control of the laser wavefront
and target position, produced proton beams with equivalent maximum energy to
manually-optimized laser pulses but using only 60% of the laser energy. This
demonstration of automated optimisation of laser-driven proton beams is a
crucial step towards deeper physical insight and the construction of future
radiation sources. | B. Loughran, M. J. V. Streeter, H. Ahmed, S. Astbury, M. Balcazar, M. Borghesi, N. Bourgeois, C. B. Curry, S. J. D. Dann, S. DiIorio, N. P. Dover, T. Dzelzanis, O. C. Ettlinger, M. Gauthier, L. Giuffrida, G. D. Glenn, S. H. Glenzer, J. S. Green, R. J. Gray, G. S. Hicks, C. Hyland, V. Istokskaia, M. King, D. Margarone, O. McCusker, P. McKenna, Z. Najmudin, C. Parisuaña, P. Parsons, C. Spindloe, D. R. Symes, A. G. R. Thomas, F. Treffert, N. Xu, C. A. J. Palmer | 2023-03-01T21:08:51Z | http://arxiv.org/abs/2303.00823v1 | # Automated control and optimisation of laser driven ion acceleration
###### Abstract
The interaction of relativistically intense lasers with organic targets represents a highly non-linear, multi-dimensional parameter space. This limits the utility of sequential 1D scanning of experimental parameters for the optimisation of secondary radiation, although to-date this has been the accepted methodology due to low data acquisition rates. High repetition-rate (HRR) lasers augmented by machine learning present a valuable opportunity for efficient source optimisation. Here, an automated, HRR-compatible system produced high fidelity parameter scans, revealing the influence of laser intensity on target pre-heating and proton generation. A closed-loop Bayesian optimisation of maximum proton energy, through control of the laser wavefront and target position, produced proton beams with equivalent maximum energy to manually-optimized laser pulses but using only 60% of the laser energy. This demonstration of automated optimisation of laser-driven proton beams is a crucial step towards deeper physical insight and the construction of future radiation sources.
proton generation, high repetition rate laser-target interaction, laser-driven particle acceleration, Bayesian optimisation
## I Introduction
Laser-target interactions have been demonstrated to provide a highly versatile source of secondary radiation, of interest for many applications [1; 2] as well as the study of fundamental science [3; 4; 5]. Specifically, laser-driven ion accelerators [6; 7] have desirable characteristics pertaining to applications in medicine [8], material science [9], nuclear fusion [10] and imaging [11; 12; 13]. The need for stable, reproducible beams that can be tuned presents a necessary, yet challenging, goal towards the realisation of many of these applications [6].
The most extensively researched mechanism driving laser-driven proton acceleration is sheath acceleration (often termed Target Normal Sheath Acceleration, or TNSA)[14]. Here, a laser pulse is focused to a relativistic intensity on the surface of a solid foil, ionising the material and heating electrons to MeV temperatures. As the accelerated electrons escape the rear target surface, they build an electrostatic sheath field which can reach \(\gtrsim 1\,\mathrm{TV/m}\) and accelerate ions to \(\gg 1\,\mathrm{MeV}\) energies over just a few microns [15]. Although target materials vary, the accelerated ions are typically dominated by protons from hydrocarbon surface contaminants which are preferentially accelerated due to their high charge-mass ratio [16].
Due to their dependence on the rear surface electrostatic sheath field, the specific characteristics of these MeV proton beams are strongly influenced by the laser-electron energy coupling, electron transport through the bulk of the target, and disruption to the target rear surface. Experiments have demonstrated the dependence of the electron and proton beam on various experimental control parameters (e.g. laser intensity and contrast or target thickness). Of particular importance is the plasma scale length at the front surface which affects the laser-electron coupling mechanisms [17; 18; 19]. In pre-plasmas with long scale lengths (\(>100\,\mathrm{\SIUnitSymbolMicro m}\)) the laser beam has been observed to filament, subsequently reducing the coupling efficiency, while for optimal scale lengths the beam can undergo relativistic self-focussing effects that enhance laser energy coupling [18; 19; 20]. The
amplified stimulated emission (ASE) pedestal and pre-pulses, common to short pulse lasers, can pre-heat the target and lead to significant plasma expansion before the main pulse arrives. For a given ASE pedestal duration, an optimal thickness exists at which this enhanced coupling and electron recirculation [21] will be advantageous for the acceleration process, while for thinner targets the inward travelling shock-wave launched by the rapid surface pre-heating can disrupt the accelerating sheath field at the rear surface [22; 23; 24]. Radiative heating from x-rays, generated in the focus of a pre-pulse incident on the target front-side, can similarly induce rear-surface expansion of thin targets, impacting TNSA for interaction parameters in which the ASE induced shock may not reach the rear surface during the acceleration window [22].
While broad trends within TNSA proton beams have been established, comparison of experimental results from different experiments highlights variation in measured beam parameters and is indicative of the nuanced relationship between laser parameters and proton beam characteristics. For example, while maximum proton energy has been observed to increase with laser intensity, the scaling follows a \(\sim\)\(I^{1/2}\) dependence for long (\(>300\) fs) duration laser pulses and a linear dependence \(\sim\)\(I\) for ultra-short (\(40-150\) fs) pulses [25; 7; 26]. A number of numerical and experimental studies have explored the impact of laser pulse duration on maximum proton energy due to TNSA in interactions with moderate laser contrast. These indicate an optimal pulse duration for proton acceleration associated with a fixed laser intensity or laser energy [27; 28; 29; 30] (e.g. optimal duration between \(100-300\) fs for laser pulse energies of 1 J). The dependence is attributed to differences in laser energy to electron coupling efficiency and acceleration time relative to the rear surface expansion timescale. For higher laser contrast, a similar trend is observed for targets with thicknesses of tens of microns [31]. More advanced temporal pulse shaping (e.g. shaping of the rising and falling edge of the pulse) has been explored recently for the interaction of high-contrast lasers with ultra-thin targets which indicate significant enhancement of proton maximum energies over those observed for best laser compression [32; 33].
With the proliferation of multi-Hz high power laser pulses [34], and the development of HRR-compatible solid-density targety [35; 36; 37; 38; 39; 40; 41], it is now possible to quickly obtain large datasets from laser-driven ion acceleration experiments. This opens the possibility to perform extensive multi-dimensional parameter scans to elucidate the interdependence of different experimental control parameters, as well as to apply machine learning techniques to optimise ion beam properties - within complex multi-dimensional parameter spaces - in automated experiments [42; 43; 44] and simulations [45; 46].
Here, we describe the first experimental demonstration of real-time Bayesian optimisation of a laser-driven ion source, using a closed-loop algorithm. The fully automated control system operated the laser, analysed the diagnostic results and made changes to the experiment control parameters. This enabled rapid and efficient optimisation of the accelerator performance through simultaneous tuning of up to six different input parameters, producing proton beams with equivalent peak energy using 57 % of the laser energy of the manually-optimised interaction.
## II Experimental Setup
The experiment (see figure 1 for setup) was performed at the Gemini TA2 facility, using a Ti:Sa laser which forms part of the Central Laser Facility at the Rutherford Appleton Laboratory. The laser pulses contained up to 500 mJ in a transform-limited pulse duration FWHM of \(\sim 40\) fs, with a central wavelength of 800 nm and a FWHM bandwidth of 30 nm. The laser was focused to a high intensity (\(I_{L}>10^{19}\) Wcm\({}^{2}\)) using an \(f/2.5\) off-axis parabolic mirror and interacted with the target at an angle of incidence of \(30^{\circ}\) with p-polarisation. The target was Kapton tape of 12 um, spooled continuously during shots using a motorised tape drive [35].
The interaction was diagnosed using a suite of particle diagnostics including a scintillator (EJ-440), positioned along the rear surface target normal, to measure the proton spatial profile, two point measurements of the proton energy spectrum using a time-of-flight (TOF) diamond detector [47] and fibre-coupled Thomson parabola spectrometer (at \(3^{\circ}\) to the target normal and along the target normal axis respectively), and a 0.15 T permanent magnet electron spectrometer in the laser-forward direction. The near-field of the specularly
Figure 1: Illustration of the experimental setup, showing the orientation of the laser-plasma interaction and the main diagnostics. The laser was focused, with an f/2.5 90\({}^{\circ}\) diamond-turned OAP, to 1.6 μm radius focal spot containing a median of \(35\pm 3\)% of pulse energy. The plane of the laser-plasma interaction was monitored by imaging self-emission at 800 nm at 60\({}^{\circ}\) to the laser propagation axis.
reflected laser light was also measured at the first and second harmonic of the drive laser.
The laser spectral phase was controlled by an acousto-optic programmable dispersive filter (DAZZLER) and measured using a small central sample of the compressed pulse and a SPIDER diagnostic. The laser parameters (6 wavefront aberrations generated through Zernike polynomials, 2nd-, 3rd- and 4th-order temporal phase, energy and polarisation) and target position relative to the laser focus were controlled using a fully automated control and acquisition system. This enabled data scans consisting of bursts of shots (up to 20) at fixed input values in parameter space. Following a burst of shots, the control code performed analysis of the measured data from the online diagnostics and adjusted laser or target parameters for the next burst. Although the controls enabled adjustment of the laser temporal pulse shape, the data presented here corresponds to pulse shapes close to best compression (\(\sim 40\,\mathrm{fs}\)) with variations due to the day-to-day variation in laser tuning.
For high-energy, multi-Hz laser facilities, prolonged high-repetition rate operation can affect the laser pulse parameters leading to degradation in peak intensity [48]. To ensure that our setup was not subject to these effects, measurements of laser parameters were made over periods of extended \(1\,\mathrm{Hz}\) operation. These measurements concluded that the effect of prolonged HRR operation on the quality of the temporal pulse shape was negligible, with the standard deviation of the random fluctuation in pulse FWHM measured as \(\sim~{}3\,\mathrm{fs}\) in over 1400 shots.
## III Automated grid scans
With the automated setup, parameter scans can be readily obtained by following a pre-programmed procedure. In doing this, the control algorithm moved through a series of equally spaced locations, taking a number of repeat shots at each configuration to quantify shot-to-shot fluctuations. Figure 2 shows proton and electron spectra for a 1D scan of target position through the laser focus with a \(12\,\mathrm{\SIUnitSymbolMicro m}\) Kapton tape and a pulse length of \(\tau_{\mathrm{FWHM}}=49\pm 3\,\mathrm{fs}\). The burst averaged 95th percentile proton energy (hereafter referred to as the maximum energy) and average electron energies are overlaid in red with the standard deviation for each burst indicated by the error bars. The proton and electron spectra are seen to extend to higher energies as the target position approaches the laser best focus as would be expected due to the increasing laser intensity at the target surface. While the electron spectra peak around the best focus, where the laser intensity is highest, a characteristic dip in the maximum proton energy and flux is observed. Around best focus (\(|z_{T}|<25\,\mathrm{\SIUnitSymbolMicro m}\)), a comparatively small number of protons are still observed at high (\(\approx 3.5\,\mathrm{MeV}\)) energy, with the spectrum dominated by lower (sub-MeV) energies, as seen in figure 2c. A second signal is also seen in the TOF spectrum, appearing as a band peaking at \(\approx 0.5\,\mathrm{MeV}\) per nucleon in figure 2a. This is most likely due to heavy ions that were accelerated to lower velocities due to their lower charge to mass ratio. Together with the spectrally peaked proton spectrum, this appears similar to observations of 'buffered' proton acceleration for higher intensities and thinner targets [49], suggesting that the sheath field is dynamic during the acceleration process.
An increase in the laser focal spot size has previously been observed to increase the number of accelerated particles, although with a reduced maximum energy
Figure 2: a) Proton and b) electron energy spectra from the rear side of the target during an automated target position scan (\(z_{T}\)) with a \(12\,\mathrm{\SIUnitSymbolMicro m}\) Kapton tape and an on-target laser energy of \((438\pm 32)\,\mathrm{mJ}\). c) Average proton spectra (and standard deviation) for different \(z_{T}\) positions as indicated in the legend. The proton spectra are recorded by the time-of-flight diamond detector. Each column of the waterfall plots is the average of the ten shots from each burst. The scan is comprised of 31 bursts at different target positions spaced at \(7.3\,\mathrm{\SIUnitSymbolMicro m}\) intervals along the laser propagation axis. Negative values of \(z_{T}\) are when the target plane is closer to the incoming laser pulse and \(z_{T}=0\) is the target at the best focus of the laser pulse. The red data points, connected with a guide line, indicate the burst-averaged 95th percentile energy as well as the standard deviation on this value across the burst.
[50]. While this is consistent with our measurements, the strong suppression of proton flux at the highest intensity may indicate that the acceleration process is further compromised at the highest laser intensities by the contrast levels of our laser, with the prepulses and amplified spontaneous emission (ASE) causing adverse pre-heating of the target. Similar disruption has previously been attributed to rear surface deformation by ASE-driven shock break-out, which can effectively steer a high energy component of the proton beam emission towards the laser axis [23; 24], modifying the spectrum measured at a single angular position, or the presence of a long scale length plasma on the rear-surface which has been shown to suppress the production of ions through TNSA in experiments and simulations [51; 52; 53].
A laser pulse contrast measurement (using an Amplitude Sequoia) showed an ASE intensity contrast of better than \(10^{-9}\) up to \(t=-20\) ps, after which the laser-intensity in the coherent pedestal [54] increased exponentially. Individual prepulses with a relative intensity of \(10^{-6}\) were also observed between \(t=-50\) ps and \(t=-65\) ps. The measured contrast, starting at \(t=-150\) ps, was used to perform 2D cylindrical hydrodynamic modelling of the target evolution ahead of the arrival of the peak intensity. The modelling was performed using the FLASH code (v4.6.2). The ASE and coherent pedestal from \(t=-150\) ps to \(t=-1\) ps was coupled to the target electrons using ray-tracing with inverse bremsstrahlung heating, and Lee-More conductivity and heat exchange models were used. This indicated the formation of an approximately 2 um exponential scale-length pre-plasma. For the target thicknesses used in the experiment (\(\gg 1\) um), the measured pre-pulse was not large enough for the generation of a shock moving quickly enough to perturb the density step of the target rear surface. This matches previous results [22; 24] at similar interaction conditions, which indicate that the ablation launched density shock does not have time to affect the rear surface during the acceleration process. This dip in signal at highest intensities is a surprising result for targets with tens of microns thickness. It is more commonly observed for experiments using ultra-thin targets where it is attributed to ASE shock-breakout [55]. For the interaction presented here, the rear surface may be affected by poor long-timescale contrast (before the start of the measurement window at \(t=-150\) ps), or through x-ray heating of the target bulk [22]. Determination of the specific processes driving the disruption of proton acceleration for this interaction requires additional measurements of long-timescale contrast and pre-plasma scale length, and is beyond the scope of this optimisation demonstration.
The laser intensity can be varied by adding wavefront abberations to the laser, which changes the focal spot shape as well as the peak intensity. Figure 3a-d show the burst-averaged electron and proton flux as well as the relative specular reflectivity at the fundamental and second harmonic wavelengths for varying target plane, \(z_{T}\) (figure 3a & 3c), and 45-degree astigmatism, \(Z_{2}^{-2}\) (figure 3 b & 3d). Again, a characteristic drop was observed in the proton flux and maximum energy for best focus. Moderately decreasing the peak intensity by either defocusing or adding astigmatism decreased the electron flux and average energy, but maximised the proton acceleration. At the low intensity limit, the particle acceleration drops to zero as expected.
Evidence of disruption to the front surface during the interaction can be seen from the sharp drop in the fundamental and second harmonic laser reflectivity at high intensities. This matches previous observations of target reflectivity and harmonic generation being adversely affected for high-intensity low-contrast laser-plasma interactions as a result of the formation of a large scale length pre-plasma [56; 57; 17].
To explore the interplay between astigmatism and defocus, an automated 2D grid-scan was performed. Burst averaged measurements of the electron and proton flux, mean electron energy and maximum proton energy are displayed in figure 3 e-h. The electron generation was maximised for both zero defocus and zero astigmatism, monotonically decreasing as \(z_{T}\) and \(Z_{2}^{-2}\) were increased. The proton flux and energy are approximately maximised for a ring around the origin, indicating a threshold intensity for disrupting the proton acceleration which can be achieved either by defocus or increasing the focal spot size through optical aberrations. There also appears to be enhanced proton flux for zero defocus, \(z_{T}=0\) um, but with the application of significant astigmatism \(Z_{2}^{-2}=\pm 1.2\) um when compared with the increased flux achieved just through defocusing with no astigmatism. This indicates that the proton acceleration process is not just intensity dependent, but is also sensitive to the spatial intensity profile on-target [58].
For all results in figure 3, the laser pulse had a shorter pulse duration of \(\tau_{\mathrm{FWHM}}=46\pm 4\) fs and was skewed with a slower rising edge than for the results in figure 2, as can be seen in figure 4. The pulse shape appears to effect the range of \(z_{T}\) over which the proton acceleration was suppressed. For the case of figure 2, proton acceleration was maximised at \(z_{T}=\pm 30\) um, while with the slower rising edge used for figure 3a, the maximum occurs at \(z_{T}=\pm 67\) um, with very low flux obtained at \(z_{T}=\pm 33\) um. For a slower rising edge, there is more time for any disruption of the target to occur prior to the arrival of the peak of the pulse, meaning that a larger defocus is required to maintain proton acceleration. A lower maximum proton energy of \(0.8\pm 0.1\) MeV was observed in figure 3a compared to \(2.8\pm 0.4\) MeV for figure 2. This shows the benefit of using temporal pulse shaping to minimise target pre-heating, as it allows for efficient proton acceleration at higher intensity interac
tion leading to higher energy protons. Despite the very different interaction conditions (here - moderate laser contrast and micron thick target), this indicates a similar trend to that observed in recent experiments using high-contrast ultra-thin targets which demonstrated the enhancement of particle energies and numbers by modification of the pulse shape, away from nominal pulse compression, with a steepened rising edge in comparison to the falling edge of the pulse [32, 33].
The interaction parameter space is multi-dimensional. While 1D and 2D slices of this parameter space provide valuable insight, true mapping of the parameter space required for deeper understanding and control of sheath-accelerated proton beams, and the location of global optima, requires the inclusion of more dimensions. This is particularly important when individual parameters are coupled in complex relationships as is evident in the case of proton acceleration from figure 3. While grid scanning is feasible for mapping up to 2D slices of parameter space with multi-Hz laser systems, for higher numbers of dimensions it becomes prohibitively time consuming. Additionally, a grid-scan covers regions of parameters with high signal and no signal with the same resolution. Given that measurements in large regions of the accessible parameter space will return no appreciable proton acceleration, this is uneconomical in terms of target usage, debris production and laser operation. It is therefore desirable to use more intelligent algorithms for probing and optimisation of the beam in highly-dimensional parameter spaces.
## IV Bayesian optimisation
In Bayesian Optimisation (BO) experimental data is used to update a _prior_ model to more accurately fit observations, thereby obtaining a _posterior_ model. The model is then used to make predictions over the experimental parameter space and select the parameter set for the next measurement, with the goal of efficiently finding the optimum within the parameter space. A commonly used type of model is Gaussian process re
Figure 4: Laser pulses temporal profiles as measured by the on-shot SPIDER diagnostic for the results of the 1D scan (figure 2), 2D scan (figure 3) and optimisation (figure 5). The integrals of the signals are set by independent measurements of the on-target laser energy which were (\(438\pm 32\)) mJ (1D scan), (\(453\pm 40\)) mJ (2D scan) and (\(258\pm 22\)) mJ (optimisation). The corresponding measured FWHM pulse widths were (\(49\pm 3\)) fs, (\(45\pm 4\)) fs and (\(39\pm 1\)) fs.
Figure 3: One dimensional scans of a) & c) target z-position \(z_{T}\) and b) & d) astigmatism \(Z_{2}^{-2}\) for \(12\,\mathrm{\SIUnitSymbolMicro m}\) thickness Kapton tape and a pre-plasma laser energy of (\(453\pm 40\)) mJ. The electron and proton flux are plotted in a) & b), and the specularly reflected fundamental and second harmonics laser signals are plotted in c) & d). All fluxes are normalised to their observed maxima over the 2D parameter scans. Two dimensional scans of electron and proton flux are shown in e) & f), with the average detected electron energy and the maximum (95th percentile) proton energies shown in g) and h) respectively. The 2D scan is a result of 143 bursts of 15 shots and the datapoints are the mean of each individual burst.
gression (GPR) [59], which is a well suited technique for modelling multi-dimensional experimental data. A key advantage for experimental science is that GPR can naturally include uncertainty quantification on the data, and also can be used to estimate the uncertainty when making predictions. BO is widely used for the optimisation of noisy processes that are expensive to evaluate and for which there is no adequate analytical description, as is typically the case in complex non-linear systems.
In the field of laser-plasma acceleration BO has been used in laser-wakefield acceleration to optimise the generated electron and x-ray beam properties [43, 60] and in simulated laser-driven ion beams for maximising proton energy [45]. To demonstrate its applicability in a laser-driven ion acceleration experiment we have adapted the algorithm from Shalloo _et al._[43]. Using the automated control of the experiment and on-line analysis of the experimental diagnostics we were able to perform multi-dimensional optimisation of any fitness function which outputted a scalar property of interest, such as the maximum proton energy.
The input values passed to the model for each burst were taken from the parameters set by the control algorithm with the exception of the plane of the interaction, \(z_{T}\). It was found that errors in positioning the tape target, while significantly smaller than the Rayleigh range of the focusing laser (15 um), were large compared to the sensitivity of the plasma accelerator and so the target position input values were taken from a spatially resolved measurement of the self-emission region at the target surface, collected at 60\({}^{\circ}\) to the front surface normal of the tape. This was found to greatly improve the confidence of the model and its ability to find the optimum.
To demonstrate the BO algorithm, a 6D optimisation was performed to maximise the maximum proton energy (as measured by the TOF). Five Zernike mode coefficients - including \(Z_{2}^{-2}\), \(Z_{2}^{2}\) (oblique & vertical astigmatism), \(Z_{3}^{-1}\), \(Z_{3}^{1}\) (vertical & horizontal coma) and \(Z_{4}^{-2}\) (oblique second astigmatism) - were used to change the spatial phase of the laser pulse, affecting the focal spot shape and peak intensity. In addition, the tape surface position relative to the focal plane of the laser \(z_{T}\), was varied using a linear motorised stage and an on-line self-emission diagnostic to measure its position, as mentioned. This optimisation started from an initially flat wavefront (optimised using a feedback loop with a HASO wavefront sensor) and the tape target was initially positioned at the estimated focal plane of the laser. This represents the typical starting point for conventional optimisation of manual experiments, with the highest intensity being assumed to be optimal.
Figure 5 shows the measured maximum proton energy along with the variation of each input parameter as a function of burst number. By chance, one of the initial randomly selected points produced a large enhancement in maximum proton energy with every parameter apart from \(Z_{2}^{2}\) close to its eventual optimum value. With each additional measurement, the model gained more knowledge of the parameter space and adjusted its prediction of the global optimum of proton energy (red line) and its location in parameter space. After 61 bursts, the model optimum was \((2.30\pm 0.10)\,\mathrm{MeV}\), compared to a starting point of \((1.22\pm 0.04)\,\mathrm{MeV}\). This optimised maximum energy is close to the value shown in figure 2 and significantly greater than seen in figure 3, despite being limited to only 260 mJ of on-target laser energy,
Figure 5: Optimisation of the 95th percentile proton energy determined by the rear surface time-of-flight diagnostic through the adjustment of the laser wavefront and position of target along the laser propagation direction (\(z_{T}\)). The top panel shows the measured values of the proton energy (median and median absolute difference of each burst) as a function of the burst number (black points and error bars respectively), together with the model predicted optimum after each burst (red line and shaded region) as well as the final optimal value from the model (blue horizontal line). The variation of each control parameter (given in microns) is shown in the lower plots (black points) along with the final optimised values (blue horizontal line), also as functions of burst number. The best individual burst is indicated by the vertical magenta line in each plot and it can be seen that for all parameters the experimental parameters fall very close to the optimum value predicted by the model (e.g. they are close to the horizontal blue line). For this data series, each burst contained 20 shots, the target was 12 μm Kapton tape and the laser energy was \(258\pm 22\,\mathrm{mJ}\).
compared to \(>430\,\)mJ for the parameter scans.
The optimum found involved a shift of \(70\,\)m from the initial position (best focus), as well as the addition of significant wavefront aberrations compared to the initially flat wavefront. The focal spot fluence distribution at the target plane was calculated from on-shot measurement of the near-field phase and fluence profiles using a HASO wavefront sensor. The resulting intensity maps are shown in figure 6 for \(Z_{2}^{-2}=-1.2,0,1.2\)m at zero defocus and for the optimised wavefront found through the optimisation. The peak intensity was \(I_{0}=5\times 10^{19}\,\)Wcm\({}^{-2}\) for the case of a flat wavefront at the focal plane. Each of the modified pulses had a similar peak intensity of \(I_{0}\approx 3\times 10^{19}\,\)Wcm\({}^{-2}\), with the spot shape appearing close to an ellipse for the optimised pulse case. From analysis of the previous parameter scans it is inferred that this intensity distribution was optimal maintaining a maximum possible intensity, while limiting disruption to the rear target surface.
The reason for the focal spot shape found in this optimisation would require expensive 2-3D numerical simulations investigating how subtle changes in wavefront affect the various energy transfer processes and plasma dynamics in sheath acceleration. This would be valuable for understanding how to further optimise laser-driven proton acceleration, but is beyond the scope of this paper which is focused on the utility of online optimisation for proton beam parameter optimisation as a particle source for applications as well as its use to identify interesting regions for deeper study within a complex non-linear system.
## V Conclusion
In conclusion, we have demonstrated the automation and optimisation of laser-driven proton acceleration from a solid tape drive. The ability to take high fidelity parameter scans in one or two dimensions will be of great benefit to the field in understanding what is a highly complex and dynamic interaction. Bayesian optimisation of the generated proton beam was demonstrated, by using real-time analysis of experimental diagnostics to create a closed-loop system with limited human intervention. This can find optima that would elude manual optimisation or single parameter scans, due to the complex interplay between the large number of control parameters. In the case of this low-contrast interaction, the temporal pulse shape was shown to play an important role in determining the intensity threshold at which the proton acceleration process was disrupted. Including temporal and spatial pulse shaping simultaneously in the optimisation process may lead to further improvement.
Automated Bayesian optimisation can quickly find regimes of stable and optimised operation without requiring the constant attention of laser-plasma experts. It is anticipated that this development will be essential for efficient utilisation of laser-driven ion acceleration for its many applications in future user facilities [61]. Bayesian optimisation could also be extremely valuable for optimising radiation pressure acceleration for which laser-plasma instabilities [62; 63] typically limit the acceleration process, as well as for optimising enhanced acceleration through relativistic transparency, which has already demonstrated highly desirable near-\(100\,\)MeV proton energies [64] and for which target evolution plays a central role. Fine tuning of the laser parameters may be able to mitigate these instabilities or further tailor the target evolution respectively, significantly enhancing the accelerated proton beam.
## Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
## Acknowledgements
We acknowledge support from the UK STFC grants ST/V001639/1 with the XFEL Physical Sciences Hub and ST/P002021/1, the UK EPSRC grants EP/V049577/1 and EP/R006202/1, as well as the U.S. DOE Office of Science, Fusion Energy Sciences under FWP No. 100182, and in part by the National Science Foundation under Grant No. 1632708 and Award No. PHY - 1903414. M.J.V.S. acknowledges support from the Royal Society URF-R1221874. G.D.G. acknowledges support from the DOE NNSA SSGF program under DE-NA0003960. A.G.R.T acknowledges support from the US DOE grant DE-SC0016804. D.M. acknowledges support from the project 'Advanced research using high-intensity laser-produced photons and
particles (CZ.02.1.01/0.0/0.0/16_019/0000789)' from European Regional Development Fund (ADONIS)
Special thanks goes to the staff at the Central Laser Facility who provided laser operational support, mechanical and electrical support, computational and administrative support throughout the experiment.
|
2307.03605 | Persistent disruption of interspecific competition after ultra-low
esfenvalerate exposure | Field and mesocosm studies repeatedly show that higher tier process reduce
the predictive accuracy of toxicity evaluation and consequently their value for
pesticide risk assessment. Therefore, understanding the influence of ecological
complexity on toxicant effects is crucial to improve realism of aquatic risk
assessment. Here we investigate the influence of repeated exposure to
ecologically realistic concentrations of esfenvalerate on the similarly
sensitive species Daphnia magna and Culex pipiens in a food limited and highly
competitive environment. We show that significant perturbations in population
development are only present close to the EC50. In contrast, interspecific
competition between species is already reduced at concentrations 3-4 orders of
magnitude below the acute EC50. We conclude that extremely low, environmentally
relevant concentrations can disrupt species interactions. This toxicant
mediated alteration of competitive balances in ecological communities may be
the underlying mechanism for shifts in species distribution at ultra-low
pesticide concentrations. A realistic risk assessment should therefore consider
these processes in order to predict potential pesticide effects on the
structure of communities. | Florian Schunck, Matthias Liess | 2023-07-07T13:48:17Z | http://arxiv.org/abs/2307.03605v1 | ## Persistent disruption of interspecific competition after ultra-low esfenvalerate exposure
## Abstract
Field and mesocosm studies repeatedly show that higher tier process reduce the predictive accuracy of toxicity evaluation and consequently their value for pesticide risk assessment. Therefore, understanding the influence of ecological complexity on toxicant effects is crucial to improve realism of aquatic risk assessment. Here we investigate the influence of repeated exposure to ecologically realistic concentrations of esfenvalerate on the similarly sensitive species _Daphnia magna_ and _Culex pipiens_ in a food limited and highly competitive environment. We show that significant perturbations in population development are only present close to the EC\({}_{50}\). In contrast, interspecific competition between species is already reduced at concentrations 3-4 orders of magnitude below the acute EC\({}_{50}\). We conclude that extremely low, environmentally relevant concentrations can disrupt species interactions. This toxicant mediated alteration of competitive balances in ecological communities may be the underlying mechanism for shifts in species distribution at ultra-low pesticide concentrations. A realistic risk assessment should therefore consider these processes in order to predict potential pesticide effects on the structure of communities.
Keywords: community, microcosms, hormesis, multiple stress, co-existence, machine-learning
## Introduction
Effect assessment is centered around screening for toxic effects with single species, single substance tests (European Food Safety Authority 2013). However, a growing number of studies indicate that whenever complex biological systems are exposed to a stressor, results diverge from expectations (Fleeger et al. 2003; Knillmann, Stampfli, Beketov, et al.
2012; Knillmann, Stampfli, Noskov, et al. 2012; Liess et al. 2013; Alexander et al. 2016; Arce-Funck et al. 2016; Vaugeois et al. 2020; Allen et al. 2021). Due to this gap between single species lab experiments and mesocosm experiments, inclusion of species interactions is one of the most important aspects of strengthening risk assessment (Gessner and Tlili 2016). Unfortunately, little progress has been made, as studies considering biological stressors such as species interactions are still underrepresented in literature (He et al. 2023).
The population state is an important co-variate for toxic effects of chemicals. Intraspecific competition can delay recovery of the population structure (Liess et al. 2006, Pieters and Liess 2006, Liess and Foit 2010). In the context of ecological communities, the competitive exclusion principle states that complete competitors (i.e., those that compete for exactly the same ecological niche) cannot coexist (Gause 1936; Hardin 1960). In natural ecosystems species diversify into their own niche, however, usually some overlap between shared resources remains, allowing for co-existence of competitors (MacArthur 1958; Hawlena et al. 2022). When such communities are exposed to toxicants, altered species-species interactions can therefore be expected. This is because usually one species will have a competitive advantage if exposed to a toxicant, due to differences in the species' sensitivity. Interspecific competition can delay recovery of species after disturbances (Knillmann, Stampfli, Noskov, et al. 2012) and increase toxic effects of pesticides (Knillmann, Stampfli, Beketov, et al. 2012). Under repeated lethal exposure to toxicants, the more sensitive species is gradually excluded, even when food density is abundant (Liess et al. 2013). In a synthetic freshwater community, the exposure to acute concentrations of an insecticide lead to reduced abundance in both competitors when they had a comparable sensitivity towards the toxicant and led to compensatory dynamics if sensitivities were different (Mano and Tanaka 2016).
How do pesticides alter interactions between competing species; do they cease or do they change when concentrations are far below levels that elicit acute effects? In the field, pesticide exposure 3 orders of magnitude below the EC\({}_{50}\) results in severe degradation of community composition with the loss of sensitive species (Liess et al. 2021). Recently, it has been shown that exposing _D. magna_ populations at carrying capacity to esfenvalerate at concentrations 2 orders of magnitude below the acute EC\({}_{50}\) can lead to a long-term increases in population abundance (Schunck and Liess 2023). This reinforces the question of how sub-acute concentrations act at the ecological level of the community.
Our aim was to reveal the effects of ultra-low esfenvalerate concentrations on the population development of two competing species (_D. magna_ and _C. pipiens_) in a food limited system with high competition between the species. For this we set up laboratory nanocosms and repeatedly exposed them with esfenvalerate concentrations as low as 3.5 orders of magnitude below the acute EC\({}_{50}\). The system state was monitored over a period of 4 months through non-invasive weekly monitoring of species abundance and measurement of physico-chemical parameters to assess the influence of environmental parameters on population development. Finally, effects of esfenvalerate on the interaction strength between the competing species _D. magna_ and _C. pipiens_ were investigated with Bayesian methods.
## Methods
### Experiment Design
To study the effects of low doses of esfenvalerate on competing populations under limited availability of food and varying environmental conditions, 80 artificial 2-species systems were assembled in November 2020, under controlled temperature (20 \(\pm\) 1 \({}^{\circ}\)C) and light conditions (16:8 day-night cycle). _Daphnia magna_ and _Culex pipiens_ were selected as competitors, both of which are common invertebrates that dwell in standing freshwater and brackish water bodies (Ebert 2022). While _D. magna_ spends its entire life cycle in water, the species _C. pipiens_ emerges after an approximately 20 day underwater larval stage as an adult mosquito and can reproduce without feeding on animal blood. Both species feed on suspended particles in the water column (Merritt et al. 1992; Ebert 2022) or moved close to the sediment to graze on organic particles of growing periphyton.
Each experimental unit consisted of a 5.5 L glass beaker (Harzkristall, Derenburg, Germany), filled with 1.5 kg of washed aquarium sand of 1-2 mm diameter. The sediment layer served as a habitat for microorganisms to facilitate self-purification of the systems as well as substrate for periphyton growth. Aachener Daphnien Medium (ADaM) (Kluttgen et al. 1994) was used as the test medium for the experiment. Throughout the duration of the experiment, the medium was not exchanged and kept at a constant volume of 3.5 L by replenishing the beakers with bi-distilled water on a weekly basis. The systems were covered with a polypropylene net to prevent escape of the adult mosquitoes. Two eyelets were embedded in the netting to grant access to the systems for measurement, sampling and supply of glucose solution. Additionally, a reaction vessel was fitted in the netting and
immersed in the water column and filled with distilled water itself. This provided access for temperature monitoring without cross-contaminating the measurement device.
For 5 months, the systems were continuously colonized, while the systems were developing periphyton growth on the sediments, which served as a food source for the organisms. The systems were deliberately left to diverge from the initial homogeneous state to reflect random variation in environmental habitats. In contrast to previously conducted nanocosm experiments (Liess et al. 2006; Foit et al. 2012), no additional food was supplied to the systems after the end of the colonization period. Instead, nutrition came from periphyton growth on the sediments and suspended algae and bacteria in the water column. This set-up was chosen to mimic density dependent processes in natural systems (Halbach 1970) and enforce competition between the two test species. Only adult mosquitoes were provided with a saturated glucose solution to enable reproduction.
### Water Quality During the Pre-Exposure Period
After 5 months of colonization, population monitoring of _C. pipiens_ (once per week) and _D. magna_ (twice per week) began. Those systems, with low emergence rates of adult _C. pipiens_ were still populated with larvae and eggs to simulate spawning events for another two months. Physico-chemical parameters were very homogeneous across all systems in the pre-exposure period (temperature 20.2 \(\pm\) 0.3 \({}^{\circ}\)C, conductivity 987 \(\pm\) 84 \(\upmu\)S/cm, oxygen 10.1 \(\pm\) 0.5 mg/L, pH 7.3 \(\pm\) 0.4). Nutrient levels were similar to previously conducted studies (PO\({}_{4}\): 0.2 \(\pm\) 0.1 mg/L, NO\({}_{3}\): 0.7 \(\pm\) 0.4 mg/L, NO\({}_{2}\): 0.02 \(\pm\) 0.01 mg/L, NH\({}_{4}\): 0.03 \(\pm\) 0.04 mg/L). The median suspended biomass of 0.2 mg/L was in the range of oligotrophic lakes, suggesting that most systems were strongly limited in biomass available for feeding, however measurements had a considerable range (90%-quantile: 0.01-2.97 mg/L). The environmental parameters that characterized the systems are summarized in Table S1 for the pre-exposure period and in Table 1 for the post-exposure period.
### Exposure
After the 2-month pre-exposure period, the systems were exposed two times to the pyrethroid insecticide esfenvalerate with a recovery period of 1 month between exposures. The treatments consisted of 5 esfenvalerate exposure levels (solvent control, 0.1, 1, 10, 100 ng/L) with a treatment size of 16 replicates each. For the preparation of stock solutions, 5 mg esfenvalerate (CAS 66230-04-4, HPC Standards GmbH, Cunnersdorf, Germany) were dissolved in DMSO and diluted to a concentration of 1000 ug/L, which also served as
exposure solution for the highest exposure treatment. From this stock, exposure solutions were diluted to 100ug/L, 10ug/L and 1 ug/L. An additional solution containing DMSO was prepared to serve as the exposure solution for the solvent control. The stocks were prepared on the day preceding the exposure and were kept refrigerated overnight. On the day of exposure, 350 uL of the treatment specific exposure solutions were added to the corresponding systems containing 3.5 L ADaM medium, amounting to a solvent concentration of 0.01 % v/v.
The accuracy of the exposure concentrations was determined by measuring the concentration of the stock solutions spiked to 1 L samples of freshly prepared ADaM. In addition, 50 ml water samples from 4 randomly selected replicates of the highest exposure treatment (100 ng/L) were taken exactly 1h after exposure and 48 h after exposure. Chemical analysis of the tested samples was performed by SGS Analytics Germany GmbH, using a GC-MS. Measured concentrations of stock solutions and samples of experimental replicates are shown in Table S3 and Table S4. The measured concentrations in experimental replicates in the highest esfenvalerate treatment are very homogeneously at 38.8 \(\pm\) 11 ng/L, 1 h after exposure and are always below the limit of quantification (LOQ) of 20 ng/L after 48h. Rapid dissipation of esfenvalerate from the water column due to adsorption and photo degradation can explain the repaid decay in the first 48h hours after exposure.
### Biological assessment of Exposure Concentrations
In addition to chemical analysis of the exposure solutions, the effect of esfenvalerate on standard test organisms was assessed. This was done under standard conditions and in the nanocosm medium. These standardized experiments were conducted in parallel to the exposure of the main experiment. For each test system, 2 x 25 ml beakers were filled with 20 ml samples of the test systems 1 hour after exposure. 5 neonates (< 24h) of _D. magna_ were placed in one beaker and 5 larvae (< 96 h) of _C. pipiens_ in the other and survival was observed for 48 h. During this period, organisms were not fed to approach conditions in the test systems. In addition, the same setup was prepared for each exposure concentration, plus a test concentration of 1000 ng/L, in standard ADaM medium. Populations were monitored for survival for 2 days without feeding (according to OECD acute standard test (OECD 2004)). Figure 1a shows that _C. pipiens_ are slightly more sensitive (_Culex_ EC\({}_{50}\) = 71 ng/L, _Daphnia_ EC\({}_{50}\) = 176 ng/L) under standard conditions. When tested in the nanocosm
medium, _Culex_ had 10% higher control mortality also under non-lethal esfenvalerate concentrations (Fig.1b), while no control mortality was detected in _D. magna_.
**Figure 1. Similar sensitivity of _D. magna_ (age < 24h) and _C. pipiens_ (age < 96 h) after 48h of exposure to esfenvalerate. The reported esfenvalerate EC\({}_{50}\) for _D. magna_ from EPA database is 310 \(\pm\) 330 ng/L (Table S2). Survival data were obtained from standard tests conducted in parallel to the 1st and 2nd exposure of the nanocosm test systems with the same exposure solutions. (a) under standard conditions (_Culex_ EC\({}_{50}\) = 71 ng/L, _Daphnia_ EC\({}_{50}\) = 176 ng/L). (b) in samples of experimental units (nanocosms) taken 1h after exposure to esfenvalerate (_Culex_ EC\({}_{50}\) = 80 ng/L, _Daphnia_ EC\({}_{50}\) = 187 ng/L). The squares indicate the EC\({}_{50}\) and shaded areas show the Bayesian credible interval of the estimate. The solid line is the maximum likelihood estimate of a 3-parameter log-logistic function and the dashed line is the Bayesian fit.**
**Monitoring of species abundance**
We monitored population development of _D. magna_ by taking 3 images of each system with a Panasonic DC-FZ1000-II (Panasonic Corporation, Kadoma, Japan), twice per week with an improved image analysis technique compared to the approach developed by Foit et al. (2012).
First, motion was detected by background subtraction; this method is based on differences between two consecutive images. Background subtraction removed all static parts of the image, so that only objects moving in both images remained. Taking the element wise maximum of this difference resulted in only the moving objects of the first image. Depending on the amount of movement in the system, between 100-100 000 detection candidates were generated. Large numbers of proposals occurred when the background was even slightly moving, or the lighting conditions changed during capture. In a second step, the bounding boxes around the coordinates of the detection were analyzed for
characteristic properties and stored in a file. These data points comprised the basis for the classification. 50 randomly selected images were annotated based on the candidate proposals. After annotation, a Support Vector Machine (SVM) classifier was trained with the annotated tags from the 50 images. The resulting accuracy of the unseen test set was 98%.
Of all labeled _Daphnia_, 97% were correctly classified as such, however an arbitrary detection candidate of a moving object generated a 2% chance of false positive detection, resulting in a slight tendency for over-detection (Fig. 2a). This resulted in problems if a large number of detection candidates was generated (e.g., when the camera was slightly moved during image capture). Therefore, in a 3rd step of the analysis, the image with lowest overall difference in pixels was chosen per series. This classification method was then used to detect population abundance in 7680 images taken throughout the experiment. Validation with the true organism count obtained at the end of the experiment shows that approximately 50% of the organisms are detected (Fig. 2b), however, this divergence is consistent throughout the assessed systems, which allows to assess the relative effects in the system.
Figure 2: Validation of the detection method. The manual count indicates the number of organisms identified by visually counting _D. magna_ in an image. In contrast, the true count is the actual number of organisms in the vessels determined when the experiment was ended. The predicted count is the number of organisms estimated by the classification algorithm. **(a)** Classifier evaluation of the capacity to detect organism from images. **(b)** Validation of the method by comparison of estimated organism count from image segmentation and classification with the true count from the last day of experimentation.
The abundance of larvae of _C. pipiens_ was manually counted once per week. Since larvae of _C. pipiens_ generally remain static in their positions below the water surface, it was possible to determine accurate population counts. Also, in contrast to automatic detection methods it was easily possible to distinguish between exoskeletons of emerged larvae and their submerged siblings. In order to detect any organisms hiding in the sediments, the systems were gently moved to provoke escape reactions of _Culex_ larvae. The abundance of C. larvae directly after hatching is not included in the population count. Only organisms > 1mm were included into the analyses.
Although, the fraction of _D. magna_ in the water column was representative of the system state (Fig. 1b), future studies should use more homogeneous, dark sediments to facilitate the detection of organisms on the sediment. The automated detection of slow-moving organisms like _C. pipiens_ could be enabled by using permanently installed cameras with longer intervals between images.
### Sampling and Measurement of Environmental Parameters
Weekly, a 5.5 ml sample was taken to measure physico-chemical parameters and cell density. Every second week, additional 20 ml sample was taken to measure the nutrient status of the systems. Greatest care was taken to avoid cross contamination of the systems. Therefore, a syringe was connected with silicon tubing to each nanocosm and reused for the entire duration of the experiment. After sampling, the samples were stored cool until analysis at or measured directly and discarded thereafter. Sampling was conducted in parallel to the monitoring of the systems. Since this process took several hours, variations in the reported parameters due to daily temperature fluctuations are present in the dataset. Medium reductions due to sampling and evaporation were replenished with bi-distilled water. Major nutrient concentrations (NO\({}_{2}\), NO\({}_{3}\), NH\({}_{4}\), PO\({}_{4}\)) were measured every second week with a Photometer (PF-12plus, Macherey-Nagel, Duren, Germany). To increase the accuracy, values were calculated from re-calibrated spectral absorption measurements and estimated concentrations (Fig. S1). Physico-chemical parameters were measured with a multi-parameter device (Portavo 908 Multi, Knick Elektronische Messgerate GmbH & Co. KG, Berlin, Germany). Density of suspended cells were measured with a CASP-TTC cell counter (Scharfe Systems, Reutlingen, Germany, now OMNI Life Science, Bremen, Germany). The raw count data was passed through a filter, discarding measurements where total counts were < 2, to separate white noise from signal. Data were then smoothed and total volume in \(\mu\)L/L was calculated and estimated as suspended
biomass density in mg/L assuming a wet weight density of 1 mg/\(\upmu\)l (e.g. (Zhu et al. 2021)). The pre-exposure measurement values are reported in Table S1.
The measured levels of N and P were in the range of eutrophic lakes (compare e.g. (Sorf et al. 2015; Beklioglu et al. 2017)). However, continuously high levels of dissolved levels of oxygen and low densities of suspended cells indicate that the observed nutrient concentrations had no effect on the studied systems. Direct effects of these nutrients are also unlikely since, they were not near high enough to elicit direct effects on _D. magna_(Serra et al. 2019).
## Statistics
\begin{table}
\begin{tabular}{l l l l l l} variable & 0.0 ng/L & 0.1 ng/L & 1.0 ng/L & 10 ng/L & 100 ng/L \\ \hline Temperature [\({}^{\circ}\)C] & \(20.2\pm 0.3\) & \(20.1\pm 0.3\) & \(20.0\pm 0.3\) & \(20.0\pm 0.3\) & \(20.2\pm 0.3\) \\ Conductivity [\(\upmu\)S/cm] & \(969\pm 49\) & \(977\pm 53\) & \(991\pm 53\) & \(997\pm 96\) & \(991\pm 78\) \\ Oxygen saturation [mg/L] & \(9.39\pm 0.27\) & \(9.43\pm 0.37\) & \(9.33\pm 0.3\) & \(9.38\pm 0.32\) & \(9.34\pm 0.32\) \\ pH & \(7.12\pm 0.08\) & \(7.11\pm 0.09\) & \(7.13\pm 0.09\) & \(7.1\pm 0.08\) & \(7.15\pm 0.16\) \\ PO4 [mg/L] & \(0.45\pm 0.33\) & \(0.51\pm 0.39\) & \(0.77\pm 1.99\) & \(0.6\pm 0.89\) & \(0.65\pm 0.82\) \\ NO3 [mg/L] & \(1.42\pm 0.69\) & \(1.47\pm 0.62\) & \(1.31\pm 0.62\) & \(1.44\pm 0.77\) & \(1.5\pm 0.83\) \\ NO2 [mg/L] & \(0.03\pm 0.03\) & \(0.03\pm 0.04\) & \(0.04\pm 0.05\) & \(0.03\pm 0.05\) & \(0.05\pm 0.05\) \\ NH4 [mg/L] & \(0.0\pm 0.01\) & \(0.01\pm 0.01\) & \(0.01\pm 0.02\) & \(0.01\pm 0.04\) & \(0.03\pm 0.15\) \\ suspended biomass [mg/L] & \(0.45\pm 1.0\) & \(0.41\pm 0.51\) & \(0.31\pm 0.3\) & \(0.67\pm 1.27\) & \(1.56\pm 5.8\) \\ \end{tabular}
\end{table}
Table 1: average of environmental parameters per group during post-exposure. Values are reported as average across time and replicates and the associated standard deviation.
For the entire analysis 17 systems were excluded from subsequent analysis. The detailed reasons for removal are listed in the Supplementary Method S1. For all analyses considering the temporal dynamic of the systems, the time series were smoothed by computing centered running averages with a time window of 11 days (Fig. 3a). This was done to reduce the influence of very short termed fluctuations in the signal, which makes the analysis more robust to measurement errors, but may rarely underestimate true treatment effects such the saw tooth pattern visible in Figure 3a.
#### Disturbance Analysis to Identify Short Term Effects of Esfenvalerate
Due to the complexity of the analyzed systems, high variance between experimental replicates complicated the identification of general patterns in the time series. The following analysis was developed to robustly identify immediate effects of the tested esfenvalerate concentrations in replicated time series with high variance between replicates. In smoothed time series, 21-day long segments centered around the exposure events were isolated (Fig. 3a, gray boxes). Then linear trends were computed through the 10-day pre-exposure sections (Fig. 3b,c, solid horizontal lines) and were extrapolated to the following 11 days (Fig. 3b,c, dashed horizontal lines). Disturbances were then estimated by calculating differences between extrapolated pre-exposure trends and the true development of smoothed time series (Fig. 3b,c, red vertical lines).
The described analysis allows the estimation of low disturbances when the population development is characterized by smooth, non-volatile cycles, which we interpret as normal behavior. On the contrary, high variance in the signal will lead to strong disturbance signals and be indicative of treatment effects.
#### Correlation Analysis to Identify Changes in Interspecific Competition
In the study of interaction between species, measuring the correlations between species can give insight into their relationship (Moran 1953; Ranta et al. 1995; McCarthy 2011). Strongly positive correlations between abundance will in theory emerge, when both species equally respond to low or high levels of resource availability. In essence, when they
Figure 3: Time series smoothing and exemplary disturbance analysis of one experimental unit. **(a)** shows a running average (solid black line) that has been computed through the time series (blue dots). The lower panels show deviations from the linear pre-exposure trend that has been extrapolated (dashed line). **(b)** first exposure on the 3rd of June 2021 (day 62) **(c)** 2nd exposure on the 30th of June 2021 (day 89).
are coexisting with significant overlap of shared resources. On the other hand, if the exclusion of either one species occurs, strong negative correlations between species will be observed. Natural systems, repeatedly observed over time, will show correlation coefficients between these extremes. However, trends in either one of the directions are indicative of changes in the relationship between species and will be interpreted as such in this work.
Estimating the correlation between count data is a non-trivial task. The approximation with the Pearson correlation coefficient will underestimate negative correlations due to the constraint of count data to be non-negative. In addition, multivariate Poisson distributions have been previously restricted to positive correlations (Ghosh et al. 2021). Modern Bayesian inference frameworks (Salvatier et al. 2016) allow for flexible transformations of variables, which enabled us to approach the problem by modeling the rate parameters of Poisson distributions as exponentiated multivariate normal variables.
\[N_{s} \sim Poisson(\lambda_{s}) \tag{1}\] \[\lambda_{s} = e^{\log(\lambda_{s})}\] (2) \[\log(\lambda_{s}) \sim MultivariateNormal(\text{mu}=\mu_{s},\text{covariance}=cov)\] (3) \[cov \sim LKJ(\eta=1,\sigma_{s})\] (4) \[\mu_{s} \sim Cauchy(0,1)\] (5) \[\sigma_{s} \sim HalfCauchy(1) \tag{6}\]
Weakly informative Cauchy distributions with heavy tails (Gelman et al. 2008; McElreath 2015) were used as priors for \(\mu_{s}\) and \(\sigma_{s}\), which describes the log species occurrence rates and their intrinsic deviation. An LKJ prior with uniform probability density over the correlation between the species (\(\eta=1\)) was used as an uninformed prior for the covariance structure of the multivariate Normal. A calculation example for 3 imaginary test systems: assume the numbers of organisms (_Culex, Daphnia_) in the respective systems were (5, 10), (10, 20) and (20, 40). A correlation coefficient close to 1 would be estimated, although with large HDIs representing the uncertainty, since only 3 samples are given. In an opposing example were (1, 20), (50, 2), (0,0) are observed, a correlation coefficient near -1 would be estimated, representing the observation that at most one species was dominant. An estimation example for a simulated dataset is given in Figure S3, which shows that the correlation coefficient can be estimated very well, even with only 12 samples, which is representative for this study. The 95% highest posterior density interval (HPDI) was computed to calculate credible intervals, which are considered to be the Bayesian analog to
confidence intervals. However, in contrast to confidence intervals, a 95% credible interval includes the true parameter value with a 95% probability by definition.
Interspecific correlation coefficients, including Bayesian uncertainty estimates were recovered from the covariance matrix, which were estimated for the whole pre- and post-exposure datasets (results: Fig. 6) and for each day in the smoothed time series (see Fig. 3) to obtain trends in the interspecific correlation (results: Figs. 4, 7).
## Results
Figure 4 shows the development of population densities of the two competing species before and after exposure to esfenvalerate as a treatment average. Larvae populations of _C. pipiens_ were stable or increasing in the pre-exposure period with highly variable population densities across replicates, indicated by large credible intervals (Fig. 4a-d). This is attributed to continued stocking of low density _Culex_ populations with additional eggs and larvae in the pre-exposure period. Only when stocking was ceased in the post exposure period, negative trends were visible in the population density of _C. pipiens_. In contrast, _D. magna_ populations, which were not artificially stocked in the pre-exposure period, follow a steady decline over the entire period of the experiment (Fig. 4e-h). In general, the declining
Figure 4: Species abundance per treatment over the time of the experiment in days computed with a Bayesian model of correlated, Poisson distributed variables. The vertical lines indicate the times of exposure to esfenvalerate. Shaded areas are 95% credible intervals and indicate the uncertainty of the estimates. **(a–d)** Expected abundance _C. pipiens._ **(e–h)** Expected abundance _D. magna._
population density reflect that the systems were characterized by resource scarcity. This corresponds to the low density of suspended organic matter (median 0.21 mg/L, 90%-quantile: 0.01-2.97 mg/L). As expected, due to the necessity of artificial stocking in the pre-exposure phase and slightly but significantly higher baseline mortality in nanocosm medium (Fig. 1b), _C. pipiens_ were significantly less abundant than _D. magna_ over the entire duration of the experiment.
After exposure to esfenvalerate, average trends of populations exposed to esfenvalerate did not significantly deviate from the controls. Correspondingly, the fraction of low-density populations towards the end of the experiment (\(\leq\)10% of the pre-exposure maximum) did not differ between _C. pipiens_ and _D. magna_. Also, the physico-chemical parameters were similar across all treatments in the post exposure period (Table 1). Compared to the pre-exposure period (Table S1), oxygen saturation slightly decreased by 7%, while the medium pH, conductivity and temperature did not change. Phosphate and Nitrate concentrations approximately doubled in the post-exposure period, while Nitrite and Ammonium did not change. Only suspended biomass differed among treatments; however, the differences are smaller than the standard deviations (Table 1).
### Community Response to Esfenvalerate Exposure
Figure 5a shows that a concentration of 100 ng/L elicits a significant negative disturbance (-6.0, p = 0.02) on populations of _D. magna_. This is also visible in the volatile trajectory of Figure 4h (100 ng/L). While disturbances after exposure to 100ng/L were negative after
Figure 5: Average 11-day disturbance of competing populations calculated from the deviation of extrapolated pre-exposure trend (10 days) to observed post exposure development in a 21-day time window (see methods: disturbance analysis). **(a)**_Daphnia_ disturbance after exposures to esfenvalerate. **(b)**_Culex_ disturbance after exposure to esfenvalerate. A significant deviation from the control treatment is indicated by an asterisk.
both exposures, exposure to concentration \(\leq 10\) ng/L resulted in negative disturbances after the 1\({}^{\mathrm{st}}\) exposure and positive disturbances after the 2\({}^{\mathrm{nd}}\) exposure. On _C. pipiens_, exposure to esfenvalerate induced no significant short-term disturbances (Fig. 5b). Under standard conditions the species had similar sensitivities to esfenvalerate (EC\({}_{50}\) Culex = 80ng/L, EC\({}_{50}\) Daphnia = 180 ng/L, Fig. 1). However, these sensitivities were not reproduced on the community level, where _D. magna_ is the only species significantly disturbed by exposure to 100 ng/L esfenvalerate. This could be explained by different durations of the observed post exposure period in standard tests (2 days) and the nanocosm test systems (11 days).
### Changes in Species Correlation after Exposure to Esfenvalerate
It was a key question of this work, whether exposure to pesticide influences the interspecific competition at low concentrations. To answer this question, the correlation between abundance of both species over time was evaluated by applying Bayesian estimation of the covariance between two Poisson distributed variables.
Figure 6: **(a–d)** Population densities of observed _Culex_ and _Daphnia_ during the entire post exposure period of all experimental replicates. The colored treatments are always compared to the same control dataset (gray). The displayed data-range was truncated to increase visibility of the dataset. Not shown data are indicated by triangles at the upper or right-hand side of the panels. **(e–h)** Bayesian posterior density estimates of the _Daphnia–Culex_ correlation coefficient, fitted on the data in panels a–d with the model described in Equations 1–6. High correlations indicate that fluctuations in population
density were synchronized, while low correlations indicate that fluctuations in population density were not synchronized.
Figure 6a-d shows the pairs of population density of _C. pipiens_ and _D. magna_ at each observation in the post-exposure period. States with simultaneously high population densities of both _D. magna_ and _C. pipiens_ were rarely observed. Figure 6c shows the population densities of highly correlated species across multiple systems. The development of _C. pipiens_ and _D. magna_ populations in these replicates was synchronized, meaning that rarely one species was abundant while the other species was not. Figure 6e-h shows the estimated correlation coefficient between both species in the community. Small concentrations of esfenvalerate increased the correlation between competitors compared to the control treatment (Fig. 6e-g). This deviation is significantly positive over the entire post-exposure period in systems that were exposed to 10 ng/L (Fig. 6g). In contrast, exposure to 100 ng/L induced a slightly negative correlation shift between _Culex_ and _Daphnia._ Considering the effect of 100 ng/L on the disturbance of _Daphnia_ population (Fig. 5a), the reduction of correlation is a sign of extinction, also visible in the phase-space of Figure 6d, which shows that one or the other species become dominant, while the other is excluded.
Figure 7: Temporal development of correlation coefficient between abundance of _C. pipiens_ and _D. magna_ (smoothed time series). Competitive exclusion in the control treatments and 100 ng treatments, and synchronized behavior in the three low concentrations. For each day of the time series, the correlation coefficient was estimated that best predicted the abundance pairs of _C. pipiens_ and _D. magna_ in all systems of one treatment. The shaded area shows the 95% credible interval and indicates the uncertainty of an estimate. The vertical lines indicate the times of exposure to esfenvalerate. Linear regression models were fitted to the correlations in the post-exposure period.
To identify the temporal development of interspecific competition, correlations between competitors were computed for each day in the monitoring period by fitting the model (Eqs. 1-6) on interpolated and smoothed daily observations. Linear regressions were computed to identify the treatment trends in competition in the post-exposure periods. We observed that the significantly negative trend in the control treatment emerged shortly after the exposure (p < 0.001), i.e., shortly after addition of manual stocking of _C. pipiens_ larvae to the systems was stopped. Figure 7a-c shows that after exposure to 0.1-10 ng/L esfenvalerate correlations were significantly positive (p \(\leq\) 0.01). However, the trend in the treatment exposed to 100 ng/L esfenvalerate was significantly negative (p < 0.001), although the correlations substantially dropped only after the second exposure (Fig. 7d).
### Effects of Environmental Conditions
Neither physico-chemical parameters (e.g., temperature, oxygen) nor major nutrients varied among the treatments during the post-exposure phase (Table 1). While exposure to esfenvalerate significantly disturbed the population development of _D. magna_, the remaining unexplained variance remained large (Fig. 5a). Pre-exposure environmental parameters could not explain this variance (Fig. S2) as there were no significant correlations. Only the pH was mildly positively correlated with the disturbance residuals (p = 0.27). The concentration of major nutrients was the range of eutrophic lakes (Table S1), however no significant positive or negative correlation with the final abundance of _D. magna_ or _C. pipiens_ could be identified. Also, the correlation between pre-exposure environmental parameters and the final abundance of _D. magna_ and _C. pipiens_ was insignificant (Tables S5, S6).
## Discussion
In this study we investigated the effect of environmentally realistic esfenvalerate exposures on a 2-species community in a highly competitive environment. We showed that exposing competing species with similar sensitivities to esfenvalerate results in a substantial reduction of interspecific competition at low concentrations, indicated by increasing positive correlations between species. These effects were detected far below effect concentrations established in standard tests, conducted in parallel to the experiment. The exposure to esfenvalerate increased the correlation between _D. magna_ and _C. pipiens_ with increasing levels of exposure, beginning as low as 3 orders of magnitude below the measured EC\({}_{50}\) (Fig. 6). Species correlations of treatments exposed to 0.1, 1 and 10 ng/L
significantly increased over time during the post-exposure period. On the contrary, the concentration closest to the EC\({}_{50}\) (100 ng/L) decreased the correlation between species and also provoked significant disturbances in the population of _D. magna_.
The systems studied in this work were highly limited in suspended biomass available for nutrition (Tables 1, S1), due to the absence of external carbon inputs. For this reason, both species showed a similar declining population trend (Fig. 4). This decrease can also be attributed to low levels of primary production, approximated by the density of suspended biomass (Table S1). We assume that sufficiently large concentrations of N and P could not be converted to biomass in the studied systems. Possible reasons are strong competition of filter feeders, which prevented growth phases of phytoplankton, or insufficient lighting conditions. In the absence of suspended biomass, organisms were observed to graze on periphyton and biofilm, which were not quantified in this work but varied considerably among the experimental replicates. Resulting from this diversity, dominance and suppression of either species was approximately random, indicated by similar fractions of low density-populations, which led to negative correlations between the species' population densities. This is associated with high interspecific competition between _C. pipiens_ and _D. magna_ in the control treatments, which increased during the post-exposure period (Fig. 7) after colonization of C. pipiens was stopped. These results fit the theory that narrow environments with considerable niche overlap do not favor coexistence of competing species (Pastore et al., 2021).
Exposure to high doses of esfenvalerate disturbs population and increases risk of single species dominance
The exposure to esfenvalerate at 100 ng/L, induced significant, direct short-term disturbances in _Daphnia_ populations and decreasing correlations in the post-exposure phase. Decreasing correlations indicate the suppression of one species, which could be exploited by the dominant species if the composition of the system in terms of suspended biomass, periphyton and biofilm allowed population growth. Since the variation between biomass density and other environmental parameters across experimental replicates could not explain the residual variance of the disturbance of species after exposure nor final population densities of either species, periphyton and biofilm may well have been responsible for the heterogeneity in the systems. Due to this heterogeneity of the systems, the observation of significant population level disturbance of esfenvalerate at 100 ng/L is assumed to be very robust. The direct effect of esfenvalerate at 100 ng/L is also visible in
Figure 6d, where species abundances increasingly converge to one or the other axes. Similar dynamic behavior has been observed for sub-populations of potato beetle larvae and adults (Costantino et al. 1997); when harvesting rates of adult beetles were experimentally increased, comparable to direct mortality effects of 100 ng/L esfenvalerate in the present work, populations were pushed out of equilibrium. Although the experimental conditions are only partly comparable, the results show that disturbances of competing (sub)populations can lead the way to significant changes in the dynamic of ecological communities.
### Exposure to low doses of esfenvalerate reduces interspecific competition
From day 70, the control treatment showed a marked interspecific competition, indicated by decreasing correlations. In contrast, treatments exposed to low doses of esfenvalerate (0.1, 1, 10 ng/L) showed reduced interspecific competition, indicated by increasing correlations. This already occurred at concentrations more than 3 orders of magnitude below the EC\({}_{50}\) and reached its maximum at 10 ng/L (Fig. 6e-g, Fig. 7a-c). These observations support a recent simulation study, which predicts that the correlation between populations of different species increases if interspecific competition is absent because then development of both populations is driven only by environmental fluctuations (Lee et al. 2020). In contrast, the same study predicts decreases in interspecific correlations if high interspecific competition is present due to resulting suppression of one or the other species. Such mechanisms are precisely those observed in the present work. In another experimental study of marine environments, low pH led to altered competitive interactions between competing algae species and gradually led to a community shift (Kroeker et al. 2013). In a single species population study, exposure to 10 ng/L esfenvalerate reduced the competitiveness of _D. magna_ and led to a hormetic increase in population abundance (Schunck and Liess 2023). Such an effect did not occur in this study. Instead, we assume that the presence of a competitor caused the absence of a stimulatory population effect, suggesting that findings of hormesis are dependent on the environmental context; i.e., only emerge when the environmental conditions do not penalize trade-offs associated with stimulatory effects.
### Conclusion and Outlook
We show that concentrations 3 orders of magnitude below the EC\({}_{50}\) induced reductions in the interspecific competition between _D. magna_ and _C. pipiens_. In contrast, concentrations
near the EC\({}_{50}\) directly impacted _D. magna_ populations and led to an increased tendency of single species dominance. The work also highlights, that single species sensitivity tests are insufficient to predict ecological effects on the community level. On the contrary, non-invasive population monitoring is very a promising approach, which can complement the higher tier risk assessment of ecological effects of toxicants, since the absence of sampling removes the most error prone and disturbing part of the method. By monitoring the correlation in abundance between competing species, more subtle effects can be detected and potentially hazardous long-term effects can be identified before they occur in the field.
## Supplementary Information
### CRediT authorship contribution statement
Florian Schunck: Conceptualization, Investigation, Data curation, Formal analysis, Visualization, Writing - Original draft. Matthias Liess: Conceptualization, Investigation - Guiding analytical cognition process, Writing.
## Acknowledgements
The authors thank Franz Dussl, Oliver Kaske, Maren Luck, Mavi Kotan and Naeem Shahid from the Department of System-Ecotoxicology of the Helmholtz Centre for Environmental Science for their support and advice during the conduction of the experiment. This work was partly funded from the European Union's Horizon Europe research and innovation programme under Grant Agreement No 101057014 (PARC).
|
2310.09081 | Electron Holes in a Regularized Kappa Background | The pseudopotential method is used to derive electron hole structures in a
suprathermal plasma having a regularized $\kappa$ probability distribution
function background. The regularized character allows the exploration of small
$\kappa$ values beyond the standard suprathermal case, for which $\kappa > 3/2$
is a necessary condition. We have found the nonlinear dispersion relation
yielding the amplitude of the electrostatic potential in terms of the remaining
parameters, in particular the drift velocity, the wavenumber and the spectral
index. Periodic, solitary wave, drifting and non-drifting solutions have been
identified. In the linear limit, the dispersion relation yields generalized
Langmuir and electron acoustic plasma modes. Standard electron hole structures
are regained in the $\kappa \gg 1$ limit. | Fernando Haas, Horst Fichtner, Klaus Scherer | 2023-10-13T13:07:49Z | http://arxiv.org/abs/2310.09081v1 | # Electron Holes in a Regularized Kappa Background
###### Abstract
The pseudopotential method is used to derive electron hole structures in a suprathermal plasma having a regularized \(\kappa\) probability distribution function background. The regularized character allows the exploration of small \(\kappa\) values beyond the standard suprathermal case, for which \(\kappa>3/2\) is a necessary condition. We have found the nonlinear dispersion relation yielding the amplitude of the electrostatic potential in terms of the remaining parameters, in particular the drift velocity, the wavenumber and the spectral index. Periodic, solitary wave, drifting and non-drifting solutions have been identified. In the linear limit, the dispersion relation yields generalized Langmuir and electron acoustic plasma modes. Standard electron hole structures are regained in the \(\kappa\gg 1\) limit.
## I Introduction
The phenomenon of so-called electron holes in a plasma has received growing attention in the recent past specially due to recent spacecraft observation of such structures, see, e.g., [1; 2] General reviews on electron holes can be found e. g. in [3; 4]. For the application to space plasmas a quantitative treatment of electron holes should take into account the presence of a suprathermal, i.e. non-Maxwellian background plasma. This was already pointed out in [5; 6] and carried out in [7; 8; 9; 10]. In [7] the Maxwellian description of the trapped (hole) and untrapped (background) electron populations was substituted by one with a so-called standard kappa distribution (SKD).
The SKD is a simple generalization of a Maxwellian that was originally introduced by [11] to describe non-Maxwellian power-law distributions of suprathermal plasma species, which are frequently observed in the solar wind [12] and are formed via the interaction of the solar wind particles with the plasma turbulence [e.g., 13; 14; 15] preventing a relaxation to a Maxwellian or bi-Maxwellian. Since then the SKD has been applied successfully to numerous space plasma and laboratory scenarios. Along with these successes also various limitations of the SKD were identified: it exhibits diverging velocity moments, a positive lower limit of allowed kappa parameter values (\(\kappa>3/2\)), and a non-extensive entropy (for a recent overview see [16]). In addition, two types of SKDs were identified, namely the original one introduced by Olbert [11] with a prescribed reference speed and a modified one that can be traced to Matsumoto [17] with a temperature equal to that of the associated Maxwellian, and it was demonstrated [18] that care has to be taken in selecting one of those for a given physical system. The kappa distribution was proposed in [19]; extensive discussion on the different kappa distributions can be found in [20; 21]. Besides these principal limitations of SKDs, there is also an observational one: SKDs do not allow to describe velocity distributions which are harder than \(v^{-5}\). However, distributions with harder tails are actually observed, see, for example, [22]. At the same time, these measurements also reveal that kappa values near two or below are frequently observed. This can also be seen for solar wind electrons, see, e.g. [23]. Such low values of kappa imply unphysical features of the SKD as is discussed in [25]. Another example are solar energetic particles, see, e.g., [26]. Kappa values as low as 1.63 and two are also obtained for particle distributions in the outer heliosphere [e.g., 27; 28]. Finally, SKDs are not consistent with exponential cut-offs of observed power-law distributions of suprathermal proton in the solar wind [29].
All of these complications in employing the SKD can be avoided when one uses the _regularized kappa distribution_ (RKD) introduced non-relativistically in [30] and for the relativistic case in [31]. The RKD exhibits an exponential cut-off of the power at high velocities. Such cut-off is a result of the fact that any acceleration process can only occur on a finite spatial scale and a finite time scale. Consequently, such power law cannot extend to infinity (as in the case of the standard kappa distribution) but must cut-off. The main purpose of the present work is to adopt a regularized version of the SKD and to analyze the consequences. The RKD particularly removes all divergences in the theory and moves the lower limit for the kappa parameter to zero [25]. Both features have consequences for correspond
ingly described physical systems: in [14] it was demonstrated that an 'infrared catastrophe' is avoided when using the RKD instead of the SKD and in [32] it was shown that extending the range of kappa values to zero broadens the possible properties of solitary ion acoustic waves in a plasma with RKD electrons. Here the reference value of \(\kappa\) is adopted according to Eq. (2) for the SKD.
Since also the first generalization of the analytical treatment of electron holes in an equilibrium plasma to a suprathermal plasma was achieved by employing the SKD [7], the same constraints remain: not all moments of the velocity distribution functions exist and kappa has to be greater than \(3/2\), thereby potentially preventing the study of a physically interesting regime because harder velocity distributions are observed, see, e.g. [22; 23] and were associated with observations of various solitary waves [24]. Therefore, the present work revisits the quantitative treatment of electron holes in a suprathermal plasma, where the electron velocity distribution is described with the RKD.
The structure of the paper is as follows: in section II the one-dimensional RKD is defined, in section III various dimensionless variables are introduced, in section IV the method of the pseudopotential is applied and in section V special solutions of the resulting Poisson equation are derived. After an analysis of the corresponding dispersion relation in section VI for homogeneous trapped electrons distributions, the final section VII contains the conclusions of the study.
## II One-dimensional regularized \(\kappa\) distribution
The starting point [25; 32] is the three-dimensional isotropic regularized \(kappa\) distribution (RKD),
\[f_{3}({\bf u})=\frac{n_{0}}{(\pi\kappa\theta^{2})^{3/2}\,U\left(\frac{3}{2}, \frac{3}{2}-\kappa,\alpha^{2}\kappa\right)}\,\left(1+\frac{u^{2}}{\kappa \theta^{2}}\right)^{-\kappa-1}\,\exp\left(-\,\frac{\alpha^{2}u^{2}}{\theta^{2 }}\right)\,, \tag{1}\]
where \(n_{0}\) is the equilibrium electrons number density, \(\kappa>0\) is the spectral index, \(\theta\) is a reference speed, \(U\) is a Kummer function of the second kind (or Tricomi function) described in [25; 32; 33], \({\bf u}\) is the velocity vector with \(u=|{\bf u}|\) and \(\alpha\geq 0\) is the cutoff parameter.
In the non-regularized limit \(\alpha\to 0\) one regains the SKD
\[f_{3}({\bf u})=\frac{n_{0}\,\Gamma(\kappa+1)}{(\pi\kappa\theta^{2})^{3/2}\, \Gamma\left(\kappa-\frac{1}{2}\right)}\,\left(1+\frac{u^{2}}{\kappa\theta^{2 }}\right)^{-\kappa-1}\,,\quad\alpha\to 0\,, \tag{2}\]
where \(\Gamma\) is the gamma function, which is positive defined provided \(\kappa>1/2\). For the RKD this constraint is not imposed on \(\kappa>0\).
For the treatment of electrostatic structures it is convenient to define the one-dimensional RKD. For this purpose we use cylindrical coordinates in velocity space and write \(u^{2}=v^{2}+w^{2}\), where \(v\) is the component of the velocity parallel to the electric field and \({\bf w}\) contains only the perpendicular velocity components, with \(w=|{\bf w}|\). In the isotropic case the one-dimensional RKD is
\[f(v) = 2\pi\int_{0}^{\infty}dw\,w\,f_{3}({\bf u}) \tag{3}\] \[= \frac{2\pi\,n_{0}\,e^{-\frac{\alpha^{2}v^{2}}{\theta^{2}}}}{(\pi \kappa\theta^{2})^{3/2}\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2} \kappa\right)}\,\int_{0}^{\infty}dw\,w\,\,\,\left(1+\frac{v^{2}+w^{2}}{\kappa \theta^{2}}\right)^{-\kappa-1}\,\exp\left(-\,\frac{\alpha^{2}w^{2}}{\theta^{2 }}\right)\] \[= \frac{n_{0}\,(\alpha^{2}\kappa)^{\kappa}\,e^{\alpha^{2}\kappa}}{ (\pi\kappa\theta^{2})^{1/2}\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2} \kappa\right)}\,\Gamma\left[-\kappa,\alpha^{2}\kappa\left(1+\frac{v^{2}}{ \kappa\,\theta^{2}}\right)\right]\,,\]
where here \(\Gamma\) is the incomplete gamma function of the indicated arguments [33]. In other words, \(f(v)\) comes from the three-dimensional version after integration over the two perpendicular velocity components.
In the non-regularized limit \(\alpha\to 0\) one regains the standard one-dimensional \(\kappa\) distribution [34; 35]
\[f(v)=\frac{n_{0}\,\Gamma(\kappa)}{(\pi\kappa\theta^{2})^{1/2}\,\Gamma\left( \kappa-\frac{1}{2}\right)}\,\left(1+\frac{v^{2}}{\kappa\theta^{2}}\right)^{- \kappa}\,,\quad\alpha\to 0\,, \tag{4}\]
which is positive definite provided \(\kappa>3/2\).
In the treatment of electrostatic structures, to satisfy Vlasov's equation the distribution function is a function of the constants of motion. In the one-dimensional, time-independent case, the available constants of motion are given by
\[\epsilon=\frac{mv^{2}}{2}-e\phi\,,\quad\sigma=\mbox{sgn}(v)\,, \tag{5}\]
where \(\phi=\phi(x)\) is the scalar potential, where \(m\) is the electron mass and \(-e\) is the electron charge. The sign of the velocity \(\sigma=v/|v|\) is an additional constant of motion just in the case of untrapped particles. The energy variable \(\epsilon\) can be used to distinguish untrapped (\(\epsilon>0\)) and trapped (\(\epsilon<0\)) electrons.
In analogy with [5; 6; 36] (where the background is not in the RKD form), presently one starts from Eq. (3) making for the untrapped part the replacement \(v\to\sigma\sqrt{2\epsilon/m}+v_{0}\)
where \(v_{0}\) is a drift velocity, defining the distributions of untrapped and trapped electrons according to
\[f=f(\epsilon,\sigma)=\frac{A\,n_{0}}{\theta} \left(1+\frac{k_{0}^{2}\Psi}{2}\right)\,\left[H(\epsilon)\,\Gamma \left(-\kappa,\alpha^{2}\kappa\left(1+\frac{1}{\kappa\,\theta^{2}}(\sigma\sqrt {2\epsilon/m}+v_{0})^{2}\right)\right)\right. \tag{6}\] \[+ \left.H(-\epsilon)\,\Gamma\left(-\kappa,\alpha^{2}\kappa\left(1+ \frac{v_{0}^{2}}{\kappa\,\theta^{2}}\right)\right)\,(1-\frac{\beta\,\epsilon }{m\,\theta^{2}})\right],\] \[A = \frac{(\alpha^{2}\kappa)^{\kappa}\,e^{\alpha^{2}\kappa}}{(\pi \kappa)^{1/2}\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2}\kappa\right) }\,, \tag{7}\]
where \(H(\epsilon)\) is the Heaviside function. The quantities \(k_{0}\) and \(\Psi\) are dimensionless variables proportional respectively to the wavenumber of periodic oscillations and to the electrostatic field amplitude, as will be qualified in the following. In addition, \(\beta\) is a dimensionless quantity associated to the inverse temperature of the trapped electrons distribution. Unlike singular distributions as in [5; 6; 7; 37] here the velocity shifted hole distribution is assumed continuous at the separatrix (\(\epsilon=0\)) and an analytic function of the energy for both trapped and untrapped electrons. These choices have been made in order to focus on the role of the cutoff parameter \(\alpha\) instead of further aspects.
In the non-regularized case, using
\[(\alpha^{2}\kappa)^{\kappa}\Gamma(-\kappa,\alpha^{2}\kappa\,s)\to\frac{s^{- \kappa}}{\kappa}\,,\quad\alpha\to 0\,,\quad\kappa>0\,, \tag{8}\]
for a generic argument \(s\), and
\[U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2}\kappa\right)\to\frac{\Gamma (\kappa-1/2)}{\Gamma(\kappa+1)}\,,\quad\alpha\to 0\,,\quad\kappa>1/2\,, \tag{9}\]
from Eq. (6) one obtains
\[f = \frac{n_{0}\,(1+k_{0}^{2}\Psi/2)}{(\pi\kappa\theta^{2})^{1/2}}\, \frac{\Gamma(\kappa)}{\Gamma(\kappa-1/2)}\left[H(\epsilon)\left(1+\frac{1}{ \kappa\theta^{2}}(\sigma\sqrt{\frac{2\epsilon}{m}}+v_{0})^{2}\right)^{-\kappa}+\right.\] \[+ \left.H(-\epsilon)\left(1+\frac{v_{0}^{2}}{\kappa\theta^{2}} \right)^{-\kappa}\left(1-\,\frac{\beta\,\epsilon}{m\,\theta^{2}}\right) \right],\]
which is the \(\kappa\) version of Schamel's distribution that is given in its original form, e.g., in Eq. (4) in [38] and illustrated in Fig. 1. A slight difference in comparison to the original formulation [38; 39] is that here the trapped electrons are described by a linear function of the energy instead of a Maxwellian function.
Finally, the Poisson equation
\[\frac{\partial^{2}\phi}{\partial x^{2}}=\frac{e}{\varepsilon_{0}}(n-n_{0})\,, \quad n=n(\phi)=\int_{-\infty}^{\infty}dv\,f(\epsilon,\sigma) \tag{11}\]
is needed, where \(\varepsilon_{0}\) is the vacuum permittivity. An uniform ionic background \(n_{0}\) has been assumed.
## III Dimensionless variables
To avoid the use of a large number of parameters, it is convenient to adopt dimensionless variables. For the RKD, it comes the question on which will be the reference speed defining the velocity rescaling. It would be tempting to consider the use of a thermal speed \(v_{T}\) defined in terms of the averaged squared velocity, but it is a cumbersome expression containing Kummer functions,
\[v_{T}^{2}=\frac{<u^{2}>}{3}=\frac{1}{3}\,\frac{\int d^{3}u\,u^{2}\,f_{3}({\bf u} )}{\int d^{3}u\,f_{3}({\bf u})}=\frac{\kappa\theta^{2}}{2}\,\frac{U\left(\frac {5}{2},\frac{5}{2}-\kappa,\alpha^{2}\kappa\right)}{U\left(\frac{3}{2},\frac{3} {2}-\kappa,\alpha^{2}\kappa\right)}\,, \tag{12}\]
the factor \(1/3\) introduced to comply with the one-dimensional geometry. Therefore, for the sake of simplicity, instead of the thermal speed it is indicated to consider \(\theta\) as the reference
Figure 1: An illustration of the Schamel distribution (in arbitrary units and for the sech-potential in Eq.(13) in [38]) for the values \(\beta=-0.9,k_{0}=1.0,\psi=1.0,\kappa=0.5,\gamma=0.1\) of the parameters used for the notation in [7].
speed. In this way, the rescaled variables are
\[\tilde{x}=\frac{x}{\lambda}\,,\quad\tilde{v}=\frac{v}{\theta}\,,\quad\tilde{v_{0}}= \frac{v_{0}}{\theta}\,,\quad\tilde{\phi}=\frac{e\phi}{m\,\theta^{2}}\,,\quad \tilde{n}=\frac{n}{n_{0}}\,,\quad\tilde{f}=\frac{f}{n_{0}/\theta}\,,\quad\tilde {\epsilon}=\frac{\epsilon}{m\,\theta^{2}}\,, \tag{13}\]
where \(\lambda=[\epsilon_{0}m\theta^{2}/(n_{0}e^{2})]^{1/2}\) is a modified Debye length.
As discussed in [18] in the non-regularized context, our standard choice of \(\theta\) as a \(\kappa-\)independent parameter better fits a scenario with enhanced tail in velocity space. Alternatively one could choose \(v_{T}\) from Eq. (12) to be \(\kappa-\)independent, which would be adequate for an enhanced core.
In dimensionless variables omitting for simplicity the tildes, the one-dimensional hole RKD from Eq. (6) is
\[f(\epsilon,\sigma)=A \left(1+\frac{k_{0}^{2}\Psi}{2}\right)\,\left[H(\epsilon)\,\Gamma \left(-\kappa,\alpha^{2}\kappa\left(1+\frac{1}{\kappa}(\sigma\sqrt{2\epsilon} +v_{0})^{2}\right)\right)\right. \tag{14}\] \[+\;H(-\epsilon)\,\Gamma\left(-\kappa,\alpha^{2}\kappa\left(1+ \frac{v_{0}^{2}}{\kappa}\right)\right)\,(1-\beta\epsilon)\bigg{]}\,,\]
while Poisson's equation (11) is
\[\frac{\partial^{2}\phi}{\partial x^{2}}=n-1\,,\quad n=n(\phi)=\int_{-\infty}^{ \infty}dv\,f(\epsilon,\sigma)\,, \tag{15}\]
where \(\epsilon=v^{2}/2-\phi\) and \(\sigma=\mathrm{sgn}(v)\). In the remaining, the purpose is to evaluate the number density in Eq. (15) in terms of \(\phi\) and to characterize the possible solutions of the Poisson's equation, specially regarding the behavior according to the parameters \(\kappa,\alpha\).
## IV Pseudopotential method
From Eqs. (14) and (15) one has
\[\frac{n}{A} = \left(1+\frac{k_{0}^{2}\Psi}{2}\right)\,\left[\int_{-\infty}^{- \sqrt{2\phi}}dv\,\Gamma\left(-\kappa,\alpha^{2}\kappa\left(1+\frac{1}{\kappa} (\sqrt{2\epsilon}-v_{0})^{2}\right)\right)+\right.\] \[+ \left.\int_{\sqrt{2\phi}}^{\infty}dv\,\Gamma\left(-\kappa,\alpha^ {2}\kappa\left(1+\frac{1}{\kappa}(\sqrt{2\epsilon}+v_{0})^{2}\right)\right)+\right.\] \[+ \left.\Gamma\left(-\kappa,\alpha^{2}\kappa\left(1+\frac{v_{0}^{2} }{\kappa}\right)\right)\,\int_{-\sqrt{2\phi}}^{\sqrt{2\phi}}dv\,(1-\beta \epsilon)\right],\]
assuming \(0\leq\phi\leq\Psi\), where \(\Psi\) denotes the peak-to-peak amplitude of the electrostatic potential, so that at \(\phi=\Psi\) one has \(d\phi/dx=0\).
The integrals in Eq. (16) for the contribution of untrapped particles can be evaluated only in the weakly nonlinear limit. Expanding the integrands in a formal power series on \(\sqrt{\phi}\) the result is
\[n=1+\frac{k_{0}^{2}\,\Psi}{2}+a\,\phi+b\,\phi\sqrt{\phi}+{\cal O}(\phi^{2})\,, \tag{17}\]
keeping the term proportional to \(\Psi\) as it has the same order of magnitude of \(\phi\) where
\[a = \frac{2}{\kappa\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2} \kappa\right)}\left[U\left(\frac{1}{2},\frac{1}{2}-\kappa,\alpha^{2}\kappa \right)+\right. \tag{18}\] \[+ \left.\frac{v_{0}}{\sqrt{\pi\,\kappa}}\,P\int_{-\infty}^{\infty} \frac{ds}{s-v_{0}}\,e^{-\alpha^{2}\,s^{2}}\,\left(1+\frac{s^{2}}{\kappa}\right) ^{-\kappa-1}\right]\]
where \(P\) stands for the principal value, and
\[b = \frac{4\sqrt{2}}{3}\,\beta\,A\,\Gamma\left(-\kappa,\alpha^{2} \kappa\left(1+\frac{v_{0}^{2}}{\kappa}\right)\right)+ \tag{19}\] \[+ \frac{8\sqrt{2}e^{-\alpha^{2}v_{0}^{2}}\left[v_{0}^{2}+2\alpha^{ 2}v_{0}^{4}+\kappa\left(-1+2(1+\alpha^{2})v_{0}^{2}\right)\right]}{3\kappa^{2 }\sqrt{\pi\kappa}\left(1+v_{0}^{2}/\kappa\right)^{\kappa+2}U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2}\kappa\right)}\,.\]
It is possible to proceed in the same way to determine the average velocity \(<v>\) from
\[n\langle v\rangle=\int_{-\infty}^{\infty}dv\,v\,f(\epsilon,\sigma) \tag{20}\]
yielding
\[\langle v\rangle=-v_{0}\left(1-a\,\phi\right)+{\cal O}(\phi^{3/2}) \tag{21}\]
giving a more precise meaning of \(-v_{0}\) which is the global drift velocity only in the limit of zero field amplitude. In addition notice the trapped electrons do not contribute to the average velocity, which comes from the untrapped part only, as found from the detail of the procedure similar to Eq. (16).
Poisson's equation (15) can be rewritten in terms of the pseudopotential \(V=V(\phi)\),
\[\frac{d^{2}\phi}{dx^{2}}=n-1=-\,\frac{\partial V}{\partial\phi}\,, \tag{22}\]
where
\[-V=\frac{k_{0}^{2}\Psi\phi}{2}+\frac{a\,\phi^{2}}{2}+\frac{2\,b\,\phi^{2} \sqrt{\phi}}{5}+{\cal O}(\phi^{3})\,, \tag{23}\]
The case where the solutions are either periodic or solitary waves requires
1. \(V(\phi)<0\) in the interval \(0<\phi<\Psi\);
2. \(V(\Psi)=0\) ,
the latter implying
\[k_{0}^{2}+a+\frac{4\,b\sqrt{\Psi}}{5}=0\,, \tag{24}\]
which allows rewriting Eq. (23) as
\[-V=\frac{k_{0}^{2}\phi}{2}(\Psi-\phi)+\frac{2\,b\,\phi^{2}}{5}(\sqrt{\phi}-\sqrt{ \Psi})\,, \tag{25}\]
up to \(\mathcal{O}(\phi^{3})\).
Equation (24) is the nonlinear dispersion relation (NDR) of the problem, providing a relation between phase velocity \(v_{0}\), wavenumber \(k_{0}\) and amplitude proportional to \(\Psi\), taking into account the expressions (18) and (19) for \(a,b\). On the other hand, Eq. (22) can be integrated yielding
\[\frac{1}{2}\left(\frac{d\phi}{dx}\right)^{2}+V(\phi)=0\,, \tag{26}\]
where the integration constant was set to zero due to property (I) and since at the potential maximum \(\phi=\Psi\) the electric field is zero. Following the usage from [5; 6; 7; 36; 37; 38; 39], the proposed _Ansatz_ has tailored \(\Psi\) so that it is the root of \(V(\phi)\) in Eq. (25). Otherwise, an irrelevant additive constant would be incorporated in the pseudopotential. The same applies to Eqs. (27) and (31) below.
## V Special solutions
### Periodic solutions
As discussed in [5; 6; 7; 36; 37; 38; 39], the expansion of the number density in powers of \(\sqrt{\phi}\) starting from an _Ansatz_ such as in Eq. (14) can give periodic or localized solutions, according to specific conditions to be identified. For the sake of reference, we collect some of the known analytic solutions, remembering that of course now the coefficients are adapted to the RKD equilibrium. For localized solutions as a by-product one has decaying boundary conditions.
The quadrature of Eq. (26) yields closed form solutions in special cases. In the linear limit, for a small amplitude so that \(\sqrt{\Psi}<<k_{0}^{2}/b\), neglecting the nonlinearity term \(\sim b\), one has
\[V=\frac{k_{0}^{2}\phi}{2}(\phi-\Psi)\,. \tag{27}\]
Then from Eq. (26) immediately one has
\[\phi=\frac{\Psi}{2}[1+\cos(k_{0}\left(x-x_{0}\right))]\,. \tag{28}\]
Hence it is verified that \(k_{0}\) indeed corresponds to the wavenumber of linear oscillations with \(0\leq\phi\leq\Psi\) in this case.
Assuming \(k_{0}\neq 0\), more insight is provided by the further rescaling
\[\bar{\phi}=\frac{\phi}{\Psi}\,,\quad\bar{x}=k_{0}\,x\,,\quad\bar{V}=\bar{V}(\bar {\phi})=\frac{V}{k_{0}^{2}\Psi^{2}}\,,\quad\bar{b}=\frac{2\,b\,\sqrt{\Psi}}{5\, k_{0}^{2}} \tag{29}\]
reduces Eq. (26) to
\[\frac{1}{2}\left(\frac{d\bar{\phi}}{d\bar{x}}\right)^{2}+\bar{V}(\bar{\phi})=0\,, \tag{30}\]
where
\[-\bar{V}(\bar{\phi}) = \frac{\bar{\phi}}{2}\,(1-\bar{\phi})+\bar{b}\,\bar{\phi}^{2}\, \left(\sqrt{\bar{\phi}}-1\right) \tag{31}\] \[= \frac{\bar{\phi}}{2}\left(1-\sqrt{\bar{\phi}}\right)\,\left(1+ \sqrt{\bar{\phi}}-2\,\bar{b}\,\bar{\phi}\right)\,,\]
containing only one free parameter \(\bar{b}\). The condition (II) for periodic or localized solutions amounts to \(\bar{V}(\bar{\phi})<0\) within the interval \(0<\bar{\phi}<1\). In view of the factorization in Eq. (31) it is easy to demonstrate the condition is always satisfied for \(\bar{b}<1\). The existence of periodic solutions such that \(0\leq\bar{\phi}\leq 1\) for \(\bar{b}<1\) comes from the shape of the rescaled pseudopotential shown in Figs. 2 and 3. The case \(\bar{b}>1\) also has periodic solutions, but with a smaller amplitude as apparent from Fig. 4. The physically meaningful solutions always occur for \(\bar{V}<0\) within the interval \(0<\bar{\phi}<1\). Notice that with the further rescaling (29) the amplitude of oscillation is set to unity, as shown in the referred figures. The required weakly nonlinear analysis always supposes \(\tilde{\phi}\sim\Psi\ll 1\) or, according to Eq. (13), \(e\phi/(m\theta^{2})\ll 1\), where \(\phi\) is the physical scalar potential.
The exact quadrature of Eq. (30) with all terms has been fully discussed in [39; 40], where the pseudopotential is formally the same as in Eq. (31) after rescaling. It is given in terms of Jacobi elliptic functions showing a periodic behavior and higher order Fourier harmonics. The present work extends these results for the case of a background RKD, with the adapted coefficients.
It is apparent that the control parameter \(\bar{b}\) depending on several variables such as the effective trapped particles inverse temperature \(\beta\) determines the qualitative aspects of the oscillatory solutions. Figures 5 and 6 and 7 show in a different style how a smaller (and possibly negative) \(\bar{b}<1\) corresponds to a larger wavenumber, which is exactly \(k_{0}\) only in the linear case.
### Localized solution with \(\bar{b}=1,k_{0}\neq 0\)
The limit case \(\bar{b}=1\) with \(k_{0}\neq 0\) is special since then \(d\bar{V}/d\bar{\phi}=0\) at \(\bar{\phi}=1\), as shown in Fig. 8, yielding a localized, non-periodic solution. Moreover this case is amenable to the simple quadrature
\[\bar{\phi}=\frac{1}{4}\left[1-3\,\tanh^{2}\left(\frac{\sqrt{3}}{4}(\bar{x}- \bar{x}_{0})\right)\right]^{2}\,, \tag{32}\]
see Fig. 9. The corresponding rescaled electric field is shown in Fig. 10. The total electrostatic energy is finite since the integral \((1/2)\int_{-\infty}^{\infty}d\bar{x}(d\bar{\phi}/d\bar{x})^{2}=6\sqrt{3}/35\) converges.
### Solitary waves with \(k_{0}=0\).
On the other hand if \(k_{0}=0\) one has
\[V=\frac{2\,b\,\phi^{2}}{5}(\sqrt{\Psi}-\sqrt{\phi})\,, \tag{33}\]
yielding the solitary pulse
\[\phi=\Psi\,{\rm sech}^{4}\left[\left(\frac{-b\sqrt{\Psi}}{20}\right)^{1/2}(x-x _{0})\right]\,, \tag{34}\]
which is well defined everywhere provided \(b<0\), which can be attainable e.g. for sufficiently small \(\beta,v_{0}^{2}\).
## VI Dispersion relation
The NDR (24) provides several behaviors according to the values in parameter space. For the sake of simplicity it will be considered the case where the trapped particle distribution is homogeneous in phase space, which amounts to the dimensionless quantity \(\beta=0\) in Eq. (10). This is an increasingly better approximation for small enough amplitude so that \(e\Psi<<m\theta^{2}\)
yielding a relatively smaller trapped area in phase space. Clearly this limit situation does not correspond to "holes", since in this case the trapped particles are not in a depression in phase space as shown e.g. in Fig. 1. However, the analytic simplicity motivates the approach. Furthermore subcases can be identified: drifting, non-drifting; oscillating, non-oscillating, as follows. Our main purpose is to provide an investigation showing a regular behavior for small \(\kappa\) values, as long as \(\alpha>0\).
Figure 8: Rescaled pseudopotential from Eq. (31) for \(\bar{b}=1\).
### Non-drifting, non-oscillating
If the trapped distribution is homogeneous and non-drifting with respect to the fixed ionic background (\(v_{0}=0\)), one has from Eq. (24)
\[k_{0}^{2}+\frac{2\,U\left(\frac{1}{2},\frac{1}{2}-\kappa,\alpha^{2}\kappa \right)}{\kappa\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2}\kappa\right)} -\frac{32\sqrt{2\,\Psi}}{15\,\kappa\,\sqrt{\pi\kappa\,U\left(\frac{3}{2},\frac {3}{2}-\kappa,\alpha^{2}\kappa\right)}}=0\,. \tag{35}\]
Furthermore in the non-oscillating case \(k_{0}=0\) one can solve Eq. (35) as
\[\Psi=\frac{\pi}{2}\,\kappa\,\left[\frac{15}{16}U\left(\frac{1}{2},\frac{1}{2}- \kappa,\alpha^{2}\kappa\right)\right]^{2}\,, \tag{36}\]
which is the amplitude of the solitary wave in terms of the remaining parameters \(\kappa,\alpha\) only. Figure 11 shows the resulting amplitude. The regular behavior as \(\kappa\to 0\) is apparent. A larger \(\alpha\) implies a smaller solitary wave amplitude. In the non-regularized limit \(\alpha\to 0\) it is possible to show that from Eq. (36) one has \(\Psi\to 1.38\) as \(\kappa\rightarrow\infty\), which is beyond the weakly nonlinear assumption. From Fig. 11 one also has that the \(\alpha=0\) case only admits small amplitude holes for \(\kappa\ll 1\), which is in contradiction with the constraint \(\kappa>3/2\) for the non-regularized equilibrium. It is interesting to note that the weakly nonlinear condition \(\Psi\ll 1\) is much better fulfilled for sufficiently high \(\alpha\). Hence, such hole structures (with \(\beta=0\), non-drifting and non-oscillating) are more reliable in a RKD background. Note, however, that high \(\alpha\) values limit the extent of the power laws.
### Non-drifting, oscillating
Allowing with \(k_{0}\neq 0\) for oscillating solutions one also has a regular behavior of the amplitude as \(\kappa\ll 1\). In this limit, assuming \(\alpha>0\), it can be shown that Eq. (35) reduces to
\[k_{0}^{2}+\frac{\sqrt{\pi}\,\alpha}{\sqrt{\kappa}}-\frac{32\,\alpha\sqrt{\Psi} }{15\,\sqrt{2\,\pi}\,\kappa}=0\,,\quad\kappa\ll 1\,,\quad\alpha>0 \tag{37}\]
yielding a vanishingly small amplitude as \(\kappa\to 0\). Figure 12 shows \(\Psi\) from Eq. (35) as a function of \(\kappa\), for \(\alpha=1.5\) and different \(k_{0}\) values. It is found that a larger \(k_{0}\) yields a larger amplitude.
### Dispersion relation with \(v_{0}\neq 0\)
Allowing for drifting structures so that \(v_{0}\neq 0\), for simplicity disregarding the nonlinear term \(\sim b\sqrt{\Psi}\) and still with homogeneous trapped electrons distribution (\(\beta=0\)), one has from Eq. (24),
\[k_{0}^{2} + \frac{2}{\kappa\,U\left(\frac{3}{2},\frac{3}{2}-\kappa,\alpha^{2 }\kappa\right)}\left[U\left(\frac{1}{2},\frac{1}{2}-\kappa,\alpha^{2}\kappa \right)+\right. \tag{38}\] \[+ \frac{v_{0}}{\sqrt{\pi\,\kappa}}\,P\int_{-\infty}^{\infty}\frac{ ds}{s-v_{0}}\,e^{-\alpha^{2}\,s^{2}}\,\left(1+\frac{s^{2}}{\kappa}\right)^{- \kappa-1}\right]=0\,.\]
Figure 11: Solitary wave amplitude in the homogeneous trapped distribution, non-drifting and non-oscillating case as a function of \(\kappa\) and different \(\alpha\)’s, from Eq. (36). Upper, dotted line: \(\alpha=0.0\); mid, dashed: \(\alpha=0.5\); Lower, solid: \(\alpha=1.5\).
Setting \(v_{0}=\omega_{0}/k_{0}\), Eq. (38) produces similar thumb curves as for holes in a Maxwellian background [38], now adapted for the RKD. Figure 13 show results for different small \(\kappa\) values, in all cases with \(\alpha=0.1\). As usual, one has a high frequency (Langmuir) mode together with a slow electron-acoustic mode [41] now adapted to the RKD background, where both modes coalesce in a certain point according to the parameters. As seen, the behavior is regular even for small \(\kappa\) values. At the extremal \(k\) value where both modes coalesce, apparently the group velocity is infinite. As discussed in [42; 43], at this point taking into account the nonlinear trapping the phase velocity of the hole should replace the diverging linear group velocity.
## VII Conclusions
In the present paper electron holes have been discussed, for the first time in a suprathermal plasma described with a regularized kappa distribution. Unlike [7], for simplicity, here the background distribution function has no singular features. It was verified that the regularization of the standard kappa distribution avoids all divergent features the solutions for \(\kappa\leq 3/2\), i.e. the analysis could be extended to all positive kappa values. This allows one to study plasma backgrounds that are described with velocity power-laws harder than \(v^{-5}\) and
Figure 12: Wave amplitude in the homogeneous trapped distribution, non-drifting and oscillating case as a function of \(\kappa\) and different wavenumbers, for \(\alpha=1.5\), from Eq. (35). Lower, solid: \(k_{0}=1.0\); mid, dashed: \(k_{0}=1.5\); upper, dotted line: \(k_{0}=2.0\).
those that exhibit an exponential cut-off, which are both observed in the solar wind. Note also, that even for kappa values below two, which can technically be handled with an SKD, unphysical features related to a non-negligible contribution of particles at high velocities are unavoidable [25]. Their removal also requires the use of an RKD.
In terms of the hole distribution function for trapped and untrapped electrons, the number density has been evaluated yielding the pseudopotential in the weakly nonlinear limit. As a consequence, the most prominent solutions of the resulting Poisson equation have been found. Drifting, non-drifting, oscillating and non-oscillating solutions have been discussed. The linear dispersion relation has been also analyzed, yielding a \(\kappa\)-dependent plasma mode diagram revealing the existence of a high frequency Langmuir mode and a low frequency electron acoustic mode (Fig. 13). Unlike for the case of a pure power-law, i.e. for an SKD background, all findings based on power-laws with an exponential cut-off, i.e. based on an RKD background, remain regular even for very small \(\kappa\) values. The results are, therefore, relevant especially for those plasmas in a suprathermal equilibrium state with spectral index
\(\kappa<3/2\), for which the SKD is not appropriate, but which are observed in space plasmas [22].
###### Acknowledgements.
FH acknowledges the support by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) and the Alexander von Humboldt Foundation for a renewed research stay fellowship.
|
2307.15067 | Set-Membership Inference Attacks using Data Watermarking | In this work, we propose a set-membership inference attack for generative
models using deep image watermarking techniques. In particular, we demonstrate
how conditional sampling from a generative model can reveal the watermark that
was injected into parts of the training data. Our empirical results demonstrate
that the proposed watermarking technique is a principled approach for detecting
the non-consensual use of image data in training generative models. | Mike Laszkiewicz, Denis Lukovnikov, Johannes Lederer, Asja Fischer | 2023-06-22T06:47:56Z | http://arxiv.org/abs/2307.15067v1 | # Set-Membership Inference Attacks using Data Watermarking
###### Abstract
In this work, we propose a set-membership inference attack for generative models using deep image watermarking techniques. In particular, we demonstrate how conditional sampling from a generative model can reveal the watermark that was injected into parts of the training data. Our empirical results demonstrate that the proposed watermarking technique is a principled approach for detecting the non-consensual use of image data in training generative models.
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it, respectively. More formally, the optimization objective is given by
\[\arg\min_{E,D}\quad\mathop{\mathbb{E}}_{\begin{subarray}{c}i\sim\mathcal{I},\\ w\in\{0,1\}^{d}\end{subarray}}L_{\mathrm{BCE}}(w,D(E(i,w)))+\lambda L_{ \mathrm{Rec}}(i,E(i,w)),\]
where \(L_{\mathrm{BCE}}\) is a bitwise binary cross-entropy loss on the decoded watermark \(D(E(i,w))\) and \(L_{\mathrm{Rec}}\) is a reconstruction loss on the watermarked image \(E(i,w)\). After training \(E\) and \(D\), we can use \(E\) to embed a watermark \(w\) into every image in \(\mathcal{P}\) to produce \(\mathcal{P}^{\prime}:=\{E(i,w):\ i\in\mathcal{P}\}\).
MIA can then be performed by verifying whether \(i\in\mathcal{T}\) possess the same watermark, for instance by measuring the number of correctly predicted bits \(\#\mathrm{cor}(i):=\#\{j\in\{1,\ldots,d\}:w_{j}=D(i)_{j}\}\), or equivalently, the bitwise accuracy \(\mathrm{acc}(i):=\#\mathrm{cor}(i)/d\). Specifically, given \(\mathcal{T}\), we can compute the one-sided \(p\)-value, which we denote as \(p_{\mathrm{avg}}\), to test \(H_{0}:\#\mathrm{cor}(i)|w\sim Bin(d,p_{w})\). Similarly, we can compute the \(p\)-value for the maximum bitwise accuracy over \(\mathcal{T}\), \(p_{\mathrm{max}}\), using the fact that under \(H_{0}\) it is \(\mathbb{P}(\max_{i\in\mathcal{T}}\mathrm{acc}(i)\geq\mathrm{acc}_{\mathrm{ max}})=1-\prod_{i\in\mathcal{T}}\mathbb{P}(\mathrm{acc}(i)\leq\mathrm{acc}_{ \mathrm{max}})\). If the \(p\)-value is small, we have evidence that \(G\) was most likely trained, partly, on \(\mathcal{P}^{\prime}\). Note that in contrast to Yu et al. (2021), we do not assume that \(p_{w}=1/2\), because we have observed that \(D\) might have a bias towards producing certain watermarks, see Table 1. Instead, given a watermark \(w\), we set \(p_{w}\) as the average bitwise accuracy of \(D(i)\) for non-watermarked real images \(i\).
## 4 Experiments
In this section, we investigate how well \(G\) reproduces the watermark when the watermarked data are diluted with non-watermarked data in the training set and how detection is affected by sampling images with the same observable attributes as \(\mathcal{P}^{\prime}\). We modify \(E\) and \(D\) slightly compared to Yu et al. (2021): (1) we use the more modern ResNet blocks instead of stacking CNNs, (2) we learn to predict a small residual added to the original image and (3) we use LPIPS (Zhang et al., 2018) as \(L_{\mathrm{Rec}}\). We trained \((E,D)\) on CelebA, which comes with attribute labels for each sample. By embedding a watermark \(w\) to each sample that has attribute \(\mathfrak{a}\in\{\text{male, bushy eyebrows, eyeglasses}\}\), we end up with \(3\) different \(\mathcal{P}^{\prime}_{\mathfrak{a}}\), which constitute \(45.5\%\), \(20.5\%\), \(4.7\%\) of the training data \(\mathcal{D}\), respectively. We trained a StyleGAN2 on \(\mathcal{D}\supset\mathcal{P}^{\prime}_{\mathfrak{a}}\) for each attribute \(\mathfrak{a}\).
Detecting UnconditionallyTable 1 displays the bitwise watermark accuracies of \(100\) unconditionally sampled images from the generative model. Unsurprisingly, the accuracy decreases as the size of \(\mathcal{P}^{\prime}_{\mathfrak{a}}\) is decreased. Nonetheless, given the null hypothesis that the watermarking accuracies are i.i.d. \(Bin(100,p_{w})\), we still end up with reasonably small \(p\)-values. Furthermore, based on the resulting \(p\)-values, we observe that the average accuracy is a more discriminative metric than the maximum accuracy.
Detecting ConditionallyInstead of sampling unconditionally, we propose to choose samples from \(G\) conditioned on \(\mathfrak{a}\) to infer the membership of \(\mathcal{P}^{\prime}_{\mathfrak{a}}\). Since we generally cannot generate conditionally, we instead use an attribution predictor1 to select a subset \(\mathcal{T}_{\mathfrak{a}}\) of \(\mathcal{T}\) according to \(\mathfrak{a}\). Again, we measure the bitwise accuracies of the inferred watermarks, shown in Table 2. Note that for each \(\mathfrak{a}\), we set \(p_{w}\) as the average bitwise accuracy achieved for CelebA samples possessing \(\mathfrak{a}\). Even though the effective sample size of \(\mathcal{T}_{\mathfrak{a}}\) is reduced, we observe a clear increase in the statistical significance as illustrated by the \(p\)-values, compared to the unconditional case.
Footnote 1: [https://github.com/d-li14/face-attribute-prediction](https://github.com/d-li14/face-attribute-prediction)
## 5 Conclusion and Future Work
In this paper, we propose a set-membership inference attack for generative models based on embedding an invisible watermark in parts of the training data. The provided experiments demonstrate that generative models create samples that possess this injected watermark, which can be used to prove set-membership. Moreover, we propose to use generated samples conditioned to be similar to the watermarked training data because we observed that they reproduce the watermark significantly better. A promising direction for future work is extending this analysis to modern text-to-image models, which are conditional models by construction.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \(\mathfrak{a}\) & \(\mathrm{acc}_{\mathrm{avg}}\) & \(p_{\mathrm{avg}}\) & \(\mathrm{acc}_{\mathrm{max}}\) & \(p_{\mathrm{max}}\) \\ \hline Male & \(69.04\%\) & \(7.5e^{-271}\) & \(100.0\%\) & \(0.00\) \\ Eyebrows & \(54.91\%\) & \(1.0e^{-10}\) & \(68.00\%\) & \(0.03\) \\ Eyeglasses & \(52.65\%\) & \(0.03\) & \(67.00\%\) & \(0.07\) \\ \hline CelebA & \(51.74\%\) & \(0.50\) & \(60.00\%\) & \(0.98\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bitwise watermark accuracies and corresponding \(p\)-values for varying \(\mathcal{P}^{\prime}_{\mathfrak{a}}\). As a reference, we measure the bitwise accuracy obtained on the original CelebA dataset. The extent to which we can recover \(w\) scales negatively with the size of \(\mathcal{P}^{\prime}_{\mathfrak{a}}\).
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \(\mathfrak{a}\) & \(\mathrm{acc}_{\mathrm{avg}}\) & \(p_{\mathrm{avg}}\) & \(\mathrm{acc}_{\mathrm{max}}\) & \(p_{\mathrm{max}}\) \\ \hline Male & \(93.66\%\) & \(0.00\) & \(100.0\%\) & \(0.00\) \\ Eyebrows & \(64.33\%\) & \(9.8e^{-19}\) & \(68.00\%\) & \(4.1e^{-3}\) \\ Eyeglasses & \(60.83\%\) & \(5.1e^{-6}\) & \(67.00\%\) & \(4.4e^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Bitwise watermark accuracies and corresponding \(p\)-values when conditioning on \(\mathfrak{a}\). We can recover \(w\) up to a significant amount, which gives clear evidence for the membership of \(\mathcal{P}^{\prime}_{\mathfrak{a}}\). |
2305.04713 | Sufficient conditions for the existence of path-factors with given
properties | A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$
if every component of $H$ is isomorphic to a path of order at least $k$, where
$k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor
critical graph if $G-V'$ contains a $P_{\geq k}$-factor for any $V'\subseteq
V(G)$ with $|V'|=l$. A graph $G$ is called a $(P_{\geq k},m)$-factor deleted
graph if $G-E'$ has a $P_{\geq k}$-factor for any $E'\subseteq E(G)$ with
$|E'|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\geq
3}$-factor. In this paper, we give some sufficient conditions for a graph to be
a $(P_{\geq 3},l)$-factor critical graph or a $(P_{\geq 3},m)$-factor deleted
graph. In this paper, we demonstrate that (i) $G$ is a $(P_{\geq 3},l)$-factor
critical graph if its sun toughness $s(G)>\frac{l+1}{3}$ and $\kappa(G)\geq
l+2$. (ii) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its degree sum
$\sigma_3(G)\geq n+2l$ and $\kappa(G)\geq l+1$. (iii) $G$ is a $(P_{\geq
3},m)$-factor deleted graph if its sun toughness $s(G)\geq \frac{m+1}{m+2}$ and
$\kappa(G)\geq 2m+1$. (iv) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if
its degree sum $\sigma_3(G)\geq n+2m$ and $\kappa(G)\geq 2m+1$. | Hui Qin, Guowei Dai, Yuan Chen, Ting Jin, Yuan Yuan | 2023-05-08T13:55:36Z | http://arxiv.org/abs/2305.04713v2 | # Sufficient conditions for the existence of path-factors with given properties
###### Abstract
A spanning subgraph \(H\) of a graph \(G\) is called a \(P_{\geq k}\)-factor of \(G\) if every component of \(H\) is isomorphic to a path of order at least \(k\), where \(k\geq 2\) is an integer. A graph \(G\) is called a \((P_{\geq k},l)\)-factor critical graph if \(G-V^{\prime}\) contains a \(P_{\geq k}\)-factor for any \(V^{\prime}\subseteq V(G)\) with \(|V^{\prime}|=l\). A graph \(G\) is called a \((P_{\geq k},m)\)-factor deleted graph if \(G-E^{\prime}\) has a \(P_{\geq k}\)-factor for any \(E^{\prime}\subseteq E(G)\) with \(|E^{\prime}|=m\). Intuitively, if a graph is dense enough, it will have a \(P_{\geq 3}\)-factor. In this paper, we give some sufficient conditions for a graph to be a \((P_{\geq 3},l)\)-factor critical graph or a \((P_{\geq 3},m)\)-factor deleted graph. In this paper, we demonstrate that (i) \(G\) is a \((P_{\geq 3},l)\)-factor critical graph if its sun toughness \(s(G)>\frac{l+1}{3}\) and \(\kappa(G)\geq l+2\). (ii) \(G\) is a \((P_{\geq 3},l)\)-factor critical graph if its degree sum \(\sigma_{3}(G)\geq n+2l\) and \(\kappa(G)\geq l+1\). (iii) \(G\) is a \((P_{\geq 3},m)\)-factor deleted graph if its sun toughness \(s(G)\geq\frac{m+1}{m+2}\) and \(\kappa(G)\geq 2m+1\). (iv) \(G\) is a \((P_{\geq 3},m)\)-factor deleted graph if its degree sum \(\sigma_{3}(G)\geq n+2m\) and \(\kappa(G)\geq 2m+1\).
keywords: path-factor, sun toughness, degree sum, \((P_{\geq 3},l)\)-factor critical graph, \((P_{\geq 3},m)\)-factor deleted graph. Msc: [2020] 05C38, 05C70 +
Footnote †: journal:
## 1 Introduction
The underlying topology in parallel machines is a graph, in which processors are represented by vertices and links between processors are represented by edges. Such graphs are interconnection networks. Matching parameters have important applications on measuring the reliability of interconnection networks; see [4; 5]. Graph factors are generalizations of matchings; see [17] for some of their applications. The path factor of a graph is a hot research topic in structural graph theory, and can be viewed as a generalization of the cardinality matching problem. It is not only of profound theoretical significance, but also of extensive applied value in information science, management science, and other fields. For example, certain file transfer problems and scheduling problems in networks can be converted into path-factor problems in graphs.
In the paper, we deal with only finite simple graph, unless explicitly stated. We refer to [2] for the notation and terminologies not defined here. Let \(G=(V(G),E(G))\) be a simple graph, where \(V(G)\) and \(E(G)\) denote the vertex set and the edge set of \(G\), respectively. A subgraph \(H\) of \(G\) is called a spanning subgraph of \(G\) if \(V(H)=V(G)\) and \(E(H)\subseteq E(G)\). Given a vertex \(v\in V(G)\), let \(N_{G}(v)\) be the set of vertices adjacent to \(v\) in \(G\) and \(d_{G}(v)=|N_{G}(v)|\) be the degree of \(v\) in \(G\). The number of connected components of a graph \(G\) is denoted by \(\omega(G)\).
A subgraph \(H\) of \(G\) is called an induced subgraph of \(G\) if every pair of vertices in \(H\) which are adjacent in \(G\) are also adjacent in \(H\). For any subset \(S\subseteq V(G)\), let \(G[S]\) denote the subgraph of \(G\) induced by \(S\), and \(G-S:=G[V(G)\setminus S]\) is the resulting graph after deleting the vertices of \(S\) from \(G\). For any \(M\subseteq E(G)\), we use \(G-M\) to denote the subgraph obtained from \(G\) by deleting \(M\). Especially, we write \(G-x=G-\{x\}\) for \(S=\{x\}\) and \(G-e=G-\{e\}\) for \(M=\{e\}\).
For a family of connected graphs \(\mathcal{F}\), a spanning subgraph \(H\) of a graph \(G\) is called an \(\mathcal{F}\)-factor of \(G\) if each component of \(H\) is isomorphic to some graph in \(\mathcal{F}\). A spanning subgraph \(H\) of a graph \(G\) is called a \(P_{\geq k}\)-factor of \(G\) if every
component of \(H\) is isomorphic to a path of order at least \(k\). For example, a \(P_{\geq 3}\)-factor means a graph factor in which every component is a path of order at least three. A graph \(G\) is called a \((P_{\geq k},l)\)-factor critical graph if \(G-V^{\prime}\) contains a \(P_{\geq k}\)-factor for any \(V^{\prime}\subseteq V(G)\) with \(|V^{\prime}|=l\). A graph \(G\) is called a \((P_{\geq k},m)\)-factor deleted graph if \(G-E^{\prime}\) has a \(P_{\geq k}\)-factor for any \(E^{\prime}\subseteq E(G)\) with \(|E^{\prime}|=m\). Especially, a \((P_{\geq k},m)\)-factor deleted graph is simply called a \(P_{\geq k}\)-factor deleted graph if \(m=1\).
The concept of a sun was introduced by Kaneko [13] as follows (e.g., see Figure 1). A graph \(H\) is called factor-critical if \(H-v\) has a 1-factor for each \(v\in V(H)\). Let \(H\) be a factor-critical graph and \(V(H)=\{v_{1},v_{2},...,v_{n}\}\). By adding new vertices \(\{u_{1},u_{2},...,u_{n}\}\) together with new edges \(\{v_{i}u_{i}:1\leq i\leq n\}\) to \(H\), the resulting graph is called a sun. Note that, according to Kaneko [13], we regard \(K_{1}\) and \(K_{2}\) also as a sun, respectively. Usually, the suns other than \(K_{1}\) are called big suns. It is called a sun component of \(G\) if the component of \(G\) is isomorphic to a sun. We denote by \(sun(G)\) the number of sun components in \(G\).
For a connected graph \(G\), its _sun toughness_, denoted by \(s(G)\), was defined as follows. If \(G\) is complete, then \(s(G)=+\infty\); otherwise,
\[s(G)=\min\left\{\frac{|X|}{sun(G-X)}:X\subseteq V(G),sun(G-X)\geq 2\right\}.\]
Since Tutte proposed the well-known Tutte 1-factor theorem [21], there are many results on graph factors [3, 9, 15, 18, 19]. Some related problems about path-factor [8, 11, 12, 27] and path-factor covered graphs [6, 7, 10, 23, 24, 26, 25]
Figure 1: Suns
have also attracted a great deal of attention. More results on graph factors are referred to the survey papers and books [1, 20, 22].
Recently, Kaneko [13] gave a characterization for a graph with a \(P_{\geq 3}\)-factor, for which Kano et al. [14] presented a simpler proof.
**Theorem 1**.: _(Kaneko [13]) A graph \(G\) has a \(P_{\geq 3}\)-factor if and only if \(\text{sum}(G-X)\leq 2|X|\) for all \(X\subseteq V(G)\)._
A claw is a graph isomorphic to \(K_{1,3}\). A graph \(G\) is said to be claw-free if there is no induced subgraph of \(G\) isomorphic to \(K_{1,3}\). For a 2-connected claw-free graph \(G\), Kelmans [16] obtained a sufficient condition for the existence of \(P_{3}\)-factors in \(G-x\) for any \(x\in V(G)\) and in \(G-e\) for any \(e\in E(G)\), respectively. Motivated by the two results, we naturally consider the more general problem as following:
**Problem 2**.: _Does \(G-V^{\prime}\) have a \(P_{\geq 3}\)-factor for any \(V^{\prime}\subseteq V(G)\) with \(|V^{\prime}|=l\), or is a graph \(G\) a \((P_{\geq 3},l)\)-factor critical graph?_
**Problem 3**.: _Does \(G-E^{\prime}\) have a \(P_{\geq 3}\)-factor for any \(E^{\prime}\subseteq E(G)\) with \(|E^{\prime}|=m\), or is a graph \(G\) a \((P_{\geq 3},m)\)-factor deleted graph?_
Suppose \(G\) is of order \(n\geq 3\). Then clearly if \(G\) is Hamiltonian, \(G\) has a \(P_{\geq 3}\)-factor by deleting one edge from a Hamiltonian cycle. So our goal is to replace "\(d_{G}(u)+d_{G}(v)\geq n\) for every pair of nonadjacent vertices of \(G\)" by a weaker condition of the same flavor. Here, we use the graphic parameter _degree sum_. Let \(G\) be a graph containing at least \(k\) independent vertices, define the degree sum
\[\sigma_{k}(G)=\min_{X\subseteq V(G)}\Big{\{}\sum_{x\in X}d_{G}(x):\text{the set}X\text{ is independent and contains }k\text{ vertices}\Big{\}}.\]
Note that when \(k=2\), this corresponds to taking the minimum of \(d_{G}(u)+d_{G}(v)\) over every pair of nonadjacent vertices of \(G\), part of the crux of the statement of Ore's Theorem.
In this paper, in terms of sun toughness and degree sum, we investigate the graphs admitting path-factors in some special settings. As main results, we
obtain two sufficient conditions for graphs to be \((P_{\geq 3},l)\)-factor critical graphs and \((P_{\geq 3},m)\)-factor deleted graphs.
## 2 \((P_{\geq 3},l)\)-factor critical graph
**Theorem 4**.: _Let \(l\geq 1\) be an integer. A graph \(G\) with \(\kappa(G)\geq l+2\) is a \((P_{\geq 3},l)\)-factor critical graph if \(s(G)>\frac{l+1}{3}\)._
Proof.: If \(G\) is a complete graph, then it is easily seen that \(G\) is a \((P_{\geq 3},l)\)-factor critical graph. Hence, we assume that \(G\) is not a complete graph.
For any \(V^{\prime}\subseteq V(G)\) with \(|V^{\prime}|=l\), we write \(G^{\prime}=G-V^{\prime}\). To verify the theorem, we only need to prove that \(G^{\prime}\) contains a \(P_{\geq 3}\)-factor. On the contrary, we assume that \(G^{\prime}\) has no \(P_{\geq 3}\)-factor. Then by Theorem 1, there exists a subset \(X\subseteq V(G^{\prime})\) such that \(sun(G^{\prime}-X)>2|X|\). In terms of the integrality of \(sun(G^{\prime}-X)\), we obtain that
\[sun(G^{\prime}-X)\geq 2|X|+1. \tag{1}\]
**Claim 1**.: \(X\neq\emptyset\) _and \(sun(G^{\prime}-X)\geq 3\)._
Proof.: On the contrary, we assume that \(X=\emptyset\). On the one hand, since \(\kappa(G^{\prime})\geq\kappa(G)-l\geq 2\), \(G^{\prime}\) is a 2-connected graph such that \(|G^{\prime}|\geq 3\). Thus, \(G^{\prime}\) is not a sun, i.e., \(sun(G^{\prime})=0\). On the other hand, it follows from (1) that \(sun(G^{\prime})=sun(G^{\prime}-X)\geq 2|X|+1=1\), which contradicts \(sun(G^{\prime})=0\). Hence, \(X\neq\emptyset\) and \(|X|\geq 1\). By (1), we have that \(sun(G^{\prime}-X)\geq 2|X|+1\geq 3\).
Note that \(sun(G-V^{\prime}\cup X)=sun(G^{\prime}-X)\geq 3\). Combining this with \(s(G)>\frac{l+1}{3}\) and the definition of \(s(G)\), we obtain that
\[\frac{l+1}{3} < s(G)\] \[\leq \frac{|V^{\prime}\cup X|}{sun(G-V^{\prime}\cup X)}\] \[= \frac{|X|+l}{sun(G^{\prime}-X)}\]
\[\leq \frac{\frac{sun(G^{\prime}-X)-1}{2}+l}{sun(G^{\prime}-X)}\] \[= \frac{1}{2}+\frac{2l-1}{2sun(G^{\prime}-X)}\] \[\leq \frac{1}{2}+\frac{2l-1}{6}\] \[= \frac{l+1}{3},\]
where the last inequality follows from Claim 1. By (1), we have that \(|X|\leq\frac{sun(G^{\prime}-X)-1}{2}\) and the third inequality follows. This is a contradiction and completes the proof of Theorem 4.
**Remark 5**.: _The conditions \(s(G)>\frac{l+1}{3}\) and \(\kappa(G)\geq l+2\) in Theorem 4 cannot be replaced by \(s(G)\geq\frac{l+1}{3}\) and \(\kappa(G)\geq l+1\). We consider the graph \(G=K_{l+1}\lor 3K_{2}\), and choose \(V^{\prime}\subseteq V(K_{l+1})\) with \(|V^{\prime}|=l\). Then it is easily seen that \(s(G)=\frac{l+1}{3}\) and \(\kappa(G)=l+1\). Let \(G^{\prime}=G-V^{\prime}\). For \(X=V(K_{l+1})\setminus V^{\prime}\), we have that \(sun(G^{\prime}-X)=3>2=2|X|\). In view of Theorem 1, \(G^{\prime}\) has no \(P_{\geq 3}\)-factor. Hence, \(G\) is not a \((P_{\geq 3},l)\)-factor critical graph._
**Theorem 6**.: _Let \(l\geq 1\) be an integer. A graph \(G\) with \(\kappa(G)\geq l+1\) is a \((P_{\geq 3},l)\)-factor critical graph if \(\sigma_{3}(G)\geq n+2l\)._
Proof.: If \(G\) is a complete graph, then it is easily seen that \(G\) is a \((P_{\geq 3},l)\)-factor critical graph. Hence, we assume that \(G\) is not a complete graph.
For any \(V^{\prime}\subseteq V(G)\) with \(|V^{\prime}|=l\), we write \(H=G-V^{\prime}\). To verify the theorem, we only need to prove that \(H\) contains a \(P_{\geq 3}\)-factor.
By contradiction, suppose that \(H\) admits no \(P_{\geq 3}\)-factor.
**Claim 1.**\(n\geq l+7\).
Proof.: By \(\sigma_{3}(H)\geq\sigma_{3}(G)-3l\geq n-l\), it is easy to verify that \(n-l\geq 5\). Let \(\{u_{1},u_{2},u_{3}\}\) be an independent set of \(H\). If \(n-l=5\), let \(\{v_{1},v_{2}\}=V(H)\setminus\{u_{1},u_{2},u_{3}\}\). Since \(\Sigma_{i=1}^{3}d_{H}(u_{i})\geq\sigma_{3}(H)\geq 5\), without loss of generality, we may assume that \(d_{H}(u_{1})=d_{H}(u_{2})=2\) and \(u_{3}v_{1}\in E(H)\). Then we can obtain a path: \(u_{1}v_{2}u_{2}v_{1}u_{3}\), which can be viewed as a \(P_{\geq 3}\)-factor of \(H\), a contradiction.
Next, we consider the case that \(n-l=6\). Let \(\{v_{1},v_{2},v_{3}\}=V(H)\setminus\{u_{1},u_{2},u_{3}\}\). Since \(\Sigma_{i=1}^{3}d_{H}(u_{i})\geq\sigma_{3}(H)\geq 6\), without loss of generality, we may assume that \(d_{H}(u_{i})\geq 2\) for \(i=1,2,3\) or \(d_{H}(u_{1})=3,d_{H}(u_{2})\geq 2,d_{H}(u_{3})=1\).
* If, for \(i=1,2,3\), \(d_{H}(u_{i})\geq 2\) and \(|N_{H}(u_{i})\cap N_{H}(u_{j})|=1\) for every \(1\leq i<j\leq 3\), then without loss of generality, we may assume that \(N_{H}(u_{1})=\{v_{1},v_{2}\},N_{H}(u_{2})=\{v_{2},v_{3}\},N_{H}(u_{3})=\{v_{1 },v_{3}\}\). Then we can obtain a path: \(u_{1}v_{1}u_{3}v_{3}u_{2}v_{2}\), which can be viewed as a \(P_{\geq 3}\)-factor of \(G\), a contradiction.
* If, for \(i=1,2,3\), \(d_{H}(u_{i})\geq 2\) and \(|N_{H}(u_{i})\cap N_{H}(u_{j})|\geq 2\) holds for some \(1\leq i<j\leq 3\), then without loss of generality, we may assume that \(\{v_{1},v_{2}\}\subseteq N_{H}(u_{1})\cap N_{H}(u_{2}),N_{H}(u_{3})=\{v_{2},v_ {3}\}\). Then we can obtain a path: \(u_{1}v_{1}u_{2}v_{2}u_{3}v_{3}\), which can be viewed as a \(P_{\geq 3}\)-factor of \(G\), a contradiction.
* If \(d_{H}(u_{1})=3,d_{H}(u_{2})\geq 2\), and \(d_{H}(u_{3})=1\), then without loss of generality, we assume that \(N_{H}(u_{1})=\{v_{1},v_{2},v_{3}\},N_{H}(u_{2})=\{v_{1},v_{2}\},|N_{H}(u_{3}) \cap\{v_{2},v_{3}\}|=1\). Then we can obtain a path: \(u_{3}v_{2}u_{2}v_{1}u_{1}v_{3}\) or \(u_{3}v_{3}u_{1}v_{1}u_{2}v_{2}\), which can be viewed as a \(P_{\geq 3}\)-factor of \(G\), a contradiction.
By Theorem 1 and the integrality of \(sun(H-S)\), there exists \(S\subseteq V(H)\) such that
\[sun(H-S)\geq 2|S|+1. \tag{2}\]
**Claim 2.**\(S\neq\emptyset\) and \(sun(H-S)\geq 3\).
Proof.: Suppose that \(S=\emptyset\), then \(sun(G)=sun(H-S)\geq 1\) by (2). On the other hand, as \(H\) is connected, we have \(sun(H)\leq\omega(H)=1\). So \(H\) is a big sun with \(n-l=|V(H)|\geq 7\). Due to the definition of a big sun, there are three distinct vertices of degree one, and the vertices set is denoted by \(\{u,v,w\}\). Obviously, \(\{u,v,w\}\) is an independent set of \(H\). So, we have
\(\sigma_{3}(H)\geq n-l\geq 7\), a contradiction. Thus we obtain \(|S|\geq 1\). This together with (2) implies that \(sun(H-S)\geq 2|S|+1\geq 3.\)
Denote the set of isolated vertices of \(H-S\) by \(I(H-S)\). Next, we consider three cases.
**Case 1**. \(i(H-S)\geq 3.\)
In this case, we can choose three independent vertices \(u,v,w\in I(H-S).\) It follows that \(d_{H}(u)+d_{H}(v)+d_{H}(w)\geq\sigma_{3}(H)\geq n-l.\) This together with \(N_{H}(u)\cup N_{H}(v)\cup N_{H}(w)\subseteq S\) implies
\[|S|\geq\max\{d_{H}(u),d_{H}(v),d_{H}(w)\}\geq\frac{\sigma_{3}(H)}{3}\geq\frac{ n-l}{3}.\]
Combining (2) and the inequality above, we have that \(n-l\geq|S|+sun(G-S)\geq 3|S|+1\geq n-l+1\), which is a contradiction.
**Case 2**. \(i(H-S)=2.\)
By Claim 2, \(H-S\) has a sun component containing at least two vertices, denoted by \(C\). Let \(u,v\) be two distinct vertices in \(I(H-S)\). Due to the definition of a sun, we can choose a vertex \(w\in V(C)\) such that \(d_{C}(w)=1\). Obviously, \(\{u,v,w\}\) is independent in \(H\). It follows that \(d_{H}(u)+d_{H}(v)+d_{H}(w)\geq\sigma_{3}(H)\geq n-l.\) Since \(N_{H}(u)\cup N_{H}(v)\subseteq S\) and \(d_{S}(w)=d_{H}(w)-1\), we obtain:
\[|S| \geq \max\{d_{S}(u),d_{S}(v),d_{S}(w)\}\] \[\geq \frac{d_{S}(u)+d_{S}(v)+d_{S}(w)}{3}\] \[= \frac{\sigma_{3}(H)-1}{3}\] \[\geq \frac{n-l-1}{3}.\]
This together with (2) implies
\[n-l \geq |S|+2\times sun(G-S)-i(G-S)\] \[\geq |S|+2\times(2|S|+1)-2\] \[\geq 5|S|\]
\[\geq \frac{5n-5l-5}{3},\]
that is, \(n-l\leq 2\), a contradiction to Claim 1.
**Case 3**. \(i(H-S)\leq 1\).
By Claim 2, \(H-S\) has at least three sun components, denoted by \(C_{1},C_{2},C_{3}\). We can choose vertex \(x_{i}\in V(C_{i})\) such that \(d_{C_{i}}(x_{i})\leq 1\) for every \(i=1,2,3\). Then \(\{x_{1},x_{2},x_{3}\}\) is independent in \(H\), and thus
\[\sum_{i=1}^{3}d_{H}(x_{i})\geq\sigma_{3}(H)\geq n-l. \tag{3}\]
Note that \(d_{S}(x_{i})\geq d_{G}(x_{i})-1\) for every \(i=1,2,3\). This together with (3) implies
\[|S|\geq\max\{d_{S}(x_{i}):i=1,2,3\}\geq\frac{\sum_{i=1}^{3}d_{S}(x_{i})}{3} \geq\frac{n-l}{3}-1. \tag{4}\]
Combining (2) and (4), we obtain
\[n-l \geq |S|+2\times sun(G-S)-i(G-S)\] \[\geq |S|+2\times(2|S|+1)-i(G-S)\] \[\geq 5|S|+1\] \[\geq \frac{5n-5l}{3}-4,\]
that is, \(n-l\leq 6\). This is a contradiction, and Theorem 6 holds.
## 3 \((P_{\geq 3},m)\)-factor deleted graphs
Zhou[24] verified that a 2-connected graph \(G\) is a \(P_{\geq 3}\)-factor deleted graph if \(s(G)\geq 1\). We extend the above result and give a sufficient condition for a graph being a \((P_{\geq 3},m)\)-factor deleted graph.
**Theorem 7**.: _Let \(m\geq 1\) be an integer. A graph \(G\) with \(\kappa(G)\geq 2m+1\) is a \((P_{\geq 3},m)\)-factor deleted graph if \(s(G)\geq\frac{m+1}{m+2}\)._
Proof.: If \(G\) is a complete graph, then it is obvious that \(G\) is a \((P_{\geq 3},m)\)-factor deleted graph by \(\lambda(G)\geq\kappa(G)\geq 2m+1\) and the definition of \((P_{\geq 3},m)\)-factor deleted graph. Hence, we may assume that \(G\) is a non-complete graph.
For any \(E^{\prime}\subseteq E(G)\) with \(|E^{\prime}|=m\), we write \(G^{\prime}=G-E^{\prime}\). To verify the theorem, we only need to prove that \(G^{\prime}\) has a \(P_{\geq 3}\)-factor. By contradiction, we assume that \(G^{\prime}\) has no \(P_{\geq 3}\)-factor. Then by Theorem 1, there exists a subset \(X\subseteq V(G^{\prime})\) such that \(sun(G^{\prime}-X)>2|X|\). In terms of the integrality of \(sun(G^{\prime}-X)\), we obtain that
\[sun(G^{\prime}-X)\geq 2|X|+1. \tag{5}\]
It follows from \(\lambda(G)\geq\kappa(G)\geq 2m+1\) that \(\lambda(G^{\prime})\geq\lambda(G)-m\geq m+1\geq 2\). Hence, \(sun(G^{\prime})=0\) by the definition of sun component. Next, we claim that \(X\neq\emptyset\). Otherwise, \(X=\emptyset\), and thus \(sun(G^{\prime})=sun(G^{\prime}-X)\geq 2|X|+1=1\) by (5), which contradicts \(sun(G^{\prime})=0\). Therefore, we have that \(X\neq\emptyset\) and \(|X|\geq 1\).
We will distinguish two cases below to completes the proof of Theorem 7.
**Case 1**. \(G-X\) is not connected.
Since \(G-X\) is not connected, \(X\) is a vertex cut set of \(G\). Then we have that \(|X|\geq 2m+1\) by the connectivity of \(G\) that \(\kappa(G)\geq 2m+1\). Note that after deleting an edge in a graph, the number of its sun components increases by at most 2. Hence, we have that
\[sun(G^{\prime}-X)=sun(G-E^{\prime}-X)\leq sun(G-X)+2m. \tag{6}\]
Combining (6) with (5), we have:
\[2|X|+1\leq sun(G^{\prime}-X)\leq sun(G-X)+2m.\]
It follows that
\[sun(G-X) \geq 2|X|+1-2m\] \[\geq 2\times(2m+1)+1-2m\]
\[= 2m+3\] \[\geq 5.\]
Then by the definition of sun toughness, we obtain that
\[\frac{m+1}{m+2} \leq s(G)\] \[\leq \frac{|X|}{sun(G-X)}\] \[\leq \frac{|X|}{2|X|+1-2m}\] \[= \frac{1}{2-\frac{2m-1}{|X|}}\] \[\leq \frac{1}{2-\frac{2m-1}{2m+1}}\] \[= \frac{2m+1}{2m+3},\]
which is a contradiction to that \(m\geq 1\).
**Case 2**. \(G-X\) is connected.
It is easily seen that \(sun(G-X)\leq\omega(G-X)=1\) since \(G-X\) is connected. Recall that
\[sun(G^{\prime}-X)=sun(G-E^{\prime}-X)\leq sun(G-X)+2m. \tag{7}\]
This together with (5) implies that \(2|X|+1\leq sun(G^{\prime}-X)\leq sun(G-X)+2m\leq 1+2m\), and thus
\[|X|\leq m. \tag{8}\]
It follows from (8) and \(\kappa(G)\geq 2m+1\) that
\[\lambda(G-X)\geq\kappa(G-X)\geq\kappa(G)-|X|\geq 2m+1-m=m+1. \tag{9}\]
Using (9), we obtain that \(\lambda(G^{\prime}-X)\geq\lambda(G-X)-m\geq 1\). Hence, \(sun(G^{\prime}-X)\leq\omega(G^{\prime}-X)=1<2|X|\), which contradicts (5).
**Remark 8**.: _The sun toughness condition \(s(G)\geq\frac{4m+1}{4m+4}\) in Theorem 7 cannot be replaced by \(s(G)\geq\frac{2m+1}{3m+3}\). We consider the graph \(G=K_{2m+1}\vee(3m+3)K_{2}\)
_and choose \(E^{\prime}\subseteq E(3(m+1)K_{2})\) with \(|E^{\prime}|=m\). Then it is easily seen that \(s(G)=\frac{2m+1}{3m+3}\) and \(\kappa(G)=2m+1\). Let \(G^{\prime}=G-E^{\prime}\). For \(X=V(K_{2m+1})\subseteq V(G^{\prime})\), we have that \(sun(G^{\prime}-X)=(2m+3)+2m>2(2m+1)=2|X|\). In view of Theorem 1, \(G^{\prime}\) has no \(P_{\geq 3}\)-factor. Hence, \(G\) is not a \((P_{\geq 3},m)\)-factor deleted graph._
**Theorem 9**.: _Let \(m\geq 1\) be an integer. A graph \(G\) with \(\kappa(G)\geq 2m+1\) is a \((P_{\geq 3},m)\)-factor deleted graph if \(\sigma_{3}(G)\geq n+2m\)._
Proof.: If \(G\) is a complete graph, then it is obvious that \(G\) is a \((P_{\geq 3},m)\)-factor deleted graph by \(\lambda(G)\geq\kappa(G)\geq 2m+1\) and the definition of \((P_{\geq 3},m)\)-factor deleted graph. Hence, we may assume that \(G\) is a non-complete graph.
For any \(E^{\prime}\subseteq E(G)\) with \(|E^{\prime}|=m\), we write \(G^{\prime}=G-E^{\prime}\). To verify the theorem, we only need to prove that \(G^{\prime}\) has a \(P_{\geq 3}\)-factor. By contradiction, we assume that \(G^{\prime}\) has no \(P_{\geq 3}\)-factor. Then by Theorem 1, there exists a subset \(X\subseteq V(G^{\prime})\) such that \(sun(G^{\prime}-X)>2|X|\). In terms of the integrality of \(sun(G^{\prime}-X)\), we obtain that
\[sun(G^{\prime}-X)\geq 2|X|+1. \tag{10}\]
It follows from \(\lambda(G)\geq\kappa(G)\geq 2m+1\) that \(\lambda(G^{\prime})\geq\lambda(G)-m\geq m+1\geq 2\). Hence, \(sun(G^{\prime})=0\) by the definition of sun component. Next, we claim that \(X\neq\emptyset\). Otherwise, \(X=\emptyset\), and thus \(sun(G^{\prime})=sun(G^{\prime}-X)\geq 2|X|+1=1\) by (10), which contradicts \(sun(G^{\prime})=0\). Therefore, we have that \(X\neq\emptyset\) and \(|X|\geq 1\).
Denote by \(Sun(G-S)\) the set of sun components of \(G-S\). We will distinguish two cases below to completes the proof of Theorem 9.
**Case 1**. \(G-X\) is not connected.
Since \(G-X\) is not connected, \(X\) is a vertex cut set of \(G\). Then we have that \(|X|\geq 2m+1\) by the connectivity of \(G\) that \(\kappa(G)\geq 2m+1\). Note that after deleting an edge in a graph, the number of its sun components increases by at most 2. Hence, we have that
\[sun(G^{\prime}-X)=sun(G-E^{\prime}-X)\leq sun(G-X)+2m. \tag{11}\]
Combining (11) with (10), we have:
\[2|X|+1\leq sun(G^{\prime}-X)\leq sun(G-X)+2m.\]
It follows that
\[sun(G-X) \geq 2|X|+1-2m\] \[\geq 2\times(2m+1)+1-2m\] \[= 2m+3\] \[\geq 5.\]
Then we can choose three distinct vertices \(u,v,w\in Sun(G-X)\) such that \(\{u,v,w\}\) is an independent set of \(G\), and hence \(d_{G}(u)+d_{G}(v)+d_{G}(w)\geq\sigma_{3}(G)\geq n+2m\). This together with \(N_{G}(u)\cup N_{G}(v)\cup N_{G}(w)\subseteq X\) implies
\[|X|\geq\max\{d_{G}(u),d_{G}(v),d_{G}(w)\}\geq\frac{\sigma_{3}(G)}{3}\geq\frac{ n+2m}{3}.\]
This together with the inequality \(sun(G-X)\geq 2|X|+1-2m\) implies that \(n=|G|\geq|X|+sun(G-X)\geq 3|X|+1-2m\geq n+1\), which is a contradiction.
**Case 2**. \(G-X\) is connected.
It is easily seen that \(sun(G-X)\leq\omega(G-X)=1\) since \(G-X\) is connected. Recall that
\[sun(G^{\prime}-X)=sun(G-E^{\prime}-X)\leq sun(G-X)+2m. \tag{12}\]
This together with (10) implies that \(2|X|+1\leq sun(G^{\prime}-X)\leq sun(G-X)+2m\leq 1+2m\), and thus
\[|X|\leq m. \tag{13}\]
It follows from (13) and \(\kappa(G)\geq 2m+1\) that
\[\lambda(G-X)\geq\kappa(G-X)\geq\kappa(G)-|X|\geq 2m+1-m=m+1. \tag{14}\]
Using (14), we obtain that \(\lambda(G^{\prime}-X)\geq\lambda(G-X)-m\geq 1\). Hence, \(sun(G^{\prime}-X)\leq\omega(G^{\prime}-X)=1<2|X|\), which contradicts (10).
## 4 Open problems
A graph \(G\) is called a \(P_{\geq k}\)-factor covered graph if it has a \(P_{\geq k}\)-factor covering \(e\) for any \(e\in E(G)\), where \(k\geq 2\) is an integer. Zhang and Zhou [23] proposed the concept of path-factor covered graph, which is a generalization of matching covered graph. They also obtained a characterization for \(P_{\geq 2}\)-factor and \(P_{\geq 3}\)-factor covered graphs, respectively. Recently, Zhou and Sun [25] extended the concept of \(P_{\geq k}\)-factor covered graph to \((P_{\geq k},l)\)-factor critical covered graph, namely, a graph \(G\) is called \((P_{\geq k},l)\)-factor critical covered if \(G-D\) is \(P_{\geq k}\)-factor covered for any \(D\subseteq V(G)\) with \(|D|=l\). Similar to \((P_{\geq k},l)\)-factor critical covered graph, the concept of \((P_{\geq k},m)\)-factor deleted graph can be further extended to \((P_{\geq k},m)\)-factor deleted covered graph, that is, a graph \(G\) is a \((P_{\geq k},m)\)-factor deleted covered graph if deleting any \(m\) edges from \(G\), the resulting graph is still a \(P_{\geq k}\)-factor covered graph.
We raise the following open problems as the end of our paper.
**Problem 10**.: _What is the tight \(s(G)\) or \(\sigma_{3}(G)\) bound of \((P_{\geq k},l)\)-factor critical covered graphs?_
**Problem 11**.: _What is the tight \(s(G)\) or \(\sigma_{3}(G)\) bound of \((P_{\geq k},m)\)-factor deleted covered graphs?_
## Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11971196,12201304,12201472) and Hainan Provincial Natural Science Foundation of China(No.120QN176).
## Conflict of interests
The authors declare that there are no conflict of interests. |
2308.04937 | Machine learning unveils multiple Pauli blockades in the transport
spectroscopy of bilayer graphene double-quantum dots | Recent breakthroughs in the transport spectroscopy of 2-D material
quantum-dot platforms have engendered a fervent interest in spin-valley qubits.
In this context, Pauli blockades in double quantum dot structures form an
important basis for multi-qubit initialization and manipulation. Focusing on
double quantum dot structures, and the experimental results, we first build
theoretical models to capture the intricate interplay between externally fed
gate voltages and the physical properties of the 2-D system in such an
architecture, allowing us to effectively simulate Pauli blockades. Employing
the master equations for transport and considering extrinsic factors such as
electron-photon interactions, we thoroughly investigate all potential
occurrences of Pauli blockades. Notably, our research reveals two remarkable
phenomena: (i) the existence of multiple resonances within a bias triangle, and
(ii) the occurrence of multiple Pauli blockades. Leveraging our model to train
a machine learning algorithm, we successfully develop an automated method for
real-time detection of multiple Pauli blockade regimes. Through numerical
predictions and validations against test data, we identify where and how many
Pauli blockades are likely to occur. We propose that our model can effectively
detect the generic class of Pauli blockades in practical experimental setups
and hence serves as the foundation for future experiments on qubits that
utilize 2-D material platforms. | Anuranan Das, Adil Khan, Ankan Mukherjee, Bhaskaran Muralidharan | 2023-08-09T13:12:02Z | http://arxiv.org/abs/2308.04937v1 | Machine learning unveils multiple Pauli blockades in the transport spectroscopy of bilayer graphene double-quantum dots
###### Abstract
Recent breakthroughs in the transport spectroscopy of 2-D material quantum-dot platforms have engendered a fervent interest in spin-valley qubits. In this context, Pauli blockades in double quantum dot structures form an important basis for multi-qubit initialization and manipulation. Focusing on double quantum dot structures, and the experimental results, we first build theoretical models to capture the intricate interplay between externally fed gate voltages and the physical properties of the 2-D system in such an architecture, allowing us to effectively simulate Pauli blockades. Employing the master equations for transport and considering extrinsic factors such as electron-photon interactions, we thoroughly investigate all potential occurrences of Pauli blockades. Notably, our research reveals two remarkable phenomena: (i) the existence of multiple resonances within a bias triangle, and (ii) the occurrence of multiple Pauli blockades. Leveraging our model to train a machine learning algorithm, we successfully develop an automated method for real-time detection of multiple Pauli blockade regimes. Through numerical predictions and validations against test data, we identify where and how many Pauli blockades are likely to occur. We propose that our model can effectively detect the generic class of Pauli blockades in practical experimental setups and hence serves as the foundation for future experiments on qubits that utilize 2-D material platforms.
## I Introduction
Quantum devices implemented on 2-D material platforms have garnered significant attention due to their unique characteristics, including non-equivalent valleys, strong spin-orbit coupling, electronically tunable bandgaps, and reduced hyperfine interactions [1, 2, 3, 4, 5, 6, 7, 8]. This makes materials like bi-layer graphene (BLG) and the transition metal dichalcogenide family (TMDCs) [9] promising platforms for spin-valley qubits. Recent experiments have explored the potential of utilizing the valley degree of freedom for qubit initialization, storage, and readout. Achieving confinement of electrons to quantum dots (QDs) on 2-D materials has, however, presented several hurdles, such as defects, high effective electron masses, and challenges in achieving precise gate voltage tunability using ohmic contacts [10, 11, 12].
Initialization and read-out of qubits on any platform involves the ascertained occurrence of Pauli-blockades [13, 14, 15, 16, 17, 18, 19] in the electronic transport spectroscopy. Recent investigations have demonstrated that in 2-D systems, the spin and the valley pseudospin are intertwined, leading to a general class of Pauli blockades [20] that cannot be solely explained by independently considering the spin or valley degrees of freedom. While the theoretical understanding of the origins of multiple Pauli blockades has made progress [19, 21], leveraging this knowledge to tune quantum dots by identifying and characterizing multiple blockades still presents a significant challenge.
Utilizing machine learning for the initialization and scaling of qubits in quantum devices have highlighted the use of automation to detect and characterize Pauli spin-blockades in semiconducting qubit systems [22, 23, 24, 25]. These algorithms can swiftly determine the occurrence of a spin-blockade, enabling real-time readout of spin qubits. Given the rapid progress in spin-valley qubit research, a fully automated algorithm for identifying multiple spin-valley coupled Pauli blockades is certainly the need of the hour. In this paper, we develop a comprehensive model to understand the transport mechanism through quantum dots. We leverage this model to train a machine learning algorithm, enabling us to predict blockade regimes in the 2-D material platforms, as illustrated in Fig. 1d.
Our work builds upon a double quantum dot (DQD) transport setup [26, 27, 28, 29, 30], as depicted in Fig. 1a. The focus of our analysis is on Pauli blockades, particularly in the regime where the total occupancy of the dots is restricted to two electrons. To construct our models, we account for various factors [31, 32, 33, 34] such as intrinsic spin-orbit (SO) coupling, spin-Zeeman splitting, valley-Zeeman splitting, sequential spin (pseudo-spin) conserving inter-dot tunneling, spin-flip tunneling, photon-assisted tunneling, and Coulomb interactions. These elements are visualized in Fig. 1b and Fig. 1c, and are the essential components of our model for understanding the intricacies of the transport mechanism, and eventually making important predictions.
By simulating the effects of spin-orbit coupling, spin-flip tunneling and photon-assisted tunneling, we unveil two novel phenomena, which are not present in spin-only systems: (i) multiple resonances and (ii) multiple Pauli blockades. We demonstrate that these phenomena have distinct underlying causes. Furthermore, we carefully train our machine learning model to not only detect the presence of a Pauli blockade but also predict whether there are multiple Pauli blockades or just multiple resonance peaks in the system. The predictability of precise Pauli blockade regimes allows us to extract both spin and valley information from the state using a single-shot readout, effectively doubling the rate at which information is stored and read. We firmly believe that our research says the groundwork for future experiments involving the initialisation and scaling of qubits on 2-D material platforms.
The paper is organised as follows. In Sec. II, we construct the effective Hamiltonian for the DQD, solve for the relevant eigenstates, and obtain the equations governing the current flow and thereby obtain the transport spectroscopy. We then present the conditions under which Pauli blockades may be realized, and under which they may be lifted. In Sec. III we perform simulations on our model and generate a training data set using different regimes and parameters. We also present the cases under which multiple excitations and multiple blockades occur. We then use the generated data to train our machine learning model and then test the same on different test cases. Finally, we present our conclusions and scope for future work in Sec. IV.
Figure 1: Device schematics and workflow. **(a)** The device structure under consideration. The five finger gates and two split gates define the DQD transport spectroscopy setup. Barrier gates \(V_{B1}\) and \(V_{B3}\) control the coupling to the source and drain respectively, while \(V_{B2}\) controls the inter-dot tunneling. Plunger gates \(V_{G1}\) and \(V_{G2}\) determine the onsite energies. **(b)** The DQD system is coupled to the left and right contacts by coupling rates \(\gamma_{L}\) and \(\gamma_{R}\), respectively. Each dot is described by the onsite energies \(\epsilon_{i}\), the intra-dot interaction \(U_{ii}\), and the inter-dot interaction \(U_{ij}\). The interdot tunneling is described by \(t_{c}\), which is spin and valley conserving. Spin-flip tunneling events are captured by \(t_{flip}\). The dot tuning is controlled via gate voltages \(V_{G1}\) and \(V_{G2}\). A finite source-drain bias of \(V_{sd}\) is also applied to facilitate current flow. The cross-capacitance between the gate voltages is \(C\). **(c)** Inelastic processes are crucial to realize bias triangles in the high-bias regime. A basic mechanism for boson-assisted tunneling is shown. Intra-dot transitions between \(\epsilon_{b}\) and \(\epsilon_{2r}\) is blocked. An electron in level \(\epsilon_{b}\) can loose energy via these interactions to transit to level \(\epsilon_{1r}\) and hence open conduction channels involving states \(\epsilon_{1r}\) and \(\epsilon_{2r}\). **(d)** Flowchart of the machine learning algorithm followed. The voltages and magnetic field inputs (blue) set to the system determine the parameters in the Hamiltonian, which, coupled with the I-V characteristics of the system (orange), are fed in to extract bias triangles and determine Pauli blockade regimes (green) using a machine learning algorithm.
## II Formalism and methods
A schematic of the model under investigation is illustrated in Fig. 1a, and Fig. 1b. The model involves a DQD setup [35; 36] with source and drain electrodes shown in blue. Split gates (pink) are used to change the carrier concentration, and finger gates (yellow) are used for fine-tuning the onsite potentials on each quantum dot. Functionally, when a dc bias is applied across this device, a current can be observed. The current flow can then also be controlled by tweaking the gate voltages. We have a left (right) quantum dot \(LD(RD)\), whose onsite energy, \(\varepsilon_{1}(\varepsilon_{2})\) can be controlled using the lever arm voltage \(V_{G1}(V_{G2})\). The model abstraction for the DQD system is illustrated in Fig. 1a. The source drain bias voltage \(V_{SD}\) is applied across the electrodes labelled \(V_{S}\) and \(V_{D}\). The voltage at gate \(V_{B2}\) controls \(t\), the inter-dot tunneling, which, in our model, preserves the spin and the valley pseudo-spin. The coupling constants between each lead and the channel, \(\gamma_{1}\) and \(\gamma_{2}\), are controlled by barrier gate voltages \(V_{B1}\) and \(V_{B3}\). The backgate voltage \(V_{BB}\) can be tuned to control the overall degree of confinement.
### Model Hamiltonian
We employ a modified version of the well-studied Hubbard Hamiltonian [37; 20; 34] to model the system. This model includes several components, such as onsite energies and inter-dot tunneling. Additionally, there exists an onsite Coulomb repulsion, denoted by \(U_{1(2)}\), between pairs of electrons on each dot, and an inter-dot Coulomb repulsion, denoted by \(U_{12(2)}\). These energies \(U_{i}\) and \(U_{ij}\) can be matrices indexed by \(i(=1,2)\) and \(j(=1,2)\), allowing for a more flexible representation.
Each dot can host a conduction electron in one of the two valleys, either \(K\) or \(K^{\prime}\), and each valley can possess either spin \(\uparrow\) or spin \(\downarrow\). Consequently, a single dot offers four available states: \(\ket{K\uparrow}\), \(\ket{K\downarrow}\), \(\ket{K^{\prime}\uparrow}\), and \(\ket{K^{\prime}\downarrow}\). In the absence of an external magnetic field, these four energy states split into two Kramer pairs due to intrinsic spin-orbit (SO) coupling [38; 39; 40; 41; 42]. However, when an external magnetic field is introduced, the degeneracy of the states within each Kramer pair is broken, affected by both the spin-Zeeman and the valley-Zeeman effects. The energy shift due to spin-Zeeman splitting is given by \(h_{S}=\sigma g_{s}\mu_{B}B_{||}\), while the valley-Zeeman splitting contributes an energy shift of \(h_{V}=\tau g_{v}\mu_{B}B_{\perp}\). Here, \(\sigma\) represents the spin quantum number (\(\sigma=+\frac{1}{2}\) for spin \(\uparrow\) and \(\sigma=-\frac{1}{2}\) for spin \(\downarrow\)), and \(\tau\) denotes the valley pseudo-spin (\(\tau=+\frac{1}{2}\) for valley \(K\) and \(\tau=-\frac{1}{2}\) for valley \(K^{\prime}\)). The electron magnetic moment is denoted by \(\mu_{B}=5.79\times 10^{-5}\) eVT\({}^{-1}\), \(B_{||}\) and \(B_{\perp}\) are the parallel and perpendicular components of the external magnetic field, and \(g_{s}\) and \(g_{v}\) represent the spin and valley g-factors, respectively.
Under such considerations, the Hamiltonian takes the form
\[\hat{H}_{DQD} =\underbrace{\varepsilon_{L}\hat{n}_{L}+\varepsilon_{R}\hat{n}_ {R}}_{\text{Onsite energy}}\] \[+\underbrace{\frac{U_{ii}}{2}\left(\hat{n}_{L}^{2}-\hat{n}_{L}+ \hat{n}_{R}^{2}-\hat{n}_{R}\right)}_{\text{Onsite repulsion}}+\underbrace{U_{ij} \hat{n}_{L}\hat{n}_{R}}_{\text{Interdot repulsion}}\] \[+\underbrace{\sum_{\tau,\sigma}t\hat{c}_{R\sigma\tau}^{\dagger} \hat{c}_{L\sigma\tau}+\text{h.c.}}_{\text{Interdot sequential tunneling}}\] \[+\underbrace{\sum_{\alpha,\tau,t}t_{SO}\hat{c}_{R\uparrow\tau}^{ \dagger}\hat{c}_{L\downarrow\tau}+\hat{c}_{R\downarrow\tau}^{\dagger}\hat{c}_ {L\downarrow\tau}+\text{h.c.}}_{\text{Interdot spin-flip tunneling}}\] \[+\underbrace{\frac{\Delta_{SO}}{2}\sum_{\alpha,\tau,\sigma}\hat{c} _{\alpha\sigma\tau}^{\dagger}\left(\mathbf{\sigma}_{3}\right)_{\sigma\sigma} \left(\mathbf{\tau}_{3}\right)_{\tau\tau}\hat{c}_{\alpha\sigma\tau}}_{\text{Spin orbit coupling}}\] \[+\underbrace{|h_{S}|\sum_{\alpha,\tau,\sigma}\hat{c}_{\alpha\sigma \tau}^{\dagger}\left(\mathbf{\sigma}_{3}\right)_{\sigma\sigma}\hat{c}_{\alpha \sigma\tau}}_{\text{Spin Zeeman effect}}\] \[+\underbrace{|h_{V}|\sum_{\alpha,\tau,\sigma}\hat{c}_{\alpha \sigma\tau}^{\dagger}\left(\mathbf{\tau}_{3}\right)_{\tau\tau}\hat{c}_{\alpha \sigma\tau}}_{\text{Valley Zeeman effect}}, \tag{1}\]
where the summations are defined over \(\alpha\in\{L,R\}\), \(\sigma\in\{\uparrow,\downarrow\}\), \(\tau\in\{K,K^{\prime}\}\). The terms \(\mathbf{\sigma}_{3}\) (\(\mathbf{\tau}_{3}\)) is the z-component of the Pauli matrix for the spin (valley pseudo-spin), defined as
\[\left(\mathbf{\sigma}_{3}\right)_{\uparrow\uparrow}=1;\quad\left(\mathbf{\sigma}_{3} \right)_{\uparrow\downarrow} =\left(\mathbf{\sigma}_{3}\right)_{\downarrow\uparrow} =0;\quad\left(\mathbf{\sigma}_{3}\right)_{\downarrow\downarrow} =-1 \tag{2a}\] \[\left(\mathbf{\tau}_{3}\right)_{KK}=1;\quad\left(\mathbf{\tau}_{3}\right)_{KK^{ \prime}}=\left(\mathbf{\tau}_{3}\right)_{K^{\prime}K}=0;\quad\left(\mathbf{\tau}_{3} \right)_{K^{\prime}K^{\prime}}=-1 \tag{2b}\]
The symbol \(\hat{n}_{L}\) (\(\hat{n}_{R}\)) denotes the number operator for the number of electrons on the left (right) dot, formulated as
\[\hat{n}_{\alpha}=\sum_{\sigma,\tau}\hat{c}_{\alpha\sigma\tau}^{\dagger}\hat{c}_ {\alpha\sigma\tau}, \tag{3}\]
where \(\hat{c}_{\alpha\sigma\tau}^{\dagger}\) is the annihilation (creation) operator for an electron on dot \(\alpha\) with spin \(\sigma\) in valley \(\tau\).
In the absence of spin-flip tunneling and photon interactions, the above Hamiltonian takes a block diagonal form and separates into nine sub-spaces, each with an invariant total number of electrons \(N=0,1,\cdots,8\). In discussing the blockades in the two-electron occupancy regime, only the subspaces \(N=1\) and \(N=2\) are relevant. We use \(L_{\zeta}\) (\(R_{\zeta}\)) to denote that there is an electron in the state \(\zeta\) on the left(right) quantum dot, where \(\zeta\in\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\}\). For instance, a system with two electrons: one in the left dot in state \(K\uparrow\) and the other in the right dot in state \(K^{\prime}\downarrow\) is represented as \(|L_{K\uparrow}R_{K^{\prime}\downarrow}\rangle\). We also develop the notation \((n_{L},n_{R})\) to
represent a state with \(n_{L}\) electrons on the left dot and \(n_{R}\) electrons on the right dot. The eigenstates are not the \((n_{L},n_{R})\) states, but a superposition of \((n_{L},n_{R})\) states with \(n_{L}+n_{R}=N=\text{constant}\).
In addition to the above Hamiltonian, we also add a term for photon-assisted tunneling (PAT) as described below. These additional effects often lift the Pauli blockades by creating eigenstates that are superpositions of multiple spin and valley states. The composite Hamiltonian of the system can be modified to include the effects of boson-mediated scattering processes [43] as shown in fig. 1c. The photon relaxation occurs in the sense that electrons in the dot can change level by absorption and emission. Such an interaction Hamiltonian is described as:
\[H_{\text{ph}}=H_{B}+H_{B-D}=\sum_{q}\omega_{q}d_{q}^{\dagger}d_{q}+\sum_{q \sigma ij}g_{ph}\left(d_{q}^{\dagger}+d_{q}\right)c_{i\sigma}^{\dagger}c_{j \sigma}, \tag{4}\]
where the coupling amplitudes are independent of quantum numbers i, j and q. Hence, \(g_{q}^{ij}=g_{ph}\). Total bosonic coupling constant is thus \(\alpha_{\text{ph}}(\omega)=2\pi g_{\text{ph}}^{2}p_{\text{b}}(\omega)\), where \(\rho_{\text{b}}(\omega)\) represents the photon density of states.
### Fock subspaces of the Hamiltonian
In the presence of sequential tunneling alone, the Hamiltonian is a block diagonal matrix, each block representing the total number of electrons, \(N\) in the DQD. Exact solutions to the eigenstates have emerged recently [20]. The \(N=1\) sub-matrix of the Hamiltonian in (1) is an \(8\times 8\) matrix and consequently has eight distinct eigenstates. These eigenstates can be categorized into two groups: the bonding states and the anti-bonding states, as follows.
\[\ket{B_{\zeta}} =\xi\ket{L_{\zeta}}+\eta\ket{R_{\zeta}} \tag{5a}\] \[\ket{AB_{\zeta}} =\xi\ket{L_{\zeta}}-\eta\ket{R_{\zeta}}. \tag{5b}\]
In the above formulation, \(B\) represents the bonding states and are lower in energy, while \(AB\) represents the anti-bonding states and are higher in energy. \(\zeta\in\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\}\). The \(N=2\) sub-matrix of the Hamiltonian is a \(28\times 28\) matrix with twenty eight eigenstates. Based on their contribution towards the current through the DQD, the eigenstates can be classified into three broad categories as follows.
\[\ket{C_{\zeta_{1}\zeta_{2}}} =\alpha\left(\ket{L_{\zeta_{1}}R_{\zeta_{2}}}-\ket{L_{\zeta_{2}} R_{\zeta_{1}}}\right)\] \[\qquad+\beta\ket{L_{\zeta_{1}}L_{\zeta_{2}}}+\kappa\ket{R_{\zeta_ {1}}R_{\zeta_{2}}} \tag{6a}\] \[\ket{D_{\zeta_{1}\zeta_{2}}} =\frac{1}{\sqrt{2}}\left(\ket{L_{\zeta_{1}}R_{\zeta_{2}}}+\ket{L_ {\zeta_{2}}R_{\zeta_{1}}}\right)\] (6b) \[\ket{P_{\zeta}} =\ket{L_{\zeta}R_{\zeta}}, \tag{6c}\]
where \(\zeta\in\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\}\) and the combination \(\zeta_{1}\zeta_{2}\in\{K\uparrow K\downarrow,\ K\uparrow K^{\prime}\uparrow, K\uparrow K^{\prime}\downarrow,\ K\downarrow K^{\prime}\uparrow,\ K\downarrow K^{\prime}\downarrow,\ K\uparrow K^{\prime}\downarrow,\ K^{\prime}\uparrow K^{\prime}\downarrow\}\) and \(\zeta_{1}\) and \(\zeta_{2}\) are taken over all possible unordered combinations of \(\zeta\). We therefore have six possible combinations of \(\zeta_{1}\zeta_{2}\). Each of the states \(C\) occurs threefold, with three different sets of values of \(\alpha\), \(\beta\), and \(\kappa\). Thus, in total, we have eighteen \(C\) states, six \(D\) states, and four \(P\) states. The state labels \(C\) and \(D\) stand for "conducting" and "dark" respectively, according to their role in the equation for the current [20].
In the presence of spin-flip tunneling, the states belonging to the \(N=1\) Fock-space are superpositions of two bonding (or anti-bonding) states. The bonding superposition takes the form
\[a\ket{B_{\zeta_{1}}}+b\ket{B_{\zeta_{1}}} =\xi a\ket{L_{\zeta_{1}}}+\eta a\ket{R_{\zeta_{1}}}\] \[\qquad+\xi b\ket{L_{\zeta_{2}}}+\eta b\ket{R_{\zeta_{2}}}\]
and the antibonding state follows accordingly. In the \(N=2\) subspace, we obtain superpositions of two \(C\) or \(D\) states in a similar fashion. The eigenvalue variation of various states in the \(N=2\) manifold with respect to \(B_{\parallel}\) and \(B_{\perp}\) is shown in Figs. 2a(a) and (b) respectively. The existence of superposition increases the number of states that satisfy the transport selection rules [20; 29; 30] and provides multiple transition pathways to the same state, each having a different transition energy. This manifests as multiple resonance peaks within a bias triangle, each corresponding to a particular transition.
### Transport formulation
Despite having a solid grasp of the theory concerning current blockades in single-degree-of-freedom DQDs [44; 45; 46; 47; 13], the behavior of the current in DQDs with multiple degrees of freedom has remained a challenge, until recently [20]. The overall current in such a system arises from the intricate interplay between the probabilities of occupation for each eigenstate and the transition rates between them. To address intricate problem, we expand upon the existing master equation, which is well-known in literature [45]. The eigenstates are labelled \(\ket{N,i}\), where \(N\) denotes the total electron occupancy and \(i\) denotes the \(i^{\text{th}}\) state in the corresponding Fock state subspace with total electron occupancy \(N\). We define the quantity \(P_{i}^{N}\) to denote the probability of occupancy of the state \(\ket{N,i}\) and \(R_{(N_{1},i)\rightarrow(N_{2},j)}^{L(R)}\) to denote the rate of transition from the state \(\ket{N_{1},i}\) to the state \(\ket{N_{2},j}\) by injection or removal of an electron from the source(drain). Henceforth, we shall use the index \(\alpha\) for the source (\(\alpha=L\)) or the drain (\(\alpha=R\)). The probabilities \(P_{i}^{N}\) evolve over time as
\[\dot{P}_{i}^{N}=\sum_{j}\left[R_{(N\pm 1,j)\rightarrow(N,i)}P_{j}^{N\pm 1}-R_{(N,i)\rightarrow(N\pm 1,j)}P_{i}^{N}\right], \tag{7}\]
where
\[R_{(N_{1},i)\rightarrow(N_{2},j)}=\sum_{\alpha\in\{L,R\}}R_{(N_{1},i)\rightarrow( N_{2},j)}^{\alpha}. \tag{8}\]
### Inelastic processes
Besides resonant tunneling, inelastic processes and cotunneling often play a pivotal role in determining the current-voltage characteristics. In fact, the origin of the base current in the bias triangles is attributed to inelastic processes[48, 49, 50]. To describe relaxation processes, an additional rate term \(R^{(ph)}\) is introduced to cater to the bosonic degrees of freedom. In a similar spirit of weak tunneling consideration, we consider weak coupling to the bosonic bath. Hence, the contributions from first order \(\alpha_{ph}\) only contributes. The transition rates are given by
\[R_{(N_{1},i)\rightarrow(N_{1},j)}=\sum_{\alpha\in\{L,R\}}R_{(N_{1},i) \rightarrow(N_{1},j)}^{\alpha(ph)}. \tag{9}\]
We notice that the transitions take place among many-body states of the same manifold, i.e., within states with same occupancy number. Individual bosonic rates \(R_{(N_{1},i)\rightarrow(N_{1},j)}^{\alpha(ph)}\) can be elaborated as
\[R_{(N_{1},i)\rightarrow(N_{1},j)}^{\alpha(ph)}=b(E)|\langle(N_{1},i)|c^{ \dagger}c|(N_{1},j)\rangle|^{2}, \tag{10}\]
where \(b(x)=sgn(x)\alpha_{ph}(x)n_{b}(x)\), with the Bose function \(n_{b}(x)=\frac{1}{(exp(x/k_{B}T)-1)}\) and \(c^{\dagger}\) represents corresponding creation operator.
### Machine learning methodology
Pauli blockades are essential for qubit initialisation and readout. We demonstrate an application of the above transport formalism to generate simulated data and train a deep learning network for automated detection of Pauli blockade. We follow an approach [51] using a residual convolutional neural network which is 18 layers deep [52].
The training data set is generated by randomly varying multiple device parameters in the transport simulations. The simulations generate charge-stability plots (in terms of \(I\) vs \(V_{G1}\), \(V_{G2}\)) and a rectangular gate voltage window enclosing a pair of bias triangles is then extracted. These images are labelled as PB (Pauli blockade) and No PB (no Pauli blockade) according to the nature of their respective charge transitions. Bias triangles corresponding to positive and negative source-drain voltages are concatenated together and presented as training data to the deep learning classifier. The presence of Pauli blockade is indicated by the suppression of current in one bias direction as compared to the other.
The machine learning model is implemented in PyTorch with CUDA. All the extracted images are converted to a grayscale format to simplify the training process and remove unnecessary colour information. A test-train split of 30% is used and the simulated data is augmented by random stretching, rotation, shearing, and contrast changes. Additional augmentations from the Torchvision library including elastic transformation and sharpness adjustment are also applied to some inputs. Training is run for 20 epochs or till sufficient accuracy is reached, and a mini-batch size of 256 is used. The input data is downsized to about 68x60 pixels to make it similar to the resolution of typical
Figure 2: Energy spectrum of the \(N=2\) Fock space as is described in Sec. II.2, as a function of the magnetic field applied. **(a)** The perpendicular component of the magnetic field is held fixed at \(B_{\perp}=0.8\) T, and the parallel component is varied. The spin Zeeman effect splits the energy spectrum according to \(\Delta E=\mu_{B}g_{s}B_{||}\). **(b)** The parallel component of the magnetic field is held fixed at \(B_{||}=0.8\) mT, and the perpendicular component is varied. The valley Zeeman effect splits the energy spectrum according to \(\Delta E=\mu_{B}g_{v}B_{\perp}\).
current-voltage measurements. Softmax function is used to obtain respective probabilities for the two classes, Pauli blockade (PB) and no Pauli blockade (no PB). The sample is assigned to the class with higher predicted probability of the two (greater than 50%).
## III Results
The charge stability diagram is an important illustrative in understanding the effect of transport dynamics in the system. It is essentially a current (conductance) map plotted with respect to the gate voltages used to tune the DQD potential barriers. We carry out a variety of analyses on the data that we obtain from our physics based simulations. Some of the primary parameters that play a crucial role in obtaining the conductance plots are (i) source-drain bias (\(V_{sd}\)), (ii) photon coupling strength \(\alpha_{ph}\), and (iii) the tunneling coefficients \(t_{c}\), \(t_{so}\), and \(t_{vo}\). The other tunable parameters kept fixed are the temperature \(T_{c}\) (= 0.9K) and cross-capacitance C (= 0.2).
We first discuss the dependence of bias triangles on primary system parameters as illustrated in Fig. 3. Next, we describe the occurrence of multiple peaks and blocksades in Fig. 4. Finally, we present some results from the use of deep learning on our simulated data for detecting the occurrence of Pauli blockade.
### Origin of the bias triangles
Figures 3a and 3b show the difference in charge stability diagrams for the high and low bias regimes. In Fig. 3a, distinct triangles are visible when the bias is set high (\(V_{sd}=-1.0mV\)). However, as depicted in Fig. 3b, we observe that the distinct bias triangles are not visible for low bias (\(V_{sd}=-0.2mV\)). Initially, only a single transition (between ground states) is possible in the low-bias regime. Multiple conduction channels open up for electrons on increasing the bias, leading to conduction between \(N-1\), \(N\), and \(N+1\) states, and the corresponding excitations as permitted via the transport selection rules [20].
Figure 3: Analysis of charge stability(CS) diagrams with emphasis on the (1,1) to (0,2) transition and the effect of various physical parameter assumptions. **(a)** CS diagrams with clearly defined bias triangles can be obtained in the high source-drain bias regime(\(V_{sd}=-1.0mV\)). **(b)** At relatively lower biases(\(V_{sd}=-0.2mV\)) however, the triangles are not formed. For both the CS diagrams, the electron occupancy in the dots is marked. **(c)** Bias triangles magnified from the encircled part in (a) for (1,1) to (0,2) transition. For \(V_{sd}=-1mV\) there is no blockade. **(d)** Reversing the bias(\(V_{sd}=-1mV\)) leads to blockade and a four-fold reduction in current is seen. **(e)** On turning off PAT, further reduction in current(by 10 times) is noticed. **(f)** Current vs detuning plot obtained by taking cross-section along the white lines in figures (c), (d), and (e). The blockade mechanism is explained in Sec. III.1.
Figure 4: Occurrence of multiple excitations and multiple blockades in a single triangle. **(a)** On including spin-flip tunnelling arising out of spin-orbit coupling (\(t_{so}\)) and applying a small parallel magnetic field (\(B_{||}=80\mu T\)), we observe new transitions, that were forbidden previously. These excited transitions lead to multiple resonances that manifest as multiple current peaks in the triangles. **(b)** In the absence of spin-flip tunneling, the transitions happen between closely packed, almost degenerate states, showing no signatures of multiple excitations. **(c)** A case of multiple blockades with two excitation peaks. Figures (d), (e), and (f) are cross-sections of the CS diagrams (a), (b), and (c), respectively.
Figure 5: A plot of the energy difference between the \(N=1\) and \(N=2\) eigenstates for transitions allowed by selection rules. The plot is corresponding to the multiple excitations as shown in FIG 4a. At a detuning of around 1.78 meV, we observe the start of the first blockade, this is precisely where a \(\mathcal{D}\) transition enters the source-drain bias window in the presence of a corresponding \(\mathcal{C}\) transition. The inset shows a zoom in for detuning in the range of multiple peaks. There is another peak observed when a new \(\mathcal{C}\) transition enters the bias window, at around 1.83 meV.
We focus on transitions featuring the (0,2) and the (1, 1) configuration. At a certain bias of \(V_{sd}=-1.0mV\), Fig. 3c depicts the bias triangle with the current. The triangle's base corresponds to the ground state-ground state (GS-GS) transition between two different charge (number) configurations, leading to larger current values. On reversing the bias, as seen in Fig. 3d, it can be noticed that there is about a three to four fold drop in the current values. The PAT processes play the role of inelastic scattering crucial in forming the triangle. If the boson coupling is turned off (Fig. 3e), we notice a further decrement in the current(about 100 times smaller) than in the non-blockaded case. A comparison of the blockaded and non-blockaded cases along a cut-line is presented in Fig. 3f. In agreement with the theory, the GS-GS transitions stay the same in both forward and reverse biases with only a drop in magnitude.
### Multiple peaks and blockades
The addition of spin-flip terms in the Hamiltonian leads to further splitting of the states giving rise to more resonant lines in the triangles, with a small spin-coupled magnetic field (\(B_{||}=80\mu T\)), and a finite tunneling coefficient (\(t_{c}=10^{-4}\)). Figure 4a depicts a case with the spin-flip present (\(t_{so}=10^{-4}\)), while Fig. 4b is the same triangle with the spin-flip term absent (\(t_{so}=0\)). The contrast can be observed in that for the former case, there are many minute peaks near the base of the triangle, while for the latter, a single GS-GS transition is observed.
Under a certain set conditions of \(B_{\perp}\) (=3.2 T), we report the signature of multiple blockades. Figure 4c depicts one such case in which two peaks sandwich a large dip across the white cut line. It can be inferred that there are thus three multi-blockaded regions in the triangle. It is essential to note that though the multiple excitation cases and the multi-blockade cases discussed here seem similar in terms of the effect, the cause for their origin varies.
Multiple Pauli blockades arise when specific conditions are met that lead to a negative differential resistance (NDR) [20; 29]. Multiple blockades arise because of the spin and valley Zeeman effects, wherein, the eigenergies in the Fock space depend upon the eigenstate, i.e., the degeneracy in energy between the \(C\) and \(D\) states is broken. The master equation then yields the relation between the energies, and in turn the gate voltages, that will lead to multiple Pauli blockades.
Multiple excitations occur in the presence of effects such as strong spin-orbit coupling or photon-qubit interactions, which leads to multiple current peaks (resonances). Multiple resonances arise due to a marked difference in the transition energies between the eigenstates in the \(N\) and \(N+1\) occupancy Fock-states. In the presence of sequential tunneling alone, the eigenstates formed in the \(N=2\) Fock space are of types \(C\) or \(D\) only. The coefficients within the \(C\) or the \(D\) state govern the occurence of multiple blockades. In contrast, spin-flip tunnelling results in the eigenstates being a superposition of two \(C\) states or \(D\) states. In such cases, the coefficients with which two \(C\) (or \(D\)) states mix dictate where a resonance line can be observed. This leads to multiple resonance lines within a single bias traingle, unlike multiple bias triangles in case of pure sequential tunneling case. Figure 5 demonstrates the relationship between the energy eigenspectrum and the occurrence of blockades.
### Identification of Pauli blockades using machine learning
Figure 6 presents a few samples from the test data set along with labels and probabilities as predicted by the deep learning classifier. It can be seen that the lower image of the concatenated pair corresponds to a positive \(V_{SD}\) and the upper image to a negative \(V_{SD}\). The overall test accuracy obtained for our implementation is 96.15%. Images in the figure reflect the actual input resolution to the classifier after augmenting and downsizing them. For the purpose of assigning predicted labels and calculating test accuracy, the sample is assigned to the class with higher probability, as mentioned before.
As a way to benchmark the performance of our neural network trained on the the simulated data set, we also perform the classification on a limited set of data from experimental results which have appeared in the literature recently [53; 54; 15]. An accuracy of 94.44% (17 out of 18 test cases predicted correctly) was obtained on the experimental test data set, thus showing the robustness of our neural network classifier and the simulated training data. This also re-validates the importance of using simulations in addition to experimental data for training, when the availability of experimental data is scarce [55; 56; 57; 51].
The machine learning procedure as mentioned above can also be used to detect and classify other kinds of features in the bias triangles, such as the occurrence of multiple peaks. This is depicted in Fig. 7, where the upper row represents bias triangles showing multiple peaks, while the images in the lower row only has a single GS-GS transition. The two classes are multiple peaks present (Multi) and multiple peaks not present (No Multi). The training procedure used in this case is similar to that for Pauli blockade detection and the test accuracy obtained on simulated data is 98.36%.
## IV Conclusion
Focusing on DQD structures in the BLG platform, and the experimental results available in literature, we first built theoretical models to capture the intricate interplay between externally fed gate voltages and the physical properties of the DQD setup, allowing us to effec
tively simulate Pauli blockades. Employing the master equations for transport and considering extrinsic factors such as electron-photon interactions, we thoroughly investigated all potential occurrences of Pauli blockades. Notably, our research revealed two remarkable phenomena: (i) the existence of multiple resonances within a bias triangle, and (ii) the occurrence of multiple Pauli blockades. Leveraging our model to train a machine learning algorithm, we successfully developed an automated method for real-time detection of multiple Pauli blockade regimes. Through numerical predictions and validations against test data, we identified where and how many Pauli blockades are likely to occur. We propose that our model can effectively detect the generic class of Pauli blockades in practical experimental setups and hence serves as the foundation for future experiments on qubits that utilize 2-D material platforms.
## Acknowledgements
The author BM acknowledges the support by the Science and Engineering Research Board (SERB), Government of India, Grant No. CRG/2021/003102 and Grant No. MTR/2021/000388.
|
2306.11128 | CAMMARL: Conformal Action Modeling in Multi Agent Reinforcement Learning | Before taking actions in an environment with more than one intelligent agent,
an autonomous agent may benefit from reasoning about the other agents and
utilizing a notion of a guarantee or confidence about the behavior of the
system. In this article, we propose a novel multi-agent reinforcement learning
(MARL) algorithm CAMMARL, which involves modeling the actions of other agents
in different situations in the form of confident sets, i.e., sets containing
their true actions with a high probability. We then use these estimates to
inform an agent's decision-making. For estimating such sets, we use the concept
of conformal predictions, by means of which, we not only obtain an estimate of
the most probable outcome but get to quantify the operable uncertainty as well.
For instance, we can predict a set that provably covers the true predictions
with high probabilities (e.g., 95%). Through several experiments in two fully
cooperative multi-agent tasks, we show that CAMMARL elevates the capabilities
of an autonomous agent in MARL by modeling conformal prediction sets over the
behavior of other agents in the environment and utilizing such estimates to
enhance its policy learning. | Nikunj Gupta, Somjit Nath, Samira Ebrahimi Kahou | 2023-06-19T19:03:53Z | http://arxiv.org/abs/2306.11128v2 | # CAMMARI: Conformal Action Modeling in
###### Abstract
Before taking actions in an environment with more than one intelligent agent, an autonomous agent may benefit from reasoning about the other agents and utilizing a notion of a guarantee or confidence about the behavior of the system. In this article, we propose a novel multi-agent reinforcement learning (MARL) algorithm _c_ammarl_, which involves modeling the actions of other agents in different situations in the form of confident sets, i.e., sets containing their true actions with a high probability. We then use these estimates to inform an agent's decision-making. For estimating such sets, we use the concept of conformal predictions, by means of which, we not only obtain an estimate of the most probable outcome but get to quantify the operable uncertainty as well. For instance, we can predict a set that provably covers the true predictions with high probabilities (e.g., 95%). Through several experiments in two fully cooperative multi-agent tasks, we show that _c_ammarl_ elevates the capabilities of an autonomous agent in MARL by modeling conformal prediction sets over the behavior of other agents in the environment and utilizing such estimates to enhance its policy learning. All developed codes can be found here: [https://github.com/Nikunj-Gupta/conformal-agent-modelling](https://github.com/Nikunj-Gupta/conformal-agent-modelling).
## 1 Introduction
Developing systems of autonomous agents capable of effective multi-agent interactions can be very useful in modern cooperative artificial intelligence (AI). For instance, service robots, surveillance agents, and many more similar applications require profound collaboration among agents (and with humans), without prior coordination. Now, to enable complex, constructive behaviors to emerge from unsupervised interactions among agents, an essential skill for an agent to have is the ability to reason about other agents in the environment. There has been considerable research addressing this problem of _agent_ or _opponent modeling_[2]. Generally, it involves constructing models of other agents that learn useful attributes to inform its own decision-making (such as the future actions of the other agents, or their current goals and plans) from current or past interaction history (such as the previous actions taken by other agents in different situations).
We are interested in the particular aspect of an interactive, autonomous agent which involves learning an additional, independent model to make predictions about the actions of the other agents in the environment, supplemental to its reinforcement learning-based policy to make decisions related to its downstream task. An autonomous agent can then incorporate those estimates to inform its decision-making and optimize its interaction with the other agents. While there exist several methods for developing such models for other agents [2], there is currently no method or theory to the best of our knowledge which would allow an agent to consider the correctness or confidence of the predictions of the learned model.
Conformal Predictions.Conformal predictions or inference is a fitting method for generating statistically accurate uncertainty sets for the predictions from machine learning classifiers. It is steadily gaining popularity owing to its explicit and non-asymptotic guarantees over the produced sets [5]. In other words, we can obtain conformal sets that provably contain the true predictions with high probabil
ities, such as 95%, chosen in advance. This can be very useful and successful in high-risk learning settings, especially in decision-making in medical applications from diagnostic information, for instance, which demand quantifying uncertainties to avoid in-sufferable model failures. What if we only prefer to use the predictions when the model is confident? For example, doctors may only consider a predicted medical diagnosis when the model is at least 95% accurate, or may want to use the predicted set with high credence to consider ruling out relevant possibilities. So, in this article, we aim to enhance the capabilities of an agent in a multi-agent reinforcement learning (MARL) setting by modeling and using conformal prediction sets (or the latent representations learned in the process1) over the behavior of an autonomous system. In particular, we model other agents' actions in the form of confident sets, i.e., sets that contain other agents' true actions with a high probability. We hypothesize that these estimated conformal sets would inform our _learning_ agent's decision-making and elevate its performance in MARL. Figure 1 shows the high-level idea of our proposed model for learning agents in any given environment.
Footnote 1: More details in Appendix A.
In this work, we aim to introduce a novel framework to train an autonomous agent that enhances its decision-making by modeling and predicting _confident conformal_ actions of other agents in the environment -- the cammarl algorithm (Section 3), and then empirically demonstrate that conformal action modeling used in cammarl indeed can help make significant improvements in cooperative policies learned by reinforcement learning agents in two multi-agent domains (Section 5).
## 2 Related Works
Decision-making without reasoning about other agents in the environment can be very challenging, for instance, due to weak or no theoretical guarantees, non-stationarity (single agent's perspective), and inefficient coordination for a considerable coherent joint behavior [25]. Modeling other agents in an environment is not new and has been studied in the past [2, 3]. However, our proposal of predicting conformal sets of actions of the other agents in the environment (with high probability) is novel and has not been attempted to the best of our knowledge.
Learning world models.Model-based reinforcement learning (MBRL) has certainly shown its advantages in data efficiency, generalization, exploration, counterfactual reasoning, and performance in many tasks and domains [14, 15, 18, 28, 31, 38] in single-agent RL, and now, it has also started to attract attention in MARL [44]. However, most of the current works in model-based MARL do not yet focus on teammate or opponent modeling. Some recent works [36, 46] incorporated dynamics modeling and a prediction module to estimate the actions of other agents within the construction of the environment model. However, these prediction models were trained without accessing the true trajectories from the other agents which can be problematic in several use cases.
Learning agent models.A widely popular technique to reason about other agents in the environment is to learn representations of different properties of other agents. For instance, learning to reconstruct the actions of other agents from their partial observations [16, 32, 26, 1], modeling an agent or its policy using encoder-decoder-based architectures [12, 48], learning latent representations from local information with or without utilizing the modeled agent's trajectories [34, 45] or modeling the forward dynamics of the system through relational reasoning using graph neural networks [42]. _Theory-of-Mind Network_ or TomNet learned embeddings cor
Figure 1: Our proposed methodology of informing an autonomous agent’s decision-making by means of conformal predictions of action sets of other agents in the environment. Two agents (\(\mathcal{N}_{self}\), \(\mathcal{N}_{other}\)) receive their own partial observations from the environment (\(o_{self}\), \(o_{other}\)) and take their actions (\(a_{self}\), \(a_{other}\)). An independent conformal action prediction model \(\mathcal{C}\) learns to output a conformal action set, \(\{a^{\prime}_{other}\}\), corresponding to \(\mathcal{N}_{other}\) which are then used as additional inputs for training by \(\mathcal{N}_{self}\) to inform its policy and perform its action \(a_{self}\).
responding to other agents in the environment for meta-learning [39]. Some works also constructed I-POMDPs to utilize recursive reasoning [2] assuming unrestricted knowledge of the observation models of other agents. Nevertheless, cammarl involves no form of reconstruction of other agent's policy or rewards, or state models. Any of these techniques can be used with cammarl which, however, is not the objective of this work. Also, unlike cammarl, many of these aforementioned techniques evaluate in fully-observable environments or rely upon direct access to other agents' experience trajectories even during execution. This can be infeasible in various settings.
Multi-agent reinforcement learning (MARL).Numerous deep MARL research works that focus on partial observability in fully cooperative settings indirectly involve reasoning about the intentions of teammates or opponents in an environment [11]. For instance, many works allow agents to communicate, enabling them to indirectly reason about the others' intentions [8, 9, 13, 20, 30, 41, 47]. On the other hand, some studied the emergence of cooperative and competitive behaviors among agents in varying environmental factors, for instance, task types or reward structures [22]. Recent work in hierarchical reinforcement learning also attempts to develop a hierarchical model to enable agents to strategically decide whether to cooperate or compete with others in the environment and then execute respective planning programs [19]. However, none of these works study the improvement in an autonomous agent's decision-making via directly modeling the other agents in the environment or predicting their actions or current or future intentions.
Inverse reinforcement learning (IRL).Research in the field of IRL also relates to our work because we share the key motive of inferring other agents' intentions and then use it to learn a policy that maximizes the utility of our learning agent [6]. However, IRL addresses this by deducing the reward functions of other agents based on their behavior, assuming it to be nearly optimal. On the other hand, in cammarl we directly model the other agent's actions based on their observations and use these estimates to indirectly infer their goal in an online manner.
Conformal prediction.Estimating well-grounded uncertainty in predictions is a difficult and unsolved problem and there have been numerous approaches for approximating it in research in supervised learning [10]. Recent works in conformal predictions [4, 7, 21, 27, 35] have now significantly improved upon some of the early research [33, 37, 43], for instance in terms of predicted set sizes, improved efficiency, and providing formal guarantees. For this article, we adapt the core ideas from _Regularized Adaptive Prediction Sets (RAPS)_[4] to our setting owing to its demonstrated improved performance evaluation on classification benchmarks in supervised learning [4].
## 3 The cammarl Algorithm
### Mathematical Model
Formally, we consider two agents in the environment -- learning agent denoted by _self_ and the other agent denoted by _other_. The partially observable Markov game [23] for our setting can then be defined using the following tuple2:
Footnote 2: The agent is a _self_ agent, which is a _self_ agent, which is a _self_ agent.
\[\left\langle\mathcal{N}_{i},\mathcal{S},\mathcal{A}_{i},\mathcal{O}_{i}, \mathcal{T},\mathcal{C},\pi_{\theta_{i}},r_{i}\right\rangle_{i\in\{self,other\}}\]
With the set \(\mathcal{S}\) describing the possible true states (or full observations) of the environment, two agents, \(\mathcal{N}_{self}\) and \(\mathcal{N}_{other}\), observe the environment locally using their sets of observations \(\mathcal{O}_{self}\) and \(\mathcal{O}_{other}\) respectively, and act using their set of actions, \(\mathcal{A}_{self}\) and \(\mathcal{A}_{other}\). Each agent \(i\) can select an action \(a_{i}\in\mathcal{A}_{i}\) using their policy \(\pi_{\theta_{i}}\), and their joint action \(\mathbf{a}\in\mathcal{A}_{self}\times\mathcal{A}_{other}\) then imposes a transition to the next state in the environment according to the state transition function \(\mathcal{T}\), defined as a probability distribution on the subsequent state based on current state and actions, \(\mathcal{T}:\mathcal{S}\times\mathcal{A}_{self}\times\mathcal{A}_{other}\times \mathcal{S}\rightarrow[0,1]\). The agents use their individual reward function \(r_{i}(s,a):\mathcal{O}_{i}\times\mathcal{A}_{i}\rightarrow\mathbb{R}\). Both agents aim to maximize their own total expected rewards \(R_{i}=\sum_{t=0}^{T}\gamma^{t}r_{i}^{t}\) where \(\gamma\in[0,1)\) as the discount factor and T is the time horizon.
In cammarl, at each time step \(t\), we also use a conformal prediction model defined as a set-valued function \(\mathcal{C}:\mathbb{R}^{d}\to 2^{\mathcal{A}_{other}}\)
\[\mathcal{C}(o_{other}^{t})\rightarrow\{A_{other}^{i}\}\]
which outputs a conformal action predictive set \(\{A_{other}^{t}\}\) for each input of \(\mathcal{N}_{other}\)'s local observation \(o_{other}^{t}\in\mathcal{O}_{other}\) at the time.
```
1:\(N_{self}\), \(N_{other}\leftarrow\) Initialize Actor-Critic networks for each -- \(\mathcal{N}_{self}\) and \(\mathcal{N}_{other}\)
2:\(conformalModel\leftarrow\) Initialize the conformal model to predict \(\mathcal{N}_{other}\)'s conformal action sets
3:\(b_{self}\), \(b_{other}\), \(b_{conformal}\leftarrow\) experience replay memory buffers for \(\mathcal{N}_{self}\), \(\mathcal{N}_{other}\) & \(conformalModel\)
4:for\(episode=1,2,\dots\)do
5: Fetch observations \(o_{self}\), \(o_{other}\) from environment
6:for\(timesteps=1,2,\dots,T\)do
7:\(conformalActions\gets conformalModel(o_{other})\)\(\triangleright\) Predict \(\mathcal{N}_{other}\)'s conformal set of actions
8:\(o_{self}\gets o_{self}\) + \(conformalActions\)\(\triangleright\) Concatenate conformal actions to \(o_{self}\)
9: Run policies \(\pi_{\theta_{self}}\) and \(\pi_{\theta_{other}}\) in the environment
10: Collect trajectories of \(\mathcal{N}_{self}\) and \(\mathcal{N}_{other}\) in \(b_{self}\) and \(b_{other}\)
11: Collect \(\mathcal{N}_{other}\)'s state-action mappings in \(b_{conformal}\)
12:if update interval reachedthen
13: Train \(conformalModel\) using \(b_{conformal}\)
14: Train \(N_{self}\) on \(b_{self}\) using PPO
15: Train \(N_{other}\) on \(b_{other}\) using PPO
16:endif
17:endfor
18:endfor
```
**Algorithm 1**Conformal action modeling in marl
### Conformal Action Modeling
Now we formally describe our proposed algorithm -- _Conformal Action Modeling-based Multi-Agent Reinforcement Learning_ or cammarl. Our objective is to inform an \(\mathcal{N}_{self}\)'s decision-making by modeling \(\mathcal{N}_{other}\)'s actions in the environment as conformal prediction sets which contain the true actions with a high probability (for example, 95%). More specifically, \(\mathcal{N}_{self}\) uses a separate conformal action prediction model to obtain sets of \(\mathcal{N}_{other}\)'s actions at each timestep that contains latter's true action in the environment at a given time step with high prespecified probabilities.
Algorithm 1 describes the complete workflow of training of agents in cammarl. We begin by initializing the actor-critic networks for both the agents in the environment, the conformal model, and the memory buffers for each of these (lines 1-3). Now, at the beginning of each episode in the environment, both agents receive their own partial observations (line 5). Next, the conformal model predicts the actions of \(\mathcal{N}_{other}\) in the form of a set, which is then provided as an additional input to \(\mathcal{N}_{self}\) (lines 7-8), whereas \(\mathcal{N}_{other}\) has access to only its own partial observation, \(o_{other}\). Both agents now take actions in the environment and continue collecting their experiences (lines 9-11). The agents and the conformal model periodically train using their respective experience memory (lines 12-16).
Figure 2 shows a detailed illustration of our conformal action modeling that materializes internally at each time step. The conformal predictor \(\mathcal{C}\) collects \(\mathcal{N}_{other}\)'s state-action pairs and periodically learns and updates a neural network classifier, \(f(\cdot):\mathbb{R}^{d}\rightarrow\mathbb{R}^{|\mathcal{A}_{other}|}\) (where \(d\) is the number of dimensions in
Figure 2: A detailed illustration of conformal action modelling and inference in cammarl to generate prediction sets of \(\mathcal{N}_{other}\)’s actions using conformal predictors.
\(\mathcal{N}_{other}\)'s local observation and \(|\mathcal{A}_{other}|\) is the number of possible discrete actions available for \(\mathcal{N}_{other}\)), to predict action from a given state. Then, we adapt RAPS conformal calibration [4] to our setting. Considering \(\mathbf{o}\in O_{other}\) as feature vectors, we use the updated \(f\) to compute action probabilities \(\hat{\pi}_{o}\in\mathbb{R}^{\,|\mathcal{A}_{other}|}\). The probabilities are then ordered from most probable to least probable followed by the estimation of the predictive action set for the given feature inputs. To promote small predictive sets we also add a regularization term as also proposed in RAPS. Formally, for a feature vector \(\boldsymbol{o}\) and the corresponding possible prediction \(\boldsymbol{a}\), to estimate a set which includes all the actions that will be included before \(\boldsymbol{a}\), let us define the total probability mass of the set of actions that are more probable than \(\boldsymbol{a}\):
\[\rho_{o}(a)=\sum_{a^{\prime}=1}^{|\mathcal{A}_{other}|}\hat{\pi}_{o}(a^{\prime })\mathbbm{1}_{\{\hat{\pi}_{o}(a^{\prime})\geq\hat{\pi}_{o}(a)\}}\]
Also, if we define a function to rank the possible action outcomes based on their probabilities \(\hat{\pi}\) as
\[z_{o}(a)=|\{a^{\prime}\in\mathcal{A}_{other}:\{\hat{\pi}_{o}(a^{\prime})\geq \hat{\pi}_{o}(a)\}\}|\]
we can then estimate a predictive action set as follows:
\[\mathcal{C}^{*}(\mathbf{o}):=\left\{\mathbf{a}:\rho_{\mathbf{o}}(\mathbf{a}) \!+\!\hat{\pi}_{\mathbf{o}}(\mathbf{a})\!\cdot\!u\!+\!\lambda\!\cdot\!(z_{ \mathbf{o}}(\mathbf{a})\!-\!k_{reg})^{+}\leq\tau\right\}\]
where \((x)^{+}\) denotes the positive portion of \(x\), and \(\lambda,k_{reg}\geq 0\) are regularization hyperparameters to incentivize small set sizes. Here, \(u\sim uniform\) [0, 1] (used to allow for randomized procedures) and the tuning parameter \(\tau\) (the cumulative sum of the classifier scores after sorting and penalization) to control the size of the sets are identical to as used in RAPS for supervised tasks [4].
To summarize, in cammarl, \(\mathcal{N}_{self}\) gets to use estimates of \(\mathcal{N}_{other}\)'s actions at each time step to make informed decisions in the environment. Instead of modeling exact actions with no uncertainty estimation, we prefer to produce an action set carrying desirable guarantees of containing \(N_{other}\)'s true action with high probability, integrate it into an agent's downstream task, and enable improved decision-making and collaboration with \(\mathcal{N}_{other}\).
## 4 Environments
In this section, we discuss the cooperative tasks with two agents used in this study (Figure 3). We note here that though we work in fully cooperative settings in this article, cammarl as an idea can be generalized to competitive or mixed settings too.
Cooperative Navigation (CN) [24, 29].In this task, agents are expected to learn to navigate and cover all the landmarks cooperatively and without colliding. Each agent can perceive the other agents and landmarks within its reference frame (in the form of relative positions and velocities) and can take discrete actions to move around (left, right, up, down, stay) in the environment. The agents receive a team reward (so \(r_{self}\) and \(r_{other}\) are the same in this case) which is calculated as the minimum of the distance of the agents' and landmarks' (\(x_{i}\), \(y_{i}\)) positions in the grid world. This reward, based on their proximity (or distance) from each of the landmarks, forces the need for cooperation in order to succeed in the task. Furthermore, agents are penalized upon collisions. Formally, the reward function in this environment can be defined as
\[r=\left[-1*\sum_{l=1}^{L}min_{\{n\in\mathcal{N}\}}(distance(n,l))\right]-c\]
where \(|\mathcal{N}|\) is the number of agents and L are the landmarks in the environment. Here, c is the number of collisions in an episode and the agents are penalized with -1 for each time two agents collide. Figure 2(a) shows an illustration of this environment.
Level-based foraging (LBF) [1].In this environment, \(\mathcal{N}_{self}\) and \(\mathcal{N}_{other}\) are part of a 12\(\times\)12 grid world which contains four randomly scattered food locations, each assigned a level. The agents also have a level of their own. They attempt to collect food which is successful only if the sum of the levels of the agents involved in loading is greater than or equal to the level of the food. This is a challenging environment, requiring agents to learn to trade
Figure 3: Multi-agent cooperative environments used in this study: (a) OpenAI MPE’s **cooperative navigation** with 2 agents (N=2) and 2 landmarks (L=2) and (b) a 12 \(\times\) 12 **level-based foraging** grid-world with 2 cooperative players and 4 food locations.
off between collecting food for their own and cooperating with the other agent to acquire higher team rewards. Moreover, this environment has sparse rewards making it difficult to learn and operate independently in the environment. In particular, each agent is rewarded equal to the level of the food it managed to collect, divided by its level (its contribution). Figure 2(b) shows an illustration of this environment.
## 5 Experiments and Results
To show the benefits of conformal action set prediction, we compare cammarl with the performances of agents in different settings with varying pieces of information made available to \(\mathcal{N}_{self}\) during training.
No-Other-Agent-Modeling (NOAM).At first, we train \(\mathcal{N}_{self}\) without allowing it to model \(\mathcal{N}_{other}\). This baseline, as expected, underperforms when compared to any other settings (where any kind of agent modeling is allowed). It is indicative of a lower bound to the learning performance of our model where no kind of benefit from agent modeling is utilized by \(\mathcal{N}_{self}\). Results are shown in Figure 4. We call this baseline -- _No-Other-Agent-Modeling_ or _NOAM_.
True-Action-Agent-Modeling (TAAM).Advancing from the inputs available in NOAM, we implement TAAM by allowing \(\mathcal{N}_{self}\) to additionally utilize \(\mathcal{N}_{other}\)'s true actions to train. This baseline helps us evaluate cammarl against works that estimate other agents' actions in the environment and use those predictions to enhance the decision-making of their controlled autonomous agents. By giving the true actions as inputs, this baseline can act as an upper bound to such works (for example, [1, 12, 26, 32, 48, 16]). Figure 4 shows TAAM's performance curve. Using additional information, it does reasonably better than NOAM in both tasks.
True-Observation-Agent-Modeling (TOAM).As discussed in Section 2, learning world models often involves reconstructing observation as an additional task while learning task-related policies [14, 15, 18]. Inspired by this research, we implement the TOAM baseline where we allow access to \(\mathcal{N}_{other}\)'s true observations to \(\mathcal{N}_{self}\) during training and execution. In other words, we augment \(\mathcal{N}_{self}\)'s partial observations with the other agent's local observations too. This baseline can act as an upper bound to the performances of research works that learn to reconstruct states for agents [14, 15, 36, 44, 46]. As expected, \(\mathcal{N}_{self}\) performs considerably better than NOAM in both environments (Figure 4). Here, the difference in returns in TOAM and TAAM can be attributed to the fact that the local observations of other agents include more information that can be useful to infer their behavior. For instance, in CN, knowing the relative positions of other agents with respect to the landmarks can be more useful to infer which landmark that agent might be approaching when compared to knowing its current (or history) actions.
Global-Information-Agent-Modeling (GIAM).On the other extreme, we also implement _GIAM_, where \(\mathcal{N}_{self}\) trains with complete access to both (1) \(\mathcal{N}_{other}\)'s true action trajectories (\(a_{other}\)), and (2) \(\mathcal{N}_{other}\)'s true observations (\(o_{other}\)) as additional information. Figure 4 shows that GIAM achieves higher returns compared to all other settings in both environments. This is intuitive because it benefits from more information. GIAM is conditioned on \(\mathcal{N}_{other}\)'s true experiences and consequently demands access to them even during execution. This can be infeasible in real-world scenarios, however, theoretically represents an upper bound on the performance of agents in cammarl and other settings.
Exact-Action-Prediction (EAP).Building over the inputs of TOAM, we construct another but stronger baseline, EAP, in which \(\mathcal{N}_{self}\) uses an additional neural network classifier to model a probability distribution over \(\mathcal{N}_{other}\)'s actions. In other words, instead of predicting conformal sets of actions (like in cammarl), in this baseline, \(\mathcal{N}_{self}\) tries to model \(\mathcal{N}_{other}\)'s actions from the latter's observations without accounting for any uncertainty quantification. This baseline is inspired by works that explicitly model the other agent's actions in the environments and utilize them to inform their controlled agent's decision-making (for instance, [12, 16, 48]). Hence, here, a cross-entropy loss is used to train the added sub-module that predicts the \(\mathcal{N}_{other}\)'s actions along with a PPO loss to train \(\mathcal{N}_{self}\)'s policy network. Figure 4 shows that cammarl agents are able to distinctly perform better than EAP in LBF, however, interestingly, the performance curve for this baseline nearly overlaps with cammarl in CN. Also, in LBF, the curves for TOAM and EAP
seem to significantly overlap. We speculate that in a complicated task like LBF, estimating the exact action of \(\mathcal{N}_{other}\) can be difficult, and with unaccounted uncertainty in the predictions, \(\mathcal{N}_{self}\) suffers from a lower return. In CN, which is comparatively simpler, the closeness of returns in EAP and cammarl seem reasonable as even the conformal model predictions eventually start predicting the most probable action with higher probabilities and hence a set of size one (more on this in Section 6).
cammarl.Now, we implement cammarl, where the conformal action prediction model periodically trains on collected observations of \(\mathcal{N}_{other}\) and predicts a corresponding conformal set of actions. \(\mathcal{N}_{self}\) uses these estimates of \(\mathcal{N}_{other}\)'s actions along with its own observations and then decides upon its actions in the environment. Figure 4 shows that cammarl agents obtain returns that are much closer to the upper bound, _GIAM_, than the lower bound, _NOAM_. Furthermore, cammarl's better performance compared to TOAM in both environments can be attributed to the fact that it can be difficult to predict the \(\mathcal{N}_{other}\)'s intentions by only using \(o_{other}\) without any information pertaining to its actions in those situations. And, in TAAM, \(\mathcal{N}_{self}\) is expected to implicitly encode information regarding \(\mathcal{N}_{other}\)'s observations from its own local observations or in the latent space and map it to \(\mathcal{N}_{other}\)'s true actions. We speculate that this could be a strong assumption and consequently very difficult, hence, cammarl agents outperform TAAM too. Note here that the sets output by the conformal action prediction model are of varying sizes in each iteration. Now, to be able to use these dynamically changing inputs for \(\mathcal{N}_{self}\) in cammarl, we convert the output sets to a corresponding binary encoding (by firing up the bits in a zero vector at indices corresponding to the actions predicted by the model). We discuss some more ways to be able to use conformal prediction sets with dynamic sizes and compare cammarl's performances in all variations later in Appendix A.
In summary, through experiments in two complex cooperative tasks, we show that (1) cammarl indeed works, (2) it outperforms common settings like NOAM, TOAM, and TAAM which assume the availability of other agents' true trajectories during training and execution (generally infeasible in real-world scenarios), (3) Its performance is closest to our upper bound of performance (GIAM), (4) cammarl agents learn their policies faster than the other baselines, and (5) cammarl can be preferred over strong benchmarks such as EAP owing to its higher interpretability due to the theoretical guarantees of conformal predictions in terms of coverage [4] (discussed more in Section 6).
Figure 4: Comparison of agent performances (in terms of reward accumulation) in CN and LBF in different settings with varying pieces of information available to \(\mathcal{N}_{self}\) during training. cammarl’s performance is very close to the upper bound, GIAM, and is considerably better than the other extreme, NOAM. It also outperforms the other defined benchmarks (TAAM, TOAM, & EAP) in both tasks, along with the benefit of uncertainty quantification of its estimates. Interestingly, in CN, cammarl can be seen to learn arguably faster, but all methods converge to similar results, whereas in LBF, it actually seems to converge to a better policy. The curves are averaged over five independent trials and smoothed using a moving window average (100 points) for readability.
## 6 Discussion
In this section, we dig deeper and try to analyze the inner components of cammarl. In particular, we plot some observable trends during the training of cammarl's agents in both the tasks (Figure 5) and discuss each of them here.
Set Sizes.We collected the set sizes produced in cammarl throughout the training and report them in Figure 4(a) and 4(e). Smaller sets are preferred, as they carry specific information which can be more useful practically. The curves show a decreasing trend in the set sizes in cammarl in both CN and LBF respectively when tracked over the number of updates of the conformal prediction model during training. This is a good sign for cammarl, as it shows that the conformal predictions are becoming more precise with continued training over time.
Coverage.As also discussed earlier, it is desirable for the predicted sets to provide 1 - \(\alpha\) coverage for a pre-defined user-specified \(\alpha\) such as 10%. Formally, to map a feature vector, \(o_{other}\in\mathcal{O}_{other}\), to a subset of discrete responses, \(a^{\prime}_{other}\in\mathcal{A}_{other}\), it is useful to define an uncertainty set function, \(\mathcal{C}(o_{other})\), such that \(P(a^{\prime}_{other}\in C(o_{other}))\geq 1-\alpha\). Figure 4(b) and 4(f) shows the increasing trend of confidence coverage in cammarl.
Model accuracy and loss.In Figure 4(c) and Figure 4(d) we show the conformal model's accuracy and loss respectively for CN and LBF in Figure 4(g) and Figure 4(h). The model accuracy, as expected, increases with more data coming in to train over time and the loss correspondingly decreases.
## 7 Conclusion
In this article, we propose a novel MARL algorithm, cammarl, which calls for confident reasoning about other artificial agents in the environment and benefiting from inferences about their behavior. Through experiments in two cooperative multi-agent tasks, CN and LBF, we showed that guiding an agent's decision-making by inferring other agents' actions in the form of conformal sets, indeed helps in achieving better performances of the learning agents. By using conformal prediction, we were also able to ensure the estimation of predictive sets that covered the real predictions of the intentions of other agents with a very high pre-specified probability of 95%. There could be several future directions from here. First, cammarl is certainly generalizable to bigger networks or more simple classifiers, and also analyzing its changing performance on varying buffer sizes can help in better comprehending its efficiency. Second, it would be interesting to investigate the cammarl's scalability to a system of many agents (say 100 or 1000) or on more complicated multi-agent environments such as tasks requiring a higher need for coordination. Finally, in this work, we restricted the agents to infer the behavior of other agents only via conformal sets; it would be interesting to study the cases where more ways of sharing information or modeling agents' behavior are additionally allowed.
Acknowledgements.The authors would like to thank the Digital Research Alliance of Canada for compute resources and CIFAR for research funding. The authors are also grateful for the detailed comments from Somjit Nath and all reviewers.
Figure 5: Analysing conformal prediction in cammarl over time during the training by looking at trends in conformal sets sizes, coverage of highly probable predictions, model loss and accuracy during training of cammarl agents. |
2304.07133 | LoRe: A Programming Model for Verifiably Safe Local-First Software | Local-first software manages and processes private data locally while still
enabling collaboration between multiple parties connected via partially
unreliable networks. Such software typically involves interactions with users
and the execution environment (the outside world). The unpredictability of such
interactions paired with their decentralized nature make reasoning about the
correctness of local-first software a challenging endeavor. Yet, existing
solutions to develop local-first software do not provide support for automated
safety guarantees and instead expect developers to reason about concurrent
interactions in an environment with unreliable network conditions.
We propose LoRe, a programming model and compiler that automatically verifies
developer-supplied safety properties for local-first applications. LoRe
combines the declarative data flow of reactive programming with static analysis
and verification techniques to precisely determine concurrent interactions that
violate safety invariants and to selectively employ strong consistency through
coordination where required. We propose a formalized proof principle and
demonstrate how to automate the process in a prototype implementation that
outputs verified executable code. Our evaluation shows that LoRe simplifies the
development of safe local-first software when compared to state-of-the-art
approaches and that verification times are acceptable. | Julian Haas, Ragnar Mogk, Elena Yanakieva, Annette Bieniusa, Mira Mezini | 2023-04-14T13:52:02Z | http://arxiv.org/abs/2304.07133v2 | # LoRe: A Programming Model for Verifiably Safe Local-First Software
###### Abstract
Local-first software manages and processes private data locally while still enabling collaboration between multiple parties connected via partially unreliable networks. Such software typically involves interactions with users and the execution environment (the outside world). The unpredictability of such interactions paired with their decentralized nature make reasoning about the correctness of local-first software a challenging endeavor. Yet, existing solutions to develop local-first software do not provide support for automated safety guarantees and instead expect developers to reason about concurrent interactions in an environment with unreliable network conditions.
We propose _LoRe_, a programming model and compiler that automatically verifies developer-supplied safety properties for local-first applications. _LoRe_ combines the declarative data flow of reactive programming with static analysis and verification techniques to precisely determine concurrent interactions that violate safety invariants and to selectively employ strong consistency through coordination where required. We propose a formalized proof principle and demonstrate how to automate the process in a prototype implementation that outputs verified executable code. Our evaluation shows that _LoRe_ simplifies the development of safe local-first software when compared to state-of-the-art approaches and that verification times are acceptable.
Local-First Software, Reactive Programming, Invariants, Consistency, Automated Verification This work was funded by the German Federal Ministry of Education and Research together with the Hessen State Ministry for Higher Education (ATHENE), the German Research Foundation (SFB 1053), and the German Federal Ministry for Economic Affairs and Climate Action project SafeFBDC (01MK21002K).
## 1 Introduction
Applications that enable multiple parties connected via partially unreliable networks to collaboratively process data prevail today. An illustrative example is a distributed calendar application with services to add or modify appointments, where a user may maintain multiple calendars on different devices, may share calendars with other users, back them up in a cloud;
calendars must be accessible to users in a variety of scenarios, including offline periods, e.g., while traveling - yet, planning appointments may require coordination between multiple parties. The calendar application is representative for other collaborative data-driven software such as group collaboration tools, digital (cross-organizational) supply chains, multiplayer online gaming, and more.
The dominating software architecture for such applications is centralized: data is collected, managed, and processed centrally in data centers, while devices on the edge of the communication infrastructure serve primarily as interfaces to users and the outside world. This architecture simplifies the software running on edge devices since concerns like consistent data changes to ensure safety properties are managed centrally. However, this comes with issues including loss of control over data ownership and privacy, insufficient offline availability, poor latency, inefficient use of communication infrastructure, and waste of (powerful) computing resources on the edge.
To address these issues, local-first principles for software development have been formulated [21], calling for moving data management and processing to edge devices instead of confining the data to clouds. But for programming approaches that implement these principles to be viable alternatives to the centralized approach, they must support automatically verifiable safety guarantees to counter for the simplifying assumptions afforded by a centralized approach. Unfortunately, existing approaches to programming local-first applications such as _Yjs_[36] or _Automerge1_ do not provide such guarantees. They use _conflict-free replicated data types (CRDTs)_[40] to store the parts of their state that is shared across devices and rely on callbacks for modeling and managing state that changes in both time and space. CRDTs have been invented in the context of geo-replicated databases and are available as off-the-shelf databases [39] or libraries [20, 14]. But the strongest consistency level ensured by CRDTs, causal consistency [28], is not enough to maintain invariants that require coordination among the participants, e.g., the invariant that employees should not enter more than the available vacation days in a use of the calendar app in a business setting. At the same time, we need to delimit the scope of coordination so as to "maximize combinations of availability and consistency that make sense for a particular application" [9]. To find the set of interactions that actually need coordination, in a local-first setting, one must reason about possible interleavings of their data flows "end-to-end", i.e., from the interface to the outside world, through device-local data-dependency chains, to remote devices and back. The unpredictability of the interactions triggered by the outside world, concurrently at different devices, paired with the absence of a central authority and the prevailing implicit dependencies in current callback-centred programming models, makes such reasoning without automated support a challenging, error-prone endeavour.
Footnote 1: [https://automerge.org/](https://automerge.org/)
To close this gap, we propose a programming model for local-first applications that features explicit safety properties and automatically enforces them. The model has three core building blocks: _reactives_, _invariants_, and _interactions_. _Reactives_ express values that change in time, but also in space by being replicated over multiple devices. They enable systematic treatment of complex state, dependencies, and concurrent changes, enabling developers to reason in terms of functional composition. _Invariants_ are formula in first-order logic specifying safety properties that must hold at all times when the application interacts with the outside world, or values of reactives are observable. _Interactions_ interface to the outside world and encapsulate changes to all reactives affected by interactions with it (state directly changed by the interactions, device-local values derived from the changed state, and
shared state at remote devices). They serve as language-managed cross-device data flows that automatically use best-in-class consistency. On one device, interactions are logically processed instantaneously, i.e., no intermediate states are observable, similar to atomicity in databases. We use automatic verification with invariants as verification obligations to identify interactions that need coordination across devices, for which the compiler generates the coordination protocol; all other interactions become visible in causal order. This way, the compiler makes an application-specific availability-safety trade-off.
The availability-safety trade-off has been explored in the systems and database community under the term "coordination avoidance" and there exist approaches that leverage user-specified safety invariants to synthesize distributed objects or to correctly configure the consistency level for each database operation [7, 44, 10]. Our work draws inspiration from these approaches, but differs in two key aspects: (1) Instead of geo-replicated databases, we target a peer-to-peer local-first setting with unique challenges; specifically, we do not assume any centralized authority and feature offline availability and interactions with the outside world. (2) Instead of programming applications against the interface to a distributed datastore/object, which fosters designs split into two separate tiers - an automatically managed monolithic data store and the application tier that uses the store's API - we provide a language-integrated mixed consistency with whole program guarantees, i.e., we verify the safety of derived data "all the way down". This is necessary in a local-first setting, where most of the complexity arises from the interactions with the external world and are handled as part of the application logic, not as part of the data store.
In summary, we make the following contributions:
1. A programming model for local-first applications with verified safety properties (Section 2), called LoRe. While individual elements of the model, e.g., CRDTs or reactives, are not novel, they are repurposed, combined, and extended in a unique way to systematically address specific needs of local-first applications with regard to ensuring safety properties.
2. A formal definition of the model including a formal notion of invariant preservation and confluence for interactions, and a modular verification that invariants are never violated. In particular, our model enables invariants that reason about the sequential behaviour of the program. In case of potential invariant violation due to concurrent execution, LoRe automatically adds the necessary coordination logic (Section 3).
3. A verifying compiler2 that translates LoRe programs to Viper [33] for automated verification and to Scala for the application logic including synthesized synchronization to guarantee the specified safety invariants (Section 4). Footnote 2: The source code of our prototype implementation is available at [https://github.com/atg-tud/LoRe](https://github.com/atg-tud/LoRe).
4. An evaluation of LoRe in two case studies (Section 5). Our evaluation validates two claims we make about the programming model proposed, (a) It facilitates the development of safe local-first software, and (b) it enables an efficient and modular verification of safety properties. It further shows that the additional safety properties offered by our model do not come with prohibitive costs in terms of verification effort and time.
## 2 LoRe in a Nutshell
We introduce the concepts of LoRe along the example of a distributed calendar for tracking work meetings and vacation days. LoRe is an external DSL that compiles to Scala (for execution) and Viper IR [33] (for verification); its syntax is inspired by both. A LoRe
program defines a distributed application that runs on multiple physical or virtual devices.3 Listing 1 shows a simplified implementation of the calendar example application in LoRe. As any LoRe program, it consists of replicated state (Source reactives in Lines 2-3), local values derived from them (Derived reactives in Lines 5-6), interactions (Lines 8-15), and invariants (Lines 20-23).
Footnote 3: We assume that every device is running the same application code (i.e., the same binary), and different types of devices (such as client and server) are modeled by limiting them to execute a subset of the defined interactions.
### Reactives
_Reactives_ are the composition units in a LoRe program. We distinguish two types of them: _source_ and _derived_ reactives, declared by the keywords Source and Derived, respectively. Source reactives are values that are directly changed through interactions. Their state is modeled as _conflict-free replicated data types_ (CRDTs) [40, 37] and is replicated between the different devices collaborating on the application. Derived reactives represent local values that are automatically computed by the system from the values of other reactives (source or derived). Changes to source reactives automatically (a) trigger updates of derived reactives
Figure 1: The data-flow graph of the calendar application.
and (b) cause devices to asynchronously send update messages to the other devices, which then merge the changes into their local state. Together, local propagations and asynchronous cross-device update messages ensure that users always have a consistent view of the overall application state. All reactives are statically declared in the program source code. LoRe then statically extracts knowledge about the data flow for modular verification and to minimize the proof goals (cf. Sec 4.1). We discuss the technical implications of static reactives in Section 7.
Listing 1 shows two source reactives, work and vacation (Line 2 and 3), each modeling a calendar as a set of appointments. The work calendar tracks work meetings, while the vacation calendar contains registered vacation days. When defining a source reactive, programmers have to choose a CRDT for the reactive's internal state. LoRe offers a selection of pre-defined CRDTs including various standard data types such as sets, counters, registers and lists. Further data types can be supported by providing a Viper specification for that data type. In this case, an _add-wins-set_ (a set CRDT where additions have precedence over concurrent deletions) is selected for both source reactives. Appointments from both calendars are tracked in the all_appointments derived reactive (Line 5), while the remaining_vacation reactive (line 6) tracks the number of remaining vacation days. The _data-flow graph_ of the application, where nodes are reactives and edges represent the derivation relation - in the direction of data flow - is visualized in Figure 1. This data-flow graph is created by the LoRe compiler and managed by its runtime.
### Interactions
Changes to the state of the system, e.g., adding appointments to a calendar, happen through explicit _interactions_. Each interaction has two sets of type parameters: the types of source reactives that it modifies and the types of parameters that are provided when the interaction is applied. For example, the add_appointment interaction in Line 8 modifies a reactive of type Calendar and takes a parameter of type Appointment. The semantics of an interaction I are defined in four parts: (1) requires (Line 9) defines the preconditions that must hold for I to be executed, (2) executes (Line 11) defines the changes to source reactives, (3) ensures (Line 12) defines the postconditions that must hold at the end of I's execution, (4) modifies (Line 13) defines the source reactives that I changes. The parameters of requires, executes, and ensures are functions that take the modified reactives and the interaction parameters as input (cal is of type Calendar and a is of type Appointment). The splitting of the definition of interactions in four parts allows for modularization and reuse. For instance, add_appointment is only a partial specification of an interaction, missing the modifies specification. Both add_work (Line 15) and add_vacation (Line 13) specify complete interactions by adding modifies to add_appointment; they are independent interactions that differ only in their modifies set.
Interactions encapsulate reactions to input from the outside world (e.g., the callback in Line 18 that is triggered by the UI and applies the arguments to add_vacation). Applying an interaction checks the preconditions, and - if they are fulfilled - computes and applies the changes to the source reactives, and propagates them to derived reactives - all in a "transactional" way in the sense that all changes to affected reactives become observable at-once ("atomically"). Only source reactives are replicated between devices, while derived reactives are computed by each device individually. LoRe grantees that executing interactions does not invalidate neither postconditions nor invariants.
### Invariants and Conflicts
LoRe expects the developer to use _invariants_, introduced with the keyword invariant, to specify application properties that should always hold. Invariants are first-order logic assertions given to a verifier based on the Viper verification infrastructure [33]. Invariants can help uncover programming bugs and reveal where the eventually-consistent replication based on CRDTs could lead to safety problems.
For illustration, consider the invariants for the calendar application in Lines 20 and 23. The invariant in Line 20 requires that all appointments must start before they end. Notice, how the invariant can be defined without knowing the amount of calendars and the actual structure of the data-flow graph by simply referring to the all_appointments reactive. This invariant represents a form of input validation, and is directly ensured by add_appointment interactions because the precondition on the arguments requires the added appointment to start before it ends (Line 9). In absence of this precondition, the LoRe compiler would reject the program and report a safety error due to a possible invariant violation. The invariant in Line 23 requires that employees do not take more vacation days than available to them. Again, this is locally enforced by the precondition of the add_vacation interaction, which ensures that new entries do not exceed the remaining vacation days. But there is nothing stopping two devices from concurrently adding vacation entries, which in sum violates the invariant. Figure 2 illustrates such a situation: A user plans a vacation of 20 days on the mobile phone (device \(D_{1}\)) and later schedules a 12-day vacation on a desktop (device \(D_{2}\)), at a time when \(D_{1}\) was offline. Thus, both interactions happened concurrently and after merging the states the calendar contains a total of 32 days of vacation, violating the remaining_vacation invariant.
This example illustrates a conflict between (concurrent) execution of interactions - in this case, two executions of the add_vacation Interaction must be coordinated (synchronized) in order to avoid invariant violations. The LoRe compiler reports conflicting interactions to the developer and automatically synthesizes the required coordination code for the execution of such interactions (see Section 4.3). In a local-first setting, it is of paramount importance to minimize the required coordination to allow offline availability. Reporting of conflicts due to invariants helps developers to explore different situations and make informed decisions about the safety guarantees of their program. When they find that their program requires too much synchronization, they can lower the guarantees by adapting their invariants.
## 3 Programming Model
This section formally presents the syntax and semantics of the programming model and discusses how we verify that execution of a program preserves safety guarantees specified by invariants. The definition of the program execution is split into a big-step semantics
Figure 2: Concurrent execution of interactions may cause invariant violations. In this example, device \(D_{1}\) adds a vacation of 20 days to the calendar, while \(D_{2}\) concurrently adds a vacation of 12 days. Given a total amount of 30 available vacation days, this leads to a negative amount of remaining vacation once the devices synchronize.
for handling reactive updates on a single device and a labelled transition system to model execution of the overall distributed system. LoRe guarantees that given a valid program state (that ensures the safety invariants), any step taken in the labelled transition system preserves the validity of the program. To preserve safety without sequentializing the whole distributed program execution, the formal semantics relies on an oracle that tells us when two interactions have a conflict. We then give a proof of safety preservation given our oracle. Using the insights of this proof, Section 4 shows how we use the verification infrastructure to compute this oracle.
### Syntax and Evaluation Semantics
#### Syntax
Figure 3 shows the abstract syntax for LoRe. A distributed program \(P\) is defined as a tuple, whose elements are a set of interactions \(A\), a set of derived reactives \(\delta\), a set of invariants \(I\), and a list of devices \((D_{1}\mid\cdots\mid D_{n})\), using \(|\) as a separator in reference to parallel composition. We write \(D\in P\) to state that \(D\) is one of the devices in \(P\), where a device \(D=\langle\sigma,L\rangle\) consists of an assignment of source reactives \(\sigma\) and a set of locks \(L\). For the following definitions, we use curly braces as part of the meta syntax to denote that an expression occurs zero or more times in any order. This is used for top-level definitions in LoRe, and we treat such expressions as having set semantics.
Every Interaction\(((r_{1},\ldots,r_{n}),l_{pre},l_{post},t_{exec})\) is composed of a list of affected reactives \((r_{1},\ldots,r_{n})\), pre- and postconditions \(l_{pre}\) and \(l_{post}\), and the interaction body \(t_{exec}\). We introduce an inner term language \(t\) for the bodies of reactives and interactions, which is a simple lambda-calculus extended with access to reactives. We write \(v=\sigma(r)\) to refer to the current value of a source reactive, if the expression val \(r=\text{Source}(v)\) is present in \(\sigma\)
Figure 3: Abstract syntax of LoRe programs.
and \(t=\delta(r)\) to refer to the body of a derived reactive if the expression \(\text{val}\ r=\text{Derived}(t)\) is present in \(\delta\). We use a first-order logic language \(l\), which embeds \(\lambda\)-terms, for defining invariants, pre-, and postconditions.
\(A\), \(\delta\), and \(I\) are static and only devices \(D=\langle\sigma,L\rangle\) may change during execution by updating the values of source reactives \(\sigma\) or their currently held locks \(L\). Semantically, each interaction \(a\in A\) has a one-to-one correspondence to a lock, thus we syntactically represent locks the same as interactions with \(L\subseteq A\).
#### Term evaluation
We use a big-step semantics for term evaluation. Figure 4 shows the rules for reactive evaluation - we omit rules related to standard lambda-calculus and logic evaluation. We write \(t\Downarrow_{\sigma}v\) to state that \(t\) evaluates to \(v\) given a current assignment of source reactives \(\sigma\). Note that evaluation depends on \(\delta\), but we omit writing it in the rules as it is fixed for each program and thus always clear from context. Rule ValueSource retrieves the current value of a source reactive from the store \(\sigma\). Rule ValueDerived evaluates the expression of a derived reactive, which may depend on other reactives, thus the rule potentially results in evaluating many derived and source reactives in the data-flow graph.
Figure 4: Semantics for reactive term evaluation.
Figure 5: Semantics for interactions and device communication.
#### Semantics for interactions and device communication
Figure 5 presents the semantics for interactions and device communication together with auxiliary functions. We use three auxiliary functions _update_, _merge_ and _conflicts_. Function _update_ takes a set of source reactives \(\sigma\) and returns a new set with updated values.
Function _merge_ takes two stores \(\sigma_{1}\) and \(\sigma_{2}\) and merges them by pair-wise merging the values of each source reactive. Merging of stores relies on a merge function \(\mathit{merge}_{r}\) for each source reactive \(r\), which is defined by the CRDT of this reactive. The merge function on stores is commutative, associative and idempotent, because \(\mathit{merge}_{r}\) also has these properties by the definition of CRDTs. The merge function implies a partial order on states \(\sigma_{1}\leq\sigma_{2}\) if \(\mathit{merge}(\sigma_{1},\sigma_{2})=\sigma_{2}\). For syntactic convenience, we lift the order to devices \(D_{1/2}=\langle\sigma_{1/2},L_{1/2}\rangle\), where we write \(D_{1}\leq D_{2}\) if \(\sigma_{1}\leq\sigma_{2}\).
The _conflicts_ function takes an interaction and returns the set of interactions that conflict with it. If an interaction \(a_{1}\) has no conflicts, then \(\mathit{conflicts}(a_{1})=\emptyset\), if there is an interaction \(a_{2}\) that conflicts with \(a_{1}\), then \(\{a_{1},a_{2}\}\subseteq\mathit{conflicts}(a_{1})\) and \(\{a_{1},a_{2}\}\subseteq\mathit{conflicts}(a_{2})\). This ensures that whenever a device wants to execute an interaction with conflicts, it has to hold the locks for the interaction itself and for all the conflicting interactions. The _conflicts_ function serves as an oracle to prevent concurrent execution of interactions. Semantically, it defines the synchronization requirements of the program. Specifying an empty conflict set for every interaction, induces no synchronization, thus providing only causal consistency, while specifying an interaction as conflicting with every other interaction yields sequential consistency, disallowing any concurrent executions. We show in Section 4 how to compute a conflict function that prevents only those concurrent executions that would result in invariant violations on merged state.
We use a labelled transition system to model program execution, where transitions are labelled by interactions or synchronizations, which both occur non-deterministically. The Interact rule defines the semantics for applying an interaction \(a\) with an argument \(v_{arg}\). The definition assumes that \(l_{post}\), \(l_{pre}\), and \(t_{exec}\) are functions; otherwise \(a\) is ill-defined and cannot be executed. The execution of \(a\) moves a device from state \(\langle\sigma,L\rangle\) to state \(\langle\sigma^{\prime},L\rangle\) if: i) the precondition \(l_{pre}\) applied to \(v_{arg}\) evaluates to true before the execution, ii) the device executing \(a\) holds the locks of all \(\mathit{conflicts}(a)\), iii) the postcondition \(l_{post}\) applied to \(v_{arg}\) evaluates to true after the execution, iv) evaluating \(t_{exec}\) returns a tuple4 of new values \((v_{1},\dots,v_{n})\) for each source reactive \((r_{1},\dots,r_{n})\), which are used to update the store of source reactives \(\sigma\).
Footnote 4: Using a suitable encoding of tuples in the lambda calculus.
The rule Sync models communication between devices. Synchronization happens between a sending device \(D_{s}=\langle\sigma_{s},L_{s}\rangle\) and a receiving device \(D_{r}=\langle\sigma_{r},L_{r}\rangle\), where the former transfers state and locks to the latter. A set of locks \(L\subseteq L_{s}\) is removed from \(D_{s}\) and added to the locks of \(D_{r}\). The state \(\sigma_{r}\) is merged with the sent state \(\sigma_{s}\). By combining state updates and lock exchange in a single transition, we ensure that for every interaction \(a\) that needs coordination, a device \(D_{r}\), which receives the locks for \(a\), also receives the effects of the last application of \(a\). Moreover, merging the state of all source reactives in a single step ensures causal consistency for the entire state of the application, not only for single reactives (i.e., single CRDTs).
An interaction only changes a single device, hence, we can abbreviate applications of Interact to \(D\xrightarrow[a]{\text{\scriptsize{$u_{arg}$}}}D^{\prime}\) to express that a device \(D\) executed an interaction regardless of
the state of the other devices. Similarly, we write \(D_{s}\xrightarrow[swap]{D_{r}}D_{r}^{\prime}\) to express that \(D_{s}\) was synchronized into \(D_{r}\) producing \(D_{r}^{\prime}\) (and \(D_{s}^{\prime}\), which we do not need to state explicitly, as it is fully defined by the other 3 devices).
### Verifying Program Safety
LoRe guarantees that program execution is safe: A program in a state where all safety invariants hold, transitions only into states where the invariants still hold. A given program state that satisfies all safety invariants is called valid. Formally:
[Validity] Given \(P\) with invariants \(I\) and devices \(D_{1},\ldots,D_{n}\), we say that \(P\) is valid, written \(\mathit{valid}(P)\), if \(\mathit{valid}(D_{i}),i\in\{1,...n\}\). A device \(D=\langle\sigma,L\rangle\) is _valid - _written valid\((D)\) - if for any \(\mathit{Invariant}(l)\in I\), \(l\Downarrow_{\sigma}\) true._
[Safety] A program \(P\) is _safe_ if for any possible transition \(P\Rightarrow P^{\prime}\), \(\mathit{valid}(P)\Rightarrow\mathit{valid}(P^{\prime})\).
To enable automatic and modular verification, we break safety checking into two simpler properties on invariants that can be automatically checked by Viper: _invariant preservation_ and _confluence_. Invariant preservation ensures the safety of individual interactions, whereas confluence (adapted from Bailis et al. [5] and other works that build on the CALM theorem [2, 16]) relates two interaction executions (of the same or different interactions). We show how to mechanically prove these properties using Viper in Section 4. The rest of this section formally introduces all mentioned properties and proves soundness of our approach by showing that safety preservation follows from invariant preservation of all interactions, and confluence of all interactions that are not marked as conflicting.
[Invariant Preservation] An interaction \(a\) is _invariant preserving_, written preserving\((a)\), if \((\mathit{valid}(D)\wedge D\xrightarrow[swap]{\xrightarrow[swap]{v_{\mathit{args}}}} \mathit{D}^{\prime})\Rightarrow\mathit{valid}(D^{\prime})\), i.e., given a valid device \(D\) and some argument \(v_{\mathit{args}}\), the execution of \(a\) in \(D\) produces a valid device \(D^{\prime}\).
[Confluence] Interactions \(a_{1}\) and \(a_{2}\) are _confluent_, written \(\mathit{confluent}(a_{1},a_{2})\), if for any valid devices \(D_{i}=\langle\sigma_{i},L_{i}\rangle\) and \(D_{j}=\langle\sigma_{j},L_{j}\rangle\), and any argument values \(v_{1}\) and \(v_{2}\)
\[D_{i}\xrightarrow[swap]{v_{1}}\langle\sigma_{i}^{\prime},L_{i} \rangle\wedge D_{j}\xrightarrow[swap]{v_{2}}\langle\sigma_{j}^{\prime},L_{j} \rangle\ \wedge\] \[D_{m_{1}}=\langle\mathit{merge}(\sigma_{i},\sigma_{j}^{\prime}), L_{i}\cup L_{j}\rangle\ \wedge\] \[D_{m_{2}}=\langle\mathit{merge}(\sigma_{i}^{\prime},\sigma_{j}), L_{i}\cup L_{j}\rangle\ \wedge\] \[\mathit{merge}(\sigma_{i}^{\prime},\sigma_{j}^{\prime})=\sigma^{ \prime}\ \wedge\ L_{i}\cap L_{j}=\emptyset\] \[\implies D_{m_{1}}\xrightarrow[swap]{v_{1}}\langle\sigma^{\prime},L_{i} \cup L_{j}\rangle\wedge\] \[D_{m_{2}}\xrightarrow[swap]{v_{2}}\langle\sigma^{\prime},L_{i} \cup L_{j}\rangle\]
Confluence states that applying interactions \(a_{1},a_{2}\) on two devices leads to the same results, no matter if a synchronization happens in-between or after the interactions. Thus, concurrent execution of the two interactions leads to the same result as sequential execution. Note that \(a_{1}\) and \(a_{2}\) are always trivially confluent if they affect disjoint subsets of source reactives and \(a_{1}\) can never invalidate \(a_{2}\)'s precondition and vice versa. In this case, sequential execution of the two interactions (in any order) always has the same result as concurrent execution. We leverage this insight to reduce the amount of confluence proofs generated by our implementation in Section 4.1.
**Definition 5** (Initial Program).: _The initial program \(P^{0}\) has devices \((D_{1}^{0}\mid\cdots\mid D_{n}^{0})\) that all have the same initial state of source reatives and all locks are with device \(D_{1}^{0}\), i.e., \(D_{1}^{0}=\langle\sigma,A\rangle\wedge D_{i}^{0}=\langle\sigma,\emptyset\rangle, \forall i\in\{2,\ldots,n\}\)._
**Lemma 6** (Correct locking).: _The locking mechanism ensures for any program execution \(P^{0}\Longrightarrow\cdots\Longrightarrow P^{m}\) that conflicting interactions are sequentially ordered. Specifically, for any two conflicting transitions \(\langle\sigma_{1},L_{1}\rangle\xrightarrow[a_{1}]{v_{1}}\langle\sigma_{1}^{ \prime},L_{1}\rangle\) and \(\langle\sigma_{2},L_{2}\rangle\xrightarrow[a_{2}]{v_{2}}\langle\sigma_{2}^{ \prime},L_{2}\rangle\), with conflicts\((a_{1})\cap\textit{conflicts}(a_{2})\neq\emptyset\), the starting state of either \(a_{1}\) or \(a_{2}\) must include all changes produced by the other: \(\sigma_{2}^{\prime}\leq\sigma_{1}\) or \(\sigma_{1}^{\prime}\leq\sigma_{2}\)._
Proof.: We first show by induction, that any program state that is reachable from the initial program \(P_{0}\) assigns each lock to exactly one device. This is true for the initial program, interact transitions do not modify the lock assignment, and the sync rule removes the same set of locks from the sending devices that are added to the receiving device.
Then, we prove by contradiction that non-overlapping locks ensure Lemma 6. Assume two conflicting interactions as above, but neither \(\sigma_{2}^{\prime}\leq\sigma_{1}\) nor \(\sigma_{1}^{\prime}\leq\sigma_{2}\). Conflicting interactions require the same lock \(l\in\textit{conflicts}(a_{1})\cap\textit{conflicts}(a_{2})\). This lock must be present on both devices executing the conflicting interactions, i.e., \(l\in L_{1}\) and \(l\in L_{2}\). Thus, the lock must be transferred from \(D_{1}^{\prime}=\langle\sigma_{1}^{\prime},L_{1}\rangle\) to \(D_{2}=\langle\sigma_{2},L_{2}\rangle\) (or symmetrically from \(D_{2}^{\prime}\) to \(D_{1}\)), using any number of transitions. By commutativity, associativity and idempotence of the _merge_ function, each sync transition ensures that the receiving device \(D_{r}\) has a state \(\sigma_{r}^{\prime}\) with \(\sigma_{s}\leq\sigma_{r}^{\prime}\) where \(\sigma_{s}\) denotes the state of the sending device. This implies that either \(\sigma_{2}^{\prime}\leq\sigma_{1}\) or \(\sigma_{1}^{\prime}\leq\sigma_{2}\), which contradicts the assumption.
**Theorem 7** (Soundness).: _Given a program \(P\) with interactions \(A\) and invariants \(I\), if the verifier has shown that (i) \(\forall a\in A\), preserving\((a)\), and (ii) \(\forall a_{1},a_{2}\in A\), confluent\((a_{1},a_{2})\)\(\vee\)\(a_{2}\in\textit{conflicts}(a_{1})\), then \(P\) is safe._
Proof.: To prove soundness, we show that for any sequence \(\mathcal{C}\) of transitions \(P^{0}\Longrightarrow\cdots\Longrightarrow P^{m}\), \(\textit{valid}(P^{0})\) implies \(\textit{valid}(P^{m})\). To show \(\textit{valid}(P^{m})\), we show that \(\forall D_{i}^{m}\in P.\)\(\textit{valid}(D_{i}^{m})\). We show that an arbitrary \(D^{m}\in P^{m}\) is valid, by showing that we can construct a serialization order \(\mathcal{S}_{i}\) of interaction transitions \(D_{1}^{0}\xrightarrow[a_{1}]{v_{1}}\ldots\xrightarrow[a_{l}]{v_{l}}D^{\prime}\) that starts from the initial device state with all locks, consists exclusively of applications of the Interact rule, and yields a device \(D^{\prime}=\langle\sigma_{i}^{m},L\rangle\) with the same state of source reatives \(\sigma_{i}^{m}\) as \(D_{i}^{m}\).5 Given such \(\mathcal{S}_{i}\) and \(\forall a\in\mathcal{S}_{i}.\)\(preserving(a)\), we can conclude that \(\textit{valid}(D^{\prime})\). In turn, \(\textit{valid}(D^{\prime})\) implies \(\textit{valid}(D_{i}^{m})\) because validity only depends on \(\sigma_{i}^{m}\). In other words, we show that every possible device state that is a result of interactions and synchronizations, could also be constructed through a sequence of only local interactions.
Footnote 5: Note that the length of \(\mathcal{C}\) and \(\mathcal{S}_{i}\) may differ, and \(\mathcal{S}_{i}\) does not necessarily include all interactions of \(\mathcal{C}\).
_Serial order construction._ Let \(D^{m}\in P^{m}\) be an arbitrary device that we choose to serialize. As a convention, we use \(D\) (without the superscript) to refer to that device, in any program state belonging to \(\mathcal{C}\) or \(\mathcal{S}\). e.g., an interaction on \(D\) would be a step of the form \(D^{k}\xrightarrow[a_{1}]{v_{1}}D^{k+1}\). We construct the serial order \(\mathcal{S}\) for \(D\) stepwise from the concurrent order \(\mathcal{C}\), by picking transitions at the end of \(\mathcal{C}\) and prepend them to \(\mathcal{S}\). We say that the last (rightmost) transition in \(\mathcal{C}\) is the focused transition \(T\). Below we consider all possible rule applications that trigger \(T\) in a case-by-case way.
**Case 1:**\(T\) is an interaction \(D\xrightarrow[a_{2}]{v_{1}}D^{\prime}\). We prepend \(T\) to \(\mathcal{S}\) and discard it from \(\mathcal{C}\).
**Case 2:**\(T\) is an interaction \(D_{i}\mathrel{\mathop{\Rightarrow}\limits^{v}_{a}}D^{\prime}_{i}\), \(D_{i}\neq D\). We discard \(T\) from \(\mathcal{C}\), because by being the last transition in \(\mathcal{C}\), it does not affect \(D\).
**Case 3:**\(T\) is a synchronization \(D_{s}\mathrel{\mathop{\Rightarrow}\limits^{D_{r}}_{sync}}D^{\prime}_{r}\) from any \(D_{s}\) (possibly \(D\)) to \(D_{r}\neq D\). We discard \(T\) from \(\mathcal{C}\) because we know that \(D^{\prime}_{r}\) will not synchronize with \(D\) after \(T\).
**Case 4:**\(T\) is a synchronization \(D_{i}\mathrel{\mathop{\Rightarrow}\limits^{D}_{sync}}D^{\prime}\) from any \(D_{i}\neq D\) to \(D\). To handle this case, we consider possible states of \(D_{i}\) and \(D\) before \(T\) occurred, especially whether devices had concurrent changes since their last synchronization:
**Case 4.1:**\(D_{i}\leq D\), i.e., there are no changes to \(D_{i}\) compared to \(D\). We discard \(T\), because it only transfers some lock, and locks are irrelevant for the final serialization order.
**Case 4.2:**\(D\leq D_{i}\), i.e., there are no changes to \(D\) compared to \(D_{i}\). Similar to case 4.1, but now all relevant changes are on \(D_{i}\). Because \(T\) is a sync, we know that the state of \(D\) after \(T\) is equal to \(D_{i}\) (except locks). We discard \(T\) and continue the construction with \(D_{i}\) as the chosen device.
**Case 4.3:** Both \(D\) and \(D_{i}\) have concurrent changes and the transition that produced the state of \(D\) preceding \(T\) is the application of an interaction \(a\). Due to Lemma 6 all interactions on \(D_{i}\) that are concurrent to \(a\) are confluent with \(a\). Thus, we can use the confluence definition to reorder \(\mathcal{C}\) to put the application of \(a\) at the rightmost end of \(\mathcal{C}\), directly after the synchronization \(T\). After the reordering, we handle \(a\) according to case 1.
**Case 4.4:** Both \(D\) and \(D_{i}\) have concurrent changes and the state of \(D\) was produced by a synchronization \(T_{k}\), \(D_{k}\mathrel{\mathop{\Rightarrow}\limits^{D^{\prime}}_{sync}}D\), that synchronizes a third device \(D_{k}\neq D_{i}\) into \(D\). In other words, the effect of the last two transitions is a merge between \(D_{k}\), \(D_{i}\), and \(D\) and we (potentially) have concurrent changes from all three for which we must find a single serialization order. On a high-level, we (arbitrarily) choose to order concurrent interactions on \(D\) first, \(D_{i}\) second, and \(D_{k}\) third by changing the transitions to first synchronize \(D_{k}\) into \(D_{i}\) and then \(D_{i}\) into \(D\). In other words, we change \(T_{k}\) (the sync from \(D_{k}\) to \(D\)) in \(\mathcal{C}\) to \(T^{\prime}_{k}=D_{k}\mathrel{\mathop{\Rightarrow}\limits^{D^{\prime}}_{sync}}D_ {i}\).6 This does not change the result state of \(D\) - it is still a result of merging the states of \(D\), \(D_{i}\), and \(D_{k}\), where the change in merge order does not matter because merging is associative and commutative.
Footnote 6: This disregards changes in transferred locks, but at this point those are irrelevant for the serialization order.
_Closing remarks._ Our construction terminates, because (a) there is a finite amount of transitions in \(C\) and (b) each case above, except 4.4, reduces the size of \(C\). Case 4.4 does not reduce the size of \(\mathcal{C}\), but it is impossible to indefinitely repeat it, as this would entail that there are indefinitely many synchronizations from \(D_{k}\) into \(D\), which is impossible as \(\mathcal{C}\) is assumed to be finite.
Also, the construction handles all situations, because cases 1 and 2 cover all possible applications of interactions, while cases 3 and 4 cover all applications of synchronizations - these are the only two rules that produce transitions. We know that the distinction in case 4 is complete, because cases 4.1 and 4.2 handle all situations where there are no concurrent changes7 on one of the devices. If there are concurrent changes, then cases 4.3 and 4.4 again exhaustively cover that situation. In particular, the one exclusion from case 4.4, specifically the situation that \(D_{k}=D_{i}\) is covered by case 4.2, because then \(D_{i}\) is synchronized into \(D\) twice, without any changes to \(D\) in between.
Footnote 7: Technically, both devices could apply confluent interactions that lead to the same result state – these are irrelevant due to idempotency of merges and thus can be discarded from the serialization.
In summary, by successively replacing and discarding transitions according to the cases above, we can generate a sequential order for any device \(D_{i}^{m}\in P^{m}\), from which follows (using invariant preservation) that \(\mathit{valid}(D_{i}^{m})\). As we can construct such a serialization (which must not be the same) for all devices, we know that \(\mathit{valid}(P^{m})\).
## 4 Implementation
Figure 6 depicts the architecture of LoRe's verifying compiler. The input to the compiler is a program with its specifications expressed by the invariants, e.g., the program in Listing 1. The output consists of the conflicting interactions and a safe executable program. To verify safety, we prove invariant preservation (Def 3) for all interactions (a failed proof results in a compilation error) and try to prove invariant confluence (Def 4) for all pairs of interactions. We employ an analysis of the data-flow graph to minimize confluence proof obligations to those invariant pairs that may actually conflict. Non-confluent interaction pairs are included in the conflict output. We use Viper for the verification step, but any other verification could be used and would still benefit from our minimization of the proof obligations. The implementation uses REScala, CRDTs, and a token-based consensus protocol for generating a safe executable program. But they could be replaced by other implementations of data-flow programming, eventual consistency, and consensus with the same guarantees. The rest of this section describes the pipeline from Figure 6 in detail - from left to right, top to bottom.
### Graph Analysis
Checking all pairs of interactions for confluence would result in an exponential amount of proof obligations. To avoid this, we employ a graph analysis to quickly detect pairs of interactions that cannot conflict, because they change completely separate parts of the data-flow graph. The graph analysis algorithm (Figure 7) checks every interaction to determine its reachable reactives: The source reactives that are affected by the interaction together with all transitively derived reactives that depend on them (Line 12). Next, the algorithm determines the _overlaps_ between interactions and invariants (Line 4). An interaction and an invariant _overlap_ if any reactive occurring in the invariant is part of the interaction's reachable reactives. Two interactions _overlap_ if there is at least one invariant they both overlap with. Only overlapping interactions produce proof obligations.
For illustration, consider the add_work interaction. It modifies the work reactive, and -transitively - all_appointments. Hence, the reachable reactives are \(\{\mathit{work},\mathit{all\_appointments}\}\) and only the first but not the second invariant in Listing 1 overlaps. Thus, neither the remaining_vacation reactive, nor the invariant on this reactive will be part of the proof obligation for the add_work interaction.
Figure 6: Overview of LoRe’s automated compilation and verification procedure.
### Automated Verification
#### Translation to Viper's Intermediate Language.
Listing 2 illustrates how we represent the data-flow graph and the safety invariants of our calendar example (Listing 1) in Viper's intermediate verification language [33]. We represent the data-flow graph as a mutable object with one field per source reactive (Line 3). In Viper, objects are implicitly defined by which fields the program accesses. Derived reactives (Line 6) and invariants (Line 10) are expressed as Viper macros - pure functions that describe the invariant or body of the reactive, and receive the reactives they depend on as function inputs.
Given these definitions, we synthesize one Viper _method_ per interaction, based on the formal evaluation semantics in Section 3. Viper verifies programs on a per-method basis where methods represent a sequential computation annotated with pre- and postconditions. Listing 3 shows the Viper method for the add_work interaction. Pre- and postconditions of interactions are simply translated to pre- and postconditions of the Viper methods (Line 7 and
Figure 7: Pseudocode of the algorithm used to determine which invariants overlap with a transaction.
12) while the executes part of each interaction is represented by the method body (Line 19). Reading derived reactives is modeled by inlining the respective Viper macro, which results in our big-step evaluation semantics. Additionally, we include every overlapping invariant as pre- and postconditions (Lines 10 and 17) so that the verifier can prove invariant preservation. Evaluating invariants is expressed by nested application of the previously defined macros (Listing 2) to the source reactives, corresponding to the propagation of values though the data-flow graph. Viper uses explicit permissions for shared state, thus we explicitly pass a reference to the data-flow graph (Line 2); it declares write permissions for all reactives that are modified by the interaction and read permissions for the reactives that are only accessed as part of the invariants (Line 5)8.
Footnote 8: Viper uses _fractional permissions_ where an acc(n) statement with any \(n<1\) corresponds to a read-permission and a statement with \(n=1\) corresponds to a write-permission.
#### Proving Invariant Preservation and Confluence.
Given the Viper encoding of each interaction - which includes overlapping invariants - Viper directly outputs verification results that correspond to invariant preservation (Def 3). Any verification errors at this stage are errors in the supplied specification or programming bugs and must be addressed by the programmer.
To check for confluence the compiler creates a Viper method for each overlapping pair of interactions. Such a method models the specification for invariant confluence (Definition 4). Any verification errors here indicate that the two interactions are non-confluent. As our soundness proof requires either confluence of interactions, or their inclusion in the set of conflicting interactions, safety is ensured by marking all non-confluent pairs of interactions as conflicting. Developers should still check the conflicts to ensure that they are not due to a bug or specification error.
### Synchronization at Runtime
Our compiler generates an executable application by converting the data-flow graph to a distributed REScala program [31, 32]. REScala supports all reactive features we require and integrates well with our CRDT-based replication, but has no mechanism for synchronization. LoRe's formal synchronization semantics (cf. Section 3) could be implemented using any existing form of coordination, such as a central server, a distributed ledger, a consensus algorithm, or a distributed locking protocol. Which choice is suitable, depends on the target application, network size, and the reliability of the chosen transport layer. We use a simple distributed locking protocol for our implementation: Each interaction has an associated lock (represented as a simple token). Whenever a device wants to execute an interaction, it acquires the tokens of all conflicting interactions. If multiple devices request the same token concurrently, the token is given to the device with the lowest ID that requested it. This ensures deadlock freedom; fairness is left for future work. After performing the interaction, the resulting state changes are synchronized with the other devices and the tokens are made available again. Timeouts ensure that whenever a device crashes or becomes unavailable for a longer period of time, its currently owned tokens are released and any unfinished interactions by the device are aborted.
## 5 Evaluation
Our evaluation aims to validate two claims about LoRe's programming model:
1. [label=C0]
2. It facilitates the development of safe local-first software.
3. It enables an efficient and modular verification of safety properties.
We base our validation on two case studies. First, we implemented the standard TPC-C benchmark [42] as a local-first application in LoRe. TPC-C models an order fulfillment system with multiple warehouses in different districts, consisting of five _database transactions_, New-Order, Payment, Order-Status, Delivery, Stock-Level, alongside twelve _consistency conditions_. This case study enables comparing LoRe's model with traditional database-centred development of data processing software and showcasing the benefits of LoRe's verifiable safety guarantees on standard _consistency conditions_. Second, we implemented the running calendar example (Section 2) using Yjs [36]. This case study allows comparing LoRe with an existing framework for local-first applications that we consider a representative of the state-of-the-art.
### Does LoRe facilitate the development of safe local-first software?
#### Local-first TPC-C
In the LoRe implementation of TPC-C, each warehouse holds a local copy of the data on which transactions - modeled as interactions - are executed before being synchronized with other warehouses.
_Reactives._ Figure 8 shows the reactive graph of the application. The structure roughly follows the database structure described in TPC-C. We represent each database table by a source reactive and model derived database values as derived reactives. For example, the DistrictYTD reactive in Listing 4 represents the year-to-date (YTD) balance of districts. After reading the districts reactive (Line 2), we perform the following steps for each district: We read the relevant entries in the payment history (Line 3), calculate the sum of the YTD values (Line 4), and return the result as a single entry mapping from district to YTD (Line 5).
```
1valdistrictYTD:Derived[Map[District,YTD]]=Derived{
2districts.map[d=>
3valpayments=paymentHistory.filter(ph=>sameDistrict(ph,d))
4valytd=payments.sum()
5(d,ytd)
6}.toMap()}
```
Listing 4: The LoRe representation of the derived DistrictYTD reactive (simplified).
###### Acknowledgements.
We implement TPC-C transactions as interactions. For illustration, consider the payment interaction in Listing 5, which is applied whenever a customer pays a certain amount to the system. Payments are not associated to a specific order and are simply stored in the payment history. Table modifications in TPC-C have multiple arguments, which we encapsulate into an argument of type PaymentArgs (Line 1). Applying the interaction (Line 3) retrieves the customer object matching the payment arguments by executing a function that accesses the customers reactive (Line 4). Line 6 updates and returns the new history. The preconditions (Line 8) encodes assumptions about the arguments, notably that a customer for whom we add the payment actually exists, and the postcondition (Line 12) describes the effect of adding a new payment.
###### Invariants.
TPC-C defines 12 consistency conditions, of which 9 are consistency constraints between tables and derived tables. Their correctness is automatically ensured by LoRe without further specification. For example, the consistency condition 9 [42]:
Entries in the DISTRICT and HISTORY tables must satisfy the relationship:
D_YTD = sum (H_AMOUNT) for each district defined by
(D_W_ID, D_ID) = (H_W_ID, H_D_ID).
directly corresponds to the definition of the DistrictYTD reactive in Listing 4 and is thus always true by design. Only the remaining 3 conditions require translating the natural language specification into first-order logic formulae to be used as invariants. As an example, consider consistency condition 5 from the TPC-C specification [42]:
For any row in the ORDER table, O_CARRIER_ID is set to a null value if and only if there is a corresponding row in the NEW-ORDER table [...].
This condition cannot be represented as a derived reactive because the carrier ID is not derived from other values but set explicitly whenever an order is shipped. Listing 6 displays the encoding in LoRe - an (almost) literal translation of the specification.
_Comparison._ The implementation effort for the core functionality of TPC-C in LoRe is comparable to a traditional design relying on a relational database model. While modelling the application using reactives might require some adaption from developers not familiar with data-flow programming, we found that using derived reactives led to a more concise and less error-prone design when compared to storing derived values in separate tables. This observation is supported by the fact that we only need to explicitly address 3 out of 12 consistency conditions of TPC-C. We were able to phrase the remaining 3 conditions as invariants by directly translating the natural language formulations into logical specifications. To prove them, we additionally needed to specify pre- and postconditions of interactions corresponding to transactions (see Figure 5). Other than that, LoRe relieves the TPC-C developer from any considerations of transaction interleavings that could potentially violate the conditions as well as from implementing the synchronization logic, both tedious and error-prone processes. Moreover, unlike TPC-C, which cares only about the consistency of the database, LoRe treats consistency constraints uniformly from (shared) state to UI. This is enabled by the reactive programming paradigm, which also guarantees 9 out of 12 consistency conditions by design.
#### Yjs-based Calendar
We now compare the LoRe implementation of the distributed calendar to an implementation using the state of the art local-first framework Yjs [36]. Like other solutions for local-first software, Yjs uses a library of CRDTs (usually maps, sets, sequences / arrays, and counters) composed into nested trees - called a _documents_ - used to model domain objects.
_Source and Derived Variables._ For illustration, consider Listing 7, showing how one could implement the domain model of the calendar application. Lines 2 and 3 initialize two CRDTs for the work and vacation calendar. Yjs has no abstraction for derived values and only provides callbacks for reacting to value changes, e.g., Lines 7-15 declare callback methods that update the derived variables in case the Yjs document changes.
_Safety Guarantees._ Using callbacks to model and manage complex state that changes both in time and in space has issues. It requires that developers programmatically update the derived values once the sources get updated, via local interactions or on receiving updates from other devices, with no guarantees that they do so consistently. It yields a complex control-flow and requires intricate knowledge of the execution semantics to ensure atomicity of updates, let alone to enforce application-level safety properties. Frameworks like Yjs do not offer support for application invariants and thus force developers to integrate custom
Figure 8: The data-flow graph of the TPC-C benchmark application.
safety measures at each possible source of safety violations. As an example, consider Listing 8, showing how the addVacation interaction could be implemented in Yjs. Lines 2 and 11 check the preconditions of the interaction, but as discussed in Section 2.3, such local checks are insufficient to maintain safety. In general, global reasoning without tool support quickly becomes infeasible, and even more so for designs based on callbacks. Once updates inducing invariant violations are propagated, it is difficult to "undo" them. Thus, programmers are required to either provide and coordinate compensation actions or to integrate mechanisms for strong synchronization on top of CRDT-based eventual consistency. Both approaches are difficult to get right, putting safety and/or availability at risk.
In summary, while the replication capabilities of systems like Yjs are valuable for local-first applications, these systems still require the developer to do state management manually. The prevailing use of callbacks and implicit dependencies makes reasoning about the code challenging for both developers and automatic analyses. In contrast, LoRe allows declarative definitions of derived values, with positive effects on reasoning [38, 12]. Moreover, LoRe integrates application invariants as explicit language constructs, which allows for a modular specification and verification and relieves developers from having to consider every involved interaction whenever the specification changes.
### Does LoRe enable efficient and modular verification of safety properties?
Safety invariants in LoRe are _global_ in the sense that they must not be violated by _any_ part of the program. But their enforcement is based on verifying individual local properties. We limit the need for verification to potential conflicts that we derive from the reactive data-flow graph. This optimization requires no further reasoning from the programmer and relies solely on the properties of the programming model. Programmers can add new functionality to the application (i.e., specify interactions) and only have to reason about the properties of that new functionality (i.e., specify its invariants) and the system ensures global safety - at only the cost of the amount of overlap with existing functionality. This allows a modular
programming style where invariants and interactions are written by different developers and changes to the program are made incrementally.
To empirically evaluate the performance of LoRe's verifier, we quantify how long it takes to verify different combinations of interactions and invariants of our two case studies. The results are shown in Table 1. The calendar example has two additional types of interactions, which we have not shown in Section 2: removing and changing calendar entries. This leads to a total of 6 interactions (3 per calendar reactive). As explained in the previous section, for TPC-C we only had to verify consistency conditions 3, 5, and 7. A "-" in the table denotes that no overlaps exist (see Section 4.1), and therefore no verification was needed. The verification tasks were performed on a desktop PC with an AMD Ryzen 5 3600 CPU using Viper release 2022.2 and the _silicon_ verification backend [43].
_Results._ Verification times differed depending mainly on the complexity and length of the transactions and invariants under consideration. Differences become apparent especially when looking at the results for TPC-C. Proofs involving the _New Order_ interaction, which is the most "write-heavy" interaction of TPC-C that changes many source reactives at once, generally took longer to verify than others. Different invariants also led to higher or lower verification times depending on which interaction they were paired with. Comparing the _Delivery_ and _New Order_ interactions, consistency condition 3 took longer with _New Order_. These results reflect the intuition that it is harder to prove preservation in cases where the interaction makes more changes that are relevant to the invariant. Consistency condition 3 requests that order numbers per district are continuous; it makes sense that this takes longer to check after a new order has been added by the _New Order_ interaction.
In summary, every interaction/invariant combination in our case studies could be verified in less than a minute. Of course, more complex examples could lead to verification times of up to a few minutes, as expected for an SMT-based verification tool [25, 45, 4]. But each interaction/invariant combination has to be verified only once and independently of other combinations. Large-scale applications can be verified step-by-step by splitting them into smaller pieces. This allows for an incremental development, where only certain parts of programs have to be (re-)verified, when they have been changed or added.
## 6 Related Work
Our work relates to three areas: distributed datatypes, formal reasoning, and language-based approaches. Sections below relate work from each area to respective aspects of our approach.
\begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{2}{c}{**Distributed Calendar**} \\ \hline
**Interaction** & **Invariant** \\ \hline Add vacation & 3.27 & 5.15 \\ Remove vacation & 4.86 & 4.13 \\ Change vacation & 4.61 & 4.99 \\ Add work & 3.92 & – \\ Remove work & 4.37 & – \\ Change work & 5.61 & – \\ \hline \hline \end{tabular}
\begin{tabular}{l c c c} \hline \hline \multicolumn{4}{c}{**TPC-C**} \\ \hline \multicolumn{4}{c}{**Transparency**} \\ \hline \multicolumn{4}{c}{**Transparency**} \\ \hline \hline \multirow{3}{*}{New Order Delivery} & 3 & 5 & 7 \\ \cline{2-4} & 11.75 & 7.63 & 6.36 \\ \cline{1-1} & 3.99 & 3.92 & 3.87 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Seconds to verify combinations of interactions and invariants of the two example applications.
### Consistency Through Distributed Data Types
_Conflict-Free Replicated Datatypes (CRDTs)[40, 37]_ are a building block for constructing systems and applications that guarantee eventual consistency. Gomes et al. [15] and Nair et al. [35] propose frameworks for formally verifying the correctness of CRTDs. CRDTs are used in distributed database systems such as _Riak_[22] and _AntidoteDB_[1]. These databases make it possible to construct applications that behave under mixed consistency, but unlike our approach, they leave reasoning about the consequences of different consistency guarantees to the programmer.
Several works suggest distributed data types that extend the capabilities of CRDTs. _Mergeable Replicated Data Types (MRDTs)_[19] automatically synthesize merge functions from relational specifications of the data type. _Katara_[24] synthesizes a verified CRDT from the specification of a sequential data type. De Porre et al. [11, 10] suggest _strong eventually consistent replicated objects (SECROs)_ relying on a replication protocol that tries to find a valid total order of all operations. Similarly, _Hamasz_[17] combines the specification of a sequential object with high-level invariants to synchronize a replicated object that satisfies the invariants. All approaches above tie consistency and safety properties to specific datatypes/objects. This is not sufficient to guarantee end-to-end correctness of an entire local-first application - consistency bugs can still manifest in derived information (e.g., in the user interface).
### Automated Reasoning about Consistency Levels
Our formalization is in part inspired by the work of Balegas et al. [7, 8] on _Indigo_. The work introduces a database middleware consisting of transactions and invariants to determine the ideal consistency level - called _explicit consistency_. They build on the notion of _invariant-confluence_ for transactions that cannot harm an invariant which was first introduced by Bailis et al. [5]. While they work on a database level, we show how to integrate this reasoning approach into a programming language (Section 3). An important difference between our _invariant-confluence_ and the one by Balegas et al. [7] is that our approach also verifies local preservation of invariants, whereas their reasoning principle assumes invariants to always hold in a local context. In a more recent work called _IPA_, Balegas et al. [6] propose a static analysis technique that aims at automatically repairing transaction/invariant conflicts without adding synchronization between devices. We consider this latter work complementary to ours. Whittaker and Hellerstein [44] also build on the idea of invariant-confluence and extend it to the concept of _segmented invariant-confluence_. Under segmented invariant-confluence, programs are separated into segments that can operate without coordination and coordination only happens in between the segments. The idea is similar to our definition of _conflicting interactions_, however, their procedure cannot suggest a suitable program segmentation, but requires developers to supply them.
The _SIEVE_ framework [26] builds on the previous work on _Red/Blue-Consistency_[27] and uses invariants and program annotations to infer where a Java program can safely operate under CRDT-based replication (_blue_) and where strong consistency is necessary (_red_). They do so by relying on a combination of static and dynamic analysis techniques. Compared to _SIEVE_, our formal reasoning does not require any form of dynamic analysis. _Blazes_[2] is another analysis framework that uses programmer supplied specifications to determine where synchronization is necessary to ensure eventual consistency. Contrary to _Blazes_, LoRe ensures that programs are "by design" at least eventually consistent, while also allowing the expression and analysis of programs that need stronger consistency. _Q9_[18] is a bounded
symbolic execution system, which identifies invariant violations caused by weak consistency guarantees. Similar to our work, _Q9_ can determine where exactly stronger consistency guarantees are needed to maintain certain application invariants. However, its verification technique is bound by the number of possible concurrent operations. LoRe can provide guarantees for an unlimited amount of devices with an unlimited amount of concurrent operations (see Chapter 3).
### Language Abstractions for Data Consistency
We categorize language-based approaches based on how they achieve consistency and on the level of programmer involvement.
_Manual Choice of Consistency Levels._ Approaches in this category expose consistency levels as language abstractions. Li et al. [27] propose _RedBlue Consistency_ where programmers manually label their operations to be either blue (eventually consistent) or red (strongly consistent). In MixT [30], programmers annotate classes with different consistency levels and the system uses an information-flow type system to ensure that the requested guarantees are maintained. However, this still requires expert knowledge about each consistency level, and wrong choices can violate the intended program semantics. Other approaches [34, 23] expect programmers to choose between _consistency_ and _availability_, again leaving the reasoning duty about consistency levels to the programmer. Compared to LoRe, languages in this category place higher burden on programmers: They decide which operation needs which consistency level, a non-trivial and error-prone selection.
_Automatically Deriving Consistency from Application Invariants._ Approaches in this category relieve programmers from reasoning about consistency levels based on some form of programmer annotations about application invariants. _CAROL_[25] uses CRDTs to replicate data and features a refinement typing discipline for expressing safety properties similar to our _invariants_. Carol makes use of pre-defined datatypes with _consistency guards_ used by the type system to check for invariant violations. The compatibility of datatype operations and consistency guards is verified ahead of time using an algorithm for the Z3 SMT solver. This approach hides much of the complexity from the programmer, but the abstraction breaks once functionality that is not covered by a pre-defined datatype is needed. Unlike Carol, LoRe does not rely on predefined consistency guards, but allows the expression of safety properties as arbitrary logical formulae. Additionally, _CAROL_ only checks the concurrent interactions of a program for invariant violations, whereas LoRe verifies the overall application including non-distributed parts. Sivaramakrishnan et al. [41] propose _QUELEA_, a declarative language for programming on top of eventually consistent datastores. It features a contract-language to express application-level invariants and automatically generates coordination strategies in cases where invariants could be violated by concurrent operations. _QUELEA_'s contract-language requires programmers to express the desired properties using low-level visibility relations, which can be challenging to get right for non-experts. LoRe avoids this intermediate reasoning and automatically derives the right level of consistency for satisfying high-level safety invariants to enable end-to-end correctness.
_Automating Consistency by Prescribing the Programming Model._ Languages in this category seek to automate consistency decisions by prescribing a certain programming model such that certain consistency problems are impossible to occur. In _Lasp_[29], programmers model the data flow of their applications using combinator functions on CRDTs. Programs written in _Lasp_ always provide eventual consistency but contrary to LoRe, _Lasp_ does not allow arbitrary compositions of distributed datatypes. _Bloom_[3] provides programmers with ways to write programs that are _logically monotonic_ and therefore offer automatic eventual
consistency. Both _Lasp_ and _Bloom_, however, are not meant to formulate programs that need stronger consistency guarantees. LoRe is similar to _Lasp_ and _Bloom_ in the sense that we also prescribe a specific - reactive - programming style. However, our programming model is less restrictive and allows arbitrary compositions of distributed datatypes. This is enabled by leveraging the composability properties of reactive data-flow graphs. Secondly, LoRe provides a principled way to express hybrid consistency applications with guarantees stronger than eventual consistency. Drechsler et al. [13] and Mogk et al. [31, 32] also use a reactive programming model similar to ours to automate consistency in presence of multi-threading respectively of a distributed execution setting. However, they do not support a hybrid consistency model. Drechsler et al. [13] enable strong consistency (serializability) only, while Mogk et al. [31, 32] support only eventual consistency.
## 7 Conclusion and Future Work
In this paper, we proposed LoRe, a language for local-first software with verified safety guarantees. _LoRe_ combines the declarative data flow of reactive programming with static analysis and verification techniques to precisely determine concurrent interactions that could violate programmer-specified safety properties. We presented a formal definition of the programming model and a modular verification that detects concurrent executions that may violate application invariants. In case of invariant violation due to concurrent execution, LoRe automatically enforces the necessary amount of coordination. LoRe's verifying compiler translates LoRe programs to Viper [33] for automated verification and to Scala for the application logic including synthesized synchronization to guarantee the specified safety invariants. An evaluation of LoRe's programming model in two case studies confirms that it facilitates the development of safe local-first applications and enables efficient and modular automated reasoning about an application's safety properties. Our evaluation shows that verification times are acceptable and that the verification effort required from developers is reasonable.
In the future, it would be desirable to integrate existing libraries of verified CRDTs [15] or even solutions that allow ad-hoc verification of CRDT-like datatypes [35, 24]. This would enable us to support a wider range of data types or even allow programmers to use custom distributed datatypes, which can be verified to be eventually consistent. Furthermore, our current data-flow analysis is limited to static data-flow graphs. While static reasoning about dynamic graphs is impossible in the general case, most applications make systematic use of dynamic dependencies, and we believe it would be feasible to support common cases.
|
2310.05863 | Fine-grained Audio-Visual Joint Representations for Multimodal Large
Language Models | Audio-visual large language models (LLM) have drawn significant attention,
yet the fine-grained combination of both input streams is rather
under-explored, which is challenging but necessary for LLMs to understand
general video inputs. To this end, a fine-grained audio-visual joint
representation (FAVOR) learning framework for multimodal LLMs is proposed in
this paper, which extends a text-based LLM to simultaneously perceive speech
and audio events in the audio input stream and images or videos in the visual
input stream, at the frame level. To fuse the audio and visual feature streams
into joint representations and to align the joint space with the LLM input
embedding space, we propose a causal Q-Former structure with a causal attention
module to enhance the capture of causal relations of the audio-visual frames
across time. An audio-visual evaluation benchmark (AVEB) is also proposed which
comprises six representative single-modal tasks with five cross-modal tasks
reflecting audio-visual co-reasoning abilities. While achieving competitive
single-modal performance on audio, speech and image tasks in AVEB, FAVOR
achieved over 20% accuracy improvements on the video question-answering task
when fine-grained information or temporal causal reasoning is required. FAVOR,
in addition, demonstrated remarkable video comprehension and reasoning
abilities on tasks that are unprecedented by other multimodal LLMs. An
interactive demo of FAVOR is available at
https://github.com/BriansIDP/AudioVisualLLM.git, and the training code and
model checkpoints will be released soon. | Guangzhi Sun, Wenyi Yu, Changli Tang, Xianzhao Chen, Tian Tan, Wei Li, Lu Lu, Zejun Ma, Chao Zhang | 2023-10-09T17:00:20Z | http://arxiv.org/abs/2310.05863v2 | # Fine-grained Audio-Visual Joint Representations for Multimodal Large Language Models
###### Abstract
Audio-visual large language models (LLM) have drawn significant attention, yet the fine-grained combination of both input streams is rather under-explored, which is challenging but necessary for LLMs to understand general video inputs. To this end, a fine-grained audio-visual joint representation (FAVOR) learning framework for multimodal LLMs is proposed in this paper, which extends a text-based LLM to simultaneously perceive speech and audio events in the audio input stream and images or videos in the visual input stream, at the frame level. To fuse the audio and visual feature streams into joint representations and to align the joint space with the LLM input embedding space, we propose a causal Q-Former structure with a causal attention module to enhance the capture of causal relations of the audio-visual frames across time. An audio-visual evaluation benchmark (AVEB) is also proposed which comprises six representative single-modal tasks with five cross-modal tasks reflecting audio-visual co-reasoning abilities. While achieving competitive single-modal performance on audio, speech and image tasks in AVEB, FAVOR achieved over 20% accuracy improvements on the video question-answering task when fine-grained information or temporal causal reasoning is required. FAVOR, in addition, demonstrated remarkable video comprehension and reasoning abilities on tasks that are unprecedented by other multimodal LLMs. An interactive demo of FAVOR is available at [https://github.com/BriansIDP/AudioVisualLLM.git](https://github.com/BriansIDP/AudioVisualLLM.git), and the training code and model checkpoints will be released soon.
## 1 Introduction
Text-based large language models (LLM) (Brown et al., 2020; Touvron et al., 2023; Chiang et al., 2023; Anil et al., 2023; Du et al., 2022) have demonstrated remarkable performance in various natural language processing tasks, especially achieving human-level capabilities in reasoning and comprehension (OpenAI, 2023). Meanwhile, instruction fine-tuning (Chung et al., 2022; Ouyang et al., 2022; Peng et al., 2023), where data is organised as pairs of user instruction (or prompt) and reference response, has emerged as a training paradigm that enables LLMs to perform various tasks by following open-ended natural language instructions from non-expert users.
Recently, there has been a burgeoning research interest in equipping LLMs with visual and auditory perception abilities. While most recent studies have been focusing on incorporating a single specific type of input, such as image (Li et al., 2023; Alayrac et al., 2022; Dai et al., 2023), video (Maaz et al., 2023; Chen et al., 2023; Zhao et al., 2022; Zeng et al., 2023), audio (Gong et al., 2023) or speech (Zhang et al., 2023; Rubenstein et al., 2023) separately. These investigations often employ a trained modality alignment module that aligns the representation space of the input modality with the text one. Subsequently, work has started looking at incorporating multiple simultaneous input modalities (Su et al., 2023; Zhang et al., 2023; Lyu et al., 2023; Zhao et al., 2023; Chen et al., 2023a). Despite the sequential nature of video and audio inputs, most aforementioned work treated video as a sampled subset of individual images and audio as a fixed-length spectrogram. As a
result, these models tend to ignore information and causal relations when the input sequence length increases. Moreover, speech, as a crucial aspect of auditory input in videos that in particular relies on fine-grained information extraction, is considerably under-explored in multimodal LLM research.
To this end, this paper proposes FAVOR, a **f**ine-grained **a**udio-**v**isual **j**oint representation learning framework for LLM-based multimodal understanding and reasoning with audio-visual input sequences consisting of images, audio events, speech, and video. It takes audio-visual sequences at certain frame rates as inputs and, if paired, temporally synchronises them using a synchronisation module. Such a frame-level synchronisation allows a more thorough and fine-grained interaction between audio and visual modalities across time, which is particularly beneficial for videos with speech. Since the input sequences have variable lengths, FAVOR divides the sequence into a number of fixed-length sliding windows and aligns the synchronised sequence within each window to the LLM input text representation space. In order to capture the causal relations among consecutive video frames within a window, a causal Q-Former structure is proposed that introduces a causal attention module to Q-Former (Li et al., 2023).
FAVOR is comprehensively evaluated using an audio-visual evaluation benchmark (AVEB) proposed in this paper, which integrates 11 tasks including 6 different types of open-source tasks with single-modal inputs, as well as 5 cross-modal inference tasks. While achieving competitive performance on single-modal tasks, FAVOR also achieved large performance improvements on cross-modal tasks compared to single-modal models, e.g. over 10% absolute accuracy improvement on audio-visual sound source detection. Notably, benefiting from the fine-grained nature, FAVOR achieved a remarkably 25% accuracy improvement in video QA tasks compared to the strong InstructBLIP baseline. The main contribution of this paper can be summarised as follows:
* This paper proposes the FAVOR learning framework for multimodal LLMs. To the best of our knowledge, FAVOR is the first approach that is capable of performing cross-modal cognitive tasks involving audio, speech, image and video inputs with high temporal resolution.
* This paper proposes the causal Q-Former structure which comprises a causal encoder module. Further with a novel diversity training loss, causal Q-Former is capable of handling audio-visual sequence input efficiently with a small number of training examples.
* This paper introduces the AVEB benchmark comprising single-modal and cross-modal tasks to quantitatively evaluate the performance of audio-visual LLMs.
## 2 Related Work
Our work is based on the Q-Former structure to fuse the audio and visual modalities and to align with the text representation space (Li et al., 2023; Dai et al., 2023). While Q-Former has been primarily proposed for visual information extraction, it also performs remarkably in extracting auditory features for automatic speech recognition (ASR) (Yu et al., 2023). In addition to Q-Former, various types of modality aligners have been studied, such as the cross-attention mechanism (Alayrac et al., 2022), pre-trained multimodal embeddings, (Girdhar et al., 2023) and temporal and spatial pooling (Maaz et al., 2023). Different from standard Q-Former approaches, our causal Q-Former used in the FAVOR framework pays particular attention to the sequential nature of the input feature streams with the model structure and training methods dedicated to audio-visual understanding.
The work most closely related to ours is Video-LLaMA (Zhang et al., 2023), Macaw-LLM (Lyu et al., 2023) and X-LLM (Chen et al., 2023), as all of them used LLMs for cross-modal understanding based on general non-silent video inputs (referred to as audio-visual sequence in this paper). X-LLM supports video and Chinese speech inputs, but cannot understand audio events and music. Video-LLaMA employs an additional video Q-Former to encode features of several equally-spaced frames extracted using a BLIP2 (Li et al., 2023) image encoder. Macaw-LLM adopted a similar approach and used three separate encoders for image, video and non-speech audio events. Both Video-LLaMA and Macaw-LLM consider only non-speech audio events, and the audio encoders in the two models are the ImageBind (Girdhar et al., 2023) and Whisper (Radford et al., 2023) model encoders respectively. While both methods involve the fusion of audio and visual feature streams, the two streams are sparsely pooled and processed rather independently, which removes fine-grained audio-visual interactions at each time step. Compared to Video-LLaMA and Macaw-LLM, FAVOR reserves fine-grained modality interactions and can understand speech inputs that are common in general non-silent videos. This leads to an emphasis on causal modality synchronisation across time and allows more content-based cross-modal interactions.
## 3 Methodology
In this section, we present the proposed FAVOR learning framework, which is designed to handle audio and visual input sequences synchronously at high temporal resolution for LLMs. This section introduces the model structure, including the causal attention module and an optional diversity loss.
### Model Architecture
The structure of FAVOR is shown in Fig. 1. Key components that realise the fine-grained audio-visual representation learning are the temporal synchronisation module and the causal Q-Former. First, visual and audio inputs are encoded using the corresponding pre-trained encoders. The visual encoder in FAVOR converts the input image into a certain number of vectors via the image encoder in InstructBLIP (Li et al., 2023). When video input is given, the visual encoder encodes each video frame separately as a sequence of images at a 2 Hz frame rate, and the output image features are concatenated along the temporal dimension to form a sequence of visual frames. The audio encoder used is the Whisper ASR model encoder (Radford et al., 2023) that converts the input speech and audio events into a sequence of vectors at a 50 Hz frame rate.
When both audio and visual inputs are present, the two encoded feature sequences are sent to the temporal synchronisation module to obtain the time-synchronised feature sequences, as shown in Fig. 1. Since video is sampled at a lower frame rate than audio, the audio and visual frames are synchronised at each video frame (_i.e._ every 0.5 seconds), with zero padding to make both sequences have equal lengths. Note that higher frequencies of visual frames are also supported in the FAVOR framework which requires higher computation and storage costs. The synchronised audio frame \(\mathbf{h}_{t}^{\mathrm{AV}}\) and visual frame \(\mathbf{h}_{t}^{\mathrm{V}}\) are then concatenated along the feature dimension to obtain the combined audio-visual feature frame \(\mathbf{h}_{t}^{\mathrm{AV}}\). That is,
\[\mathbf{h}_{t}^{\mathrm{AV}}=\mathrm{Concat}(\mathbf{h}_{t}^{\mathrm{A}}, \mathbf{h}_{t}^{\mathrm{V}}), \tag{1}\]
where \(\mathrm{Concat}(\cdot)\) represents the concatenation along the feature dimension. Note that in cases when only one input modality is present, the other modality is filled with a sequence of zero padding of the same sequence length. While an image alone is treated as a single frame, when paired audio input exists, such as images with spoken captions (Hsu et al., 2020), each image is duplicated as if it were a video input with a matched length to the audio input.
Figure 1: The fine-grained audio-visual joint representation (FAVOR) learning framework for multimodal LLMs. The temporal synchronisation module does not contain trainable parameters, and the audio and visual feature encoders are not updated during training.
In order to handle variable-length inputs, the combined feature sequences are first divided into fixed-length windows spanning, _e.g._ every 5 or 10 seconds. Then, a causal Q-Former based on the same \(N\) trainable input query tokens \(\mathbf{q}_{1},\ldots,\mathbf{q}_{N}\) is applied to convert each sliding window and generate \(N\) output query vectors carrying the audio-visual information. As shown in Eqn. (2),
\[\mathbf{h}_{w,1}^{\text{Q}},...,\mathbf{h}_{w,N}^{\text{Q}}=\text{Q-Former}_{ \text{causal}}(\mathbf{h}_{t}^{\text{AV}},\ldots,\mathbf{h}_{t+k}^{\text{AV} };\mathbf{q}_{1},\ldots,\mathbf{q}_{N}), \tag{2}\]
where \(w\) is the window index and \(k\) is the number of video frames in that window, and \(\text{Q-Former}_{\text{causal}}(\cdot)\) denotes the causal Q-Former computation described in detail later in Section 3.2. The output query representations, \(\mathbf{h}_{w,1}^{\text{Q}},...,\mathbf{h}_{w,N}^{\text{Q}}\), are projected to the LLM input dimension before sending to the LLM. Therefore, if the input sequence length of causal Q-Former is \(T\), the number of sliding windows \(W\) becomes \(\lceil T/k\rceil\), and the overall output sequence length from causal Q-Former will be \(W\times N\). Through end-to-end training, the output audio-visual representations of causal Q-Former are trained to align with the LLM input token space. Therefore, the use of sliding windows enables the LLM input token sequence length \(W\times N\) to vary based on \(T\) and can achieve a good trade-off between the degree of information reserved and the computation and storage costs.
Finally, the instruction prompt, such as questions or task descriptions will be appended to the concatenated output queries of all windows to form the input to the LLM. The response sequence \(\hat{\mathbf{Y}}\) can be generated as follows:
\[\hat{\mathbf{Y}}=\operatorname*{argmax}_{\hat{\mathbf{Y}}}P(\mathbf{Y}| \mathbf{h}_{1,1}^{\text{Q}},\ldots,\mathbf{h}_{1,N}^{\text{Q}},\ldots,\mathbf{ h}_{W,1}^{\text{Q}},\ldots,\mathbf{h}_{W,N}^{\text{Q}},\mathbf{c}_{1},\ldots, \mathbf{c}_{M}), \tag{3}\]
where \(\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{M}\) are the contents of the prompt.
### Q-Former with Causal Self-Attention
The proposed causal Q-Former structure is shown in Fig. 2. To capture the causal temporal correlation among frames that are extracted independently, an additional causal self-attention module is added to the standard Q-Former structure, indicated by the red block in Fig. 2.
With the causal attention module, the encoding of one specific frame also includes the information of all previous frames carried in an auto-regressive way. This is particularly beneficial for causal reasoning questions, such as the "what happens next" questions (Xiao et al., 2021). Such questions are sometimes difficult to learn using only the positional embeddings.
### System Training and Diversity Loss
The training data of video tasks, such as video question-answering (QA), usually only requires one or two keyframes, and the output queries tend to repeatedly capture the same information. Therefore, a novel diversity loss is proposed to encourage the causal Q-Former to extract more diverse aspects of the input sequence. Specifically, the diversity loss is formulated as:
\[\mathcal{L}_{\text{diverse}}=\sum_{w=1}^{W}\sum_{i=1}^{N}\sum_{j=1}^{N}\text{ sim}(\mathbf{h}_{w,i}^{\text{Q}},\mathbf{h}_{w,j}^{\text{Q}}), \tag{4}\]
where \(W\) and \(N\) are the total number of windows and the number of output queries of each window respectively, and \(\text{sim}(\cdot)\) is the cosine similarity between two vectors. Cosine similarity is adopted
Figure 2: The causal attention module in the causal Q-Former with a block-wise triangular causal mask (grey cells are masked). The number of features per frame here is 2 as an example.
since it is widely used for semantic similarity measurements, and in FAVOR, the output queries are aligned with a semantic space of the LLM input token representations. This choice is also supported by the fact that the modulus of the output query tokens is very similar due to the layer normalisation operation of the causal Q-Former. By encouraging the audio-visual frames to be orthogonal to each other, the diversity loss forces the output query representations to be more spread in the text representation space. Overall, the system is trained in an end-to-end fashion using the cross-entropy (CE) loss and the diversity loss, as shown below:
\[\mathcal{L}=\mathcal{L}_{\text{CE}}+\lambda\mathcal{L}_{\text{diverse}}, \tag{5}\]
where \(\lambda\) is the factor controlling the importance of the diversity loss, and the CE loss is calculated using the reference answer as the target.
## 4 Experimental Setup
### Audio-Visual Evaluation Benchmark (AVEB)
In this paper, we propose the AVEB benchmark for audio-visual LLM evaluation, which evaluates single-modal perception ability via selected representative tasks while particularly focusing on multi-modal inference. AVEB contains 6 single-modal tasks, including automatic speech recognition (ASR) (Panayotov et al., 2015), audio captioning (AC) (Kim et al., 2019), image captioning (IC) (Young et al., 2014), optical character recognition (OCR) (Singh et al., 2019), visual question answer (VQA) (Hudson and Manning, 2019), and video question answer (Video QA) (Xu et al., 2017), together with 5 audio-visual tasks including audio-visual speech recognition (AVSR) (Sanabria et al., 2018), audio-visual scene-aware dialogue (AVSD) (Alamri et al., 2019), image spoken question answering (ISQA), audio-visual matching (AVM) (Hsu et al., 2020) and audio-visual sound source detection (AVSSD) (Chen et al., 2020; Zhao et al., 2023). Related datasets are indicated in the actions. More details about the test datasets can be found in Appendix A.
ASR and AC are evaluated using word error rate (WER) and SPIDEr (Liu et al., 2017), a combination of SPICE and CIDEr respectively. The evaluation of IC uses CIDEr following (Dai et al., 2023), and METEOR, as LLMs tend to use a diverse range of words with similar meanings. OCR, VQA and Video QA are measured using top-1 accuracy. For OCR, the scoring follows (Singh et al., 2019) where each hit in the reference answer contributes 1/3 to the total hit. For VQA and Video QA, it is counted as correct if the reference answer exactly exists in the generated answer using a word-by-word matching1. In particular, during inference only, Video QA is formulated as an in-context multiple-choice task where the choices are given in the prompt, and one hit is counted only when the generated answer exactly matches the reference. The same measurement is taken for ISQA and AVM. Furthermore, for AVSD and AVSSD, as the reference answer is a full sentence, ChatGPT-assisted scoring is used to determine whether the generated answer is equivalent to the reference answer (see the prompt design in Appendix B).
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Task** & **Test set** & **Num. of samples** & **Metrics** \\ \hline ASR & LibriSpeech test-clean & 2620 utterances & WER \\ AC & AudioCaps test & 938 audio clips & SPIDEr \\ IC & Flickr30k test & 1000 images & CIDEr / METEOR \\ OCR & TextVQA test & 1000 images & Accuracy \\ VQA & QQA testyle balanced & 1000 images & Accuracy \\ Video QA & NEXT-QA test & 1000 clips & Accuracy \\ AVSR & How2 dev5 & 500 clips & WER \\ AVSD & AVSD val & 200 clips 2000 turns & Accuracy \\ ISQA & TextVQA + GQA & 2000 images & Accuracy \\ AVSSD & VGGSS & 850 video clips & Accuracy \\ AVM & SpokenCOCO val2014 + VGGSS & 1000 pairs 500 each & Accuracy \\ \hline \hline \end{tabular}
\end{table}
Table 1: AVEB details, including the number of samples used for evaluation and metrics reported. Since TextVQA, GQA, NExT-QA, AVSD and VGGSS test sets are large, randomly sampled subsets with enough samples for statistical significance were used in AVEB for efficient evaluation.
While all other tasks already exist with open-source test sets, this paper particularly proposes ISQA and AVM tasks where audio-visual interaction is necessary. ISQA is the task where the question is in the audio and the answer can be found in the image. This test set is derived from the data used for OCR and VQA, where the questions are synthesised using a commercial text-to-speech synthesis system with a diverse range of speakers and styles. The text prompt is always "answer the question in the audio about the image", while the LLM is required to first understand the question in the speech, and then answer it by looking at the image. AVM is the task of determining whether the given spoken description in the SpokenCOO dataset (Hsu et al., 2020) matches the image, or whether the given audio clip is compatible with the given video chosen from the VGGSS dataset (Chen et al., 2020). AVSSD is another task that requires a strong binding of audio and visual modalities, as a single modality usually only provides partial information about the sound.
### Model Configurations
To validate the FAVOR learning framework, the Vicuna (Chiang et al., 2023) models (including 7B and 13B models, and 13B is the default option if not specified) are used as the LLM, Whisper (Radford et al., 2023) large-v2 encoder as the audio encoder and InstructBLIP (Dai et al., 2023) vision Transformer (ViT) plus Q-Former as the visual encoder. The visual encoder outputs 32 feature vectors for each video frame (every 0.5 seconds), and the audio encoder outputs 50 feature vectors per second. The causal Q-Former has two Transformer blocks with 768-dim hidden states. The output query representations are projected to 5120-dim before being sent to the LLM. The LLM is adapted using the low-rank adaptation (LoRA) (Hu et al., 2022) method with a rank of 32. Only the parameters of the attention query, key and value projections and feed-forward network weights are updated, which comprised 0.4% of the total number of LLM parameters.
Whisper and InstructBLIP are used as the single-modality baseline systems for comparison. As FAVOR adopted video data with different styles and focuses, to eliminate the discrepancy in training data and achieve fair comparisons, InstructBLIP is further fine-tuned on the same image and video training data as FAVOR. For each video clip, five equally-spaced frames were used resulting in 160 output queries. This is the same as the number of output queries used for 25-second videos in FAVOR. Video-LLaMA (Zhang et al., 2023b) was used as the multimodal baseline where only Vicuna-7B checkpoint was released for audio-visual input2.
Footnote 2: [https://github.com/DAMO-NLP-SG/Video-LLaMA.git](https://github.com/DAMO-NLP-SG/Video-LLaMA.git).
### Training Data and Specifications
FAVOR directly uses multi-task instruction fine-tuning to train the model parameters of causal Q-Former and LoRA. Training data contains both single-modal and audio-visual paired data. For audio-only tasks, LibrSpeech train-clean-100 and train-clean-360 sets are used for ASR, and AudioCaps are used for AC. For visual-only tasks. A mixture of LLAVA-150k (Liu et al., 2023) image QA data, OCRVQA OCR data (Mishra et al., 2019), TextCaps Sidorov et al. (2020) image caption data, NexT-QA video QA training data (Xiao et al., 2021), COCO train2014 data for IC (Lin et al., 2014) as well as 11k samples from VideoChat (Li et al., 2023b) are used. For audio-visual tasks, randomly selected 600-hour Ego4D video captioning data (Grauman et al., 2022), how2 300-hour training set AVSR data and AVSD training set are used. In order to further stimulate modality interaction during training, 5,000 images with spoken captions are used in the training set for the AVM task. Details about the training datasets can be found in Appendix A.
In addition to all the training datasets mentioned above, in order to explicitly encourage the model to generically combine both modalities, a storytelling fine-tuning set is designed. The dataset is gathered by prompting GPT-3.5 with reference audio caption or transcription, together with video caption, and asking GPT-3.5 to generate a coherent story combining both information (see details in Appendix C). The model is fine-tuned on this data for only 100 steps with a very small learning rate without causing any loss in the benchmark performance.
It is worth noting that in order to compare FAVOR with the original InstructBLIP on image tasks directly, Flickr30k for IC, TextVQA for OCR and GQA for VQA in the benchmark are not included in the training, and hence the model performed zero-shot learning on them. Moreover, since ISQA uses synthesised speech, this is also not a trained task and the model performed zero-shot learning.
## 5 Experimental Results
### Main Results
The main results of using FAVOR on AVEB tasks are summarised in Table 2 and Table 3 for single-modal and audio-visual tasks respectively. While other models can only perform a subset of AVEB tasks, FAVOR achieves competitive performance on all tasks compared to the single-modal counterparts, with remarkably better performance on audio-visual tasks. In particular, as the first work that integrates audio, speech, image and video modality into LLMs, FAVOR effectively achieves audio-visual co-reasoning which is reflected by the performance on ISQA, AVSSD and AVM tasks.
On audio-based tasks in Table 2, FAVOR obtains a similar WER compared to Whisper large-v2 and mixed results compared to the audio-only FAVOR. Further, with the aid of visual information, FAVOR achieves a lower WER on AVSS than both models in Table 3. On visual tasks, FAVOR demonstrates the best results on IC, OCR and Video QA, and on-par results on VQA with Instruct-BLIP fine-tuned on the same training set. In particular, the fine-grained causal modelling of video in FAVOR yields over 20% improvements compared to InstructBLIP even though the latter is fine-tuned on the same set of video data.
On the audio-visual tasks in Table 3, while outperforming all the baseline systems in every task, FAVOR demonstrated a strong audio-visual co-reasoning ability based on the audio-visual matching (AVM) dataset results and is the only system to our knowledge that can perform speech-image co-reasoning based on image-spoken QA (ISQA). Audio-visual co-reasoning (including speech-image co-reasoning) is an important yet challenging ability which requires the model to comprehend the visual content as well as both speech and non-speech sounds in the audio, and to capture the correlation between what it "hears" and "sees". Such tasks were almost infeasible for any other audio-visual models so far, since they were unable to understand both speech and non-speech sounds
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Systems** & **ASR \(\downarrow\)** & **AC \(\uparrow\)** & **Video QA \(\uparrow\)** & **IC \(\uparrow\)** & **OCR \(\uparrow\)** & **VQA \(\uparrow\)** \\ \hline Whisper large-v2 & 2.9\% & - & - & - & - \\ InstructBLIP 13B & - & - & 21.0\% & 84.5 / 26.0 & 36.5\% & **48.9\%** \\ InstructBLIP 13B fine-tuned & - & - & 24.7\% & 78.9 / 26.1 & 36.7\% & 45.6\% \\ Video-LLMA 7B & - & - & 22.5\% & 22.0 / 16.6 & 16.4\% & 15.1\% \\ \hline FAVOR 13B (ours, audio-only) & **2.7**\% & 39.7 & - & - & - \\ FAVOR 13B (ours, visual-only) & - & - & 44.8\% & 74.0 / 26.5 & 34.2\% & 45.6\% \\ FAVOR 7B (ours, audio-visual) & 4.1\% & 39.1 & 42.5\% & 78.1 / 26.3 & 34.6\% & 45.3\% \\ FAVOR 13B (ours, audio-visual) & 3.3\% & **42.6** & **49.3**\% & **86.0 / 27.5** & **37.8**\% & 45.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: AVEB single-modal task results. If specified, InstructBLIP is fine-tuned on the training data of FAVOR (“InstructBLIP fine-tuned”). IC is reported in CIDEr/METEOR. When using audio-only and visual-only inputs, the other modality is masked during training and inference. Tasks unable to be performed are marked with “-”.
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Systems** & **AVSR \(\downarrow\)** & **AVSD \(\uparrow\)** & **ISQA \(\uparrow\)** & **AVSSD \(\uparrow\)** & **AVM \(\uparrow\)** \\ \hline Whisper large-v2 & 8.3\% & - & - & - & - \\ InstructBLIP 13B & - & 41.4\% & - & 1.1\% & - \\ InstructBLIP 13B fine-tuned & - & 52.1\% & - & 20.3\% & - \\ Video-LLaMA 7B & - & 27.6\% & - & 41.9\% & 52.3\% \\ \hline FAVOR 13B (ours, audio-only) & 8.3\% & - & - & 34.7\% & - \\ FAVOR 13B (ours, visual-only) & - & 53.3\% & - & 23.5\% & - \\ FAVOR 7B (ours, audio-visual) & 8.7\% & 51.2\% & 24.5\% & 50.5\% & 74.3\% \\ FAVOR 13B (ours, audio-visual) & **8.1**\% & **54.5**\% & **32.3**\% & **51.1**\% & **77.1**\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: AVEB audio-visual task results. If specified, InstructBLIP is fine-tuned on the training data of FAVOR (“InstructBLIP fine-tuned”). When using audio-only and visual-only inputs, the other modality is masked in both training and testing. Tasks unable to be performed are marked with “-”.
and did not model the audio-visual correlations in fine-grain. Various audio-visual emergent abilities in addition to the audio-visual co-reasoning ability, as discussed in Section 5.5.
### Ablation Studies
Detailed ablation studies are performed for each proposed component in FAVOR as shown in Table 8 for single-modal tasks and Table 9 for multimodal tasks in Appendix D. This section particularly focuses on the use of causal Q-Former and audio-visual synchronisation on video and audio-visual tasks, as summarised in Table 4.
First, the effect of the causal attention module is most clearly reflected by the performance on video QA, ISQA and AVSSD, as it both boosted the temporal causality modelling as well as provided a better audio-visual fusion before applying the cross-attention in the Q-Former. Second, the use of a sliding window is important to achieve good results on speech input as shown in the AVSSN results. Without the sliding window, a fixed number of output queries are used no matter how long the audio is, which results in more deletion errors. Besides, using sliding window also benefits the video QA task as they encourage the localised causal relationship to be captured. Furthermore, the use of synchronisation is crucial for audio-visual co-reasoning to work as supported in particular by the ISQA and AVSM results. Without synchronisation, modality alignment is done rather independently and the correlation between audio and video is only modelled among high-level features that are aligned in the text space. This may easily omit information about the concurrency of audio and visual contents, such as how a specific part of speech relates to a specific visual scene. On the other hand, synchronisation enables a temporally aligned cross-modal interaction which allows such concurrency to be captured, resulting in enhanced performances on audio-visual tasks.
### Analysis on the Sliding Window Size
As mentioned in Section 3.1, the trade-off between the sliding window size and the model performance is shown in Figure 3. Specifically, (a) and (b) show the influence of the numbers of frames \(k\) in a window while keeping the ratio \(N/k\) a constant (_i.e._ keeping the total output queries \(W\times N\) unchanged) and the same frame rate. This is trained on 10% of the full training data for quick experiments. Although using shorter windows benefits ASR, as fewer output tokens are used to encapsulate all the visual information within that window, performance on video QA is degraded. On the other hand, larger windows heavily reduce the ASR performance as the monotonic alignment in ASR is especially difficult to learn with 10% of the training data.
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Systems** & **Video QA** & **AVSR** & **AVSD** & **ISQA** & **AVSSD** & **AVM** \\ \hline Complete FAVOR & **49.3**\% & 8.1\% & **54.5**\% & **32.3**\% & **51.1**\% & **77.1**\% \\ FAVOR without causal encoder & 42.8\% & **8.0**\% & 54.1\% & 20.9\% & 37.1\% & 74.8\% \\ FAVOR without sliding window & 44.8\% & 8.5\% & 53.6\% & 29.7\% & 45.3\% & 74.5\% \\ FAVOR without synchronisation & 47.4\% & 8.4\% & 53.4\% & 17.2\% & 50.5\% & 72.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ablation studies on the core components of FAVOR based on video and audio-visual tasks. Each row represents removing one component with other parts remaining the same.
Figure 3: Influence of the window sizes and the frames per second (FPS) to the model performance on speech and video tasks. (a) and (b): results by training and evaluating using different window sizes \(k\) on 10% of data. (c): the influence of FPS using the best model on full data.
Figure 3 (c) shows the influence of the number of frames per second (FPS) on the model performance during inference, and hence the best model trained on the full set is used with the same number of frames per window. While low accuracy is observed when the frame rate is low, increasing FPS beyond 1.0 only receives marginal improvements at the cost of having many more output queries sent to the LLM. The choice of 2.0 FPS is chosen as it made the audio and visual sequences have the most similar lengths, and hence easier for synchronisation.
### Analysis of the Diversity Loss
Analysis of the effect of diversity loss is also performed using 10% of the training data as shown in Figure 4, and examples of cosine similarity matrices among output queries are shown in Appendix E. For ASR, the model is trained to include all the speech information in the audio sequence and the cosine similarity varies according to the length of the speech. For videos, the cosine similarity is close and does not vary too much for different video lengths, and hence diversity loss effectively acts as a way to encourage more diversified information to be captured. However, when a high \(\lambda\) is employed, diverse information causes confusion in the model and results in a more severe hallucination problem (_e.g._ high insertion rate in WER) with heavily degraded model performance.
### Discussions on Incorporating Speech and Speech-Video Interactions
Speech is an important source of information for video that should always be considered for audio-visual LLM to perform a comprehensive understanding. Unlike audio events, the speech content can hardly be inferred from the visual modality, making it particularly indispensable to comprehend any videos involving people talking. Moreover, the co-occurrence of speech and video events, which is modelled by the fine-grained temporal synchronisation in FAVOR, is required to understand the audio-visual temporal relations, _e.g._ "What did A say" (more examples in Appendix F).
One of the major contributions of FAVOR is to incorporate speech in a multimodal LLM and effectively combine both speech and video content to generate responses. In addition to the ISQA and AVM tasks that have already reflected the co-reasoning ability, the advantage of FAVOR can be more clearly demonstrated by the emergent abilities (shown in Appendix F). For instance, in response to questions about why a movie clip is funny or romantic, FAVOR combines the video, dialogue between characters and background audio or music to generate a more encompassing and convincing answer. Besides, FAVOR is able to understand the scene better by using knowledge from the speech, such as the species of a particular fish introduced in a documentary.
## 6 Conclusion
This paper proposed FAVOR, a fine-grained audio-visual joint representation learning framework for multimodal LLMs. On the proposed AVEB benchmark for audio-visual evaluation, FAVOR achieved competitive performance on audio and visual single-modal tasks with a remarkable 20% absolute accuracy improvement on the causal reasoning video QA task compared to the baselines. FAVOR demonstrated audio-visual, and particularly strong speech-visual co-reasoning abilities, with remarkable cross-modal emergent abilities demonstrated via examples.
Figure 4: Variations of model performance due to the diversity loss factor, _i.e._\(\lambda\) in Eqn. (4), on (a) AVSR measured in %WER, (b) Video QA measured in %Accuracy and (c) AVSSD measured in %Accuracy. Variations of average cosine similarities are also shown under different \(\lambda\)’s.
## 7 Reproducibility Statement
To make the experiments and models reproducible, the benchmark details are provided in the supplementary materials, and a demo page is provided in the abstract for a convenient try-out of the model. The details of the training and test data are summarised in Section 4 and Appendix A. Key hyper-parameter settings were discussed in the result section. The complete training and inference code together with model checkpoints will be released upon acceptance.
|
2308.10942 | Illuminating Nucleon Gluon Interference via Calorimetric Asymmetry | We present an innovative approach to the linearly polarized gluons confined
inside the unpolarized nucleon in lepton-nucleon scattering. Our method
analyzes the correlation of energy flows at azimuthal separations $\phi$. The
interference of the spinning gluon with both positive and negative helicities
translates into a $\cos(2\phi)$ asymmetry imprinted on the detector. Unlike the
conventional transverse momentum dependent (TMD) probes, the $\cos(2\phi)$
asymmetry in this approach is preserved by rotational symmetry, holds to all
orders, and is free of radiation contamination, thus expected to provide the
exquisite signature of the nucleon linearly polarized gluons. | Xiao Lin Li, Xiaohui Liu, Feng Yuan, Hua Xing Zhu | 2023-08-21T18:00:02Z | http://arxiv.org/abs/2308.10942v1 | # Illuminating Nucleon Gluon Interference via Calorimetric Asymmetry
###### Abstract
We present an innovative approach to the linearly polarized gluons confined inside the unpolarized nucleon in lepton-nucleon scattering. Our method analyzes the correlation of energy flows at azimuthal separations \(\phi\). The interference of the spinning gluon with both positive and negative helicities translates into a \(\cos(2\phi)\) asymmetry imprinted on the detector. Unlike the conventional transverse momentum dependent (TMD) probes, the \(\cos(2\phi)\) asymmetry in this approach is preserved by rotational symmetry, holds to all orders, and is free of radiation contamination, thus expected to provide the exquisite signature of the nucleon linearly polarized gluons.
_Introduction_. Quarks and gluons that are confined within nucleons will be examined in unprecedented detail at the next generation QCD facilities [1; 2; 3]. Extracting their fundamental properties requires the analysis of scattering data to reveal their distributions inside nucleons. It is now widely recognized that even within an unpolarized nucleon, partons can exhibit polarization, leading to a scattering cross-section of the schematic form
\[\sigma \propto |\hat{\mathcal{M}}|+\rangle+\hat{\mathcal{M}}|-\rangle|^{2} \tag{1}\] \[= \sum_{i=+,-}\langle i|\hat{\mathcal{M}}^{\dagger}\hat{\mathcal{M} }|i\rangle+\left(\langle+|\hat{\mathcal{M}}^{\dagger}\hat{\mathcal{M}}|- \rangle+c.c.\right),\]
in which \(|i\rangle\) denotes the helicity state of the parton out of the hadron, and \(\hat{\mathcal{M}}\) is the transition operator. Thus far experimental and theoretical studies have placed extensive focus on the first trace term, which brings about the most familiar unpolarized parton distributions, such as the collinear parton distribution functions (PDFs) and the unpolarized transverse momentum-dependent PDFs (TMDs). While these unpolarized distributions have provided us with valuable insights into the dynamics of strong interactions, the off-diagonal terms contain the intrinsic quantum effects. The operator \(\hat{\mathcal{M}}\) acts as a screen in helicity space, leading to a double-slit interference phenomenon when both \(i=+\) and \(i=-\) are allowed, as illustrated in Fig. 1. A similar effect in the context of final states has drawn recent discussions in jet physics [4; 5; 6] and top physics [7], leading to an interesting application of 2D conformal symmetry in 4D collider physics [8; 9].
The knowledge of the off-diagonal contribution requires the quantum description of a nucleon by the density matrix \(\rho_{ij}=|i\rangle\langle j|\), with \(i,j=+/-\), where \(\frac{1}{2}\mathrm{Tr}\rho\) gives the unpolarized distributions. Out of that, the entropy of a nucleon in the helicity space can also be defined, \(S=-\rho\ln\rho\). Furthermore, the concepts, such as the positivity of \(\rho\) and the maximum entropy principle, may be introduced and tested in the hadron structure studies. In the following, we focus on the hadron structure associated with the gluon distribution, and we refer to the off-diagonal contribution as the linearly polarized gluon distribution.
In order to observe the effect, one has to introduce a transverse reference direction that goes beyond the conventional collinear PDFs. In the literature, this has been demonstrated in the generalized parton distribution (GPD) framework [10; 11; 12; 13] and the TMD framework. In the GPD formalism, the associated distribution is also called the transversity or helicity-flip gluon GPD [14; 15; 16], where the momentum transfer between the initial and final state hadrons plays the role as a reference direction. Similarly, in the TMD formalism, the transverse momentum of the gluon helps to define the linearly polarized gluon distribution [17; 18]. An anticipated outcome of this novel gluon distribution is a \(\cos(2\phi)\) azimuthal angular asymmetry in the associated hard scattering processes [14; 15; 16; 19; 20; 21; 22; 23; 24; 25]. While, in the TMD framework, the asymmetry could receive additional \(\cos(n\phi)\) corrections not associated with the nucleon target's parton polarization [22; 23; 25], which overshadows the naive \(\cos(2\phi)\)
Figure 1: Nucleon structure as a double split experiment in the helicity space.
expectation.
In this paper, we apply recently proposed nucleon energy-energy correlator (NEEC) [26], which is a novel extrapolation of the energy-energy correlator [27] from final-state jet substructure [28; 29] to the nucleon structure, to introduce a fresh strategy to identify the linearly polarized gluon distribution in an unpolarized hadron. One standout feature of our proposal is that the gluon operator in the NEEC follows the (helicity-dependent) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution [30] and a collinear factorization is applied to compute the associated differential cross sections and the relevant azimuthal angular asymmetries. This contrasts heavily with the linearly polarized gluon distribution in the TMD formalism, where soft gluon radiation plays an important role. The customary approach to the linearly polarized gluon involves the gluonic TMD distributions by observing the \(\cos(2\phi)\) asymmetry between the slight imbalance \(\vec{k}_{T}\) and the leading jet momentum \(\vec{P}_{J,T}\) in di-jet/di-hadron production in the DIS process [19; 20; 21; 22; 23]. However, soft radiations that are unrelated to the nucleon target's partons can also generate significant anisotropy in the form of \(\cos(n\phi)\) (where \(n=1,2,\dots\)) [22; 23; 25] and thus obscure the physics. Additionally, Sudakov logarithms that lead to a significant suppression in the non-perturbative region [31] complicates the analysis of other TMD probes such as Higgs production [32; 33] in hadronic collisions.
Unlike its predecessors, the linearly polarized gluon contribution in the NEEC is formulated in the collinear factorization for the inclusive process, without the contamination from soft radiation, eliminating the need for the Sudakov resummation. The \(\cos(2\phi)\) signature is preserved by rotational symmetry and persists to all orders. Therefore this methodology provides a unique opportunity to observe the interference effects from the spinning gluon inside the hadrons.
_Helicity dependent NEEC._ The NEEC was introduced in [26] as a new quantity for the nucleon structures, which complements the TMDs and has been demonstrated as an efficient portal to the onset of gluon saturation [34]. The operator definition of the unpolarized NEEC can be found in [26; 30], which involves the asymptotic energy flow operator \(\hat{\mathcal{E}}(\theta_{a})\) that records the energy deposition in the calorimeter at a fixed angle \(\theta_{a}\), normalized to the proton energy, but with the azimuthal position \(\phi_{a}\) integrated over.
If we keep the azimuthal dependence and measure the energy flow into the solid angle \((\theta_{a},\phi_{a})\), the related flow direction \(n_{a}^{\alpha}=(1,\sin\theta_{a}\cos\phi_{a},\sin\theta_{a}\sin\phi_{a},\cos \theta_{a})\) supplies a chance to map out the intrinsic Lorentz structure of the gluon field in terms of the helicity dependent NEEC,
\[f_{g,\mathrm{EEC}}^{\alpha\beta}(x,\vec{n}_{a})=\int\frac{dy^{-} }{4\pi xP^{+}}e^{-ixP^{+}\frac{y^{-}}{2}}\] \[\qquad\times\langle P|\mathcal{F}^{+\alpha}\left(y^{-}\right) \mathcal{L}^{\dagger}[\mathbf{\infty},y^{-}]\hat{\mathcal{E}}(\vec{n}_{a}) \mathcal{L}[\mathbf{\infty},0]\mathcal{F}^{+\beta}(0)|P\rangle\] \[=-g_{T}^{\alpha\beta}f_{g,\mathrm{EEC}}+\left(\frac{n_{a,T}^{ \alpha}n_{a,T}^{\beta}}{n_{a,T}^{2}}-\frac{g_{T}^{\alpha\beta}}{2}\right)d_{ g,\mathrm{EEC}}\,, \tag{2}\]
where the first equation furnishes the operator definition of the helicity-dependent gluon NEEC in which \(\mathcal{F}\) is the gauge field strength tensor, and \(\mathcal{L}\) is the gauge link. If we average the gluon helicity, we recover the unpolarized gluon NEEC [30]. In the second equation, \(g_{T}^{\alpha\beta}=g^{\alpha\beta}-\frac{P^{\alpha}\bar{n}^{\beta}+\bar{n}^ {\alpha}P^{\beta}}{\bar{n}.P}\), with \(\bar{n}\cdot P=P^{0}+P^{z}\equiv P^{+}\), \(\bar{n}_{a}=\sin\theta_{a}(\cos\phi_{a},\sin\phi_{a})\) and \(n_{a,T}^{\alpha}=(0,\bar{n}_{a},0)\) is the transverse component of the light ray vector \(n_{a}^{\alpha}\). The second equation is the most general parameterization of \(f_{g,\mathrm{EEC}}^{\alpha\beta}\) to satisfy rotational covariance around the \(z\)-axis.
The coefficient \(f_{g,\mathrm{EEC}}(\theta_{a})\) is the unpolarized NEEC [30], while \(d_{g,\mathrm{EEC}}(\theta_{a})\) is the _linearly polarized gluon NEEC_ originated from the interference between different helicity states. To see this, we parameterize the gluon polarization vectors as \(\epsilon_{\pm}^{\star\alpha}=\frac{1}{\sqrt{2}}\left(0\,,1\,,\mp i\,,0\right)\), it is then straightforward to check that \(\epsilon_{\pm,\alpha}\epsilon_{\pm,\beta}^{\star}f_{g,\mathrm{EEC}}^{\alpha \beta}=f_{g,\mathrm{EEC}}\), and \(\epsilon_{\mp,\alpha}\epsilon_{\pm,\beta}^{\star}f_{g,\mathrm{EEC}}^{\alpha \beta}=\frac{1}{2}e^{\mp 2i\phi_{a}}d_{g,\mathrm{EEC}}\), which manifests that the linearly polarized gluon NEEC is a consequence of helicity interference. Since the energy flow measurement \(\bar{E}(\vec{n}_{a})\) is isotropic in the azimuthal plane, the nontrivial \(\phi_{a}\) dependence of the \(f_{g,\mathrm{EEC}}^{\alpha\beta}\) probes directly the polarization of the gluon field inside the nucleon.
When \(P^{+}\theta_{a}\gg\Lambda_{\mathrm{QCD}}\), the NEEC can be further matched onto the collinear PDFs. At \(\mathcal{O}(\alpha_{s})\), the matching of the unpolarized NEEC can be found in [30]. The linearly polarized gluon NEEC is calculated from the polarized splitting function and found to be
\[d_{g,\mathrm{EEC}}(x,\theta_{a}^{2})=\frac{\alpha_{s}}{4\pi^{2}} \frac{2}{\theta_{a}^{2}}\int\frac{dz}{z}(1-z)\frac{1-z}{z}\] \[\qquad\qquad\times\frac{x}{z}\left[C_{F}f_{q}\left(\frac{x}{z} \right)+C_{A}f_{g}\left(\frac{x}{z}\right)\right]\,, \tag{3}\]
Here we have averaged over the initial parton color and spin. The evolution of \(f_{g,\mathrm{EEC}}\) follows the DGLAP evolution [30]. The \(d_{g,\mathrm{EEC}}\) obeys the helicity-dependent DGLAP equation that resums logarithms \(\alpha_{s}^{n}\ln^{n-1}\theta_{a}/\theta_{a}\) and will be carried out in future work.
_Measurement of the Energy Correlator._ We consider the unpolarized DIS process in the Breit Frame, in which the incoming proton is along the \(z\)-axis and the virtual photon generates no transverse momentum with its momentum \(q^{\mu}=(0,0,0,-Q)\). We measure the energy flows that deposit in 2 arbitrary pixels on the calorimeter located at \(\vec{n}_{a}=\sin\theta_{a}(\cos\phi_{a},\sin\phi_{a})\) and \(\vec{n}_{b}=\sin\theta_{b}(\cos\phi_{b},\sin\phi_{b})\). Here, \(\theta\)'s and \(\phi\)'s are polar and
azimuthal angles, respectively. The polar angles are measured with respect to the \(z\)-axis and the azimuthal angles are measured from the plane spanned by the proton and the leptons, as shown in Fig. 2. We then construct \(\vec{d}=\vec{n}_{b}-\vec{n}_{a}\).
We require one of the pixels much closer to the proton beam axis, i.e., suppose it is \(a\), then \(\theta_{a}\ll\theta_{b}\) and when \(Q\theta_{a}\sim\mathcal{O}(\Lambda_{\text{QCD}})\) we probe the NEEC of the proton [26]. The other pixel, suppose it is \(b\), is in the central region. Since \(\theta_{a}\ll\theta_{b}\), the measurement of the energy flow along \(\vec{n}_{b}\) guarantees the inclusive di-jet configuration in the central region to balance the transverse momentum.
To probe the interference of the gluon helicities, we look at the azimuthal angle difference \(\phi=\phi_{d}-\phi_{a}\) between \(\vec{n}_{a}\) and \(\vec{d}\)1, see Fig. 2. We note that when \(\theta_{a}\to 0\), \(\vec{d}\rightarrow\vec{n}_{b}\) and \(\phi\rightarrow\phi_{b}-\phi_{a}\). More specifically, we measure the energy-weighted cross-section
Footnote 1: One can also measure the azimuthal difference \(\phi^{\prime}\) between \(\vec{n}_{b}\) and \(\vec{n}_{a}\). The difference between \(\phi\) and \(\phi^{\prime}\) vanishes as \(\theta_{a}/\theta_{b}\to 0\). However, the power correction to the factorization in Eq. (7) could be significant if we use \(\phi_{b}-\phi_{a}\) directly. A similar strategy is used to suppress the power correction in [5].
\[\Sigma(x_{B},Q^{2},\cos\theta_{a,b},\phi) \tag{4}\] \[= \sum_{ij}\int d\sigma(x_{B},Q^{2})\frac{E_{i}}{E_{P}}\frac{E_{j}} {E_{P}}\delta(\vec{n}_{a}-\vec{n}_{i})\delta(\vec{n}_{b}-\vec{n}_{j})\] \[\times\mathcal{F}(\phi;\vec{n}_{a,b})\,,\]
where \(\mathcal{F}(\phi;\vec{n}_{a,b})\) imposes the phase space measurement to construct \(\phi\). Here we note that we integrated over the azimuthal angles of the lepton, and the \(\phi_{a,b}\). The only azimuthal angle we observe in this measurement is \(\phi\).
The general form of the cross-section \(\Sigma\) is given by
\[\Sigma=\frac{4\pi\alpha^{2}e_{q}^{2}}{Q^{4}}l_{\mu\nu}\Sigma^{\mu\nu}(x_{B}, \cos\theta_{a,b},\phi)\,, \tag{5}\]
where \(\alpha\) is the electrical fine structure constant and \(l_{\mu\nu}=g_{\mu\nu}(-2l\cdot l^{\prime})+4l^{\mu}l^{\nu}\), where the Ward identity \(q_{\mu}\Sigma^{\mu\nu}=0\) has been applied. Here \(-2l\cdot l^{\prime}=Q^{2}\) and \(l^{\mu}=Q\frac{1+y}{y}(1,\frac{\sqrt{1+y}}{1+y},0,\frac{y}{1+y})\). The \(\Sigma^{\mu\nu}\) is the cross section for \(\gamma^{*}P\to X\) with the energy correlators measured. When \(\theta_{a}\ll\theta_{b}\), the calculation of the \(\Sigma^{\mu\nu}\) can be performed within the collinear NEEC factorization, closely follows [30], which gives
\[\Sigma^{\mu\nu} = \frac{y^{2}}{16\pi Q^{2}}\int d\Phi_{X}\mathcal{M}_{q}^{\mu} \mathcal{M}_{q}^{\nu\dagger}f_{q,\text{EEC}}(x,\vec{n}_{a}) \tag{6}\] \[+\,\mathcal{M}_{g,\alpha}^{\mu}\mathcal{M}_{g,\beta}^{\nu\dagger} \,f_{g,\text{EEC}}^{\alpha\beta}(x,\vec{n}_{a})\,.\]
The factorization theorem is illustrated in Fig. 3. Here, \(\Phi_{X}\) stands for the phase space of the final state partons, including the integration over the incoming parton momentum fraction \(\int\frac{dx}{x}\), with the energy \(E_{j}/E_{P}\) weighting and the angle \(\phi\) measurement in Eq. (4) included. The \(E_{i}/E_{P}\) and \(\vec{n}_{a}\) measurements have been absorbed into the definition of the NEEC. \(\mathcal{M}_{i}^{\mu}\) is the matrix element for the partonic \(\gamma^{*}i\to jj+X\) production and can be calculated order by order in \(\alpha_{s}\), whose leading-order (LO) contribution is illustrated in Fig. 3. The subscript \(q\) (\(g\)) indicates the quark (gluon)-initiated partonic process. Here we have used the fact that in perturbative QCD (and also QED), the massless quark helicity is conserved, therefore only the unpolarized quark NEEC \(f_{q,\text{BEC}}\) is involved.
It can be shown that to all orders, Eq. (6) fulfills the general form such that, up to power corrections \(\sim\mathcal{O}(\theta_{a}/\theta_{b})\),
\[l_{\mu\nu}\Sigma^{\mu\nu}=\int\frac{dz}{z}\Bigg{[}\sum_{i=q,g} \hat{H}_{i}(z,y,c_{b})\frac{x_{B}}{z}f_{i,\text{EEC}}\left(\frac{x_{B}}{z}, \theta_{a}^{2}\right) \tag{7}\] \[+ \frac{1}{2}\cos(2\phi)\Delta\hat{H}_{g}(z,y,c_{b})\frac{x_{B}}{z} d_{g,\text{EEC}}\left(\frac{x_{B}}{z},\theta_{a}^{2}\right)\Bigg{]}\,,\]
Figure 2: The measurement proposed as a probe of the gluon polarization in the DIS process. The energy flow into different pixels (in red blocks) at \(\vec{n}_{a}\) and \(\vec{n}_{b}\) are recorded, for \(\theta_{a}\ll\theta_{b}\). \(\phi\) angles are measured from the plane where lies the leptons. The measurement of \(E_{i}(\vec{n}_{a})\) induces the NEEC.
where we abbreviated \(c_{b}=\cos\theta_{b}\) and suppressed the scale dependence. Here, \(z\equiv\frac{x_{B}}{x}\) and the factor \(\frac{x_{B}}{x}\) is originated from \(E_{j}/E_{P}\). To understand how we get the form of Eq. (7), we first note that the parton momentum \(p^{\alpha}=\frac{Q}{z}(1,0,0,1)\) that initiates the interaction is determined by \(z\). Given that we are inclusive over the final state energy, then the partonic cross section \(\hat{H}\) and \(\Delta\hat{H}\) can only be functions of \(z\), \(y\), and \(\vec{n}_{b}\approx\vec{d}\) in the small \(\theta_{a}\) limit. The quark channel is un-polarized and hence \(\phi\) independent. As for the gluon channel, given the tensor structure of \(f^{\alpha\beta}_{g,\rm BEC}\) in Eq. (2), any Lorentz structures of \(\int d\Phi_{X}l_{\mu\nu}{\cal M}^{\mu}_{g,\alpha}{\cal M}^{\nu,\dagger}_{g,\beta}\) constructed out of the longitudinal vectors vanishes when contracted with \(f^{\alpha\beta}_{g,\rm BEC}\). Therefore, its non-vanishing contribution to \(l_{\mu\nu}\Sigma^{\mu\nu}\) must require the all-order form
\[-g_{T}^{\alpha\beta}A(z,c_{b})+\left(\frac{n^{\alpha}_{b,T}n^{\beta}_{b,T}}{n ^{2}_{b,T}}-\frac{g_{T}^{\alpha\beta}}{2}\right)B(z,c_{b})\,. \tag{8}\]
The form is determined by the reason that \(\int d\Phi_{X}l_{\mu\nu}{\cal M}^{\mu}_{g,\alpha}{\cal M}^{\nu,\dagger}_{g,\beta}\) is rotational covariant around the \(z\)-axis and can only be constructed out of \(g_{\alpha\beta}\), \(p^{\alpha}\), \(q^{\alpha}\) and \(d^{\alpha}\approx n^{\alpha}_{b}\), while neither \(p^{\alpha}\) nor \(q^{\alpha}\) but only \(n^{\alpha}_{b}\) acquires a transverse component. Furthermore, the \(\phi_{b}\) integration eliminates possible \(\phi_{b}\) dependence within \(A\) and \(B\). Now contracting Eq. (8) with \(f^{\alpha\beta}_{g,\rm BEC}\), we arrive at the final form in Eq. (7), where the unpolarized gluon contribution comes from the contraction of the \(-g_{T}^{\alpha\beta}\) structures, and the \(\cos(2\phi)\) from the \(B\) term. Here, we have applied \(\phi=\phi_{d}-\phi_{a}\approx\phi_{b}-\phi_{a}\) in the \(\theta_{a}\to 0\) limit.
We conclude from Eq. (7) that the NEEC-based measurement provides a unique chance to probe the linearly polarized gluons since
* Eq. (7), (2) and (8) hold to all-orders with all radiation effects such as parton shower being taken into account. Therefore, unlike the TMDs, to all orders, the NEEC probe involves only one azimuthal structure \(\cos(2\phi)\), due to the absence of soft radiations and hence no cross-talk between \({\cal M}_{\mu}{\cal M}^{\dagger}_{\nu}\) and \(f^{\mu\nu}_{\rm BEC}\). Each of the \({\cal M}_{\mu}{\cal M}^{\dagger}_{\nu}\) and \(f^{\mu\nu}_{\rm BEC}\) can only depend on one of the azimuthal angles (\(\phi_{a}\) or \(\phi_{d}\approx\phi_{b}\)). Therefore, \(\phi\) enters only through the tensor structure in Eq. (2) and Eq. (8), which uniquely determines the \(\cos(2\phi)\) asymmetry. In contrast, the TMD soft radiation with momentum \(k\) could simultaneously connect all directions, for instance, both the proton \(P\) and the leading jet \(P_{J}\) in the dijet process, and thus depends on all azimuthal angles. On that account, additional azimuthal dependence due to the eikonal factor \(\frac{1}{k\cdot P_{k}\cdot P_{J}}\propto\frac{1}{\cdot\cdot+\cos(\phi)} \rightarrow\sum_{n}c_{n}\cos(n\phi)\)[22; 23; 25], contaminates the naive \(\cos(2\phi)\) expectation. In this sense, the NEEC is a cleaner probe of the rotating gluons inside the nucleon target.
* Furthermore, the NEEC factorization in Eq. (7) suffers no Sudakov suppression [30; 26; 34] in the non-perturbative signal region when \(\theta_{a}Q\sim{\cal O}(\Lambda_{\rm QCD})\), which is quite different from the TMD case. On the contrary, the region is enhanced by the DGLAP evolution of the NEEC [30].
* The energy flow measurement \(E(\vec{n}_{b})\) in the central region can be replaced by a jet constructed using a standard jet algorithm. The precision of this measurement can be improved by using the tracking information [35; 36; 37; 38]. To further enhance the sensitivity to the gluon \(f_{\rm EEC}\), we can tag the heavy quark species (charm/\(b\)-energy flow) for \(E(\vec{n}_{b})\) measurement [39; 40].
_Numerics._ Now we present numerical studies. Our main objective of this study is to examine the all-order \(\cos(2\phi)\) structure in Eq. (7), through a perturbative calculation at higher orders. We use the nlojet++[41] to generate tri-jet production in DIS at NLO (\({\cal O}(\alpha_{s}^{2}+\alpha_{s}^{3})\), up to four jets). Since the exact NEEC is not known, we model it by restricting \(0.005<\theta_{a}<0.02\). In this calculation, the strong coupling constant \(\alpha_{s}\) is evaluated at \(Q^{2}\) and \(\alpha=1/128.0\). Through our numerical study, we aim to provide a non-trivial test of the all-order factorization structure derived in Eq. (7), offering an initial insight into what could be expected from the measurement.
In Fig. 4, we show the normalized nlojet++ \(\phi\) distribution result (in circle dots) at NLO \({\cal O}(\alpha_{s}^{2}+\alpha_{s}^{3})\) in the Breit frame for \(E_{l}=18\,{\rm GeV}\), \(E_{P}=275\,{\rm GeV}\), \(Q^{2}=100\,{\rm GeV}^{2}\). We choose \(x_{B}=0.01\). We set \(1.0<\theta_{b}<1.5\). We fit the un-normalized distribution with \(a+b\cos(2\phi)\) (solid curve) to observe an excellent \(\cos(2\phi)\) asymmetry to agree with our expectation from Eq. (7). Since both loop corrections and real emission up to 4 jet contributions are involved at this order, Fig. 4 acts as a highly non-trivial test of the all-order formalism we derived in Eq. (7).
In this calculation, we find the asymmetry induced by the linearly polarized gluon is
\(b/a\approx\ 2.69\,\%\), as can be seen from Eq. (7). Here, \(\Delta\hat{H}_{g}\) and \(\hat{H}\) can be calculated perturbatively. Therefore, in reality, measuring the azimuthal asymmetry could tell directly the "amount" of the linearly polarized gluons within the nucleon, once the unpolarized \(f_{\rm{EEC}}\) is measured. We note that there could be logarithmic \(\ln\theta_{a}\) correction to the normalized distribution showed in Fig. 4 which can be resummed through the evolution of \(f_{i,\rm{EEC}}\) and \(d_{g,\rm{EEC}}\) following the strategy in [30]. However, resummation does not change the tensor structure in Eq. (2) and thus leaves the \(\cos(2\phi)\) pattern unaffected. Meanwhile, in this fixed order simulation, the strong coupling constant \(\alpha_{s}\) is evaluated at \(Q\) which will underestimate the size of the non-perturbative contribution. Nevertheless, since the primary objective of this study is to examine the \(\cos(2\phi)\) structure rather than determine the exact size of \(d_{g,\rm{EEC}}\) (which in any case should be determined by future experiments), we have chosen to leave the \(\ln\theta_{a}\) resummation for a subsequent study. We emphasize that Fig. 4 encompasses all channels, while, in practical measurements, the heavy flavor tagging for the energy deposition in the central region [39] could enhance the sensitivity to the gluon channel and improve the significance of the observed azimuthal asymmetry.
Finally, we delve into the squeezed limit, defined as \(\theta_{a}\ll\theta_{b}\ll 1\). We require \(0.1<\theta_{b}<0.3\), \(x_{B}=0.03\), and the results for the NLO normalized cases are shown in Fig. 5. Once again, we observe the presence of azimuthal asymmetry of size \(b/a\approx-6.09\%\) indicating the persistence of the \(\cos(2\phi)\) structure. We note that the sign of the asymmetry flips in Fig. 4 and Fig. 5, due to different kinematics regimes and the significant modification of large logarithms \(\ln\theta_{b}\). We will carry out a detailed study in a future publication.
_Conclusion._ Our findings indicate that it is possible to directly investigate the linearly polarized gluons through the observation of helicity-dependent NEEC in the DIS process. This method measures the energy deposition asymmetry in the calorimeter, which arises from the spinning gluon confined inside the nucleon and manifests as a \(\cos(2\phi)\) correlation. Importantly, the \(\cos(2\phi)\) signature is preserved by rotational symmetry and holds at all orders. Consequently, the shape of the asymmetry remains robust against radiation/parton shower contamination and free of Sudakov suppression. The size of the asymmetry is to be determined by future experimental analysis and will provide us with a unique opportunity to determine the significance of gluon polarization at the current and future electron-ion facilities. Moreover, the absence of a polarized beam requirement suggests that it may be feasible to experimentally verify the factorization formalism using the available HERA data. Looking ahead, we plan to present the evolution of the helicity-dependent NEEC, as outlined in [30], to make all-order predictions for the azimuthal distribution. However, we do not anticipate any qualitative modifications to the results presented in this study.
_Acknowledgement._ We are grateful to Dingyu Shao and Jian Zhou for their useful discussions. This work is supported by the Natural Science Foundation of China under contract No. 12175016 (X. L. L. and X. L.), No. 11975200 (H. X. Z.) and the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 (F.Y.).
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2308.01697 | A strict maximum principle for nonlocal minimal surfaces | In the setting of fractional minimal surfaces, we prove that if two nonlocal
minimal sets are one included in the other and share a common boundary point,
then they must necessarily coincide.
This strict maximum principle is not obvious, since the surfaces may touch at
an irregular point, therefore a suitable blow-up analysis must be combined with
a bespoke regularity theory to obtain this result.
For the classical case, an analogous result was proved by Leon Simon. Our
proof also relies on a Harnack Inequality for nonlocal minimal surfaces that
has been recently introduced by Xavier Cabr\'e and Matteo Cozzi and which can
be seen as a fractional counterpart of a classical result by Enrico Bombieri
and Enrico Giusti.
In our setting, an additional difficulty comes from the analysis of the
corresponding nonlocal integral equation on a hypersurface, which presents a
remainder whose sign and fine properties need to be carefully addressed. | Serena Dipierro, Ovidiu Savin, Enrico Valdinoci | 2023-08-03T11:32:43Z | http://arxiv.org/abs/2308.01697v1 | # A strict maximum principle
###### Abstract.
In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide.
This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result.
For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabre and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti.
In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.
OS was supported by the NSF grant DMS-2055617. EV was supported by the Australian Laureate Fellowship FL190100081.
stationary hypersurfaces under suitable assumptions and relying on an "extrinsic" Harnack inequality.
The goal of this article is to establish a strict maximum principle in the framework of nonlocal minimal surfaces. This result can be seen as a fractional counterpart of the main result in [10] and also leverages a recent Harnack Inequality for nonlocal minimal surfaces introduced by Xavier Cabre and Matteo Cozzi in [11].
The setting that we consider is that of nonlocal minimal surfaces, as introduced in [10]. Namely, given a (bounded and Lipschitz) domain \(\Omega\subset\mathbb{R}^{n}\) and a (measurable) set \(E\subseteq\mathbb{R}^{n}\), we say that \(E\) is \(s\)-minimal in \(\Omega\) if, whenever a (measurable) set \(F\) coincides with \(E\) outside \(\Omega\), it holds that
\[\operatorname{Per}_{s}(E,\Omega)\leqslant\operatorname{Per}_{s}(F,\Omega),\]
where
\[\operatorname{Per}_{s}(E,\Omega) := \iint_{(E\cap\Omega)\times(E^{c}\cap\Omega)}\frac{dx\,dy}{|x-y|^{n +s}}\] \[\qquad+\iint_{(E\cap\Omega)\times(E^{c}\cap\Omega^{c})}\frac{dx \,dy}{|x-y|^{n+s}}+\iint_{(E\cap\Omega^{c})\times(E^{c}\cap\Omega)}\frac{dx\, dy}{|x-y|^{n+s}}.\]
Here above and in what follows, \(s\in(0,1)\) is a fractional parameter and the superscript "c" denotes the complement set in \(\mathbb{R}^{n}\).
Nonlocal minimal surfaces are a fascinating, and very difficult topic of investigation. While their regularity is known in the plane (see [11]) and up to dimension \(7\) when \(s\) is close to \(1\) (see [10]), the full regularity theory of these objects is not well-understood. Though no examples of singular sets are presently known, it is commonly believed that nonlocal minimal surfaces do develop singularities, therefore, for the validity of a strict maximum principle, in general it does not suffice to take into consideration only regular points.
In this setting, our main result is the following:
**Theorem 1.1**.: _Let \(r>0\) and \(p\in\mathbb{R}^{n}\). Let \(E_{1}\), \(E_{2}\) be \(s\)-minimal sets in \(B_{r}(p)\), with \(E_{1}\subseteq E_{2}\). Assume that \(p\in(\partial E_{1})\cap(\partial E_{2})\)._
_Then, \(E_{1}=E_{2}\)._
We recall that a better understanding of nonlocal minimal surfaces is important also in connection with hybrid heat equations (see [11]), geometric motions (see [12, 13, 14, 15, 16, 17, 18] ), nonlocal soap bubble problems (see [14, 15, 16]), capillarity problems (see [16, 17]), long-range phase coexistence models (see [11, 17]), etc.
Interestingly, the nonlocal perimeter functional interpolate the classical perimeter as \(s\nearrow 1\) (see [1, 18, 19, 20]) with a suitably weighted Lebesgue measure as \(s\searrow 0\) (see [11, 2]). In this spirit, on the long run, it is expected that nonlocal minimal surfaces can serve as auxiliary tools to better understand classical minimal surfaces as well (see [11, 12]) and can provide new perspectives on classical geometric objects by bringing forth tools alternative to, and different than, differential geometry (see [17]).
For other types of nonlocal minimal surfaces in which the interaction kernel is of integrable type, see [19].
The rest of this paper focuses on the proof of Theorem 1.1. Roughly speaking is that, by a blow-up and dimensional reduction argument, one reduces to the case in which the two \(s\)-minimal sets under consideration touch at a point which exhibit the same limit cone for both surfaces.
One then writes a geometric equation coming from the vanishing of the nonlocal mean curvature of one of these surfaces and a suitable translation of the other one and then scales to have a normalized picture in which the separation between these two sheets is of order one (assuming, for a contradiction, that they do not coincide to start with).
Some caveat is in place, since one has to switch from the notion of solution in the smooth or viscosity sense to the one in the weak sense: for this, some energy and capacity estimates will be required.
In this scenario, after a limit procedure, one aims at applying a suitable Harnack Inequality to obtain a contradiction between a linear separation close to the origin and the assumption that the two surfaces share the same1 tangent cone.
Footnote 1: As usual in geometric measure theory (see e.g. [10]), here and in what follows by the “same tangent cone” we mean that any blow-up sequence possesses a subsequence along which the two dilated surfaces share the same tangent cone in the limit.
However, a number of difficulties emerge, since the equations under consideration present some complicated kernels, are not defined everywhere, and produce a remainder which needs to be specifically analyzed (in particular, we will need to check that this remainder is bounded and has a sign, to be able to infer uniform Holder estimates on the normalized oscillation, pass them to the limit, obtain a limit inequality, and apply to it an appropriate version of the Harnack Inequality).
Interestingly, the regularity theory utilized in this process is twofold: first we use Holder estimates for a nonlocal equation which is not symmetric to perform a compactness argument on a sequence of normalized solutions which encode the distance between the original surfaces and thus obtain a positive supersolution on the limit cone for a fractional Jacobi operator, then we apply to this setting the geometric Harnack Inequality to obtain the desired conclusion.
A slightly more detailed and technical sketch of the proof will be presented in the forthcoming Section 2: after this, we focus on the rigorous arguments needed to make the actual proof work. More specifically, we will present in Section 3 a suitable geometric equation which will play a major role in our analysis. The interplay between viscosity and weak solutions of this equation will be discussed in Section 4, through some capacity and cut-off arguments (some technical estimates being deferred to Appendix A).
The geometric Holder estimates needed to pass to the limit the rescaled configuration in which the sheets are separated by an order one will be presented in Section 5 (actually, these estimates are of general flavor and can find application in other geometric problems as well).
To apply these Holder estimates one will also need a uniform bound on solutions of fractional equations in a geometric setting and this argument is contained in Section 6. In turn, in our case, this uniform bound will be a consequence of an integral bound, which is presented in Section 7.
The proof of Theorem 1.1 is thus completed in Section 8 (the details about the good set of regular boundary points utilized in the proof being contained in Appendix B).
## 2. The plan of the proof
The proof will combine the blow-up methods introduced in [13] with the fine analysis of the nonlocal setting needed in our context. The idea of the proof is to consider blow-up sequences of the minimal surfaces which approach the same limit cone at the origin and take a normal parameterization of these surfaces away from the singular set. We then look at the difference between these parameterizations, say \(w\) (or, more precisely, \(w_{k}\), since it depends on the step). The inclusion of the two sets guarantees that \(w\) has a sign, say \(w\geqslant 0\) (and, by contradiction, we can suppose that \(w>0\) at some point).
The function \(w\) may behave very badly in the vicinity of the singular set, thus we perform our analysis in the complement of a neighborhood of the singular set (we will shrink this neighborhood at the end of the proof, relying on the regularity theory of the nonlocal minimal surfaces). Moreover, since the convergence of \(w=w_{k}\) is only local, we restrict our attention to a given ball \(B_{R}\) (we will send \(R\to+\infty\) in the end).
Roughly speaking, the gist is to choose a point \(x_{\star}\) (in \(B_{R}\) and away from a given small neighborhood of the singular set) and a sequence of sets so that
\[2w_{k}(x_{\star})\geqslant t^{-1}w_{k}(tx_{\star}) \tag{2.1}\]
for all \(t\in(0,1]\): it is indeed possible to fulfill this inequality (or a slight modification of it) since the two minimal surfaces share the same tangent cone (hence their difference is "sublinear" at the origin, see Appendix B for full details about this technical construction). We can also normalize \(w_{k}\) at \(x_{\star}\) to be \(1\), i.e. look at the normalized function \(v_{k}:=\frac{w_{k}}{w_{k}(x_{\star})}\).
In this way, we see that \(w=w_{k}\) (and therefore \(v=v_{k}\)) satisfies a suitable nonlocal equation of geometric type (in view of Theorem 3.4) and we show that \(v\) remains locally bounded (owing to Lemma 6.1) and therefore Holder continuous in compact sets that avoid the singular set (thanks to Lemma 5.1).
We can thereby pick a convergent subsequence to a limit function \(v_{\infty}\) which is a positive supersolution to a suitable Jacobi operator on the cone (first in the viscosity sense and then in the distributional sense, due to Lemma 4.1). From this, we will apply the weak Harnack Inequality of [10] and deduce that \(v_{\infty}\) is locally bounded below by a universal constant and we contradict in this way the inequality in (2.1) (normalized and passed to the limit).
The details to carry over this plan are very technical and several complications arise from the nonstandard form of the integro-differential equations involved (such as the domain of integration not being flat and the kernels involved not being symmetric) and by the fact of dealing with non-standard domains of definitions (such as large balls with neighbourhoods of singular sets removed). These important technicalities occupy the rest of this paper.
## 3. A geometric equation
Now we consider the case in which two \(s\)-minimal sets, one included into the other, can be written locally as a graph in terms of another smooth hypersurface (concretely, in our case, the regular part of their common limit cone, though the arguments presented are of general flavor).
In this scenario, we obtain a geometric inequality for the difference between the parameterizations of the nonlocal minimal surfaces, which is reminiscent of the Jacobi field for nonlocal minimal surfaces (see e.g. [1, 13]). The advantage of this approach in general is that these Jacobi fields leverage additional cancellations with respect to the two individual equations for the nonlocal mean curvature of the two surfaces. The specialty of
our setting is however that the two nonlocal minimal surfaces are nice graphs with respect to the limit cone only away from the possible singularities of the cone (hence, in our application, we will have to consider graphs converging to the limit cone away from its singularities and carefully estimate the error terms).
Actually, the methods that we develop work in a greater generality, which is also useful to appreciate the geometric structures arising in the limit construction. Thus, we split our calculations in different regions of the space (namely, near a regular point in Lemma 3.1, in an intermediate ring in Lemma 3.2, and far away in Lemma 3.3) and we collect these calculations into the general form given by Theorem 3.4.
This type of results are influenced by the classical theory of minimal surfaces but present significant differences with respect to the classical case. For instance, in the local setting, one can apply directly the theory of Jacobi fields and obtain a linear equation describing the normal displacement of a surface approaching a limit cone (see equation (7) in [20]). Instead, in the nonlocal setting, remote interactions highly complicate the structure of the equations and a fine theory of cancellations is needed to obtain the convergence of the desired approximation and effective estimates on the remainder terms.
In the forthcoming calculations, we use the notation
\[\widetilde{\chi}_{E}(x):=\chi_{\mathbb{R}^{n}\setminus E}(x)-\chi_{E}(x)= \begin{cases}1&\text{ if }x\in\mathbb{R}^{n}\setminus E,\\ -1&\text{ if }x\in E.\end{cases}\]
Also, here below we will take into consideration suitable kernels defined on a hypersurface, specifically, a certain regular subset of \(\partial E_{1}\), with \(E_{1}\subseteq\mathbb{R}^{n}\): in this situation, given \(x_{0}\) on this hypersurface and a vector \(y\) (with small norm) in the linear tangent hyperplane at \(x_{0}\) we will denote by \(y_{x_{0}}^{+}\) the point on the hypersurface whose projection onto the affine tangent hyperplane at \(x_{0}\) coincides with \(x_{0}+y\) and by \(y_{x_{0}}^{-}\) the point on the hypersurface whose projection onto the affine tangent hyperplane at \(x_{0}\) coincides with \(x_{0}-y\), see Figure 1.
**Lemma 3.1**.: _Let \(E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}\)._
_Suppose that there exist \(\overline{X}\in\mathbb{R}^{n}\) and \(\delta>0\) such that \((\partial E_{1})\cap B_{\delta}(\overline{X})\) is a hypersurface of class \(C^{2}\). Let \(\nu\) be the unit external normal to \(E_{1}\) in \(B_{\delta}(\overline{X})\)._
_Suppose that_
\[(E_{2}\setminus E_{1})\cap B_{\delta}(\overline{X})=\Big{\{}x+t\nu(x),\text{ with }x\in\partial E_{1},\;t\in\big{[}0,w(x)\big{)}\Big{\}}\cap B_{\delta}( \overline{X}),\]
Figure 1. Projections along the tangent hyperplane.
_for some function \(w\) of class \(C^{2}\) with values in \([0,\delta/4]\)._
_Given \(x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})\), let_
\[\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu(x_{0})\qquad\text{and}\qquad X_{0}:=x_{0}+ w(x_{0})\nu(x_{0}).\]
_Then, if \(\delta\) and \(\|w\|_{C^{2}(B_{\delta}(\overline{X}))}\) are small enough,_
\[\frac{1}{2}\int_{B_{\delta/4}(\overline{X})}\frac{\widetilde{ \chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\] \[\qquad=\int_{(\partial E_{1})\cap B_{\delta/4}(\overline{X})} \left(w(x_{0})-w(x)\right)K_{1}(x,x_{0})\,d\mathcal{H}_{x}^{n-1}\] \[\qquad\qquad-w(x_{0})\int_{(\partial E_{1})\cap B_{\delta/4}( \overline{X})}\left(1-\nu(x_{0})\cdot\nu(x)\right)K_{2}(x,x_{0})\,d\mathcal{ H}_{x}^{n-1}+O(w^{2}(x_{0})), \tag{3.1}\]
_where \(K_{j}:((\partial E_{1})\cap B_{\delta/4}(\overline{X}))^{2}\to[0,+\infty]\), with \(j\in\{1,2\}\), satisfies, for a suitable \(C\geqslant 1\), that_
\[\text{for all $x\neq x_{0}$, we have that }\ \frac{1}{C|x-x_{0}|^{n+s}}\leqslant K_{j}(x,x_{0}) \leqslant\frac{C}{|x-x_{0}|^{n+s}},\] \[\text{and, for all $y\in B_{\delta}\setminus\{0\}$, }K_{j}(y_{x_{0}}^{+},x_{0})-K_{j}(y_{x_{0}}^{-},x_{0})=O(|y|^{1-n-s}). \tag{3.2}\]
_The "big \(O\)" terms here above depend only on the curvatures of \(\partial E_{1}\) in \(B_{9\delta/10}(\overline{X})\), on \(\|w\|_{C^{2}(B_{\delta}(\overline{X}))}\), on \(n\) and \(s\)._
_Also, the kernel \(K_{j}\) approaches, up to a normalizing constant, the kernel \(|x-x_{0}|^{-n-s}\) when \(\|w\|_{C^{2}(B_{\delta}(\overline{X}))}\) tends to zero._
Proof.: Given \(X\in B_{\delta}(\overline{X})\), we write \(X=x+t\nu(x)\), with \(x\in\partial E_{1}\) and \(t\in\mathbb{R}\). If \(\delta>0\) is small enough, the map linking \(X\) to \((x,t)\) is a diffeomorphism and the intersection with \(\partial E_{1}\) occurs only for \(t=0\).
Thus, we denote the corresponding geometric Jacobian determinant by \(J(x,t)\) (set to zero when \(x+t\nu(x)\not\in B_{\delta/4}(\overline{X})\)) and we have that
\[\begin{split}&\int_{B_{\delta/4}(\overline{X})}\frac{ \widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^ {n+s}}\,dX\\ &\qquad=\iint_{((\partial E_{1})\cap B_{\delta/4}(\overline{X})) \times\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{ \widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J(x,t)\,d\mathcal{H}_{x}^{n-1}\,dt.\end{split} \tag{3.3}\]
Now, given \(x\in(\partial E_{1})\cap B_{\delta/4}(\overline{X})\), we have that
\[\widetilde{\chi}_{E_{2}}(x+t\nu(x))=\begin{cases}1&\text{ if }t\geqslant w(x),\\ -1&\text{ if }t<w(x).\end{cases}\]
Also, \(\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=-1\) if and only if \(x+t\nu(x)\in\widetilde{E}_{1}\), and so if and only if \(x(t):=x+t\nu(x)-w(x_{0})\nu(x_{0})\in E_{1}\).
Now, the signed distance of \(x(t)\) to the tangent hyperplane of \(E_{1}\) at \(x\) is equal to
\[d(x):=(x(t)-x)\cdot\nu(x)=t-w(x_{0})\nu(x_{0})\cdot\nu(x).\]
Moreover, the projection of \(x(t)-x\) onto the tangent plane is
\[(x(t)-x)-\big{(}(x(t)-x)\cdot\nu(x)\big{)}\nu(x)=t\nu(x)-w(x_{0})\nu(x_{0})- \big{(}t-w(x_{0})\nu(x_{0})\cdot\nu(x)\big{)}\nu(x)\]
\[=-w(x_{0})\nu(x_{0})+w(x_{0})\nu(x_{0})\cdot\nu(x)\nu(x)=w(x_{0})\Big{(}\nu(x_{0}) \cdot\nu(x)\nu(x)-\nu(x_{0})\Big{)},\]
which has length equal to
\[w(x_{0})\Big{|}\nu(x_{0})\cdot\nu(x)\nu(x)-\nu(x_{0})\Big{|}=w(x_{0})\sqrt{1-( \nu(x_{0})\cdot\nu(x))^{2}}.\]
Since \(\partial E_{1}\) detaches at most quadratically from its tangent hyperplane, we have that
\[\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=\begin{cases}1&\text{ if }t \geqslant\widetilde{w}(x),\\ -1&\text{ if }t<\widetilde{w}(x),\end{cases}\]
where
\[\widetilde{w}(x):=w(x_{0})\nu(x_{0})\cdot\nu(x)+O\big{(}w^{2}(x_{0})|\nu(x)- \nu(x_{0})|^{2}\big{)}.\]
From these observations, we infer that, for all \(x\in(\partial E_{1})\cap B_{\delta/4}(\overline{X})\),
\[\int_{\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))}{|x-x_ {0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J(x,t)\,dt=h_{1}(w(x))-h_{2}(w(x))\] \[\text{and}\quad\quad\int_{\mathbb{R}}\frac{\widetilde{\chi}_{ \widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J( x,t)\,dt=h_{1}(\widetilde{w}(x))-h_{2}(\widetilde{w}(x)),\]
where
\[h_{1}(\xi) :=\int_{\xi}^{+\infty}\frac{J(x,t)\,dt}{|x-x_{0}+t\nu(x)-w(x_{0}) \nu(x_{0})|^{n+s}}\] \[\text{and}\,\,\,h_{2}(\xi) :=\int_{-\infty}^{\xi}\frac{J(x,t)\,dt}{|x-x_{0}+t\nu(x)-w(x_{0}) \nu(x_{0})|^{n+s}}. \tag{3.4}\]
We insert this information into (3.3) and we find that
\[\int_{B_{\delta/4}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}( X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\] \[\qquad=\int_{(\partial E_{1})\cap B_{\delta/4}(\overline{X})} \Big{(}h_{1}(w(x))-h_{1}(\widetilde{w}(x))-h_{2}(w(x))+h_{2}(\widetilde{w}(x) )\Big{)}\,d\mathcal{H}_{x}^{n-1}\] \[\qquad=\int_{(\partial E_{1})\cap B_{\delta/4}(\overline{X})} \Big{(}h(w(x))-h(\widetilde{w}(x))\Big{)}\,d\mathcal{H}_{x}^{n-1}, \tag{3.5}\]
where
\[h:=h_{1}-h_{2}. \tag{3.6}\]
We also observe that
\[-h^{\prime}(\xi)=-h^{\prime}_{1}(\xi)+h^{\prime}_{2}(\xi)=\frac{2J(x,\xi)}{|x- x_{0}+\xi\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}. \tag{3.7}\]
Therefore, letting
\[A(x_{0}):=\text{Id}+w(x_{0})\text{{I}I}(x_{0})\] \[\text{and}\quad\quad\sigma(x):=A(x_{0})(x-x_{0})=\big{(}\text{Id} +w(x_{0})\text{{I}I}(x_{0})\big{)}(x-x_{0}),\]
where \(I\!I\) denotes the second fundamental form along \(\partial E_{1}\), and using the change of variable \(\theta:=\frac{\xi-w(x_{0})}{|\sigma(x)|}\), we have that
\[\begin{split}&\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\\ =&-\frac{1}{2}\int_{a+w(x_{0})}^{b+w(x_{0})}h^{\prime }(\xi)\,d\xi\\ =&\int_{a+w(x_{0})}^{b+w(x_{0})}\frac{J(x,\xi)\,d\xi} {|x-x_{0}+\xi\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\\ =&\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{|\sigma( x)|\,J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}\,d\theta}{\Big{|}x-x_{0}+ \big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})\Big{|}^{n+s }}.\end{split} \tag{3.8}\]
We also remark that, as \(|x-x_{0}|\to 0\), the vector \(x-x_{0}\) becomes tangent to \(\partial E_{1}\), therefore
\[(x-x_{0})\cdot\nu(x)=O(|x-x_{0}|^{2}).\]
Moreover,
\[\nu(x)-\nu(x_{0})=I\!I(x_{0})(x-x_{0})+O(|x-x_{0}|^{2}).\]
For this reason,
\[\begin{array}{ll}&\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0}) \nu(x_{0})\\ =&\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\big{(}\nu(x_{0})+I\!I(x_{0})(x-x_{ 0})+O(|x-x_{0}|^{2})\big{)}-w(x_{0})\nu(x_{0})\\ =&w(x_{0})I\!I(x_{0})(x-x_{0})+\theta|\sigma(x)|\nu(x_{0})+O(|x-x_{0}|^{2}). \end{array}\]
As a result,
\[\begin{array}{ll}&\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu (x)-w(x_{0})\nu(x_{0})\Big{|}^{2}\\ =&\Big{|}\big{(}\text{Id}+w(x_{0})I\!I(x_{0})\big{)}(x-x_{0})+\theta|\sigma(x )|\nu(x_{0})+O(|x-x_{0}|^{2})\Big{|}^{2}\\ =&\Big{|}\sigma(x)+\theta|\sigma(x)|\nu(x_{0})+O(|x-x_{0}|^{2})\Big{|}^{2}\\ =&(1+\theta^{2})|\sigma(x)|^{2}+2\theta|\sigma(x)|\sigma(x)\cdot\nu(x_{0})+O (|x-x_{0}|^{3}).\end{array}\]
Actually, since
\[\begin{array}{ll}&\sigma(x)\cdot\nu(x_{0})=\big{(}w(x_{0})I\!I(x_{0})(x-x_{ 0})\big{)}\cdot\nu(x_{0})+O(|x-x_{0}|^{2})\\ &\qquad=w(x_{0})\big{(}\nu(x)-\nu(x_{0})\big{)}\cdot\nu(x_{0})+O(|x-x_{0}|^{2} )\\ &\qquad=w(x_{0})\big{(}\nu(x)\cdot\nu(x_{0})-1\big{)}+O(|x-x_{0}|^{2} )\\ &\qquad=-\frac{w(x_{0})}{2}\big{|}\nu(x)-\nu(x_{0})\big{|}^{2}+O(|x-x_{0}|^{2} )\\ &\qquad=O(|x-x_{0}|^{2}),\end{array}\]
we find that
\[\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0 })\Big{|}^{2}=(1+\theta^{2})|\sigma(x)|^{2}+O(|x-x_{0}|^{3}).\]
This gives that
\[\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0 })\nu(x_{0})\Big{|}^{-(n+s)}\] \[= \Big{(}(1+\theta^{2})|\sigma(x)|^{2}+O(|x-x_{0}|^{3})\Big{)}^{- \frac{n+s}{2}}\] \[= (1+\theta^{2})^{-\frac{n+s}{2}}|\sigma(x)|^{-(n+s)}\Big{(}1+O(|x -x_{0}|)\Big{)}^{-\frac{n+s}{2}}\] \[= (1+\theta^{2})^{-\frac{n+s}{2}}|\sigma(x)|^{-(n+s)}\Big{(}1+O(|x -x_{0}|)\Big{)}\]
and consequently, in view of (3.8),
\[\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\] \[=\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{|\sigma(x)|\,J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}\,\big{(}1+O(|x-x_{0}|)\big{)}}{(1+\theta^{2 })^{\frac{n+s}{2}}|\sigma(x)|^{n+s}}\,d\theta. \tag{3.9}\]
Now we take
\[a:=w(x)-w(x_{0})\qquad\text{and}\qquad b:=\widetilde{w}(x)-w(x_{0}).\]
We observe that
\[|w(x)-w(x_{0})|=O\big{(}\|\nabla w\|_{L^{\infty}(B_{\delta}(\overline{X}))}|x- x_{0}|\big{)}\]
and
\[|\widetilde{w}(x)-w(x_{0})| \leqslant w(x_{0})\big{|}\nu(x_{0})\cdot\nu(x)-1\big{|}+O\big{(}w^{2}(x_{0 })|x-x_{0}|^{2}\big{)}\] \[\leqslant O\big{(}w(x_{0})|x-x_{0}|^{2}\big{)}.\]
This says that, in our setting,
\[\frac{a}{|\sigma(x)|}=O\big{(}\|\nabla w\|_{L^{\infty}(B_{\delta}(\overline{X} ))}\big{)}\qquad\text{and}\qquad\frac{b}{|\sigma(x)|}=O\big{(}w(x_{0})|x-x_{0 }|\big{)}. \tag{3.10}\]
Therefore, with reference to the integral in (3.9), in this setting we have that \(\theta=O\big{(}\|w\|_{C^{1}(B_{\delta}(\overline{X}))}\big{)}\) and therefore
\[J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}=J\big{(}x,w(x_{0})\big{)}+O\big{(} |x-x_{0}|\big{)}=J\big{(}x_{0},w(x_{0})\big{)}+O\big{(}|x-x_{0}|\big{)},\]
with the latter quantity depending on the curvatures of \(\partial E_{1}\) in \(B_{9\delta/10}(\overline{X})\), on \(\|w\|_{C^{2}(B_{\delta}(\overline{X}))}\), on \(n\) and \(s\).
Gathering this and (3.9) we conclude that
\[\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\] \[=\frac{|\sigma(x)|\,\big{(}J(x_{0},w(x_{0}))+O(|x-x_{0}|)\big{)} }{|\sigma(x)|^{n+s}}\,\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{d\theta}{(1+ \theta^{2})^{\frac{n+s}{2}}}\] \[=\frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}\,\left( b\Phi\left(\frac{b}{|\sigma(x)|}\right)-a\Phi\left(\frac{a}{|\sigma(x)|} \right)\right),\]
where
\[\Phi(\alpha):=\frac{1}{\alpha}\int_{0}^{\alpha}\frac{d\theta}{(1+\theta^{2}) ^{\frac{n+s}{2}}}.\]
Now we let
\[\mathcal{A}_{1}(x,x_{0}):=\Phi\left(\frac{w(x)-w(x_{0})}{|\sigma(x)|}\right)\qquad \text{and}\qquad\mathcal{A}_{2}(x,x_{0}):=\Phi\left(\frac{\widetilde{w}(x)-w(x _{0})}{|\sigma(x)|}\right) \tag{3.11}\]
and we see that
\[\frac{h(w(x))-h(\widetilde{w}(x))}{2}\] \[= \frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}\left(( \widetilde{w}(x)-w(x_{0}))\mathcal{A}_{2}(x,x_{0})-(w(x)-w(x_{0}))\mathcal{A}_ {1}(x,x_{0})\right)\] \[= \frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}\] \[\times\Big{[}\Big{(}w(x_{0})(\nu(x_{0})\cdot\nu(x)-1)+O(w^{2}(x_{ 0})|x-x_{0}|^{2})\Big{)}\mathcal{A}_{2}(x,x_{0})-(w(x)-w(x_{0}))\mathcal{A}_ {1}(x,x_{0})\Big{]}\] \[= \Big{(}w(x_{0})(\nu(x_{0})\cdot\nu(x)-1)+O(w^{2}(x_{0})|x-x_{0}|^ {2})\Big{)}K_{2}(x,x_{0})-(w(x)-w(x_{0}))K_{1}(x,x_{0}),\]
where
\[K_{j}(x,x_{0}):=\frac{\big{(}J(x_{0},w(x_{0}))+O(|x-x_{0}|)\big{)}\,\mathcal{ A}_{j}(x,x_{0})}{|\sigma(x)|^{n+s}}.\]
By inserting this into (3.5), we have thereby obtained the desired result in (3.1), but we need to check (3.2) in order to ensure the necessary cancellations to make sense of the integrals involved.
To this end, we observe that the Jacobian \(J(x_{0},t)\) approaches \(1\) as \(t\to 0\). Moreover,
\[\Phi(\alpha)\leqslant\frac{1}{\alpha}\int_{0}^{\alpha}\,d\theta=1\]
and, if \(\alpha\in(-1,1)\),
\[\Phi(\alpha)\geqslant\frac{1}{\alpha}\int_{0}^{\alpha}\frac{d\theta}{2^{\frac{ n+s}{2}}}=\frac{1}{2^{\frac{n+s}{2}}}.\]
This, (3.10) and (3.11) entail that, in the range of interest, also \(\mathcal{A}_{j}(x,x_{0})\in\left[\frac{1}{2^{\frac{n+s}{2}}},1\right]\).
Besides, for small \(w(x_{0})\), we have that \(|\sigma(x)|\in\left[\frac{|x-x_{0}|}{2},2|x-x_{0}|\right]\) and these considerations establish the first claim in (3.2).
Additionally,
\[|\sigma(x_{0}\pm y)|=|\pm A(x_{0})y|=|A(x_{0})y|.\]
Furthermore,
\[\Phi(\alpha)=\frac{1}{\alpha}\int_{0}^{\alpha}\left(1-\frac{n+s}{2}\,\theta^{ 2}+O(\theta^{4})\right)\,d\theta=1-\frac{n+s}{6}\,\alpha^{2}+O(\alpha^{4}),\]
giving that, if \(\alpha\in(-1,1)\), then \(|\Phi^{\prime}(\alpha)|\leqslant C\) and accordingly
\[|\Phi(\alpha)-\Phi(\beta)|\leqslant C|\alpha-\beta|.\]
We also have that
\[w(y_{x_{0}}^{+})-w(y_{x_{0}}^{-})=O(|y|^{2})\]
and
\[\widetilde{w}(y_{x_{0}}^{\pm})-w(x_{0})=w(x_{0})\big{(}\nu(x_{0})\cdot\nu(y_{x _{0}}^{\pm})-1\big{)}+O(|y|^{2})=O(|y|^{2}).\]
As a consequence,
\[\left|\mathcal{A}_{1}(y_{x_{0}}^{+},x_{0})-\mathcal{A}_{1}(y_{x_{0} }^{-},x_{0})\right|\] \[\qquad=\left|\Phi\left(\frac{w(y_{x_{0}}^{+})-w(x_{0})}{|A(x_{0})y |}\right)-\Phi\left(\frac{w(y_{x_{0}}^{+})-w(x_{0})}{|A(x_{0})y|}\right)\right|\] \[\qquad\leqslant C\left|\frac{w(y_{x_{0}}^{+})-w(y_{x_{0}}^{-})}{|A (x_{0})y|}\right|\leqslant C|y|\]
and
\[\left|\mathcal{A}_{2}(y_{x_{0}}^{+},x_{0})-\mathcal{A}_{2}(y_{x_ {0}}^{-},x_{0})\right|\] \[\qquad=\left|\Phi\left(\frac{\widetilde{w}(y_{x_{0}}^{+})-w(x_{0} )}{|A(x_{0})y|}\right)-\Phi\left(\frac{\widetilde{w}(y_{x_{0}}^{-})-w(x_{0})} {|A(x_{0})y|}\right)\right|\leqslant C|y|.\]
The proof of (3.2) is thereby complete.
The fact that \(K_{1}\) and \(K_{2}\) can be seen as perturbations of the kernel \(|x-x_{0}|^{-n-s}\) when \(\left\|w\right\|_{C^{2}(B_{\delta}(\overline{X}))}\) tends to zero follows since in this asymptotics \(\sigma(x)\) approaches \(|x-x_{0}|\), \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) approach \(\Phi(0)=1\) and the Jacobian term approaches \(J(x,0)=1\).
**Lemma 3.2**.: _Let \(\delta\in(0,1)\), \(R>1+2\delta\), \(\overline{X}\in\mathbb{R}^{n}\) and \(E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}\)._
_Assume that there exists \(\mathcal{G}\subseteq B_{2R}(\overline{X})\) such that \((\partial E_{1})\cap\mathcal{G}\) is a hypersurface of class \(C^{2}\) with unit external normal \(\nu\)._
_Suppose that_
\[(E_{2}\setminus E_{1})\cap\mathcal{G}=\left\{x+t\nu(x),\text{ with }x\in \partial E_{1},\;t\in\left[0,w(x)\right)\right\}\cap\mathcal{G},\]
_for some function \(w\) of class \(C^{2}\) with values in \([0,\delta/4]\)._
_Assume also that_
(3.12) _the Hausdorff distance between \(\partial E_{1}\) and \(\partial E_{2}\) in \(B_{2R}(\overline{X})\) is less than \(\frac{\delta}{20}\)._
_Let \(x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})\), \(\nu_{0}\in\partial B_{1}\),_
\[\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu_{0}\qquad\text{and}\qquad X_{0}:=x_{0}+w( x_{0})\nu_{0}.\]
_Let also \(\mathcal{G}^{\prime}\Subset\mathcal{G}\) and assume that_
\[B_{2R}(\overline{X})\setminus\mathcal{G}^{\prime}\subseteq\bigcup_{j\in \mathbb{N}}B_{r_{j}}(p_{j}), \tag{3.13}\]
_for some balls \(\{B_{r_{j}}(p_{j})\}_{j\in\mathbb{N}}\), with_
\[\eta:=\sum_{j\in\mathbb{N}}r_{j}^{n-1}<1. \tag{3.14}\]
_Then, if \(\delta\) is small enough, and \(\eta\) and \(\left\|w\right\|_{L^{\infty}(B_{R}(\overline{X})\cap\mathcal{G})}\) are small compared to \(\delta\) and \(1/R\),_
\[\frac{1}{2}\int_{B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E }_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\] \[\qquad=\int_{((\partial E_{1})\cap\mathcal{G}^{\prime})\setminus B _{\delta/4}(\overline{X})}\left(w(x_{0})-w(x)\right)K_{1}(x,x_{0})\,d\mathcal{H }_{x}^{n-1}\]
\[-w(x_{0})\int_{((\partial E_{1})\cap\mathcal{G}^{\prime})\setminus B_{ \delta/4}(\overline{X})}\big{(}1-\nu_{0}\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d \mathcal{H}_{x}^{n-1}\] \[+\frac{1}{2}\int_{(B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X}))\setminus\mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)- \widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+O\left(w(x_{0})\eta\right),\]
_where the "big \(O\)" term here above depends only on \(\delta\), \(R\), \(n\) and \(s\)._
_In addition, \(K_{j}:(((\partial E_{1})\cap\mathcal{G}^{\prime})\setminus B_{\delta/4}( \overline{X}))^{2}\to[0,+\infty]\), with \(j\in\{1,2\}\), satisfies (3.2) and approaches, up to a normalizing constant, the kernel \(|x-x_{0}|^{-n-s}\) when \(\|w\|_{C^{2}(B_{R}(\overline{X})\cap\mathcal{G})}\) tends to zero._
Proof.: As in the proof of Lemma 3.1, given \(X\in(B_{R}(\overline{X})\setminus B_{\delta/4}(\overline{X}))\cap\mathcal{G}^ {\prime}\), we write \(X=x+t\nu(x)\), with \(x\in\partial E_{1}\) and \(t\in\mathbb{R}\).
By (3.12), we see that
\[\text{ if }|t|\geqslant\!\frac{\delta}{10},\text{ then }\widetilde{\chi}_{E_{2}}(x+t\nu(x))- \widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=0.\]
In this setting, if \(\delta>0\) is small enough,
\[\begin{split}&\int_{(B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X}))\cap\mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)- \widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\ =&\iint_{((\partial E_{1})\cap(B_{R}(\overline{X}) \setminus B_{\delta/4}(\overline{X}))\cap\mathcal{G}^{\prime})\times\mathbb{R }}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{ 1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,J(x,t)\,d\mathcal{H }_{x}^{n-1}\,dt,\end{split} \tag{3.15}\]
where \(J(x,t)\) denotes the geometric Jacobian determinant (set to zero when \(x+t\nu(x)\not\in B_{R}(\overline{X})\)).
Given \(x\in(\partial E_{1})\cap\mathcal{G}^{\prime}\), we have that
\[\widetilde{\chi}_{E_{2}}(x+t\nu(x))=\begin{cases}1&\text{ if }t\geqslant w(x),\\ -1&\text{ if }t<w(x).\end{cases} \tag{3.16}\]
Moreover, \(\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=-1\) if and only if \(x+t\nu(x)\in\widetilde{E}_{1}\), and so if and only if \(x(t):=x+t\nu(x)-w(x_{0})\nu_{0}\in E_{1}\).
Now, the signed distance of \(x(t)\) to the tangent hyperplane of \(E_{1}\) at \(x\) is equal to
\[d(x):=(x(t)-x)\cdot\nu(x)=t-w(x_{0})\nu_{0}\cdot\nu(x).\]
Moreover, the projection of \(x(t)-x\) onto the tangent plane is
\[(x(t)-x)-\big{(} (x(t)-x)\cdot\nu(x)\big{)}\nu(x)=t\nu(x)-w(x_{0})\nu_{0}-\big{(}t -w(x_{0})\nu_{0}\cdot\nu(x)\big{)}\nu(x)\] \[=w(x_{0})\nu_{0}\cdot\nu(x)\nu(x)-w(x_{0})\nu_{0}\]
which has length equal to
\[\ell(x):=\sqrt{w^{2}(x_{0})-(w(x_{0})\nu_{0}\cdot\nu(x))^{2}}.\]
We stress that \(\ell(x)\leqslant w(x_{0})\), hence if \(w(x_{0})\) is small enough then \(B_{\ell(x)}(x)\) lies in \(\mathcal{G}\) and accordingly \(\partial E_{1}\) detaches at most quadratically from its tangent hyperplane in this region. This gives that
\[\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=\begin{cases}1&\text{ if }t \geqslant\widetilde{w}(x),\\ -1&\text{ if }t<\widetilde{w}(x),\end{cases} \tag{3.17}\]
where
\[\widetilde{w}(x):=w(x_{0})\nu_{0}\cdot\nu(x)+O\big{(}w^{2}(x_{0})\big{)}.\]
By using (3.16) and (3.17), and in the notation of (3.4) and (3.6) (with \(\nu(x_{0})\) replaced by \(\nu_{0}\) here), we deduce that, if \(x\in(\partial E_{1})\cap(B_{R}(\overline{X})\setminus B_{\delta/4}(\overline{ X}))\cap\mathcal{G}^{\prime}\), then
\[\int_{\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))- \widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_ {0}|^{n+s}}\,J(x,t)\,dt=h(w(x))-h(\widetilde{w}(x))\]
and therefore
\[\begin{split}\iint_{((\partial E_{1})\cap(B_{R}(\overline{X}) \setminus B_{\delta/4}(\overline{X}))\cap\mathcal{G}^{\prime})\times\mathbb{ R}}&\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{ \widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,J(x,t )\,d\mathcal{H}_{x}^{n-1}\,dt\\ =\int_{(\partial E_{1})\cap(B_{R}(\overline{X})\setminus B_{ \delta/4}(\overline{X}))\cap\mathcal{G}^{\prime}}\Big{(}h(w(x))-h( \widetilde{w}(x))\Big{)}\,d\mathcal{H}_{x}^{n-1}.\end{split} \tag{3.18}\]
Thus, we define
\[K(x,x_{0}):=-\frac{h(w(x))-h(\widetilde{w}(x))}{2(w(x)-\widetilde{w}(x))}\]
and we remark that this kernel satisfies (3.2). Indeed, we can take here \(K_{1}=K_{2}=K\) and, since in our setting \(|x-x_{0}|\geqslant\frac{\delta}{100}\), we only need to check the second claim in (3.2).
And this claim holds true, because, by (3.7),
\[-\frac{h(w(x))-h(\widetilde{w}(x))}{w(x)-\widetilde{w}(x)}=-\int_ {0}^{1}h^{\prime}(tw(x)+(1-t)\widetilde{w}(x))\,dt\] \[=\int_{0}^{1}\frac{2J(x,tw(x)+(1-t)\widetilde{w}(x))}{|x-x_{0}+( tw(x)+(1-t)\widetilde{w}(x))\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,dt,\]
which, when \(|x-x_{0}|\geqslant\frac{\delta}{100}\) is bounded from above and below by \(\frac{1}{|x-x_{0}|^{n+s}}\), up to multiplicative constants.
We also deduce from (3.18) that
\[\begin{split}\frac{1}{2}\int_{(B_{R}(\overline{X})\setminus B_{ \delta/4}(\overline{X}))\cap\mathcal{G}^{\prime}}&\frac{ \widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0} |^{n+s}}\,dX\\ =\int_{(\partial E_{1})\cap(B_{R}(\overline{X})\setminus B_{ \delta/4}(\overline{X}))\cap\mathcal{G}^{\prime}}\big{(}\widetilde{w}(x)-w(x )\big{)}\,K(x,x_{0})\,d\mathcal{H}_{x}^{n-1}.\end{split} \tag{3.19}\]
Now we deal with the term
\[\begin{split}&\int_{(B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X}))\setminus\mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)- \widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\ =&\int_{(B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X}))\setminus\mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)- \widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+\int_{(B_{R}(\overline{X}) \setminus B_{\delta/4}(\overline{X}))\setminus\mathcal{G}^{\prime}}\frac{ \widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0 }|^{n+s}}\,dX.\end{split}\]
Specifically, to estimate the latter term, we define \(F_{1}\) as the symmetric difference between \(E_{1}\) and \(\widetilde{E}_{1}\) and we point out that, by (3.13),
\[\begin{split}\left|\int_{(B_{R}(\overline{X})\setminus B_{\delta/4} (\overline{X}))\setminus\mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}( X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|& \leqslant 2\int_{((B_{R}(\overline{X})\setminus B_{\delta/4}( \overline{X}))\setminus\mathcal{G}^{\prime})\cap F_{1}}\frac{dX}{|X-X_{0}|^{n +s}}\\ &\leqslant\frac{C}{\delta^{n+s}}\sum_{j\in\mathbb{N}}\big{|}B_{r_{ j}}(p_{j})\cap F_{1}\big{|}.\end{split} \tag{3.20}\]
To complete this estimate, it is useful to observe that, for all \(r>0\), all \(\tau\in\mathbb{R}^{n}\) and all measurable sets \(L\), we have that
\[\begin{split}\Big{|}B_{r}&\cap\big{(}(L+\tau) \setminus L\big{)}\Big{|}=\left|\int_{B_{r}\cap(L+\tau)}dx-\int_{B_{r}\cap L} dx\right|=\left|\int_{(B_{r}-\tau)\cap L}\,dx-\int_{B_{r}\cap L}\,dx\right|\\ &=\Big{|}L\cap\big{(}(B_{r}-\tau)\setminus B_{r}\big{)}\Big{|} \leqslant\big{|}(B_{r}-\tau)\setminus B_{r}\big{|}\leqslant\big{|}B_{r+\tau} \setminus B_{r}\big{|}\leqslant Cr^{n-1}\tau.\end{split}\]
Plugging this information into (3.20) and exploiting (3.14), we conclude that
\[\left|\int_{(B_{R}(\overline{X})\setminus B_{\delta/4}(\overline{X}))\setminus \mathcal{G}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{ \widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|\leqslant\frac{Cw(x_{0})}{ \delta^{n+s}}\sum_{j\in\mathbb{N}}r_{j}^{n-1}\leqslant\frac{C\eta w(x_{0})}{ \delta^{n+s}}.\]
The desired result follows from this inequality and (3.19) (the asymptotics of the kernel being similar to those discussed in Lemma 3.1).
**Lemma 3.3**.: _Let \(\delta\in(0,1)\), \(R>1+2\delta\), \(\overline{X}\in\mathbb{R}^{n}\) and \(E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}\)._
_Let \(x_{0}\in B_{\delta/8}(\overline{X})\), \(\nu_{0}\in\partial B_{1}\), \(w_{0}\in[0,\delta/4]\)_
\[\widetilde{E}_{1}:=E_{1}+w_{0}\nu_{0}\qquad\text{and}\qquad X_{0}:=x_{0}+w_{0 }\nu_{0}.\]
_Then,_
\[\frac{1}{2}\int_{\mathbb{R}^{n}\setminus B_{R}(\overline{X})}\frac{\widetilde {\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\, dX=\frac{1}{2}\int_{\mathbb{R}^{n}\setminus B_{R}(\overline{X})}\frac{ \widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX +O\left(\frac{w_{0}}{R^{1+s}}\right),\]
_where the "big \(O\)" term here above depends only on \(n\) and \(s\)._
Proof.: We observe that
\[\begin{split}&\left|\int_{\mathbb{R}^{n}\setminus B_{R}( \overline{X})}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde {E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|\\ &\qquad=\left|\int_{\mathbb{R}^{n}\setminus B_{R}(\overline{X}) }\frac{\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX-\int_{\mathbb{R}^{n} \setminus B_{R}(\overline{X}-w_{0}\nu_{0})}\frac{\widetilde{\chi}_{E_{1}}(X)}{| X+w_{0}\nu_{0}-X_{0}|^{n+s}}\,dX\right|\\ &\qquad\leqslant\left|\int_{\mathbb{R}^{n}\setminus B_{R}( \overline{X})}\frac{\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX-\int_{ \mathbb{R}^{n}\setminus B_{R}(\overline{X}-w_{0}\nu_{0})}\frac{\widetilde{\chi }_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|+\frac{Cw_{0}}{R^{1+s}}\\ &\qquad\leqslant\frac{Cw_{0}}{R^{1+s}},\end{split}\]
up to renaming \(C\), as usual, from line to line.
Now we recall the notion of nonlocal mean curvature of a set \(E\) at a point \(X\in\partial E\), namely
\[H_{E}^{s}(X):=\int_{\mathbb{R}^{n}}\frac{\widetilde{\chi}_{E}(Y)}{|X-Y|^{n+s}} \,dY.\]
Putting together Lemmata 3.1, 3.2 and 3.3, we conclude that:
**Theorem 3.4**.: _Let \(E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}\). Let \(\delta\in(0,1)\), \(R>1+2\delta\) and \(\overline{X}\in\mathbb{R}^{n}\)._
_Suppose that \((\partial E_{1})\cap\mathcal{G}\) is a hypersurface of class \(C^{2}\), for some \(\mathcal{G}\subseteq\mathbb{R}^{n}\) such that \(\mathcal{G}\ni B_{2\delta}(\overline{X})\), and let \(\nu\) be the unit external normal to \(E_{1}\) in \(\mathcal{G}\)._
_Suppose that_
\[(E_{2}\setminus E_{1})\cap\mathcal{G}=\Big{\{}x+t\nu(x),\text{ with }x\in \partial E_{1},\;t\in\big{[}0,w(x)\big{)}\Big{\}}\cap\mathcal{G},\]
_for some function \(w\) of class \(C^{2}\) with values in \([0,\delta/4]\)._
_Assume also that the Hausdorff distance between \(\partial E_{1}\) and \(\partial E_{2}\) in \(B_{2R}(\overline{X})\) is less than \(\delta/20\)._
_Given \(x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})\), let_
\[\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu(x_{0})\qquad\text{and}\qquad X_{0}:=x_{0 }+w(x_{0})\nu(x_{0}).\]
_Let also \(\mathcal{G}^{\prime}\Subset\mathcal{G}\) with \(\mathcal{G}^{\prime}\Supset B_{2\delta}(\overline{X})\) and assume that_
\[B_{2R}(\overline{X})\setminus\mathcal{G}^{\prime}\subseteq\bigcup_{j\in \mathbb{N}}B_{r_{j}}(p_{j}),\]
_for some balls \(\{B_{r_{j}}(p_{j})\}_{j\in\mathbb{N}}\), with_
\[\eta:=\sum_{j\in\mathbb{N}}r_{j}^{n-1}<1. \tag{3.21}\]
_Then, if \(\delta\) is small enough, and \(\eta\) and \(\|w\|_{C^{2}(B_{R}(\overline{X})\cap\mathcal{G})}\) are small compared to \(\delta\) and \(1/R\),_
\[\begin{split}&\frac{H_{E_{2}}^{s}(X_{0})-H_{\widetilde{E}_{1}}^{s} (X_{0})}{2}=\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}}\big{(}w(x_{0})-w( x)\big{)}\,K_{1}(x,x_{0})\,d\mathcal{H}_{x}^{n-1}\\ &\qquad-w(x_{0})\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}} \big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d\mathcal{H}_{x}^{n-1} \\ &\qquad+\frac{1}{2}\int_{\mathbb{R}^{n}\setminus\mathcal{G}^{ \prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{ 0}|^{n+s}}\,dX+O\left(w(x_{0})\eta\right)+O\left(\frac{w(x_{0})}{R^{1+s}} \right),\end{split} \tag{3.22}\]
_where \(K_{j}:((\partial E_{1})\cap\mathcal{G}^{\prime})^{2}\to[0,+\infty]\), with \(j\in\{1,2\}\), satisfies, for a suitable \(C\Supset 1\), that_
\[\begin{split}&\text{for all }x\neq x_{0}\text{, we have that }\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant K_{j}(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n +s}},\\ &\text{and, for all }y\in B_{\delta}\setminus\{0\}\text{, }K_{j}(y_{x_{0}}^{+},x_{0})-K_{j}(y_{x_{0}}^{-},x_{0})=O(|y|^{1-n-s}).\end{split} \tag{3.23}\]
_All the "big \(O\)" terms here above depend only on the curvatures of \(\partial E_{1}\) in \(B_{9\delta/10}(\overline{X})\), on \(\|w\|_{C^{2}(B_{R}(\overline{X})\cap\mathcal{G})}\), on \(\delta\), \(R\), \(n\) and \(s\), except for the last one in (3.22), which depends only on \(n\) and \(s\)._
_Also, the kernel \(K_{j}\) approaches, up to a normalizing constant, the kernel \(|x-x_{0}|^{-n-s}\) when \(\|w\|_{C^{2}(B_{R}(\overline{X})\cap\mathcal{G})}\) tends to zero._
## 4. Cut-off arguments
Having obtained a geometric equation for a nonnegative function in Theorem 3.4, our objective would be to apply a Harnack Inequality to it (actually, to a suitable limit of it, and this will indeed be implemented in the forthcoming Section 8). This is however rather delicate, since in principle, in our setting, the equation will only be valid in a pointwise sense
along the regular part of a limit \(s\)-minimal cone, but we do not have any information about the equation along the singular points of this cone.
Also, the equation obtained needs to be set into a distributional framework, which, again, is not for free since the quantities involved may explode at singular points, not allowing for a formulation in a suitable energy space.
These difficulties can be overcome by an appropriate cut-off argument, based on a fine covering of the singular points (if any) of the limit cone. Namely, the regularity theory of \(s\)-minimal cones (see [13, 14]) will allow us to confine these singular points within a small region, which can be covered by suitable balls that produce a negligible energy term.
Similar difficulties arose in [19, Section 6] and this issue was solved there in Lemma 6.3 via a bespoke capacity argument. We provide here a self-contained approach. The technical result that we use goes as follows:
**Lemma 4.1**.: _Let \(\mathcal{R}\subset\mathbb{R}^{n}\) and \(S:=\overline{\mathcal{R}}\setminus\mathcal{R}\). Assume that \(\mathcal{R}\) is locally a \(C^{2}\)-smooth hypersurface and that \(S\) is closed and has Hausdorff dimension \(d\leqslant n-3\)._
_Assume also that, for every \(r>0\) and \(p\in\mathbb{R}^{n}\),_
\[\mathcal{H}^{n-1}(\mathcal{R}\cap B_{r}(p))\leqslant Cr^{n-1}, \tag{4.1}\]
_for some \(C>0\)._
_Let \(u:\mathcal{R}\to[0,+\infty)\) be such that, for every \(x\in\mathcal{R}\),_
\[\int_{\mathcal{R}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1} \geqslant 0. \tag{4.2}\]
_Given \(Q>0\), let \(u_{Q}:=\min\{u,Q\}\)._
_Then, for every \(\zeta\in W^{1,\infty}(\mathbb{R}^{n},[0,+\infty))\) with support of \(\zeta\big{|}_{\mathcal{R}}\) contained in \(\mathcal{R}\), we have that_
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y))(\zeta(x)-\zeta( y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}\geqslant 0. \tag{4.3}\]
_Moreover, for every \(R_{0}>0\),_
\[\iint_{(\mathcal{R}\cap B_{R_{0}})\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y)) ^{2}}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}<+\infty \tag{4.4}\]
_and, for every \(\phi\in C_{0}^{\infty}(\mathbb{R}^{n},[0,+\infty))\),_
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y))(\phi(x)-\phi(y) )}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}\geqslant 0. \tag{4.5}\]
For the reader's facility, the rather technical proof of Lemma 4.1 will be given in Appendix A.
## 5. Holder estimates for fractional operators in a geometric setting
Here we present a Holder regularity result in a geometric setting. Namely, differently from the cases already treated in the literature, the equation considered here is defined by an integral on a portion of a hypersurface and the kernel is not necessarily symmetric, but only symmetric up to a suitable remainder.
**Lemma 5.1**.: _Let \(\mathcal{R}\) be a portion of a \(C^{2}\) hypersurface in \(B_{2}\) that is \(C^{2}\)-diffeomorphic to \(B_{2}\cap\{x_{n}=0\}\). Let \(\nu\) be the unit normal vector field of \(\mathcal{R}\) and \(f\in L^{\infty}(B_{2})\)._
_Let \(K:\mathcal{R}\times\mathcal{R}\to[0,+\infty]\) be such that_
\[\begin{split}&\text{for all $x\neq x_{0}$, we have that }\ \ \frac{1}{C|x-x_{0}|^{n+s}}\leqslant K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}}, \\ &\text{and, for all $y\in B_{\delta}\setminus\{0\}$, }|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant C|y|^{1-n-s}, \end{split} \tag{5.1}\]
_for some \(C\geqslant 1\)._
_Assume that, for all \(x_{0}\in\mathcal{R}\cap B_{1}\), the function \(v\in L^{\infty}(\mathcal{R})\) is a solution of_
\[\int_{\mathcal{R}}\big{(}v(x_{0})-v(x)\big{)}\,K(x,x_{0})\,d\mathcal{H}_{x}^{n -1}=f(x_{0}).\]
_Then, \(v\) is Holder continuous in \(\mathcal{R}\cap B_{1/2}\). More precisely,_
\[\|v\|_{C^{\alpha}(\mathcal{R}\cap B_{1/2})}\leqslant C_{0}\Big{(}\|f\|_{L^{ \infty}(B_{2})}+\|v\|_{L^{\infty}(\mathcal{R})}\Big{)}, \tag{5.2}\]
_where \(\alpha\in(0,1)\) and \(C_{0}>0\) depend only on \(n\), \(s\), the regularity parameters of \(\mathcal{R}\) and the structural constant \(C\) of the kernel \(K\) in (5.1)._
Proof.: We let \(u:=v\chi_{B_{3/4}}\) and we see that, for all \(x_{0}\in\mathcal{R}\cap B_{2/3},\)
\[\begin{split}&\int_{\mathcal{R}}\big{(}u(x_{0})-u(x)\big{)}\,K(x,x _{0})\,d\mathcal{H}_{x}^{n-1}=\int_{\mathcal{R}}\big{(}v(x_{0})-v(x)\chi_{B_{3 /4}}(x)\big{)}\,K(x,x_{0})\,d\mathcal{H}_{x}^{n-1}\\ &\qquad=f(x_{0})+\int_{\mathcal{R}\setminus B_{3/4}}v(x)\,K(x,x _{0})\,d\mathcal{H}_{x}^{n-1}=:g(x_{0})\end{split}\]
and
\[\begin{split}&|g(x_{0})|\leqslant\|f\|_{L^{\infty}(B_{2})}+C\|v \|_{L^{\infty}(\mathcal{R})}\int_{\mathcal{R}\setminus B_{3/4}}\frac{d \mathcal{H}_{x}^{n-1}}{|x-x_{0}|^{n+s}}\\ &\qquad\qquad\leqslant C\Big{(}\|f\|_{L^{\infty}(B_{2})}+ \mathcal{H}^{n-1}(\mathcal{R})\|v\|_{L^{\infty}(\mathcal{R})}\Big{)}=:C_{ \star},\end{split}\]
up to renaming \(C\) from line to line.
We now take \(B_{2}^{\prime}:=B_{2}\cap\{x_{n}=0\}\) and a \(C^{2}\)-diffeomorphism \(\phi:\mathcal{R}\to B_{2}^{\prime}\). We can also adapt the diffeomorphism so that \(\phi(\mathcal{R}\cap B_{2/3})\) contains \(B_{3/4}^{\prime}\). We define \(U(y):=u(\phi^{-1}(y))\) and we observe that, for all \(y_{0}=\phi(x_{0})\in B_{3/4}^{\prime}\),
\[\begin{split}& G(y_{0}):=g(\phi^{-1}(y_{0}))=\int_{\mathcal{R}} \big{(}u(\phi^{-1}(y_{0}))-u(x)\big{)}\,K(x,\phi^{-1}(y_{0}))\,d\mathcal{H}_{x} ^{n-1}\\ &\qquad=\int_{B_{2}^{\prime}}\big{(}U(y_{0})-U(y)\big{)}\,K_{*}(y,y_{0})\,d\mathcal{H}_{y}^{n-1},\end{split} \tag{5.3}\]
where
\[K_{*}(y,y_{0}):=K(\phi^{-1}(y),\phi^{-1}(y_{0}))\,J(y), \tag{5.4}\]
for a suitable Jacobian function \(J\), which is bounded and bounded away from zero.
Thus, by (5.1), we have that \(K_{*}(y,y_{0})\) is bounded from above and below, up to constants, by \(|\phi^{-1}(y)-\phi^{-1}(y_{0})|^{-(n+s)}\), which in turn is comparable to \(|y-y_{0}|^{-(n+s)}\), since \(\phi\) is a diffeomorphism.
Additionally, if
\[L(y_{0},z):=K_{*}(y_{0}+z,y_{0})-K_{*}(y_{0}-z,y_{0}),\]
we have that
\[\begin{split}|L(y_{0},z)|&\leqslant|K(\phi^{-1}(y_{0}+z),\phi^{-1}(y_{0}))-K(\phi^{-1}(y_{0}-z),\phi^{-1}(y_{0}))|\,J(y_{0}+z)\\ &\qquad\qquad\qquad+|K(\phi^{-1}(y_{0}-z),\phi^{-1}(y_{0}))|\,|J( y_{0}+z)-J(y_{0}-z)|\\ &\leqslant C|K(\phi^{-1}(y_{0}+z),\phi^{-1}(y_{0}))-K(\phi^{-1}(y _{0}-z),\phi^{-1}(y_{0}))|\\ &\qquad\qquad\qquad\qquad+C|z|^{-n-s}\,|J(y_{0}+z)-J(y_{0}-z)|\\ &\leqslant C|z|^{1-n-s},\end{split} \tag{5.5}\]
up to renaming \(C\) line after line.
Actually, up to renaming \(G\) by an additional bounded function, if we extend \(U\) by zero outside \(B_{2}^{\prime}\) we can rewrite (5.3) in the form
\[\int_{\mathbb{R}^{n-1}}\left(U(y_{0})-U(y)\right)K_{*}(y,y_{0})\,d\mathcal{H }_{y}^{n-1}=G(y_{0}), \tag{5.6}\]
for all \(y_{0}\in B_{3/4}^{\prime}\).
We can thereby apply the regularity theory for integro-differential equations (see e.g. [20, Theorem 5.4 and Remark 4.4], or [13, Theorem 3.1], or [12]) and obtain the desired Holder estimate for \(U\), and therefore for \(u\), and then for \(v\).
## 6. An upper bound in a geometric setting
In the proof of our main result, as it will be presented in Section 8, a somewhat delicate issue comes from the possibility of extracting a convergent sequence from the renormalized distance between minimal sheets. Roughly speaking, the plan would be to remove an arbitrarily small neighborhood of the singular set and focus on an arbitrarily large ball, obtain estimates in this domain which are uniform with respect to the sequences under consideration (with constants possibly depending on the neighborhood of the singular set and on the radius of the large ball), pass the sequence to the limit, and only at the end of the argument shrink the neighborhood of the singular set and invade the space by larger and larger balls.
In this setting, Lemma 5.1 is instrumental to provide the necessary compactness. However, to use this result, one needs a bound in \(L^{\infty}\), as dictated by the right-hand side of (5.2).
Such a bound does not come completely for free, not even in the situation in which the distance between minimal sheets is normalized to be \(1\) at some point, due to the possible divergence of this function at the singular set and the correspondingly large values that this function could, in principle, attain in the domain under consideration. To get around such a difficulty, we present here an \(L^{\infty}\) bound in terms of an integral estimate (as made precise by (6.3) here below). This integral control will be then checked in our specific case via a sliding method (as made precise by the forthcoming Lemma 7.1).
This strategy will thus allow us to pass to the limit locally away from the singular set: summarizing, one needs to just establish an integral bound (that will come from Lemma 7.1), to deduce from it a bound in \(L^{\infty}\) (coming from the next result), and thus obtain a bound in \(C^{\alpha}\) (coming from Lemma 5.1), which in turns provides the desired compactness property (by the Arzela-Ascoli Theorem).
**Lemma 6.1**.: _Let \(\mathcal{R}\) be a portion of a \(C^{2}\) hypersurface in \(B_{2}\) that is \(C^{2}\)-diffeomorphic to \(B_{2}\cap\{x_{n}=0\}\). Let \(\nu\) the unit normal vector field of \(\mathcal{R}\)._
_Let \(K:\mathcal{R}\times\mathcal{R}\to[0,+\infty]\) be such that_
\[\begin{split}&\text{for all $x\neq x_{0}$, we have that }\ \ \frac{1}{C|x-x_{0}|^{n+s}}\leqslant K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}}, \\ &\text{and, for all $y\in B_{\delta}\setminus\{0\}$, }|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant C|y|^{1-n-s}, \end{split} \tag{6.1}\]
_for some \(C\geqslant 1\)._
_Assume that, for all \(x_{0}\in\mathcal{R}\cap B_{1}\), the function \(v\) satisfies_
\[\int_{\mathcal{R}}\left(v(x_{0})-v(x)\right)K(x,x_{0})\,d\mathcal{H}_{x}^{n-1 }-a(x_{0})v(x_{0})\leqslant M, \tag{6.2}\]
_for some \(M\geqslant 0\) and \(a\in L^{\infty}(\mathbb{R}^{n})\), and that_
\[\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}\leqslant M^{\prime}, \tag{6.3}\]
_for some \(M^{\prime}\geqslant 0\)._
_Then, \(v\) is bounded from above in \(\mathcal{R}\cap B_{1/2}\), with_
\[\sup_{\mathcal{R}\cap B_{1/2}}v\leqslant C_{0}\left(1+M+M^{\prime}\right),\]
_with \(C_{0}>0\) depending only on \(n\), \(s\), \(\|a\|_{L^{\infty}(\mathbb{R}^{n})}\), the regularity parameters of \(\mathcal{R}\) and the structural constant \(C\) of the kernel \(K\) in (6.1)._
Proof.: The argument presented here extends to the geometric framework and for more general kernels a method of proof utilized in [1, Lemma 5.2]. Specifically, recalling the notation \(B_{r}^{\prime}:=B_{r}\cap\{x_{n}=0\}\), as in (5.4) we consider a \(C^{2}\)-diffeomorphism \(\phi:\mathcal{R}\to B_{2}^{\prime}\), define \(V(y):=v(\phi^{-1}(y))\) and look at the corresponding integral equation driven by the kernel \(K_{*}\)
We look at the function \(B_{1}^{\prime}\ni y\mapsto\Psi(y):=\Theta\,(1-|y|^{2})^{-n}\), with \(\Theta>0\) suitably large, and we slide the graph of \(\Psi\) by above in \(B_{1}^{\prime}\) till we touch the graph of \(V\). That is, we find \(t\in\mathbb{R}\) such that \(V\leqslant\Psi+t\) in \(B_{1}^{\prime}\) with equality holding at some \(q\in B_{1}^{\prime}\). We can assume that \(t\geqslant 0\) (otherwise, \(v\leqslant\Psi+t\leqslant\Psi\) and this would give the desired bound).
Let also \(d:=1-|q|\). Let \(r\in\left(0,\frac{d}{2}\right)\) to be taken suitably small. We consider the tangent plane of the barrier \(\Psi+t\) at \(q\), namely the linear function
\[\ell(y):=\nabla\Psi(q)\cdot(y-q).\]
It would be desirable to freely subtract tangent planes in the operators, but this is not possible in our framework, due to the lack of symmetry of the kernel, therefore we need to take care of an additional remainder. Namely,
\[\left|2\int_{B_{r}^{\prime}(q)}\ell(y)\,K_{*}(y,q)\,d\mathcal{H} _{y}^{n-1}\right|=\left|2\nabla\Psi(q)\cdot\int_{B_{r}^{\prime}(q)}(y-q)\,K_{* }(y,q)\,d\mathcal{H}_{y}^{n-1}\right|\] \[\qquad=\left|\nabla\Psi(q)\cdot\int_{B_{r}^{\prime}}z\big{(}K_{*} (q+z,q)-K_{*}(q-z,q)\big{)}\,d\mathcal{H}_{z}^{n-1}\right|.\]
Now, as observed in (5.5), it follows from (6.1) that
\[\left|K_{*}(q+z,q)-K_{*}(q-z,q)\right|\leqslant C|z|^{1-n-s}\]
As a result,
\[\left|2\int_{B^{\prime}_{r}(q)}\ell(y)\,K_{*}(y,q)\,d\mathcal{H}^{n- 1}_{y}\right|\leqslant|\nabla\Psi(q)|\,\int_{B^{\prime}_{r}}|z|\,\big{|}K_{*}(q+ z,q)-K_{*}(q-z,q)\big{|}\,d\mathcal{H}^{n-1}_{z}\] \[\leqslant C|\nabla\Psi(q)|\,\int_{B^{\prime}_{r}}|z|^{2-n-s}\,d \mathcal{H}^{n-1}_{z}\leqslant C|\nabla\Psi(q)|\,r^{1-s}.\]
Consequently, if \(q:=\phi(p)\) and \(\mathcal{R}_{r}(p):=\phi^{-1}(B^{\prime}_{r}(q))\),
\[\begin{split}&\int_{\mathcal{R}_{r}(p)}\big{(}v(p)-v(x)\big{)} \,K(x,p)\,d\mathcal{H}^{n-1}_{x}\\ &\qquad=\int_{B^{\prime}_{r}(q)}\big{(}V(q)-V(y)+\ell(y)\big{)}\,K _{*}(y,q)\,d\mathcal{H}^{n-1}_{y}-\int_{B^{\prime}_{r}(q)}\ell(y)\,K_{*}(y,q) \,d\mathcal{H}^{n-1}_{y}\\ &\qquad\geqslant-\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y )\big{)}\,K_{*}(y,q)\,d\mathcal{H}^{n-1}_{x}-C|\nabla\Psi(q)|\,r^{1-s}.\end{split} \tag{6.4}\]
Now we use the notation of positive and negative parts of a function, namely \(g=g_{+}-g_{-}\), where \(g_{+}:=\max\{g,0\}\) and \(g_{-}:=\max\{-g,0\}\), to see that, if \(y\in B^{\prime}_{r}(q)\),
\[|y|\leqslant|q|+|y-q|\leqslant 1-d+r\leqslant 1-\frac{d}{2}\]
and, as a consequence,
\[\big{(}V(y)-V(q)-\ell(y)\big{)}_{+} \leqslant \max\{\Psi(y)-\Psi(q)-\ell(y),0\}\] \[\leqslant \frac{C|y-q|^{2}}{d^{n+2}}.\]
For this reason,
\[\begin{split}&\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y) \big{)}_{+}\,K_{*}(y,q)\,d\mathcal{H}^{n-1}_{y}\leqslant\frac{C}{d^{n+2}}\int _{B^{\prime}_{r}(q)}|y-q|^{2}\,K_{*}(y,q)\,d\mathcal{H}^{n-1}_{y}\\ &\qquad\qquad\leqslant\frac{C}{d^{n+2}}\int_{0}^{r}\frac{d\rho}{ \rho^{s}}=\frac{Cr^{1-s}}{d^{n+2}}.\end{split} \tag{6.5}\]
Moreover,
\[\big{(}V(y)-V(q)-\ell(y)\big{)}_{-} \geqslant V(q)-V(y)+\ell(y)\] \[\geqslant \Psi(q)+t-V^{+}(y)+\ell(y)\]
and therefore
\[\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{*}(y, q)\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\geqslant\frac{1}{C}\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q) -\ell(y)\big{)}_{-}\,\frac{d\mathcal{H}^{n-1}_{y}}{|y-q|^{n+s}}\] \[\qquad\geqslant\frac{1}{Cr^{n+s}}\int_{B^{\prime}_{r}(q)}\big{(} V(y)-V(q)-\ell(y)\big{)}_{-}\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\geqslant\frac{1}{Cr^{n+s}}\int_{B^{\prime}_{r}(q)}\big{(} \Psi(q)+t-V^{+}(y)+\ell(y)\big{)}\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\geqslant\frac{1}{Cr^{n+s}}\left(\big{(}\Psi(q)+t\big{)}r^ {n-1}-\int_{B^{\prime}_{r}(q)}V^{+}(y)\,d\mathcal{H}^{n-1}_{y}\right)-\frac{C |\nabla\Psi(q)|}{r^{s}}\] \[\qquad\geqslant\frac{\Psi(q)+t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_ {\mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}^{n-1}_{x}-\frac{C|\nabla\Psi(q)|}{ r^{s}}.\]
Note that
\[\Psi(q)=\Theta\,(1-|q|^{2})^{-n}=\Theta\,(1+|q|)^{-n}(1-|q|)^{-n}\geqslant 2 ^{-n}\Theta d^{-n}\]
and
\[|\nabla\Psi(q)|=\frac{2n\Theta|q|}{(1-|q|^{2})^{n+1}}\leqslant\frac{2n\Theta} {d^{n+1}},\]
giving that
\[\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{* }(y,q)\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}- \frac{C}{r^{n+s}}\int_{\mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}^{n-1}_{x}- \frac{C\Theta}{d^{n+1}r^{s}}.\]
Notice that
\[\frac{\Theta}{Cd^{n}r^{1+s}}-\frac{C\Theta}{d^{n+1}r^{s}}=\frac{\Theta}{Cd^{ n}r^{s}}\left(\frac{1}{r}-\frac{C^{2}}{d}\right)>\frac{\Theta}{Cd^{n}r^{1+s}}\]
up to renaming \(C\), if \(r\) is smaller than a small constant times \(d\), and therefore
\[\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{* }(y,q)\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}- \frac{C}{r^{n+s}}\int_{\mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}^{n-1}_{x}.\]
Combining this estimate and (6.5), we infer that
\[\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}\,K_{*}(y, q)\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\leqslant-\frac{\Theta}{Cd^{n}r^{1+s}}-\frac{t}{Cr^{1+s}}+ \frac{C}{r^{n+s}}\int_{\mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}^{n-1}_{x}+ \frac{Cr^{1-s}}{d^{n+2}}.\]
From this and (6.4) we conclude that
\[\begin{split}&\int_{\mathcal{R}_{r}(p)}\left(v(p)-v(x)\right)K(x,p) \,d\mathcal{H}_{x}^{n-1}\\ &\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}- \frac{C}{r^{n+s}}\int_{\mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}- \frac{Cr^{1-s}}{d^{n+2}},\end{split} \tag{6.6}\]
where, as customary, we have renamed constants line after line.
Also,
\[v(p)=V(q)=\Psi(q)+t\geqslant\Psi(q)\geqslant 0\]
and therefore
\[\int_{\mathcal{R}\setminus\mathcal{R}_{r}(p)}\left(v(p)-v(x) \right)K(x,p)\,d\mathcal{H}_{x}^{n-1}\geqslant-C\int_{\mathcal{R}\setminus \mathcal{R}_{r}(p)}v^{+}(x)\,\frac{d\mathcal{H}_{x}^{n-1}}{|x-p|^{n+s}}\] \[\qquad\geqslant-\frac{C}{r^{n+s}}\int_{\mathcal{R}\setminus \mathcal{R}_{r}(p)}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}\geqslant-\frac{C}{r^{n+s} }\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}.\]
This, (6.2) and (6.6) yield that, if \(\Theta\) is large as specified above,
\[M \geqslant \int_{\mathcal{R}}\left(v(p)-v(x)\right)K(x,p)\,d\mathcal{H}_{x}^ {n-1}-a(p)v(p)\] \[\geqslant \frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}} \int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}-\frac{Cr^{1-s}}{d^{n+2}}-a( p)(\Psi(q)+t)\] \[\geqslant \frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s} }\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}-\frac{Cr^{1-s}}{d^{n+2}}-a( p)\Psi(q)\] \[\geqslant \frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s} }\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}-\frac{Cr^{1-s}}{d^{n+2}}- \frac{C\Theta}{d^{n}}.\]
Notice that for \(r\) small enough, the latter term can be reabsorbed. In particular, taking \(r=\alpha d\), for a small \(\alpha\in(0,1)\), we find that
\[M \geqslant \frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s} }\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}-\frac{Cr^{1-s}}{d^{n+2}}\] \[= \frac{\Theta}{C\alpha^{1+s}d^{n+1+s}}+\frac{t}{C\alpha^{1+s}d^{1+ s}}-\frac{C}{\alpha^{n+s}d^{n+s}}\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1} -\frac{C\alpha^{1-s}}{d^{n+1+s}}\] \[\geqslant \frac{t}{C\alpha^{1+s}d^{1+s}}+\frac{\Theta}{\alpha^{1+s}d^{n+1 +s}}\left[\frac{1}{C}-\frac{Cd}{\alpha^{n-1}\Theta}\int_{\mathcal{R}}v^{+}(x )\,d\mathcal{H}_{x}^{n-1}-\frac{C\alpha^{2}}{\Theta}\right].\]
The smallness of \(\alpha\) having played its role, we omit it from the notation from now on, absorbing it into the constants. In particular, if
\[\Theta:=C\left(1+\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}\right),\]
with \(C\) large enough, we have that
\[M\geqslant\frac{t}{Cd^{1+s}}+\frac{\Theta}{Cd^{n+1+s}}\geqslant\frac{t}{Cd^ {1+s}}.\]
All in all, we have found that, if \(\xi\in\mathcal{R}\cap B_{1/2}\),
\[v(\xi)=V(\phi(\xi)) \leqslant \Psi(\phi(\xi))+t\]
\[\leqslant C\left(1+M+\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1}\right)(1- |\phi(\xi)|^{2})^{-n}\] \[\leqslant C\left(1+M+\int_{\mathcal{R}}v^{+}(x)\,d\mathcal{H}_{x}^{n-1} \right),\]
up to renaming \(C\).
## 7. An integral bound in a geometric setting
In retrospect, the Holder regularity theory established in Lemma 5.1 relied on a specific hypothesis, namely an \(L^{\infty}\) bound, which in turn was reduced to an integral bound by means of Lemma 6.1.
However, in our setting, integral bounds do not come completely for free. As already mentioned, the complication arises from the fact that we need to apply this regularity theory to a normalized parameterization between minimal sheets: thus, in view of the possible divergence of normal parameterizations at singular points, \(L^{\infty}\) bounds, and even \(L^{1}\) bounds, may be nontrivial. The next result fills this gap and ensures an \(L^{1}\) bound, which, in our application, will lead to an \(L^{\infty}\) bound (via Lemma 6.1) and thus to a Holder estimate (owing to Lemma 5.1), from which one will deduce suitable compactness properties of the normalized distance of minimal sheets.
**Lemma 7.1**.: _Let \(\mathcal{R}\) be a portion of a \(C^{2}\) hypersurface in \(B_{2}\) that is \(C^{2}\)-diffeomorphic to \(B_{2}\cap\{x_{n}=0\}\). Let \(\nu\) be the unit normal vector field of \(\mathcal{R}\)._
_Let \(K:\mathcal{R}\times\mathcal{R}\to[0,+\infty]\) be such that_
\[\text{for all $x\neq x_{0}$, we have that }\ \frac{1}{C|x-x_{0}|^{n+s}}\leqslant K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s }},\] \[\text{and, for all $y\in B_{\delta}\setminus\{0\}$, $|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant C|y|^{1-n-s}$}, \tag{7.1}\]
_for some \(C\geqslant 1\)._
_Assume that, for all \(x_{0}\in\mathcal{R}\cap B_{1}\), the function \(v\) satisfies_
\[\int_{\mathcal{R}}\left(v(x_{0})-v(x)\right)K(x,x_{0})\,d\mathcal{H}_{x}^{n-1 }-a(x_{0})v(x_{0})\in[-M_{0},M_{0}], \tag{7.2}\]
_for some \(M_{0}>0\) and \(a\in L^{\infty}(\mathbb{R}^{n},[0,+\infty))\)._
_Assume also that \(v\) is nonnegative and that \(v(x_{\star})=1\), for some \(x_{\star}\in\mathcal{R}\cap B_{1}\)._
_Then,_
\[\int_{\mathcal{R}}v(x)\,d\mathcal{H}_{x}^{n-1}\leqslant C_{0},\]
_with \(C_{0}>0\) depending only on \(n\), \(s\), \(M_{0}\), \(\|a\|_{L^{\infty}(\mathbb{R}^{n})}\), the regularity parameters of \(\mathcal{R}\) and the structural constant \(C\) of the kernel \(K\) in (7.1)._
Proof.: We straighten \(\mathcal{R}\) by a diffeomorphism \(\phi:\mathcal{R}\to B_{2}^{\prime}:=B_{2}\cap\{x_{n}=0\}\) and set \(V(y):=v(\phi^{-1}(y))\). We let \(y_{\star}:=\phi(x_{\star})\) and slide the parabola \(-M|y-y_{\star}|^{2}\) from below till we touch the graph of \(V\) at some point \(y_{\sharp}\). Notice that
\[M|y_{\sharp}-y_{\star}|^{2}\leqslant V(y_{\sharp})+M|y_{\sharp}-y_{\star}|^{ 2}\leqslant V(y_{\star})=v(x_{\star})=1.\]
Thus, the parameter \(M>0\) is fixed (once and for all) sufficiently large such that the touching point \(y_{\sharp}\) lies in \(B_{3/2}^{\prime}\).
We remark that
\[P(y):=V(y_{\sharp})+M|y_{\sharp}-y_{\star}|^{2}-M|y-y_{\star}|^{2}\leqslant V(y)\]
and in particular \(P(y_{\sharp})=V(y_{\sharp})\geqslant 0\).
Hence, if \(\widetilde{V}:=V-P\), we have that \(\widetilde{V}(y_{\sharp})=0\) and thus, recalling the notation in (5.4),
\[\int_{B_{2}^{\prime}} \widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}=\int _{B_{2}^{\prime}}\left(\widetilde{V}(y)-\widetilde{V}(y_{\sharp})\right)K_{*}( y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}\] \[=\int_{B_{2}^{\prime}}\left(V(y)-V(y_{\sharp})\right)K_{*}(y,y_{ \sharp})\,d\mathcal{H}_{y}^{n-1}-\int_{B_{2}^{\prime}}\left(P(y)-P(y_{\sharp} )\right)K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}\] \[\leqslant\int_{B_{2}^{\prime}}\left(V(y)-V(y_{\sharp})\right)K_{ *}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}+C_{0}\] \[=\int_{\mathcal{R}}\left(v(x)-v(x_{\sharp})\right)K(x,x_{\sharp}) \,d\mathcal{H}_{x}^{n-1}+C_{0},\]
with \(C_{0}\) bounded in terms of \(n\), \(s\), the regularity parameters of \(\mathcal{R}\) and the structural constant \(C\) of the kernel \(K\) in (7.1).
From this observation and (7.2), we infer that
\[\int_{B_{2}^{\prime}}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{ n-1}\leqslant M_{0}-a(x_{\sharp})v(x_{\sharp})+C_{0}\leqslant C_{0},\]
with \(C_{0}\) now depending also on \(M_{0}\).
Since \(\widetilde{V}\geqslant 0\), we thereby conclude that
\[\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d \mathcal{H}_{y}^{n-1}\leqslant C_{0}.\]
We point out that, if \(y\in B_{1/100}^{\prime}(y_{\sharp})\),
\[K_{*}(y,y_{\sharp})\geqslant\frac{1}{C|y-y_{\sharp}|^{n+s}}\geqslant\frac{10 0^{n+s}}{C},\]
and thus
\[\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V}(y)\,d\mathcal{H}_{y}^{n-1} \leqslant\frac{C}{100^{n+s}}\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V} (y)\,K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}\leqslant C_{0}.\]
As a result, up to keeping renaming \(C_{0}\),
\[\int_{B_{1/100}^{\prime}(y_{\sharp})}V(y)\,d\mathcal{H}_{y}^{n-1}\leqslant\int _{B_{1/100}^{\prime}(y_{\sharp})}P(y)\,d\mathcal{H}_{y}^{n-1}+C_{0}\leqslant C _{0}. \tag{7.3}\]
Furthermore,
\[\int_{B_{1/100}^{\prime}(y_{\sharp})}\left(V(y_{\sharp})-V(y) \right)K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}\] \[\qquad\leqslant\int_{B_{1/100}^{\prime}(y_{\sharp})}\left(P(y_{ \sharp})-P(y)\right)K_{*}(y,y_{\sharp})\,d\mathcal{H}_{y}^{n-1}\leqslant C_{0}\]
and therefore one deduces from (7.2) that
\[C_{0} +\int_{B^{\prime}_{2}\setminus B^{\prime}_{1/100}(y_{\sharp})} \left(V(y_{\sharp})-V(y)\right)K_{*}(y,y_{\sharp})\,d\mathcal{H}^{n-1}_{y}\] \[\geqslant\int_{B^{\prime}_{2}}\left(V(y_{\sharp})-V(y)\right)K_{* }(y,y_{\sharp})\,d\mathcal{H}^{n-1}_{y}\] \[=\int_{\mathcal{R}}\left(v(x_{\sharp})-v(x)\right)K(x,x_{\sharp} )\,d\mathcal{H}^{n-1}_{x}\] \[\geqslant a(x_{\sharp})v(x_{\sharp})-C_{0}\] \[\geqslant-C_{0}.\]
This gives that
\[\int_{B^{\prime}_{2}\setminus B^{\prime}_{1/100}(y_{\sharp})}V(y) \,K_{*}(y,y_{\sharp})\,d\mathcal{H}^{n-1}_{y}\] \[\qquad\leqslant V(y_{\sharp})\int_{B^{\prime}_{2}\setminus B^{ \prime}_{1/100}(y_{\sharp})}\,K_{*}(y,y_{\sharp})\,d\mathcal{H}^{n-1}_{y}+C_ {0}\] \[=C_{0}\big{(}V(y_{\sharp})+1\big{)}\] \[\qquad\leqslant C_{0}\big{(}V(y_{\star})-M|y_{\sharp}-y_{\star}|^ {2}+1\big{)}\] \[\qquad\leqslant C_{0},\]
up to keeping renaming \(C_{0}\), and consequently
\[\int_{B^{\prime}_{2}\setminus B^{\prime}_{1/100}(y_{\sharp})}V(y)\,d\mathcal{ H}^{n-1}_{y}\leqslant C_{0}.\]
This and (7.3) yield the desired result.
We now present a useful variant2 of Lemma 7.1. Its utility in our context is that the geometric equation that we deal with will present an integral term coming from the nonlocal mean curvature of two minimal sets that we cannot reabsorb into "smooth" objects (due to the fact that the domain of integration is a "bad set", say \(\mathcal{B}\), containing far away points and points close to the singular set, for which any regularity information is missing). The next result however will provide a uniform control of such additional term.
Footnote 2: Actually, not only the proofs of Lemmata 7.1 and 7.2 are similar, but one could state just one single, albeit more complicated, result to condensate Lemmata 7.1 and 7.2 into a single statement. For the sake of simplicity, however, we prefer to keep the two statements separate.
**Lemma 7.2**.: _Let \(\mathcal{R}\) be a portion of a \(C^{2}\) hypersurface in \(B_{2}\) that is \(C^{2}\)-diffeomorphic to \(B_{2}\cap\{x_{n}=0\}\). Let \(\nu\) be the unit normal vector field of \(\mathcal{R}\)._
_Let \(K:\mathcal{R}\times\mathcal{R}\to[0,+\infty]\) be such that_
(7.4) _for all \[x\neq x_{0}\], we have that \[\ \frac{1}{C|x-x_{0}|^{n+s}}\leqslant K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n +s}},\] \[\text{and, for all }y\in B_{\delta}\setminus\{0\}\], \[|K(y^{+}_{x_{0}},x_{0})-K(y^{-}_{x_{0}},x_{0})|\leqslant C|y|^{1-n-s},\]
_for some \(C\geqslant 1\)._
_Assume that, for all \(\widetilde{x}\in\mathcal{R}\cap B_{3/2}\), the nonnegative function \(w\) satisfies_
\[\int_{\mathcal{R}}\left(w(\widetilde{x})-w(x)\right)K(x,\widetilde{x})\,d \mathcal{H}_{x}^{n-1}-\int_{\mathcal{B}}\frac{\Phi(X)}{|X-\widetilde{X}|^{n+s} }\,dX\geqslant-\mu, \tag{7.5}\]
_where \(\mathcal{B}\) is some measurable set of \(\mathbb{R}^{n}\setminus\mathcal{R}\), \(\Phi\in L^{\infty}(\mathbb{R}^{n},[0,1])\), \(\mu\geqslant 0\), and \(\widetilde{X}:=\widetilde{x}+w(\widetilde{x})\nu(\widetilde{x})\)._
_Assume that there exists a point \(x_{0}\) in \(\mathcal{R}\cap B_{3}\) whose distance from \(\mathcal{B}\) is bounded from below by some \(r_{0}>\widetilde{C}\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})}\), with \(\widetilde{C}>1\)._
_Then, there exists \(C_{0}>0\), depending only on \(n\), \(s\), \(r_{0}\), the regularity parameters of \(\mathcal{R}\) and the structural constant \(C\) of the kernel in (7.4), such that, if \(\widetilde{C}\geqslant C_{0}\),_
\[\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX\leqslant C_{0}\,\big{(} \|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})}+\mu\big{)}, \tag{7.6}\]
_where \(X_{0}:=x_{0}+w(x_{0})\nu(x_{0})\)._
Proof.: We straighten \(\mathcal{R}\) by a diffeomorphism \(\phi:\mathcal{R}\to B_{2}^{\prime}:=B_{2}\cap\{x_{n}=0\}\) and set \(W(y):=w(\phi^{-1}(y))\). We can also suppose for simplicity that \(\mathcal{R}\cap B_{1}\) is mapped by \(\phi\) in \(B_{1}^{\prime}\) and that \(\mathcal{R}\cap B_{3/2}\) is mapped by \(\phi\) in \(B_{3/2}^{\prime}\). Let \(x_{0}\in\mathcal{R}\cap B_{1}\) with \(B_{r_{0}}(x_{0})\cap\mathcal{B}=\varnothing\). Let also \(y_{0}:=\phi(x_{0})\) and notice that \(y_{0}\in B_{1}^{\prime}\).
Given \(M>0\), we slide the parabola \(P(y):=-M|y-y_{0}|^{2}\) from below till we touch \(W\) at some point \(\widehat{y}\). The touching condition between the slid parabola and the graph of \(W\) entails that
\[W(\widehat{y})+M|\widehat{y}-y_{0}|^{2}-M|y-y_{0}|^{2}\leqslant W(y). \tag{7.7}\]
Here we choose
\[M:=C_{*}^{2}\,\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})}\max\left\{1,\frac{1}{ r_{0}^{2}}\right\}, \tag{7.8}\]
with \(C_{*}\) to be chosen sufficiently large. In this way, by evaluating (7.7) at \(y:=y_{0}\),
\[|\widehat{y}-y_{0}|\leqslant\sqrt{\frac{W(y_{0})-W(\widehat{y})} {M}}\leqslant\sqrt{\frac{W(y_{0})}{M}}=\sqrt{\frac{w(x_{0})}{M}}\] \[\qquad\leqslant\sqrt{\frac{1}{C_{*}^{2}\max\left\{1,\frac{1}{r_{0 }^{2}}\right\}}}=\frac{1}{C_{*}}\min\{1,r_{0}\}.\]
In particular, if \(C_{*}\) is large enough, we have that \(\widehat{y}\in B_{3/2}^{\prime}\) and \(|\widehat{y}-y_{0}|<\frac{r_{0}}{10}\).
In this way, the point \(\widehat{x}:=\phi^{-1}(\widehat{y})\) belongs to \(\mathcal{R}\cap B_{3/2}\). We can therefore utilize (7.5) at \(\widehat{x}\), finding that
\[\begin{split}&\int_{\mathcal{B}}\frac{\Phi(X)}{|X-\widehat{X}|^{n+s }}\,dX\leqslant\int_{\mathcal{R}}\left(w(\widehat{x})-w(x)\right)K(x, \widehat{x})\,d\mathcal{H}_{x}^{n-1}+\mu\\ &\qquad=\int_{B_{2}^{\prime}}\left(W(\widehat{y})-W(y)\right)K_{* }(y,\widehat{y})\,d\mathcal{H}_{y}^{n-1}+\mu,\end{split} \tag{7.9}\]
where \(\widehat{X}:=\widehat{x}+w(\widehat{x})\nu(\widehat{x})\) and we used the notation in (5.4).
Notice that, on the one hand,
\[|X-\widehat{X}|\leqslant|X-X_{0}|+|X_{0}-\widehat{X}|\leqslant|X-X_{0}|+|x_{0 }-\widehat{x}|+2\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})}\]
\[\leqslant|X-X_{0}|+\frac{r_{0}}{10}+2\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})} \leqslant|X-X_{0}|+\frac{3r_{0}}{10}.\]
On the other hand, if \(X\in\mathcal{B}\),
\[|X-X_{0}|\geqslant|X-x_{0}|-\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})} \geqslant r_{0}-\|w\|_{L^{\infty}(\mathcal{R}\cap B_{2})}\geqslant\frac{9r_{0 }}{10}.\]
By combining these observations, we find that
\[|X-\widehat{X}|\leqslant|X-X_{0}|+\frac{3}{10}\cdot\frac{10}{9}|X-X_{0}| \leqslant C_{0}|X-X_{0}|,\]
for some \(C_{0}>1\).
This and (7.9), up to renaming \(C_{0}\), give that
\[\frac{1}{C_{0}}\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX\leqslant \int_{B_{2}^{\prime}}\left(W(\widehat{y})-W(y)\right)K_{*}(y,\widehat{y})\,d \mathcal{H}_{y}^{n-1}+\mu.\]
As a consequence, by (7.7),
\[\frac{1}{C_{0}}\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX \leqslant M\int_{B_{2}^{\prime}}\left(|y-y_{0}|^{2}-|\widehat{y}-y _{0}|^{2}\right)K_{*}(y,\widehat{y})\,d\mathcal{H}_{y}^{n-1}+\mu\] \[\leqslant C_{0}\big{(}M+\mu\big{)}.\]
The desired result thus follows in view of (7.8).
## 8. Completion of the proof of Theorem 1.1
We can now complete the proof of Theorem 1.1 by relying on the work carried out so far and on the Harnack Inequality by Cabre and Cozzi in [1].
Proof of Theorem 1.1.: Suppose, by contradiction, that \(E_{1}\neq E_{2}\). Without loss of generality, we may suppose that, in the notation recalled in footnote 1,
\[E_{1}\text{ and }E_{2}\text{ share the same tangent cone at the origin.} \tag{8.1}\]
This is a standard procedure in geometric measure theory, based on blow-up, dimensional reduction, and regularity theory. We recall the details for the facility of the reader.
Indeed, if (8.1) does not hold, we have a converging blow-up sequence for \(E_{1}\) approaching a cone \(\mathcal{C}_{1}\) with the corresponding blow-up sequence for \(E_{2}\) approaching a cone \(\mathcal{C}_{2}\neq\mathcal{C}_{1}\). We take a rotation \(\mathcal{R}_{1}\) of \(\mathcal{C}_{1}\) such that \(E_{1}^{(1)}:=\mathcal{R}_{1}\mathcal{C}_{1}\subseteq\mathcal{C}_{2}=:E_{2}^{( 1)}\) and there exists \(p^{(1)}\in(\partial E_{1}^{(1)})\cap(\partial E_{2}^{(1)})\cap(\partial B_{1})\).
We then consider a blow-up of \(E_{1}^{(1)}\) and \(E_{2}^{(1)}\) at \(p^{(1)}\), which produces two new \(s\)-minimal cones in \(\mathbb{R}^{n}\), which will be denoted by \(\mathcal{C}_{1}^{(1)}\) and \(\mathcal{C}_{2}^{(2)}\), with \(\mathcal{C}_{1}^{(1)}\subseteq\mathcal{C}_{2}^{(2)}\).
Now, two cases can occur. If \(\mathcal{C}_{1}^{(1)}=\mathcal{C}_{2}^{(2)}\), it suffices to replace \(p\), \(E_{1}\), \(E_{2}\), \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) respectively with \(p^{(1)}\), \(E_{1}^{(1)}\), \(E_{2}^{(1)}\), \(\mathcal{C}_{1}^{(1)}\) and \(\mathcal{C}_{2}^{(1)}\): in this way we have obtained (8.1) in this new configuration.
If instead \(\mathcal{C}_{1}^{(1)}\neq\mathcal{C}_{2}^{(1)}\), we observe that both these cones are cylinders over \(\mathbb{R}^{n-1}\). In this way, by the dimensional reduction (see [1]), we have found two \(s\)-minimal cones in \(\mathbb{R}^{n-1}\), which we denote by \(\widetilde{E}_{1}^{(2)}\) and \(\widetilde{E}_{2}^{(2)}\), such that \(\widetilde{E}_{1}^{(2)}\subseteq\widetilde{E}_{2}^{(2)}\) and \(\widetilde{E}_{1}^{(2)}\neq\widetilde{E}_{2}^{(2)}\) (and, up to an isometry, \(\mathcal{C}_{j}^{(1)}=\widetilde{E}_{j}^{(2)}\times\mathbb{R}\) for \(j\in\{1,2\}\)).
We thus repeat the previous algorithm, taking a rotation \(\mathcal{R}_{2}\) in \(\mathbb{R}^{n-1}\), such that \(E_{1}^{(2)}:=\mathcal{R}_{2}\widetilde{E}_{1}^{(2)}\subseteq\widetilde{E}_{2}^{ (1)}=:E_{2}^{(1)}\) and there exists \(p^{(2)}\in(\partial E_{1}^{(2)})\cap(\partial E_{2}^{(2)})\cap(\partial B_{1})\). Then, a blow-up at \(p_{2}\) produces two \(s\)-minimal cones \(\mathcal{C}_{1}^{(2)}\) and \(\mathcal{C}_{2}^{(2)}\), which either coincide (whence we replace \(p\), \(E_{1}\), \(E_{2}\), \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) respectively with \(p^{(2)}\), \(E_{1}^{(2)}\), \(E_{2}^{(2)}\), \(\mathcal{C}_{1}^{(2)}\) and \(\mathcal{C}_{2}^{(2)}\)) or not, in which case we apply again the dimensional reduction (reducing now to \(\mathbb{R}^{n-2}\)) and proceed.
This algorithm will stop at a certain dimension, producing the same \(s\)-minimal cones after a blow-up, since, by the regularity of \(s\)-minimal cones in dimension \(2\) (see [13]), we know that halfspaces are the only nontrivial minimal cones in \(\mathbb{R}^{2}\). The proof of (8.1) is thereby complete.
Hence, from now on, we will denote by \(\mathcal{C}\) the common tangent cone of \(E_{1}\) and \(E_{2}\) along a blow-up sequence that we are now going to specify. To this end, we write \(E_{2}\) in normal coordinates with respect to \(E_{1}\) at its regular points (this is possible up to an initial dilation), that is suppose that, at a set of regular points, \(E_{2}\setminus E_{1}\) has the form \(x+t\nu(x)\), with \(x\in\partial E_{1}\) and \(t\in\big{[}0,w_{0}(x)\big{)}\), for a suitable function \(w_{0}>0\). Then, by Corollary B.2, we find a set \(\mathcal{M}_{0}\) of regular points for \(E_{1}\) and an infinitesimal sequence of points \(z_{\star,k}\in\mathcal{M}_{0}\) such that
\[\frac{w_{0}(x)}{|x|}\leqslant\frac{2w_{0}(z_{\star,k})}{|z_{\star,k}|} \tag{8.2}\]
for all \(x\in\mathcal{M}_{0}\) with \(|x|\leqslant|z_{\star,k}|/2\). In addition, by (B.1),
(8.3) the distance of any point \(x\in\mathcal{M}_{0}\) from the singular set is at least \(\theta_{0}|x|\),
with \(\theta_{0}>0\).
What is more, by (B.2), we also know that, for all \(\rho\in(0,\rho_{0}]\),
\[\mathcal{M}_{0}\cap(\partial B_{\rho})\neq\varnothing. \tag{8.4}\]
Hence, we choose
\[r_{k}:=|z_{\star,k}| \tag{8.5}\]
and consider the corresponding blow-up sequences for \(j\in\{1,2\}\) defined by
\[E_{j,k}:=\frac{E_{j}-p}{r_{k}}. \tag{8.6}\]
We now apply Theorem 3.4. More specifically, the sets \(E_{1}\) and \(E_{2}\) in the statement of Theorem 3.4 are here the sets \(E_{1,k}\) and \(E_{2,k}\), as defined in (8.6), with \(k\) large enough.
The gist of the construction is that we cover the singular set of \(\mathcal{C}\) by a set of small balls (the fact that the singular set has dimension at most \(n-3\), due to [13, 14] guarantees that these balls can be chosen to satisfy (3.21) for \(\eta\) as small as we wish).
Since, for \(k\) large enough, \(E_{1,k}\) and \(E_{2,k}\) locally lie in a small neighborhood of \(\mathcal{C}\), we have that both \(E_{1,k}\) and \(E_{2,k}\) are smooth away from the above covering of the singular set of \(\mathcal{C}\), thanks to the improvement of flatness of nonlocal minimal surfaces put forth in [11, Corollary 4.4 and Theorem 6.1] (in this way, the convergence of \(E_{1,k}\) and \(E_{2,k}\) to the limit cone \(\mathcal{C}\) occurs in the \(C^{1,\alpha}\) sense away from any small neighborhood of the singular set of \(\mathcal{C}\), and actually in the \(C^{k}\) sense for any \(k\geqslant 2\), thanks to the bootstrap regularity in [14]).
This allows us to look at a "large" set3\(\mathcal{G}\) containing regular points as in Theorem 3.4 (and at a small shrinkage of it, namely \(\mathcal{G}^{\prime}\)) such that all the singular points of \(\partial\mathcal{C}\), \(\partial E_{1}\) and \(\partial E_{2}\)
in a large ball \(B_{R}\) lie outside \(\mathcal{G}\) (also, \(\mathcal{G}\) covers all \(B_{R}\), up to a negligible covering of balls, as specified in (3.21)).
Accordingly, for \(j\in\{1,2\}\) and \(k\) sufficiently large, one can parameterize \(E_{j,k}\) as a graph of class \(C^{2}\) in the normal direction of \(\mathcal{C}\) away from a small neighborhood of the singular set.
It is however technically simpler to recenter this parameterization on the "middle surface", namely to parameterize \(E_{2}=E_{2,k}\) in terms of \(E_{1}=E_{1,k}\), and this corresponds to the normal parameterization \(w=w_{k}\) in Theorem 3.4. We stress that since both \(E_{1,k}\) and \(E_{2,k}\) converge to \(\mathcal{C}\) locally in the \(C^{2}\)-sense away from the singular points of \(\partial\mathcal{C}\), we also know that the \(C^{2}\)-norm of \(w=w_{k}\) is as small as we like, provided that \(k\) is chosen large enough (possibly in dependence also of the covering of the singular set, which is now fixed once and for all).
We can therefore utilize Theorem 3.4 in this context and conclude that the normal parameterization \(w=w_{k}\) of \(E_{2}=E_{2,k}\) with respect to \(E_{1}=E_{1,k}\) satisfies the equation in (3.22), with suitable integrable kernels, as defined in (3.23). More specifically, since \(E_{1}\) and \(E_{2}\) are \(s\)-minimal sets and therefore \(H^{s}_{E_{1}}=H^{s}_{E_{2}}\) at every regular boundary point (see [10, Theorem 5.1]), we obtain from Theorem 3.4 that
\[0=\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}}\left(w(x_{0})- w(x)\right)K_{1}(x,x_{0})\,d\mathcal{H}^{n-1}_{x}\\ -w(x_{0})\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}}\left(1- \nu(x_{0})\cdot\nu(x)\right)K_{2}(x,x_{0})\,d\mathcal{H}^{n-1}_{x}\\ +\frac{1}{2}\int_{\mathbb{R}^{n}\setminus\mathcal{G}^{\prime}} \frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s }}\,dX+O\left(w(x_{0})\eta\right)+O\left(\frac{w(x_{0})}{R^{1+s}}\right), \tag{8.7}\]
where \(\nu=\nu_{k}\) is the outer unit normal of \(\partial E_{1}=\partial E_{1,k}\) at its regular points and \(X_{0}:=x_{0}+w(x_{0})\nu(x_{0})\).
Strictly speaking, the function \(w\) does not need to be defined everywhere in \(\mathbb{R}^{n}\) for the validity of (8.7), since such an equation does not consider values of \(w\) outside certain sets, but we will implicitly suppose that \(w\) is defined everywhere for definiteness.
We now take \(\mathcal{G}^{\prime\prime}\Subset\mathcal{G}^{\prime}\) and \(x_{0}\in(\partial E_{1})\cap B_{R/2}\cap\mathcal{G}^{\prime\prime}\). In this setting, owing to Lemma 7.2 and to the fact that \(E_{1}\subseteq E_{2}\), we see that, when the size of \(w\) is small,
\[0\leqslant\int_{\mathbb{R}^{n}\setminus\mathcal{G}^{\prime}}\frac{ \widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{E_{2}}(X)}{|X-X_{0}|^{n+s}}\,dX =O(\|w\|_{L^{\infty}(B_{R}\cap\mathcal{G})}). \tag{8.8}\]
Now, since \(E_{1}\neq E_{2}\), we have that \(w\) does not vanish identically, and in fact \(w>0\) on the regular part of \(\partial E_{1}\) (otherwise, we could compute the nonlocal mean curvature of \(E_{1}\) at that point in the smooth sense, as well as the one of \(E_{2}\) in the viscosity sense and get a contradiction). Therefore we can normalize \(w\) to take value \(1\) at a suitable point. Roughly speaking, it would be desirable to pick this point to reach the supremum of \(\frac{w(x)}{|x|}\), so to obtain a "linear" separation about the minimal sheets at the origin and thus contradict (8.1): however, this choice of the normalizing point may be impossible, since the supremum of \(\frac{w(x)}{|x|}\) might well be on the singular set (where \(w\) is not even defined) or dangerously close to it. This is the reason for which we have chosen an appropriate sequence of blow-up radii \(r_{k}\) in (8.2) and (8.5).
In this setting, we define \(x_{\star,k}:=\frac{z_{\star,k}}{|z_{\star,k}|}=\frac{z_{\star,k}}{r_{k}}\) and, in light of (8.3), we stress that the distance of \(x_{\star,k}\) from the singular set is at least \(\theta_{0}\). This ensures that, up to a subsequence, \(x_{\star,k}\) converges4 to a regular point \(x_{\star,\infty}\) of the limit cone as \(k\to+\infty\).
Footnote 4: Let us stress that the fact that the limit point \(x_{\star,\infty}\) remains bounded and bounded away from the singular set is essential to apply the Harnack Inequality in [10]: this is the reason for which we can well allow intermediate constants to depend on the radius of the large ball that we are considering, on the set of small balls chosen to cover the singular set, as well as on the regularity of the \(s\)-minimal sheets away of this cover, since we will pass \(k\to+\infty\) before removing this large ball and this small cover, but we will employ the normalization at the limit point \(x_{\star,\infty}\) to bound from below the last term in (8.13).
Furthermore, by (8.4), given \(\delta\in\left(0,\frac{1}{2}\right]\), to be chosen conveniently small in what follows, we can pick a point \(y_{k}\in\mathcal{M}_{0}\cap(\partial B_{\delta r_{k}})\). In this way, by (8.3), we also have that the distance of \(y_{\star,k}:=\frac{y_{k}}{r_{k}}\) from the singular set is at least \(\delta\theta_{0}\) and therefore, up to a subsequence, \(y_{\star,k}\) converges to a regular point \(y_{\star,\infty}\) of the limit cone as \(k\to+\infty\).
We also recall (8.2) and write that
\[\frac{w_{0}(y_{k})}{\delta r_{k}}=\frac{w_{0}(y_{k})}{|y_{k}|}\leqslant\frac{ 2w_{0}(z_{\star,k})}{|z_{\star,k}|}=\frac{2w_{0}(z_{\star,k})}{r_{k}}\]
and accordingly
\[w_{k}(y_{\star,k})\leqslant 2\delta w_{k}(x_{\star,k}). \tag{8.9}\]
We now define
\[v(x)=v_{k}(x):=\frac{w_{k}(x)}{w(x_{\star,k})}\]
and we divide (8.7) by \(w(x_{\star,k})=w_{k}(x_{\star,k})\), recalling also (8.8), to find that, for all \(x_{0}\in(\partial E_{1})\cap B_{R/2}\cap\mathcal{G}^{\prime\prime}\),
\[\begin{split} 0=\int_{(\partial E_{1})\cap\mathcal{G}^{ \prime}}&\left(v(x_{0})-v(x)\right)K_{1}(x,x_{0})\,d\mathcal{H}_{ x}^{n-1}\\ &-v(x_{0})\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}}\left(1 -\nu(x_{0})\cdot\nu(x)\right)K_{2}(x,x_{0})\,d\mathcal{H}_{x}^{n-1}\\ &-\psi(x_{0})+O\left(\eta\right)+O\left(\frac{1}{R^{1+s}}\right),\end{split} \tag{8.10}\]
where \(0\leqslant\psi=O(1)\).
Also, if \(\mathcal{G}^{\prime\prime\prime}\Subset\mathcal{G}^{\prime\prime}\), we have that, in \((\partial E_{1})\cap B_{R/4}\cap\mathcal{G}^{\prime\prime\prime}\),
\[0\leqslant v\leqslant O(1). \tag{8.11}\]
Indeed, we first observe that \(v\geqslant 0\), since \(w\geqslant 0\). The upper bound in (8.11) follows from the previously developed regularity theory. Specifically, one first employs Lemma 7.1 and finds that
\[\int_{(\partial E_{1})\cap\mathcal{G}^{\prime}}v(x)\,d\mathcal{H}_{x}^{n-1} \leqslant O(1).\]
This inequality provides the uniform bound corresponding to (6.3) in the present setting. Now, the upper bound in (8.11) is a consequence of Lemma 6.1. The proof of (8.11) is thereby complete.
Accordingly, by the Holder estimates in Lemma 5.1, we deduce that the \(C^{\alpha}\)-norm of \(v\) is bounded locally uniformly in \(k\) (and we stress that, since \(w=w_{k}\), also \(v=v_{k}\)).
Therefore, up to a subsequence, \(v_{k}\) converges locally uniformly in \(B_{R/4}\cap\mathcal{G}^{\prime\prime\prime}\) to a function \(v_{\infty}\). In view of (8.10), \(v_{\infty}\) satisfies, for all \(x_{0}\in(\partial E_{1})\cap B_{R/4}\cap\mathcal{G}^{\prime\prime\prime}\),
\[0\leqslant\int_{(\partial\mathcal{C})\cap\mathcal{G}^{\prime}} \frac{v_{\infty}(x_{0})-v_{\infty}(x)}{|x-x_{0}|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\] \[-v_{\infty}(x_{0})\int_{(\partial\mathcal{C})\cap\mathcal{G}^{ \prime}}\frac{1-\nu(x_{0})\cdot\nu(x)}{|x-x_{0}|^{n+s}}\,d\mathcal{H}_{x}^{n-1 }+O\left(\eta\right)+O\left(\frac{1}{R^{1+s}}\right).\]
Actually, we can now take \(R\) as large as we wish and invade all the space outside the singular set by the "good" domains \(\mathcal{G}\), \(\mathcal{G}^{\prime}\) and \(\mathcal{G}^{\prime\prime}\). In this way, we find that, on the regular part \(\operatorname{Reg}\mathcal{C}\) of \(\partial\mathcal{C}\),
\[0\leqslant\int_{\operatorname{Reg}\mathcal{C}}\left(v_{\infty}( x_{0})-v_{\infty}(x)\right)K_{1,\infty}(x,x_{0})\,d\mathcal{H}_{x}^{n-1}\] \[-v_{\infty}(x_{0})\int_{\operatorname{Reg}\mathcal{C}}\left(1- \nu(x_{0})\cdot\nu(x)\right)K_{2,\infty}(x,x_{0})\,d\mathcal{H}_{x}^{n-1}.\]
Now, we wish to apply Lemma 4.1. To this end, we observe that (4.1) is satisfied in our case by \(\mathcal{R}:=\operatorname{Reg}\mathcal{C}\), thanks to the perimeter estimates for nonlocal minimal surfaces, see [13, equation (1.16)].
Hence, from Lemma 4.1, we deduce that
\[\iint_{(\operatorname{Reg}\mathcal{C}\cap B_{2})\times\operatorname{Reg} \mathcal{C}}\frac{(v_{Q}(x)-v_{Q}(y))^{2}}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1 }\,d\mathcal{H}_{y}^{n-1}<+\infty\]
and, for every \(\phi\in C_{0}^{\infty}(\mathbb{R}^{n},[0,+\infty))\),
\[\iint_{\operatorname{Reg}\mathcal{C}\times\operatorname{Reg}\mathcal{C}}\frac {(v_{Q}(x)-v_{Q}(y))(\phi(x)-\phi(y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d \mathcal{H}_{y}^{n-1}\geqslant 0,\]
where \(v_{Q}:=\min\{v_{\infty},Q\}\).
Moreover,
\[v_{\infty}(x_{\star,\infty})=\lim_{k\to+\infty}v_{k}(x_{\star,k})=1. \tag{8.12}\]
It follows from the Harnack Inequality in [13, Theorem 1.7] that, for every compact subset \(\mathcal{K}\) of \(\operatorname{Reg}\mathcal{C}\),
\[\inf_{\mathcal{K}\cap B_{1}}v_{Q}\geqslant c\left(\int_{\mathcal{K}\cap B_{1} }v_{Q}(x)\,d\mathcal{H}_{x}^{n-1}+\int_{\mathcal{K}\setminus B_{1}}\frac{v_{Q }(x)}{|x|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\right),\]
for some \(c>0\), depending only on \(n\) and \(s\).
Sending \(Q\to+\infty\) and using (8.12), we thus obtain
\[m:=\inf_{\mathcal{K}\cap B_{1}}v_{\infty}\geqslant c\int_{\mathcal{K}\cap B_{ 1}}v_{\infty}(x)\,d\mathcal{H}_{x}^{n-1}>0. \tag{8.13}\]
As a result, for every compact subset \(\mathcal{K}\) of \(\operatorname{Reg}\mathcal{C}\cap B_{3/4}\), we have that
\[\inf_{x\in\mathcal{K}}\frac{w_{k}(x)}{w_{k}(x_{\star,k})}=\inf_{x\in\mathcal{ K}}v_{k}(x)\geqslant\frac{m}{2},\]
as long as \(k\) is large enough (possibly in dependence of \(\mathcal{K}\)).
In particular,
\[\frac{w_{k}(y_{\star,k})}{w_{k}(x_{\star,k})}\geqslant\frac{m}{2},\]
which gives a contradiction with (8.9) when \(\delta\) is sufficiently small.
## Appendix A Proof of Lemma 4.1
Here we give a self-contained energy argument to prove Lemma 4.1.
Proof of Lemma 4.1.: For the convenience of the reader, we split this proof into independent steps.
**Step 1. Proof of (4.3).** First of all, we have that, for every \(x\in\mathcal{R}\),
(A.1) \[\int_{\mathcal{R}}\frac{u_{Q}(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1 }\geqslant 0.\]
Indeed, if \(u(x)\geqslant Q\), the claim is obvious, since in this case \(u_{Q}(x)=Q\geqslant u_{Q}(y)\).
Therefore, we may suppose that \(u(x)<Q\). In this case,
\[\int_{\mathcal{R}}\frac{u_{Q}(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d \mathcal{H}_{y}^{n-1}=\int_{\mathcal{R}}\frac{u(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d \mathcal{H}_{y}^{n-1}\] \[\qquad\qquad=\int_{\mathcal{R}\cap\{u\geqslant Q\}}\frac{u(x)-Q} {|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}+\int_{\mathcal{R}\cap\{u<Q\}}\frac{u(x) -u(y)}{|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}\] \[\qquad\qquad=\int_{\mathcal{R}\cap\{u\geqslant Q\}}\frac{u(x)-Q} {|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}-\int_{\mathcal{R}\cap\{u\geqslant Q\}} \frac{u(x)-u(y)}{|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}+\int_{\mathcal{R}}\frac {u(x)-u(y)}{|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}\] \[\qquad\qquad=\int_{\mathcal{R}\cap\{u\geqslant Q\}}\frac{u(y)-Q} {|x-y|^{n+s}}\,d\mathcal{H}_{y}^{n-1}+\int_{\mathcal{R}}\frac{u(x)-u(y)}{|x-y| ^{n+s}}\,d\mathcal{H}_{y}^{n-1}\] \[\qquad\qquad\geqslant\int_{\mathcal{R}}\frac{u(x)-u(y)}{|x-y|^{n +s}}\,d\mathcal{H}_{y}^{n-1}\]
and accordingly (A.1) follows from (4.2).
Now we prove (4.3). To this end, as a byproduct of (A.1), we find that
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y))( \zeta(x)-\zeta(y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n -1}\] \[\qquad\qquad=2\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x) -u_{Q}(y))\,\zeta(x)}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{ n-1}\geqslant 0,\]
which establishes (4.3).
**Step 2. Covering arguments.** Let now \(\varepsilon\in(0,1)\), to be taken as small as we wish here below. From our assumption on \(S\), if \(D\in(d,n-2-s]\), we know that \(S\) is contained in the union of some balls \(\{B_{r_{j}}(p_{j})\}_{j\in\mathbb{N}}\), with
(A.2) \[\sum_{j\in\mathbb{N}}r_{j}^{D}\leqslant\varepsilon.\]
We claim that we can reduce to the case in which
(A.3) the family of dilated balls \(\{B_{32r_{j}}(p_{j})\}_{j\in\mathbb{N}}\) has a finite intersection property.
For this, we start by noticing that, without loss of generality, there exists \(q_{j}\in B_{r_{j}}(p_{j})\cap S\) (if not, such a ball can be freely removed from the covering of \(S\)). Moreover, by the Besicovitch Covering Theorem, we may suppose that
(A.4) \[\text{the original family of balls }\{B_{r_{j}}(p_{j})\}_{j\in\mathbb{N}}\text{ has a finite intersection property.}\]
Thus, to prove (A.3), we argue by contradiction and suppose, say, that there are infinitely many \(j\)'s such that \(B_{32r_{j}}(p_{j})\cap B_{32r_{1}}(p_{1})\). In particular, all these \(p_{j}\)'s remain at a bounded distance from \(p_{1}\), and so do the corresponding \(q_{j}\)'s.
Therefore, up to a subsequence, we can assume that there exists \(q\in\mathbb{R}^{n}\) such that \(q_{j}\to q\) as \(j\to+\infty\). In fact, since \(S\) is closed, we know that \(q\in S\). As a consequence, there exists \(j_{\star}\in\mathbb{N}\) such that \(q\in B_{r_{j_{\star}}}(p_{j_{\star}})\). This and the convergence of the \(q_{j}\)'s yield that \(q_{j}\in B_{r_{j_{\star}}}(p_{j_{\star}})\) for infinitely many \(j\)'s.
In particular, \(q_{j}\in B_{r_{j_{\star}}}(p_{j_{\star}})\cap B_{r_{j}}(p_{j})\) for infinitely many \(j\)'s, which is in contradiction with (A.4). The proof of (A.3) is thereby complete.
**Step 3. Bump functions.** Let now \(\tau_{j}\in C_{0}^{\infty}(B_{4r_{j}}(p_{j}),[0,1])\) be such that \(\tau_{j}=1\) in \(B_{3r_{j}}(p_{j})\) and
(A.5) \[|\nabla\tau_{j}|\leqslant\frac{C}{r_{j}}\,\chi_{B_{4r_{j}}(p_{j})\setminus B_{ 3r_{j}}(p_{j})}.\]
Let \(\phi\in C_{0}^{\infty}(\mathbb{R}^{n},[0,+\infty))\) and assume that the support of \(\phi\) is contained in some ball \(B_{R_{0}}\). Since \(S\cap\overline{B_{R_{0}}}\) is compact, we may suppose that
(A.6) \[S\cap\overline{B_{R_{0}}}\subseteq\bigcup_{j=0}^{N}B_{r_{j}}(p_{j}).\]
Thus, we define
(A.7) \[\varphi:=\phi\,\tau_{\star},\qquad\text{where}\quad\tau_{\star}:=\min_{j=0, \ldots,N}\{1-\tau_{j}\}.\]
We observe that \(\varphi\) actually depends on \(\varepsilon\), since so does the family of balls for which (A.2) holds true, but, for short, we omit this dependence in the notation (yet, we will consider the limit in \(\varepsilon\) here below). Also, we note that \(\varphi\in W_{0}^{1,\infty}(\mathbb{R}^{n},[0,+\infty))\) and we also claim that
(A.8) \[\text{the support of }\varphi\big{|}_{\mathcal{R}}\text{ is contained in }\mathcal{R}.\]
For this, let \(q_{k}\in\mathcal{R}\) be such that \(\varphi(q_{k})>0\) and \(q_{k}\to q\) as \(k\to+\infty\). Then, \(q_{k}\in B_{R_{0}}\), whence \(q\in\overline{B_{R_{0}}}\). We point out that
(A.9) \[|q-p_{j}|\geqslant 2r_{j}\text{ for all }j\in\{0,\ldots,N\}.\]
Indeed, if not, say \(|q-p_{0}|<2r_{0}\), we would have that also \(|q_{k}-p_{0}|<2r_{0}\) for infinitely many \(k\)'s and therefore \(\tau_{0}(q_{k})=1\). But this would give that \(\varphi(q_{k})>0\), which is a contradiction and (A.9) is proved.
From this, we obtain that \(q\) belongs to the complement of \(B_{2r_{0}}(p_{0})\cup\cdots\cup B_{2r_{N}}(p_{N})\), and so to the complement of \(S\), from which (A.8) follows.
**Step 4. Integral estimates.** Thanks to (A.8), we can exploit (4.3) with \(\zeta:=\frac{\varphi^{2}}{u_{Q,\ell}+1}\), being \(u_{Q,\ell}\) a mollified sequence of \(u_{Q}\), with \(\ell\in\mathbb{N}\). In this way, symmetrizing the integrands
when necessary,
\[0 \leqslant \iint_{\mathcal{R}\times\mathcal{R}}(u_{Q}(x)-u_{Q}(y))\left(\frac{ \varphi^{2}(x)}{u_{Q,\ell}(x)+1}-\frac{\varphi^{2}(y)}{u_{Q,\ell}(y)+1}\right) \,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[= \iint_{\mathcal{R}\times\mathcal{R}}\varphi^{2}(x)(u_{Q}(x)-u_{Q }(y))\left(\frac{1}{u_{Q,\ell}(x)+1}-\frac{1}{u_{Q,\ell}(y)+1}\right)\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad+\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q }(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(y)+1}\,\frac{d\mathcal{H}_{x} ^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[= -\iint_{\mathcal{R}\times\mathcal{R}}\frac{\varphi^{2}(x)(u_{Q} (x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1 )}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad+\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q }(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(y)+1}\,\frac{d\mathcal{H}_{x} ^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[= -\frac{1}{2}\iint_{\mathcal{R}\times\mathcal{R}}\frac{(\varphi^{ 2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q, \ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y} ^{n-1}}{|x-y|^{n+s}}\] \[\qquad+\frac{1}{2}\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q }(x)-u_{Q}(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(x)+1}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[= -\frac{1}{2}\iint_{\mathcal{R}\times\mathcal{R}}\frac{(\varphi^{ 2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q, \ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y} ^{n-1}}{|x-y|^{n+s}}\] \[\qquad+\frac{1}{2}\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q }(x)-u_{Q}(y))(u_{Q,\ell}(x)+u_{Q,\ell}(y)+2)(\varphi(x)-\varphi(y))(\varphi(x) +\varphi(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n- 1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Thus, given \(\alpha\in(0,1)\), to be chosen suitably small in what follows, by a weighted Cauchy-Schwarz Inequality we deduce that
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(\varphi^{2}(x)+\varphi ^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n +s}}\] \[\leqslant \alpha\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q} (y))^{2}(\varphi(x)+\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac {d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad+C_{\alpha}\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q,\ell}(x)+u_{Q,\ell}(y)+2)^{2}(\varphi(x)-\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_ {Q,\ell}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{ n+s}}.\]
We also use the bound
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y))^{2 }(\varphi(x)+\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad\leqslant 4\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)-u_{ Q}(y))^{2}(\varphi^{2}(x)+\varphi^{2}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Hence, sending \(\ell\to+\infty\), and reabsorbing one term to the left-hand side via a suitable choice of \(\alpha\), we conclude that
\[\iint_{\mathcal{R}\times\mathcal{R}}\frac{(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q} (x)-u_{Q}(y))^{2}}{(u_{Q}(x)+1)(u_{Q}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n-1}\,d \mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\]
\[\leqslant C\iint_{\mathcal{R}\times\mathcal{R}}\frac{(u_{Q}(x)+u_{Q}(y)+2)^{2}( \varphi(x)-\varphi(y))^{2}}{(u_{Q}(x)+1)(u_{Q}(y)+1)}\,\frac{d\mathcal{H}_{x}^{n -1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Since \(u_{Q}\in[0,Q]\), we have that \(u_{Q}+1\in[1,Q+1]\) and therefore
(A.10) \[\iint_{\mathcal{R}\times\mathcal{R}}(\varphi^{2}(x)+\varphi^{2}( y))(u_{Q}(x)-u_{Q}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n -1}}{|x-y|^{n+s}}\] \[\qquad\leqslant C_{Q}\iint_{\mathcal{R}\times\mathcal{R}}(\varphi (x)-\varphi(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x- y|^{n+s}}.\]
**Step 5. Further integral estimates towards the proof of (4.4).** We observe that \(\varphi=\phi\) outside the set
\[U:=\bigcup_{j=0}^{N}B_{4r_{j}}(p_{j})\]
and, by (4.1) and (A.2),
(A.11) \[\mathcal{H}^{n-1}(\mathcal{R}\cap U)\leqslant\varepsilon.\]
This and (A.7) say that, as \(\varepsilon\searrow 0\), \(\tau_{\star}\) approaches \(1\), and so \(\varphi\) approaches \(\phi\), up to negligible sets in \(\mathcal{R}\) and therefore, as \(\varepsilon\searrow 0\), the left-hand side of (A.10) can be replaced (or minorized) by
(A.12) \[\iint_{\mathcal{R}\times\mathcal{R}}(\phi^{2}(x)+\phi^{2}(y))(u_{Q}(x)-u_{Q} (y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
We claim that, as \(\varepsilon\searrow 0\), the integral in the right-hand side of (A.10) can be majorized by
(A.13) \[\iint_{\mathcal{R}\times\mathcal{R}}(\phi(x)-\phi(y))^{2}\,\frac{d\mathcal{H }_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
To check this, we remark that, given \(\beta\in(0,1)\), to be taken as small as we wish in what follows,
\[(\varphi(x)-\varphi(y))^{2} = (\phi(x)\tau_{\star}(x)-\phi(y)\tau_{\star}(y))^{2}\] \[= \Big{(}(\phi(x)-\phi(y))\tau_{\star}(x)+\phi(y)(\tau_{\star}(x)- \tau_{\star}(y))\Big{)}^{2}\] \[\leqslant (1+\beta)(\phi(x)-\phi(y))^{2}\tau_{\star}^{2}(x)+C_{\beta}\phi^ {2}(y)(\tau_{\star}(x)-\tau_{\star}(y))^{2}.\]
The first term in the last line will produce the desired result in (A.13), after sending \(\beta\searrow 0\), therefore, to prove (A.13), we need to check that, for a given \(\beta>0\), the second term produces an integral that vanishes as \(\varepsilon\searrow 0\). To this end, without loss of generality we can assume that the balls selected in (A.6) satisfy \(B_{r_{j}}(p_{j})\subseteq B_{2R_{0}}\), since the others, for small \(\varepsilon\) would not intersect \(\overline{B_{R_{0}}}\) anyway. As a result, if \(x\in\mathbb{R}^{n}\setminus B_{2R_{0}}\), we have that \(\tau_{\star}(x)=1\) and so, if \(y\in B_{R_{0}}\),
\[\tau_{\star}(x)-\tau_{\star}(y)=1-\tau_{\star}(y)=(1-\tau_{\star}(y))\chi_{U}( y).\]
From this observation, (4.1) and (A.11), we infer that
\[\iint_{(\mathcal{R}\setminus B_{2R_{0}})\times\mathcal{R}}\phi^{2}(y)(\tau_{ \star}(x)-\tau_{\star}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y} ^{n-1}}{|x-y|^{n+s}}\]
\[\leqslant C\iint_{(\mathcal{R}\cap B_{2R_{0}})\times(\mathcal{R} \cap B_{R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d\mathcal{H}_{x}^{ n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\leqslant C\sum_{k=1}^{+\infty}\iint_{(\mathcal{R}\cap B_{2k+1_{ 0}}\setminus B_{2k_{0}}))\times(\mathcal{R}\cap B_{R_{0}}\cap U)}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{\big{(}(2^{k}-1)R_{0}\big{)}^{n+s}}\] \[\leqslant C\varepsilon\sum_{k=1}^{+\infty}\frac{(2^{k+1}R_{0})^{n -1}}{\big{(}(2^{k}-1)R_{0}\big{)}^{n+s}}\] \[\leqslant C\varepsilon.\]
Therefore, to estimate, as \(\varepsilon\searrow 0\), the integral on the right-hand side of (A.10), we can focus on the computation below:
\[\iint_{(\mathcal{R}\cap B_{2R_{0}})\times\mathcal{R}}\phi^{2}(y)( \tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{ H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad\leqslant C\iint_{(\mathcal{R}\cap B_{2R_{0}})\times( \mathcal{R}\cap B_{2R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
If both \(x\) and \(y\) lie outside \(U\), we have that \(\tau_{\star}(x)-\tau_{\star}(y)=1-1=0\), hence, up to renaming \(C\), we can actually reduce to
(A.14) \[\begin{split} C\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap U)\times( \mathcal{R}\cap B_{2R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\\ \qquad\leqslant C\sum_{j=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0} }\cap B_{4r_{j}}(p_{j}))\times(\mathcal{R}\cap B_{2R_{0}})}(\tau_{\star}(x)- \tau_{\star}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x -y|^{n+s}}.\end{split}\]
Furthermore, using again (4.1) and (A.11),
\[\sum_{j=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{4r_{j}}(p _{j}))\times((\mathcal{R}\cap B_{2R_{0}})\setminus B_{16r_{j}}(p_{j}))}\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad\leqslant\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\iint_{(\mathcal{ R}\cap B_{2R_{0}}\cap B_{4r_{j}}(p_{j}))\times((\mathcal{R}\cap B_{2R_{0}}) \cap(B_{4^{k+1_{r_{j}}}(p_{j})}\setminus B_{4^{k_{r_{j}}}}(p_{j})))}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}\] \[\qquad\leqslant\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\iint_{( \mathcal{R}\cap B_{4r_{j}}(p_{j}))\times(\mathcal{R}\cap B_{4^{k+1_{r_{j}}}}(p _{j}))}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{\big{(}(4^{k}-4 )r_{j}\big{)}^{n+s}}\] \[\qquad\leqslant C\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\,\frac{(4r_{ j})^{n-1}(4^{k+1}r_{j})^{n-1}}{\big{(}(4^{k}-4)r_{j}\big{)}^{n+s}}\] \[\qquad\leqslant C\sum_{j=0}^{N}r_{j}^{n-2-s}\] \[\qquad\leqslant C\varepsilon.\]
For this reason, as \(\varepsilon\searrow 0\), we can reduce the calculation in (A.14) to
(A.15) \[C\sum_{j=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j}))\times( \mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j}))}(\tau_{\star}(x)-\tau_{\star }(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Now we observe that, given real numbers \(\{a_{j}\}_{j\in\{0,\ldots,N\}}\) and \(\{b_{j}\}_{j\in\{0,\ldots,N\}}\), we have that
(A.16) \[\left|\min_{j\in\{0,\ldots,N\}}a_{j}-\min_{j\in\{0,\ldots,N\}}b_{j}\right| \leqslant\max_{j\in\{0,\ldots,N\}}|a_{j}-b_{j}|.\]
To check this, one can assume, up to swapping the two classes of numbers, that
\[a_{j_{a}}=\min_{j\in\{0,\ldots,N\}}a_{j}\geqslant\min_{j\in\{0,\ldots,N\}}b_{ j}=b_{j_{b}},\]
for suitable \(j_{a}\), \(j_{b}\in\{0,\ldots,N\}\).
Therefore,
\[\left|\min_{j\in\{0,\ldots,N\}}a_{j}-\min_{j\in\{0,\ldots,N\}}b_{j}\right|=a_{ j_{a}}-b_{j_{b}}\leqslant a_{j_{b}}-b_{j_{b}}\leqslant|a_{j_{b}}-b_{j_{b}}| \leqslant\max_{j\in\{0,\ldots,N\}}|a_{j}-b_{j}|,\]
which proves (A.16).
As a byproduct of (A.16), we see that
\[|\tau_{\star}(x)-\tau_{\star}(y)|=\left|\min_{j=0,\ldots,N}\{1- \tau_{j}(x)\}-\min_{j=0,\ldots,N}\{1-\tau_{j}(y)\}\right|\] \[\qquad\leqslant\max_{j=0,\ldots,N}\left|(1-\tau_{j}(x))-(1-\tau_ {j}(y))\right|=\max_{j=0,\ldots,N}|\tau_{j}(x)-\tau_{j}(y)|.\]
Utilizing this information, we majorize (A.15) by
(A.17) \[C\sum_{j=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j})) \times(\mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j}))}\max_{m=0,\ldots,N} (\tau_{m}(x)-\tau_{m}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y }^{n-1}}{|x-y|^{n+s}}.\]
Now we observe that, for each \(x\), \(y\in B_{16r_{j}}(p_{j})\), we have that \(\tau_{m}(x)-\tau_{m}(y)=0\) if both \(x\) and \(y\) lie in the complement of \(B_{4r_{m}}(p_{m})\). Accordingly, (A.17) reduces to
\[C\sum_{j=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j}))\times (\mathcal{R}\cap B_{2R_{0}}\cap B_{16r_{j}}(p_{j}))}\max_{\begin{subarray}{c }m=0,\ldots,N\\ 4r_{m}(m)\cap B16r_{j}(p_{j})\neq\varnothing\end{subarray}}(\tau_{m}(x)-\tau_ {m}(y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Thus, recalling the gradient bound in (A.5), we majorize this quantity by
\[C\sum_{j,m=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{ m}}(p_{m}))\times(\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m}))}\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{r_{m}^{2}\,|x-y|^{n+s-2}}\] \[\qquad\leqslant C\sum_{j,m=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0 }}\cap B_{32r_{m}}(p_{m}))\times(\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{ m}))}\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{r_{m}^{2}\,|x-y|^{n+s-2}}\]
and consequently, in light of the finite intersection property in (A.3), up to renaming constants, by
(A.18) \[C\sum_{m=0}^{N}\iint_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m}))\times (\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m}))}\frac{d\mathcal{H}_{x}^{n- 1}\,d\mathcal{H}_{y}^{n-1}}{r_{m}^{2}\,|x-y|^{n+s-2}}.\]
Now, given \(x\in\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m})\), we utilize (4.1) to see that
\[\int_{\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m})}\,\,\frac{d \mathcal{H}_{y}^{n-1}}{r_{m}^{2}\,|x-y|^{n+s-2}}\leqslant\sum_{k=0}^{+\infty} \int_{\mathcal{R}\cap(B_{64r_{m}/2^{k}}(x)\setminus B_{64r_{m}/2^{k+1}}(x))}\, \,\frac{d\mathcal{H}_{y}^{n-1}}{r_{m}^{2}\,|x-y|^{n+s-2}}\] \[\qquad\leqslant\sum_{k=0}^{+\infty}\frac{(r_{m}/2^{k})^{n-1}}{r_{ m}^{2}\,(r_{m}/2^{k})^{n+s-2}}\leqslant\frac{C}{r_{m}^{1+s}}\,\sum_{k=0}^{+ \infty}\frac{1}{2^{(1-s)k}}\leqslant\frac{C}{r_{m}^{1+s}}.\]
Therefore, using again (4.1) and (A.2), the quantity in (A.18) can be bounded from above by
\[C\sum_{m=0}^{N}\int_{(\mathcal{R}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m})}\,\, \frac{d\mathcal{H}_{x}^{n-1}}{r_{m}^{1+s}}\leqslant C\sum_{m=0}^{N}r_{m}^{n-2- s}\leqslant C\varepsilon,\]
as desired: this shows that, as \(\varepsilon\searrow 0\), the integral in the right-hand side of (A.10) can be majorized by the quantity in (A.13).
**Step 6. Completion of the proof of (4.4).** In view of the bounds in (A.10), (A.12) and (A.13), we know that
\[\iint_{\mathcal{R}\times\mathcal{R}}(\phi^{2}(x)+\phi^{2}(y))(u_{Q}(x)-u_{Q}( y))^{2}\,\frac{d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}} \leqslant C\iint_{\mathcal{R}\times\mathcal{R}}(\phi(x)-\phi(y))^{2}\,\frac{d \mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}}{|x-y|^{n+s}}.\]
Assuming \(\phi=1\) in \(B_{R_{0}/2}\) (and possibly renaming \(R_{0}\)) we thereby conclude that (4.4) holds true, as desired.
**Step 7. Proof of (4.5).** We observe that the left-hand side of (4.5) is finite, thanks to (4.4). We employ (4.3) by taking \(\zeta\) as the function defined in (A.7). In this way, we have that
(A.19) \[\begin{split} 0&\leqslant\iint_{\mathcal{R}\times \mathcal{R}}\frac{(u_{Q}(x)-u_{Q}(y))(\phi(x)\tau_{\star}(x)-\phi(y)\tau_{ \star}(y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d\mathcal{H}_{y}^{n-1}\\ &=\iint_{\mathcal{R}\times\mathcal{R}}\frac{\tau_{\star}(x)(u_{Q} (x)-u_{Q}(y))(\phi(x)-\phi(y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x}^{n-1}\,d \mathcal{H}_{y}^{n-1}\\ &\qquad+\iint_{\mathcal{R}\times\mathcal{R}}\frac{\phi(y)(u_{Q} (x)-u_{Q}(y))(\tau_{\star}(x)-\tau_{\star}(y))}{|x-y|^{n+s}}\,d\mathcal{H}_{x} ^{n-1}\,d\mathcal{H}_{y}^{n-1}.\end{split}\]
The latter term vanishes as \(\varepsilon\searrow 0\), owing to (4.4), and the claim in (4.5) follows.
## Appendix B The set of "good" regular points
Here we describe the set of regular points of a nonlocal minimal set around which the boundary is a hypersurface of class \(C^{2}\) with bounded curvatures in a conveniently scale invariant setting. In the classical case, this set was introduced in [15, Lemma 1] and we provide here a nonlocal version of it.
**Lemma B.1**.: _Let \(E\) be \(s\)-minimal in \(B_{1}\) and assume that the origin belongs to its boundary._
_Given \(\theta>0\), let \(\mathcal{M}_{\theta,E}\) be the set of points \(x\) belonging to the regular part of \(\partial E\) such that_
(B.1) \[\begin{split}&(\partial E)\cap B_{\theta|x|}(x)\text{ belongs to the regular part of }\partial E\\ &\text{and possesses curvatures bounded in absolute value by }\frac{1}{\theta|x|}.\end{split}\]
_Then, there exist \(\rho_{0}\), \(\theta_{0}>0\), depending only on \(E\), \(n\) and \(s\), such that, for every \(\rho\in(0,\rho_{0}]\) and \(\theta\in(0,\theta_{0}]\), we have that_
(B.2) \[\mathcal{M}_{\theta,E}\cap(\partial B_{\rho})\neq\varnothing.\]
Proof.: Suppose not. Then, there exist sequences \(\theta_{j}\searrow 0\) and \(\rho_{j}\searrow 0\) for which \(\mathcal{M}_{\theta_{j},E}\cap(\partial B_{\rho_{j}})=\varnothing\). That is, if \(x\in\partial B_{\rho_{j}}\) belongs to the regular part of \(\partial E\) and also \((\partial E)\cap B_{\theta_{j}|x|}(x)\) belongs to the regular part of \(\partial E\), then necessarily its curvatures are not bounded in absolute value by \(\frac{1}{\theta_{j}|x|}=\frac{1}{\theta_{j}\rho_{j}}\).
Let \(F_{j}:=E/\rho_{j}\). Up to a subsequence, we know that \(F_{j}\) converges locally uniformly to a cone \(\mathcal{C}\) which is \(s\)-minimal (see [10]). In fact, thanks to the improvement of flatness of nonlocal minimal surfaces (see [10, Corollary 4.4 and Theorem 6.1]), this convergence occurs in the \(C^{1,\alpha}\) sense at the regular points of \(\mathcal{C}\), and actually in the \(C^{k}\) sense for any \(k\geqslant 2\) (see [1]).
Hence, we pick a regular point \(y_{0}\in\mathcal{C}\cap(\partial B_{1})\) (whose existence is guaranteed by the regularity theory of nonlocal minimal surfaces, see [11]). Then, we find a sequence \(y_{j}\in\partial B_{1}\) of regular points of \(F_{j}\) approaching \(y_{0}\) as \(j\to+\infty\) and \((\partial F_{j})\cap B_{r_{0}}(y_{j})\) consists of regular points with curvatures bounded in absolute value by \(M_{0}\), for suitable \(r_{0}\), \(M_{0}>0\).
Scaling back and setting \(x_{j}:=\rho_{j}y_{j}\in\partial B_{\rho_{j}}\), we find that \((\partial E)\cap B_{r_{0}\rho_{j}}(x_{j})\) lies in the regular set of \(\partial E\), with curvatures bounded in absolute value by \(\frac{M_{0}}{\rho_{j}}\), which, for large \(j\), is strictly less than \(\frac{1}{\theta_{j}\rho_{j}}\), contradiction.
With reference to Lemma B.1, we observe that, for all \(r>0\),
(B.3) \[\mathcal{M}_{\theta,\frac{E}{r}}=\frac{\mathcal{M}_{\theta,E}}{r}.\]
Then, we have:
**Corollary B.2**.: _Let \(E_{1}\), \(E_{2}\) be \(s\)-minimal sets in \(B_{1}\) and assume that the origin belongs to their boundary. Assume also that \(E_{1}\) and \(E_{2}\) have the same tangent cone at the origin._
_Let \(\theta_{0}\), \(\rho_{0}>0\) be as in Lemma B.1, used here with \(E:=E_{1}\). Let also \(\mathcal{M}\) be the set \(\mathcal{M}_{\theta_{0},E}\) in Lemma B.1 with \(E=E_{1}\)._
_Suppose that_
\[E_{2}\setminus E_{1}=\Big{\{}x+t\nu(x),\;x\in\mathcal{M},\;t\in\big{[}0,w(x) \big{)}\Big{\}},\]
_for some function \(w\) of class \(C^{2}\), where \(\nu\) denotes the external unit normal of \(E_{1}\) at its regular points._
_Then,_
(B.4) \[\lim_{r\searrow 0}\frac{\sup_{x\in(\partial B_{r})\cap\mathcal{M}}|w(x)|}{r}=0.\]
_In addition, if \(w(x)\neq 0\) for all \(x\neq 0\), then there exists an infinitesimal sequence of points \(z_{\star,k}\in\mathcal{M}\) such that_
(B.5) \[\frac{|w(x)|}{|x|}\leqslant\frac{2|w(z_{\star,k})|}{|z_{\star,k}|}\]
_for all \(x\in\mathcal{M}\) with \(|x|\leqslant|z_{\star,k}|/2\)._
_Finally, the distance of \(z_{\star,k}\) from the singular set of \(E_{1}\) is at least \(\theta_{0}|z_{\star,k}|\)._
Proof.: Before proving (B.4), we stress that the supremum in this equation makes sense, thanks to (B.2).
We now prove (B.4). Suppose not. Then, there exist \(c>0\), an infinitesimal sequence \(r_{j}>0\) and \(x_{j}\in(\partial B_{r_{j}})\cap\mathcal{M}\) such that
(B.6) \[|w(x_{j})|\geqslant cr_{j}.\]
Thus, if \(w_{j}:=\frac{w}{r_{j}}\) and \(E_{i,j}:=\frac{E_{i}}{r_{j}}\) for \(i\in\{1,2\}\), we see that
\[E_{2,j}\setminus E_{1,j}=\left\{x+t\nu_{j}(x),\;x\in\frac{\mathcal{M}}{r_{j}},\;t\in\big{[}0,w_{j}(x)\big{)}\right\},\]
where \(\nu_{j}\) denotes the external unit normal of \(E_{1,j}\).
Combining this and (B.3), we obtain
\[E_{2,j}\setminus E_{1,j}=\Big{\{}x+t\nu_{j}(x),\;x\in\mathcal{M}_{\theta_{0}, E_{1,j}},\;t\in\big{[}0,w_{j}(x)\big{)}\Big{\}}.\]
So, setting \(y_{j}:=\frac{x_{j}}{r_{j}}\in(\partial B_{1})\cap\mathcal{M}_{\theta_{0},E_{1,j}}\), we have that
(B.7) \[y_{j}+w_{j}(y_{j})\nu_{j}(y_{j})\in\partial E_{2,j}.\]
Notice in addition that, since the distance of \(x_{j}\) from the singular set of \(E_{1}\) is at least \(\theta_{0}|x_{j}|=\theta_{0}r_{j}\), we have that the distance of \(y_{j}\) from the singular set of \(E_{1,j}\) is at least \(\theta_{0}\). The curvature bound thus allows us to pass to the limit as \(j\to+\infty\), up to a subsequence, and find that \(y_{j}\to y_{\infty}\), with \(y_{\infty}\) belonging to the regular part of the tangent cone of \(E_{1}\), and also \(\nu_{j}(y_{j})\to\nu_{\infty}(y_{\infty})\), where \(\nu_{\infty}\) denotes the normal of this tangent cone.
Hence, since \(E_{2}\) shares the same tangent cone of \(E_{1}\), it follows from (B.7) that \(w_{j}(y_{j})\to 0\) as \(j\to+\infty\). But this is in contradiction with (B.6) and the proof of (B.4) is thereby complete.
Now we prove (B.5). For this, let
\[\sigma(r):=\frac{\sup_{x\in(\partial B_{r})\cap\mathcal{M}}|w(x)|}{r}.\]
Thanks to (B.4), we have that \(\sigma(r)\to 0\) as \(r\searrow 0\).
Also, we know that \(\sigma(r)>0\) for all \(r\neq 0\). Therefore, given \(\mu_{0}>1\) we pick an infinitesimal sequence \(r_{k}>0\) and choose \(r_{\star,k}\in(0,r_{k}]\) such that
\[\sigma(r_{\star,k})\geqslant\frac{\sup_{r\in(0,r_{k}]}\sigma(r)}{\mu_{0}}.\]
As a result, for all \(\mu\in(0,1)\),
\[\frac{\sup_{x\in(\partial B_{\mu r_{\star,k}})\cap\mathcal{M}}|w(x)|}{\mu r_{ \star,k}}=\sigma(\mu r_{\star,k})\leqslant\sup_{r\in(0,r_{k}]}\sigma(r) \leqslant\mu_{0}\,\sigma(r_{\star,k}).\]
Besides, by (B.2), we can pick \(z_{\star,k}\in(\partial B_{r_{\star,k}})\cap\mathcal{M}\) with
\[\frac{|w(z_{\star,k})|}{r_{\star,k}}\geqslant\frac{\sup_{x\in(\partial B_{r_{ \star,k}})\cap\mathcal{M}}|w(x)|}{\mu_{0}\,r_{\star,k}}=\frac{\sigma(r_{\star, k})}{\mu_{0}}.\]
This gives that, for all \(x\in\mathcal{M}\) with \(|x|\leqslant|z_{\star,k}|/2\), we have that
\[\frac{|w(x)|}{|x|}\leqslant\sup_{\mu\in(0,1/2]}\frac{\sup_{x\in(\partial B\mu_{ \tau_{\star,k}})\cap\mathcal{M}}|w(x)|}{\mu r_{\star,k}}\leqslant\mu_{0}\, \sigma(r_{\star,k})\leqslant\frac{\mu_{0}^{2}\,|w(z_{\star,k})|}{|z_{\star,k}|},\]
which, choosing \(\mu_{0}:=\sqrt{2}\), establishes (B.5), as desired.
Finally, the distance of \(z_{\star,k}\) from the singular set is estimated from below by the first line in (B.1).
|
2307.11201 | Choosing the Right Approach at the Right Time: A Comparative Analysis of
Causal Effect Estimation using Confounder Adjustment and Instrumental
Variables | In observational studies, unobserved confounding is a major barrier in
isolating the average causal effect (ACE). In these scenarios, two main
approaches are often used: confounder adjustment for causality (CAC) and
instrumental variable analysis for causation (IVAC). Nevertheless, both are
subject to untestable assumptions and, therefore, it may be unclear which
assumption violation scenarios one method is superior in terms of mitigating
inconsistency for the ACE. Although general guidelines exist, direct
theoretical comparisons of the trade-offs between CAC and the IVAC assumptions
are limited. Using ordinary least squares (OLS) for CAC and two-stage least
squares (2SLS) for IVAC, we analytically compare the relative inconsistency for
the ACE of each approach under a variety of assumption violation scenarios and
discuss rules of thumb for practice. Additionally, a sensitivity framework is
proposed to guide analysts in determining which approach may result in less
inconsistency for estimating the ACE with a given dataset. We demonstrate our
findings both through simulation and an application examining whether maternal
stress during pregnancy affects a neonate's birthweight. The implications of
our findings for causal inference practice are discussed, providing guidance
for analysts for judging whether CAC or IVAC may be more appropriate for a
given situation. | Roy S. Zawadzki, Daniel L. Gillen | 2023-07-20T19:29:42Z | http://arxiv.org/abs/2307.11201v2 | Choosing the Right Approach at the Right Time: A Comparative Analysis of Causal Effect Estimation using Confounder Adjustment and Instrumental Variables
###### Abstract
In observational studies, unobserved confounding is a major barrier in isolating the average causal effect (ACE). In these scenarios, two main approaches are often used: confounder adjustment for causality (CAC) and instrumental variable analysis for causation (IVAC). Nevertheless, both are subject to untestable assumptions and, therefore, it may be unclear which assumption violation scenarios one method is superior in terms of mitigating inconsistency for the ACE. Although general guidelines exist, direct theoretical comparisons of the trade-offs between CAC and the IVAC assumptions are limited. Using ordinary least squares (OLS) for CAC and two-stage least squares (2SLS) for IVAC, we analytically compare the relative inconsistency for the ACE of each approach under a variety of assumption violation scenarios and discuss rules of thumb for practice. Additionally, a sensitivity framework is proposed to guide analysts in determining which approach may result in less inconsistency for estimating the ACE with a given dataset. We demonstrate our findings both through simulation and an application examining whether maternal stress during pregnancy affects a neonate's birthweight. The implications of our findings for causal inference practice are discussed, providing guidance for analysts for judging whether CAC or IVAC may be more appropriate for a given situation.
## 1 Introduction
A common goal in observational studies is to estimate the average causal effect (ACE) of a treatment or exposure on a specific outcome such as the effect of maternal stress during pregnancy on a child's birthweight. Due to the exposure not being randomized, the presence of confounders may bias estimates of the ACE. Confounders are factors that influence both the level of exposure (or treatment assignment) and the outcome. If uncontrolled for, confounders can create extraneous differences between the exposure groups that make it difficult to isolate the causal effect. The effectiveness of confounder adjustment for causality (CAC), however, is contingent on the untestable assumption that all possible confounders, or suitable proxies, are present in the dataset and have been accounted for appropriately. In other words, we cannot use observed data to prove that the CAC is consistent for the ACE. A simple example is if a mother's obstetric risk, a known confounder for stress and birthweight, and any proxies were missing from the dataset we would like to analyze. Without _a priori_ knowledge that obstetric risk was a confounder, we would not be able to ascertain that the CAC assumptions were violated.
An alternative approach that may avoid concerns about unobserved confounders is the instrumental variable (IV) approach, or instrumental variable analysis for causation (IVAC). An IV is defined by three main conditions: (i) it influences the treatment assignment (relevance), (ii) it is not a cause of the outcome after conditioning on treatment assignment (exclusion restriction), and (iii) it is not associated with unobserved
confounders (independence). In the event that we have a variable that satisfies these conditions, we could then use variation in the IV as a proxy for variation in the treatment and measure the effect on the outcome. Importantly, by (iii), the variation in the IV is independent from unobserved confounding and, therefore, unlike CAC, IVAC does not require appropriately accounting for all possible confounding to consistently estimate the ACE.[1]
When choosing to use CAC over the IVAC, and vice-versa, we trade one set of untestable assumptions for another. For CAC, we cannot prove that we have accounted for all confounders while, for IVAC, we cannot prove we have a valid IV. For example, proving the exclusion restriction requires us to establish the lack of a direct relationship between the IV and outcome, or a null result, which is not possible with data. Furthermore, to be consistent for the ACE, the IVAC must meet certain untestable conditions surrounding treatment effect heterogeneity. We therefore have no guarantee that under either approach our produced estimate is consistent for the ACE. Despite this, we remain interested in estimating the ACE and subsequently direct our efforts towards attempting to estimate a parameter with the least possible distance from the ACE. In this vein, the key question and central premise of the current manuscript focuses on addressing the following question in practice: under the potential violation of the untestable assumptions, when is the parameter estimated by CAC closer to the ACE than that of IVAC, and vice versa?
In the literature, there exist general guidelines surrounding whether CAC is more appropriate than IVAC.[1, 2] Generally, we contend that analysts should weigh whether the potential degree of unobserved confounding outweighs the potential for violations in the IV assumptions. There is, however, little by way of theoretical research that directly compares the two approaches to assess these trade-offs. Focusing on ordinary least squares (OLS) for CAC and two-stage least squares (2SLS) for IVAC, there are two overarching goals we seek to accomplish with this paper. Firstly, we seek to analytically compare the inconsistency of 2SLS to OLS in a variety of scenarios involving assumption violations. Next, we provide a sensitivity framework to guide analysts in determining whether the inconsistency of 2SLS is more than that of OLS, and vice versa.
In order to more succinctly express the relative performance of OLS and 2SLS under assumption violations, we focus on the scenario where, for a given variable, we must decide on whether the variable should be adjusted for as a confounder in OLS, used as an IV in 2SLS, or not incorporated into the analysis at all. Note that we allow confounders to be adjusted for in 2SLS - an IV is only used in the first stage whereas confounders would need to be present in both the first and second stage.
To our knowledge, there is no existing theoretical literature directly comparing the trade-offs of pursuing CAC versus IVAC, though authors have considered the impact of assumption violations in both settings. The first area of literature relevant to the themes of our paper is bias amplification, which refers to the fact that using certain variables as confounders may increase pre-existing bias due to unobserved confounding. In the linear setting, an IV is a bias amplifier.[3, 4, 5, 6] In these papers, the authors compare the consistency for the ACE with and without adjusting for an IV. Pearl (2012) provides an extension where there is an imperfect IV (in that the exclusion restriction is violated) and shows that under certain conditions, adjusting for the imperfect IV may actually reduce bias. Nevertheless, he does not address whether this variable may be more appropriately used in 2SLS.
The bias amplification literature and, by extension, our findings have important implications on applied practice. In particular, the rise of data-driven variable selection approaches for the propensity score, or probability of receiving the intervention, in confounder methods and first-stage for IV analysis. Though IVs, by definition, influence the treatment assignment, they may amplify bias if included in the model for the propensity score.[3] For IV methods, though a variable may not be a perfect IV it may still be worth using it as such. In addition, data-driven modeling of the first-stage may, for example, shrink to zero an important confounder used to achieve the IV assumptions. The gain in the strength of the first-stage may, however, offset the penalty incurred by omitted variable bias in 2SLS. As it stands, these intuitions are difficult to incorporate into variable selection procedures. With this work, we seek to elucidate these complicated scenarios.
In another area of the literature, there has been some work related to comparing OLS to 2SLS under the violation of IV assumptions. It is a well-known result that if an IV is poorly predictive of the intervention (i.e. weak), then small violations in the exclusion restriction and independence assumption can lead to large inconsistencies for the IV estimand.[7, 8]. In addition, to assess the independence assumption, one may
compare the impact of intentionally omitting an observed confounder on OLS and 2SLS in order to compare the sensitivity of OLS to that of 2SLS in estimating the ACE.[9] The main assumption behind this procedure is that the impact of omitting an observed confounder on the consistency of OLS and 2SLS is similar to that of omitting a correlated unobserved confounder. A similar assumption and "benchmarking" procedure will be used in our sensitivity analysis.
There are several procedures in the literature regarding sensitivity analyses for violations in IV assumptions. For example, Cinelli and Hazlett (2022) provide a compelling framework and visualization scheme for omitted variable bias in 2SLS based on several partial \(R^{2}\) measures.[10] Their framework addresses the question of how large the impact of an unobserved confounder would have to be in order to qualitatively change the inferential conclusions of a study, which covers both violations of the exclusion restriction and independence assumptions. A similar procedure to this is the E-value.[11] While we use many of the same tools - notably benchmarking unobserved \(R^{2}\) measures with observed data from Cinelli and Hazlett (2022) - we do not focus on this sensitivity analysis paradigm of hypothesis testing but instead consider the relative inconsistencies of CAC and IVAC for the ACE.
Another complication to IVAC lies in treatment effect heterogeneity. In this setting, Imbens and Angrist (1994) state that, under monotonicity conditions, IVAC identifies the local average causal effect (LACE) or the causal effect of the "compliers" subpopulation (those whose treatment assignment varies with the IV).[12] If the factors that determine compliance also cause treatment effect heterogeneity, then the LACE may not be equal to the ACE. Hartwig et al. (2020) and Wang and Tchetgen Tchetgen (2018) give clear explanations of the assumptions needed for the LACE to equal the ACE.[13, 14] Essentially, the heterogeneity between the treatment and outcome should be independent of both the IV and the effect modification between the treatment and outcome. For the ease of parameterization in our paper, we will use Wang and Tchetgen Tchetgen's notion that one of two conditions must be met: (i) no unmeasured confounders are additive effect modifiers of the relationship of both the instrument and treatment or (ii) no unmeasured confounders are additive effect modifiers the treatment and the outcome.
The rest of the paper is organized as follows. First, we provide the general model setting of interest and introduce relative notation. We then present results regarding the consistency of no adjustment, OLS, and 2SLS for the scenarios of an exclusion restriction violation, independence violation, and treatment effect heterogeneity relevant to IV estimation with and without covariates. In all scenarios, we consider unobserved confounding and isolate the impact of individual assumption violations (e.g. both an exclusion restriction and independence violation). Following this, we present a sensitivity analysis procedure based on partial \(R^{2}\) and benchmarking unobserved quantities with observed quantities. The goal of this procedure is to give the analyst relevant information to assess the plausibility of whether it may be more appropriate to adjust for a variable in OLS or 2SLS. Next, in simulations, we verify our closed-form results and demonstrate the use of the sensitivity analysis procedure in a variety of scenarios. Then, we apply the procedure to a real world dataset studying a neonate's birthweight as a function of maternal stress during pregnancy. We conclude with a discussion regarding the implications of our closed form results on the practice of causal inference and, additionally, provide further guidance on how to use our sensitivity analysis procedure.
## 2 Notation and Set-Up
Our goal is to estimate the ACE of some continuous treatment \(X\) on an outcome \(Y\), denoted as \(\beta_{1}=\frac{\partial}{\partial x}E[Y|do(x)]\) in Pearl's notation.[15] We depict the causal relationships in Figure 1, a directed acyclic graph (DAG) with edge weights \(c_{0},\ldots,c_{3}\). We further consider the following structural equations where \(E[\epsilon_{1}|U,Z]=0\) and \(E[\epsilon_{2}|X,U]=0\):
\[X =\alpha_{1}U+\alpha_{2}Z+\epsilon_{1}, \tag{1}\] \[Y =\beta_{1}X+\beta_{2}U+\epsilon_{2}. \tag{2}\]
Given that all variables in Figure (1) are standardized to have mean 0 and variance 1, the edge weights are equivalent to the coefficients in Eqs. (1) and (2). For example, the ACE, \(\beta_{1}\), is the same as the edge weight \(c_{0}\). In addition, \(\alpha_{1}=c_{1}\), \(\alpha_{2}=c_{3}\), \(\beta_{1}=c_{0}\), and \(\beta_{2}=c_{2}\). This equivalence will be helpful in visually
expressing assumptions surrounding different scenarios. Furthermore, the edge weights are correlations and are bounded between \(-1\) and \(1\).
Throughout, we assume the stable unit treatment value assumption (SUTVA) of consistency and no interference. Because \(U\) is unobserved, we must estimate the the following reduced form regression where the subscript \(R\) indicates these are the values related to the reduced regression
\[Y=\beta_{1}^{R}X+\epsilon_{2}^{R}. \tag{3}\]
\(U\) is a confounder because it both influences \(X\) and \(Y\), and, therefore, because we have failed to block the path through \(U\), \(\hat{\beta}_{1}^{R}\) is inconsistent for the ACE. Alternatively, to estimate the ACE, we may utilize \(Z\), which is a valid IV if
1. \(\alpha_{2}>0\) (relevance)
2. \(Z\) is not a cause of \(Y\) conditional on \(X\) (exclusion restriction)
3. \(Z\) is not influenced by any unaccounted for confounders such that \(Z\not\perp Y|X\) (independence).
Supposing that there is no treatment effect heterogeneity or that the conditions of Hartwig et al. (2020) or Wang and Tchetgen Tchetgen (2018) [13, 14] are met, we have \(\beta_{IV}=\frac{Cov(Z,Y)}{Cov(X,Z)}=\frac{\beta_{1}\alpha_{1}}{\alpha_{1}}= \beta_{1}\), and hence 2SLS will provide a consistent estimate of the ACE.
We are interested in estimating and comparing the following estimands: the causal effect i.e. Eq. (4), one that omits \(Z\) i.e. Eq. (5), one that uses \(Z\) as a confounder i.e. Eq. (6), and one that utilizes \(Z\) as an IV i.e. Eq.(7):
\[A_{1}=\frac{\partial}{\partial x}E[Y|do(x)]=c_{0}, \tag{4}\]
\[A_{2}=\frac{\partial}{\partial x}E[Y|x], \tag{5}\]
\[A_{3}=\frac{\partial}{\partial x}E[Y|x,z], \tag{6}\]
\[A_{4}=\frac{Cov(Y,Z)}{Cov(X,Z)}=\frac{\frac{\partial}{\partial z}E[Y|z]}{ \frac{\partial}{\partial z}E[X|z]}. \tag{7}\]
We define the degree of inconsistency for \(A_{1}\) of estimates of \(A_{2}\), \(A_{3}\), and \(A_{4}\) as a set of absolute differences: \(\lambda_{2}=|A_{4}-A_{2}|\), \(\lambda_{3}=|A_{4}-A_{3}|\), and \(\lambda_{4}=|A_{4}-A_{3}|\). The crux of our findings is to compare these \(\lambda\)s. One approach "performs better" than the other if the respective \(\lambda\) is smaller. For example, 2SLS performs better than OLS if \(\lambda_{4}>\lambda_{3}\).
As a simple example of the types of calculations and comparisons that we will do in the next section, we can use the conditions of Figure 1. From Pearl (2012), we have that \(A_{2}=c_{0}+c_{1}c_{2}\) by Wright's rules of path analysis and \(A_{3}=c_{0}+c_{2}\frac{\partial}{\partial x}E[U|x,z]=c_{0}+\frac{c_{1}c_{2}}{ 1-c_{3}^{2}}\).[5] As a result of adjusting for \(Z\) as a confounder, we have bias amplification that increases with the strength of the IV (i.e. the magnitude of \(c_{3}\)). In addition, we have that \(\hat{A}_{4}\stackrel{{ p}}{{\rightarrow}}c_{0}\). The proof of this is in the first section of the Appendix. Further, by Chebyshev's inequality and assuming finite variances hold, we also have that \(\hat{A}_{2}\stackrel{{ p}}{{\rightarrow}}c_{0}+c_{1}c_{2}\) and \(\hat{A}_{3}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{1}c_{2}}{ 1-c_{3}^{2}}\). Thus, \(\lambda_{3}=\frac{c_{1}c_{2}}{1-c_{3}^{2}}>\lambda_{2}=c_{1}c_{2}>\lambda_{4}=0\) or, in words, 2SLS performs better than no adjustment, which performs better than OLS.
Figure 1: A directed acyclic graph with one confounder and one instrumental variable.
Trade-offs Under Violations in Instrumental Variable Assumptions
In this section, we present results for the trade-offs between confounder and IV methods for three scenarios: (i) violation of the exclusion restriction assumption, (ii) violation of the independence assumption, and (iii) treatment effect heterogeneity. In all scenarios, \(U\) is unobserved, which provides the realistic setting where we may be motivated to use 2SLS due to the concern of unobserved confounding. For ease of exposition, we first derive the quantities of interest without adjustment for observed confounding. We then present the quantities with these observed covariates. For ease of comparison of the quantities of interest, we further assume all regression slope parameters are positive throughout this section and we will handle the general case in the sensitivity analysis portion. Unless otherwise stated, all proofs for the propositions in this section can be found in the Appendix.
### Exclusion Restriction Violation
Figure 2 directly reproduces Figure 2 from Pearl (2012) and presents a violation of the exclusion restriction assumption for \(Z\) if \(c_{ER}\neq 0\). We use this quantity to denote the degree of violation. By traditional logic, one would define \(Z\) as a confounder because it is a cause both of \(X\) and \(Y\) and use it as such. It is not, however, unequivocally true that one should use \(Z\) as a confounder. To see this, suppose we have the following structural equations:
\[X=c_{1}U+c_{3}Z+\epsilon_{3} \tag{8}\]
\[Y=c_{0}X+c_{2}U+c_{ER}Z+\epsilon_{4}. \tag{9}\]
**Proposition 3.1**.: _Under the conditions of Eqs. (8) and (9), \(\hat{A}_{2}\xrightarrow{p}c_{0}+c_{1}c_{2}+c_{3}c_{ER}\), \(\hat{A}_{3}\xrightarrow{p}c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}\), and \(\hat{A}_{4}\xrightarrow{p}c_{0}+\frac{c_{ER}}{c_{3}}\)._
The derivations for \(A_{2}\) and \(A_{3}\) can be found in Pearl (2012) so we omit them. The proof for \(A_{4}\) can be found in the Appendix. We see that adjusting for \(Z\) decreases inconsistency compared to not adjusting for \(Z\) if \(\frac{c_{ER}}{c_{3}}\geq\frac{c_{1}c_{2}}{1-c_{3}^{2}}\). This inequality could be difficult to attain if the instrument is strong.[5] Interestingly, the left term of this inequality is the inconsistency of 2SLS and thus the IV being strong is relatively advantageous for the use of \(Z\) as an IV. We can re-arrange the inequality between \(A_{3}\) and \(A_{4}\) as \(\frac{c_{ER}(1-c_{3}^{2})}{c_{3}}\geq c_{1}c_{2}\) where \(c_{1}c_{2}\) indicates the impact of unobserved confounding in the relationship between \(X\) and \(Y\). Here, it becomes more clear that the strength of the IV can be large enough such that the degree of exclusion restriction violation (i.e. \(c_{ER}\)) is offset and is smaller than the impact of unobserved confounding.
The trade-offs between \(A_{2}\), \(A_{3}\), and \(A_{4}\) can be visualized with a 3-D contour plot. In Figure 3, letting \(c_{0}=0.3,c_{1}=0.7\), and \(c_{2}=0.7\), we can vary the values of \(c_{3}\) and \(c_{ER}\). Note the plausible coefficient values for \(c_{3}\) and \(c_{ER}\) are restricted due to the requirement that the variances for the variables to sum to one (see "Notes about Simulations" section in the Appendix). This image gives us the visual intuition that when the IV is stronger, moderate violations in the exclusion restriction violation do not preclude the use of \(Z\) as an IV. Furthermore, adjusting for \(Z\) will be inferior compared to not adjusting for \(Z\). When the IV is weak, 2SLS predictably performs very poorly in all cases.
Figure 2: Exclusion Restriction Violation using \(Z\) as an IV.
### Independence Violation
Figure 4 represents one violation of the independence assumption where the residual confounder \(U\) is a cause of \(Z\). Here, \(c_{I}\) represents the degree of violation or, in this case, the effect of \(U\) on \(Z\). Alternative specifications create a new confounder that is only associated with \(Z\) but not \(X\). Nonetheless, our parameterization provides a useful case where both the independence assumption is violated and there is confounding in the relationship between \(Z\) and \(X\). For structural equations, we can re-use Eqs 1 and 2 as well as add an extra structural equation given by
\[Z=c_{I}U+\epsilon_{5}. \tag{10}\]
**Proposition 3.2**.: _Under Eqs. (1), (2), and (10), \(\hat{A}_{2}\stackrel{{ p}}{{\rightarrow}}c_{0}+c_{1}c_{2}+c_{2}c_{ 3}c_{I}\), \(\hat{A}_{3}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{1}c_{2}(1 -c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}\), and \(\hat{A}_{4}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{2}c_{I}}{c _{3}+c_{1}c_{I}}\)._
The derivation for \(A_{2}\) is a straightforward application of Wright's path analysis so only the proofs for \(A_{3}\) and \(A_{4}\) are provided.[16] To better interpret these quantities, we can think about both paths on the DAG and remaining variance after orthogonalizing variables via the Frisch-Waugh-Lovell Theorem (FWL).[17] Looking at the pathways, \(c_{1}c_{2}\) is the backdoor path from \(X\) to \(Y\) through \(U\), \(c_{2}c_{3}c_{I}\) is the backdoor path through \(Z\), and \(c_{2}c_{I}\) is the path from \(Z\) to \(Y\) via \(U\). \(c_{3}+c_{1}c_{I}\) is the unconditional correlation between \(Z\) and \(X\), which includes both the direct and backdoor path via \(U\). \(1-c_{I}^{2}\) depicts the remaining (stochastic) variation in \(Z\) after orthogonalizing \(U\) while \(1-(c_{3}+c_{1}c_{I})^{2}\) is the remaining variance in \(X\) after orthogonalizing \(Z\).
Of particular interest is the trade-off between using \(Z\) in 2SLS and using \(Z\) in OLS. We find that adjusting
Figure 4: \(Z\) is Correlated with an Unobserved Confounder \(U\)
Figure 3: Analytical comparison of no adjustment, OLS adjusting for the IV, and 2SLS using the IV under varying IV strength and degree of exclusion restriction violation.
for \(Z\) is superior to using \(Z\) for 2SLS if \(\frac{c_{I}}{c_{3}+c_{1}c_{I}}>\frac{c_{1}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}\). We can begin to interpret this inequality with the reoccurring theme that if the IV is strong then attaining this inequality is more difficult: a strong IV will cause \(c_{3}+c_{1}c_{I}\) to be large which inflates the inconsistency in \(A_{3}\) while it decreases the consistency for \(A_{4}\). The remaining variation in \(Z\) not caused by \(U\), or \(c_{I}\), is an important quantity because as \(c_{I}\) decreases, the inconsistency in \(A_{3}\) will increase while the inconsistency in \(A_{4}\) will decrease. In this sense, a simple sensitivity analysis procedure could be to benchmark the variation in the IV that is explained by the covariates. One could use this benchmark to conjecture how much of the variation in the IV is explained by an unobserved confounder. If this quantity is small then one could plausibly assume a fair amount of variation in \(Z\) free from unobserved confounding and, thus, \(c_{I}\) is small. These trade-offs can be visualized in Figure 5 where \(c_{0}=0.3,c_{1}=0.7\), and \(c_{2}=0.7\).
### Treatment Effect Heterogeneity
We return the setting of Figure 1 where we have a perfect IV but now we omit the edge weights and introduce treatment effect heterogeneity. In this case, we do not know of any literature that gives us a salient way to to represent treatment heterogeneity using DAGs and edge weights. Therefore, the nuance is in the structural equations:
\[X=\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1}, \tag{11}\]
\[Y=\beta_{1}X+\beta_{2}U+\beta_{3}XU+\epsilon_{2}. \tag{12}\]
The estimand of interest remains the ACE or the average of the individual treatment effects, which is denoted by \(\beta_{1}\). This parameterization is consistent with a violation of Wang and Tchetgen Tchetgen (2018) assumptions 5a and 5b if \(\alpha_{3}\neq 0\) and \(\beta_{3}\neq 0\). The key parameter for measuring the degree of "assumption violation" is \(\alpha_{3}\) because we aim to quantify how large an unobserved confounder needs to be in order to modify the effect on \(Z\) in the first-stage to render \(\lambda_{4}>\lambda_{3}\) and \(\lambda_{4}>\lambda_{2}\), or that IVAC is inferior to CAC.
**Proposition 3.3**.: _Under the conditions of Eqs. (11) and (12) as well as assuming \(E[U^{3}]=0\), \(\hat{A}_{2}\xrightarrow{p}\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3} \beta_{3},\hat{A}_{3}\xrightarrow{p}\beta_{1}+\frac{\alpha_{2}\beta_{2}+\alpha _{1}\alpha_{3}\beta_{3}}{1-a_{1}^{2}},\) and \(\hat{A}_{2}\xrightarrow{p}\beta_{1}+\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}\)_
When comparing the relative trade-off between not adjusting for \(Z\) and adjusting for \(Z\), the latter is inferior if \(\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-a_{1}^{2}}>\alpha_{ 2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}\). Unlike the case of no effect modification, it is not always true that adjusting
Figure 5: Analytical comparison of no adjustment, OLS adjusting for the IV, and 2SLS using the IV under varying IV strength and degree of the independence violation.
for an IV will amplify bias. Specifically, by rearranging terms we see the inequality holds if \(2\alpha_{1}^{2}+\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1\). If \(|\alpha_{1}|\gtrapproftypree 0.7\), or the IV is strongly associated with \(X\), then this inequality will hold; however, this would not be the case if the IV is sufficiently weak and the multiplication of the coefficients for the interactions are larger than the multiplication of the coefficients representing the main effect.
In the more realistic case, if an analyst is aware \(Z\) is an IV and assuming that the IV was sufficiently strong, the main point of concern surrounds whether the LACE estimate is more inconsistent than the ACE estimate from the unadjusted OLS. This notion is true if \(\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}>\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_ {3}\beta_{3}\) or \(\frac{\alpha_{1}\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}+2\alpha_{1}^{2}<1\). Firstly, a strong IV, \(|\alpha_{1}|\gtrapproftypree 0.7\), precludes this inequality from being attained. Alternatively, this inequality could fail to be attained if the ratio of main effects to interaction effects, scaled by the IV strength, is large. Setting \(\alpha_{2}=0.15\), \(\beta_{1}=0.1\), \(\beta_{2}=0.2\), and \(\beta_{3}=0.1\), we can visualize the above inequalities in Figure 6.
A final takeaway from these results is that the strength of an IV has implications on the the OLS inconsistencies even though it is independent from \(U\) (e.g. via the term \(2\alpha_{1}\alpha_{3}\beta_{3}\) in \(A2\)). Therefore, accounting for any present IVs, even if never utilized, is important for understanding the degree of inconsistency present in confounder methodology. The results of the previous three Propositions are summarized in Table 1.
### Adding Additional Confounders
In the vast majority of data analyses, an analyst will have access to observed confounders that will be adjusted for to mitigate confounding both in OLS and 2SLS. Temporarily, we consider a single observed confounder, \(W\), that is continuous and has mean 0 and variance 1. If we have multiple confounders, \(\{W_{1},W_{2},...W_{J}\}\), that are in the same form as \(W\), we can simply redefine \(W\) as some function of the confounders, \(f(W_{1},W_{2},...W_{J})\). For example, this might take a form linear combination derived from the first principal component of the \(W\) matrix. Therefore, the edge weights and regression coefficients will be the joint effect of the confounders. Therefore, operationally, our updated results can accommodate multiple covariates.
Because we would like to use \(W\) to benchmark relationships involving \(U\), we must update the DAGs and structural equations such that \(W\) has a similar form to \(U\) in each assumption violation. Because of this, our previous results will change. In all cases, \(W\) is orthogonal to \(U\) or, in other words, \(U\) represents confounding in ACE unrelated to \(U\). With the exception of \(A_{1}\), the quantities of interest are updated to
Figure 6: Analytical comparison of no adjustment, OLS adjusting for the IV, and 2SLS using the IV under varying IV strength and degree of the treatment effect heterogeneity assumption.
\begin{table}
\begin{tabular}{l|c|c|c|c} \hline \hline
**Scenario** & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \hline No Violation & & & \\ Z & X & Y & & \\ \(X=c_{1}U+c_{3}Z+\epsilon_{1}\) & & & \\ \(Y=c_{0}X+c_{2}U+\epsilon_{2}\) & & \(c_{0}+c_{1}c_{2}\) & \(c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}\) & \(c_{0}\) \\ \hline Exclusion Restriction Violation & & & \\ Z & X & Y & & \\ \(X=c_{1}U+c_{3}Z+\epsilon_{3}\) & & & \\ \(Y=c_{0}X+c_{2}U+c_{ER}Z+\epsilon_{4}\). & & & \\ \(c_{0}+c_{1}c_{2}+c_{3}c_{ER}\) & & \(c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{3}}\) & \(c_{0}+\frac{c_{ER}}{c_{3}}\) \\ \hline Independence Assumption & & & \\ Z & X & Y & \\ \(X=\alpha_{1}U+\alpha_{2}Z+\epsilon_{1}\) & & & \\ \(Y=\beta X+\theta U+\epsilon_{2}\) & & & \\ \(Z=c_{I}U+\epsilon_{5}\). & & \(c_{0}+c_{1}c_{2}+c_{2}c_{3}c_{I}\) & \(c_{0}+\frac{c_{1}c_{2}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}\) & \(c_{0}+\frac{c_{2}c_{I}}{c_{3}+c_{1}c_{I}}\) \\ \hline Treatment Effect Heterogeneity & & & \\ \(X=\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1}\), & & & \\ \(Y=\beta_{1}X+\beta_{2}U+\beta_{3}XU+\epsilon_{2}\). & & \(\beta_{1}+\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-a_{1}^{ 2}}\) & \(\beta_{1}+\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}\) \\ \hline \end{tabular}
\end{table}
Table 1: Summary of Results
\[A_{2} =\frac{\partial}{\partial x}E[Y|x,w] \tag{13}\] \[A_{3} =\frac{\partial}{\partial x}E[Y|x,z,w]\] (14) \[A_{4} =\frac{Cov(Y,Z|W)}{Cov(X,Z|W)}=\frac{\frac{\partial}{\partial z}E[ Y|z,w]}{\frac{\partial}{\partial z}E[X|z,w]}. \tag{15}\]
#### 3.4.1 Exclusion Restriction
Figure 7 serves as an example of a covariate that has no direct impact on \(Z\) or \(U\). Nevertheless, if we were to condition upon \(X\) and nothing else, \(U\) would no longer be independent of \(W\) or \(Z\) due to the collider effect. A collider effect induces an association between two variables that point to (i.e. cause) a single variable that has been conditioned.[15] However, conditioning upon \(W\) breaks this association.
The updated structural equations are
\[X=c_{1}U+c_{3}Z+c_{5}W+\epsilon_{3} \tag{16}\]
\[Y=c_{0}X+c_{2}U+c_{ER}Z+c_{6}W+\epsilon_{4}. \tag{17}\]
**Proposition 3.4**.: _Under Eqs. (16) and (17), \(\hat{A}_{2}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{1}c_{2} +c_{4}c_{ER}}{1-c_{5}^{2}}\) and \(\hat{A}_{3}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{1}c_{2} }{1-c_{3}^{2}-c_{5}^{2}}\)_
The proof is very similar to that of Proposition 3.1 so it is omitted. As expected, adjusting for \(W\) leads to decreased variance in \(X\), which leads to a higher proportional contribution of \(U\), at the benefit of eliminating the backdoor path via \(c_{5}c_{6}\). For 2SLS, the results for \(A_{4}\) are not affected because the assumption violations vis-a-vis using \(Z\) as an IV are not influenced by \(W\). Nevertheless, in practical settings, adjusting for \(W\) will usually increase precision of \(\hat{A}_{4}\).
#### 3.4.2 Independence
Besides introducing \(W\) as a confounder in the \(X-Y\) relationship, Figure 8 extends \(W\) to be a confounder in the \(Z-Y\) and \(Z-X\) relationships. Therefore, the validity of \(Z\) as an IV is contingent on conditioning upon both \(W\) and \(Z\). Thus, the structural equations are updated to
\[Z=c_{I}U+c_{7}W+\epsilon_{5} \tag{18}\] \[X=c_{1}U+c_{3}Z+c_{5}W+\epsilon_{1},\] (19) \[Y=c_{0}X+c_{2}U+c_{6}W+\epsilon_{2}. \tag{20}\]
The results for all quantities, \(A_{2}\), \(A_{3}\), and \(A_{4}\), can now be updated per the following proposition:
Figure 7: A DAG with observed confounder and violation of exclusion restriction (\(W\)).
**Proposition 3.5**.: _Under Eqs. (19), (20), and (18):_
\[\hat{A}_{2}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{1}c_{2}+c_{2 }c_{3}c_{I}}{1-(c_{5}+c_{3}c_{7})^{2}}, \tag{21}\]
\[\hat{A}_{3}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{\frac{c_{1}c _{2}(1-c_{7}^{2}-c_{I}^{2})}{1-c_{7}^{2}}}{1-(c_{5}+c_{3}c_{7})^{2}-(1-c_{7}^{ 2})(c_{3}+\frac{c_{1}c_{I}}{1-c_{7}^{2}})^{2}}, \tag{22}\]
\[\hat{A}_{4}\stackrel{{ p}}{{\rightarrow}}c_{0}+\frac{c_{2}c_{I} }{(1-c_{7}^{2})(c_{3}+\frac{c_{1}c_{I}}{(1-c_{7}^{2})})}. \tag{23}\]
These results bear some resemblance to Proposition 3.2 when we did not have \(W\) present. For \(A_{2}\), in the numerator, because \(W\) does not mitigate the influence of \(U\) in the DAG, we still have two backdoor paths from \(X\) to \(Y\) that go through \(U\). Meanwhile, in the denominator the variance of \(X\) is reduced via controlling for \(W\), which has a direct path to \(X\) as well as an indirect path via \(Z\).
For \(A_{3}\), the quantity is similar conceptually to our findings in Proposition 3.2 except that we must additionally account for controlling for \(W\). In the numerator, \(c_{1}c_{2}(1-c_{7}^{2}-c_{I}^{2})\) represents the magnitude of unobserved confounding reduced (i.e. multiplied) by the exogenous variance of \(Z\) due to there no longer being a backdoor path to \(Y\) via \(Z\). We must account for the cost of adjusting \(W\) therefore this quantity is amplified (i.e. divided) by \(1-c_{7}^{2}\), the variance in \(Z\) free of \(W\). In the denominator, we have the remaining variance of \(X\) after adjusting for \(W\) and \(Z\). The first subtracting term represents the unconditional \(R^{2}\) between \(X\) and \(Z\) while the second takes the variation of \(Z\) free of \(W\) and multiplies it by the partial \(R^{2}\) between \(Z\) and \(X\), adjusting for \(W\). Because we haven't adjusted for \(U\), the backdoor path via \(U\) remains and, furthermore, because \(W\) does not mitigate this, \(W\) essentially acts like a bias amplifier for \(c_{1}c_{I}\). Lastly, \(A_{4}\) represents the association between \(Z\) and \(X\) via \(U\) as well as the variation exogenous from \(W\) directly from \(Z\).
#### 3.4.3 Treatment Effect Heterogeneity
Because we require our observed confounder to be of the same form of \(U\) for benchmarking purposes, we will set \(W\) as both an effect modifier of the treatment on the outcome and of the instrument on the treatment assignment with the following structural equations:
\[X =\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\alpha_{4}W+\alpha_{5}ZW+ \epsilon_{1}, \tag{24}\] \[Y =\beta_{1}X+\beta_{2}U+\beta_{3}XU+\beta_{4}W+\beta_{5}XW+ \epsilon_{2}. \tag{25}\]
The presence of \(XW\), which is observed but still endogenous, means that in OLS, we must adjust for it and in 2SLS, must provide an an additional IV. In particular, we will choose \(ZW\). Therefore we modify the
Figure 8: \(W\) mimics \(U\) in the DAG.
quantities of interest to
\[A_{2} =\frac{\partial}{\partial x}E[Y|x,w,xw] \tag{26}\] \[A_{3} =\frac{\partial}{\partial x}E[Y|x,z,w,xw]\] (27) \[A_{4} =\frac{\partial}{\partial\hat{x}}E[Y|\hat{x},\widehat{xw},w] \tag{28}\]
where \(\hat{x}\) and \(\widehat{xw}\) represent the fitted values from using \(Z\), \(ZW\), and \(W\) as regressors for \(X\) and \(XW\), respectively. Note that now the main effect is obtained after orthogonalizing \(XW\) and \(W\), or \(\widehat{XW}\) and \(W\) in 2SLS, which we can find via FWL by treating \(XW\) as any other covariate. Therefore, the corresponding interpretation of the main effect is when \(W=0\), or we are at the average value of the covariate due to centering the variable.
**Proposition 3.6**.: _Under Eqs. (24) and (25) as well as \(E[U^{3}]=0\), \(E[W^{3}]=0\), \(E[W^{4}]=3\), and \(E[U^{4}]=3\) we have \(\hat{A}_{2}\overset{p}{\rightarrow}\beta_{1}+\frac{\alpha_{2}\beta_{2}+2 \alpha_{1}\alpha_{3}\beta_{3}}{1-\alpha_{4}^{2}-\frac{4\alpha_{2}^{2}\alpha_{ 2}^{2}}{1+\alpha_{4}^{2}+2\alpha_{5}^{2}}}\) and \(\hat{A}_{4}\overset{p}{\rightarrow}\beta_{1}+\frac{\alpha_{1}\alpha_{3} \beta_{3}-\frac{2\alpha_{1}\alpha_{3}^{2}\beta_{3}}{\alpha_{1}^{2}+\alpha_{5}^ {2}}}{\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{2}^{2}\alpha_{2}^{2}}{( \alpha_{1}^{2}+\alpha_{5}^{2})}}\)._
The interpretation of \(A_{2}\) is consistent with Proposition 3.3 with the denominator reflecting the fact that we are adjusting for \(W\) and \(XW\), which reduces the remaining variance of \(X\). For \(A_{4}\), because we are no longer computing the ratio of coefficients the interpretation is not directly comparable to Proposition 3.3. Nevertheless, we can see the influence of the \(Z-U\) interaction on the inconsistency and observe that because \(\hat{XU}\) is correlated with \(\widehat{XW}\), adjusting for \(\widehat{XW}\) will reduce the influence of \(\widehat{XU}\), hence the subtraction terms.
One may notice that we have omitted the derivation for the result of \(A_{3}\) and this is because we only wish to compare \(A_{2}\) and \(A_{4}\). From our discussion of Proposition 3.3, assuming \(sign(\alpha_{2}\beta_{2})=sign(\alpha_{3}\beta_{3})\) for sake of simplicity, bias amplification will hold in the effect modification case if \(2\alpha_{1}^{2}+\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1\). When we have a strong IV, or \(|\alpha_{1}|\gtrapprox 0.7\), then this inequality holds. If we instead lower the strength of the IV to where we must consider \(\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1-2\alpha_{1}^{2}\), we would require the multiplication interaction effects to be at least as large as the main effects with this requirement increasing as the IV strength increases. We argue that in this circumstance one should avoid using \(Z\) altogether because concern over this ratio would imply that the IV is weak and the LACE is likely to be far from the ACE due to large heterogeneity. Thus, a comparison between OLS and 2SLS is not warranted. As we will describe later on, one may "benchmark" this ratio using the observed confounders and instrument as an initial check if an IV is appropriate in their circumstance for targeting the ACE.
## 4 Sensitivity Analysis
In this section, we use the closed form derivations of the previous section to develop a set of sensitivity analysis procedures that provide analysts with information surrounding whether OLS or 2SLS may be more appropriate given the observed data and hypothesized assumption violations. We focus on comparing \(A_{3}\) and \(A_{4}\), or using \(Z\) as a confounder versus as an IV, by graphically presenting the relative inconsistency \(\frac{\lambda_{4}}{\lambda_{3}}\) as measured by the degree of IV assumption violations and unobserved confounding.
The graphical depiction of the relative inconsistencies is largely motivated by Cincelli and Hazlett (2020) [18] and Cincelli and Hazlett (2022)[10] where they use a set of partial \(R^{2}\)'s to characterize how large the assumption violations in confounder and 2SLS analyses must be to render statistically significant results null. Instead of focusing on hypothesis testing, however, we examine how large unobserved confounding and the IV assumption violations could be in order for \(\frac{\lambda_{4}}{\lambda_{3}}>1\) or \(\frac{\lambda_{4}}{\lambda_{3}}<1\). Following this, like Cincelli and Hazlett, we use benchmarking to estimate the unobserved sensitivity quantities across a variety of scenarios.
To briefly summarize the benchmarking procedure, suppose we have the quantity \(R^{2}_{Y\sim U|X,W}\), or the additional gain in \(R^{2}\) by adding \(U\) to the regression of \(Y\) on treatment \(X\) and observed covariate \(W\). We must assume that for the individual confounders that make up the composite confounder \(W\), the magnitude of the \(W_{j}-X\)
relationship and \(W_{j}-Y\) relationship for \(j\in\{1,2,...J\}\) are similar to the magnitude of the \(U-X\) and \(U-Y\) relationship, respectively. As a result, we can therefore benchmark \(R^{2}_{Y\sim U|X,W}\) as conservatively as possible with a quantity such as \(\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j}}\) where \(W_{-j}\) represents all other observed confounders besides \(W_{j}\). In the remainder of this section, we will focus on the sensitivity analysis procedure in the scenario of the exclusion restriction. Then, using the same principles, present the results for the independence and heterogeneity assumption violations more succinctly.
### Exclusion Restriction Violation
**Proposition 4.1**.: \(\lambda_{4}=\frac{|c_{4}|}{|c_{3}|}=\frac{c_{1}c_{2}}{1-c_{3}^{2}-c_{5}^{2}}= \lambda_{3}\) _can be re-written as \(\frac{\sqrt{R^{2}_{Y\sim Z|X,U,W}}\sqrt{1-R^{2}_{Y\sim U|X,W}(1-R^{2}_{X\sim W +Z})}}{\sqrt{1-R^{2}_{Y\sim Z|X,W}\sqrt{R^{2}_{X\sim Z}}}sd(Z^{\perp X,W})}> \frac{\sqrt{R^{2}_{X\sim U}R^{2}_{Y\sim U|X,W,Z}}}{\sqrt{1-R^{2}_{U\sim X|W,Z}}}\) where \(sd(Z^{\perp X,W})\) is the standard deviation resulting from the residuals of regressing \(Z\) on \(X\) and \(W\)._
In order to simplify the description of the above quantity, we will decompose the terms in the above inequality into three factors: symbolically, we let
\[\phi =\frac{1-R^{2}_{X\sim W+Z}}{\sqrt{1-R^{2}_{Y\sim Z|X,W}}\sqrt{R^{ 2}_{X\sim Z}}sd(Z^{\perp X,W})} \tag{29}\] \[\gamma =\frac{\sqrt{R^{2}_{X\sim U}}\sqrt{R^{2}_{Y\sim U|X,W,Z}}}{\sqrt{ 1-R^{2}_{U\sim X|W,Z}}\sqrt{1-R^{2}_{Y\sim U|X,W}}}\] (30) \[\theta =\sqrt{R^{2}_{Y\sim Z|X,W,U}}. \tag{31}\]
Here, \(\phi\) represents all observed quantities that can be directly estimated, \(\gamma\) is a function that is monotonically increasing as the unobserved confounding increases, and \(\theta\) is the degree of the exclusion restriction violation. We will need to benchmark \(\gamma\) and \(\theta\), which we will refer to as \(\gamma^{B}\) and \(\theta^{B}\). These observed benchmarks are detailed in Table 2.
We begin our sensitivity analysis procedure with ambivalence between OLS and 2SLS, or that \(\frac{\lambda_{4}}{\lambda_{3}}=1\implies\frac{\theta\phi}{\gamma}=1\). Then, given \(\hat{\phi}\) and \(\theta^{B}\), we can solve for how large \(\gamma\), or the degree of unobserved confounding, must be in order for the equality to be maintained. We call this \(\gamma^{I}\) where "I" indicates the \(\gamma\) "implied" by assuming ambivalence. \(\gamma^{I}\) serves as an anchor point for the next step when we compute \(\gamma^{B}\).
Now using the data, we will find \(\gamma^{B}\) through benchmarking and combine this with \(\phi\) and \(\theta_{B}\) to form \(\frac{\theta^{B}\phi}{\gamma^{B}}\). We will refer to this fraction as the "inconsistency ratio," or "IR", which may be used to judge the use of 2SLS or OLS. For example, a ratio of \(<1\) implies that \(\lambda_{3}>\lambda_{4}\) and suggests that we should use 2SLS as opposed to OLS. Of course, the degree of unobserved confounding could be different than the observed benchmark. To mitigate this, we may additionally add a multiplier, \(M*\gamma^{B}\), if we believe the observed confounders underestimate (\(M<1\)) or overestimate (\(M>1\)) the degree of unobserved confounding. For example, if \(M=0.5\), we are assuming that the unobserved confounding is half as much as the benchmarked
\begin{table}
\begin{tabular}{|c|c|} \hline
**Quantity** & **Benchmark** \\ \hline \(R^{2}_{Y\sim Z|X,W,U}\) & \(R^{2}_{Y\sim Z|X,W}\) \\ \hline \(R^{2}_{X\sim U}\) & \(\max_{j}R^{2}_{X\sim W_{j}}\) \\ \hline \(R^{2}_{Y\sim U|X,W,Z}\) & \(\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},Z}\) \\ \hline \(R^{2}_{U\sim X|W,Z}\) & \(\max_{j}R^{2}_{W_{j}\sim X|\mathcal{W}_{-j},Z}\) \\ \hline \(R^{2}_{Y\sim U|X,W}\) & \(\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Benchmarking for \(\gamma\) and \(\theta\) Quantities for Exclusion Restriction
confounding. Lastly, we compute \(\theta^{I}\) or the degree of the exclusion restriction violation to satisfy \(\frac{\theta^{I}\phi^{B}}{\gamma^{B}}=1\), or returning to ambivalence.
The above quantities culminate in the graphical visualization in Figure 9. This Figure was produced under a simulation where \(\frac{\lambda_{4}}{\lambda_{3}}<1\), implying that 2SLS is a better choice than OLS in reducing inconsistency for the ACE. The x-axis shows the IR, while the y-axis tells us the degree of unobserved confounding as captured by \(\gamma\). Focusing on the y-axis, the position of the black dotted line, our anchor point, gives us the plausibility of \(\gamma^{I}\) by comparing to \(0.5\times\gamma_{B}\) (red), \(\gamma_{B}\) (purple), and \(1.5\times\gamma_{B}\) (blue). We can see that the degree of unobserved confounding required for ambivalence is roughly 60% of the benchmarked confounding. We obtain 60% by comparing the y-intercept of the purple line (\(\approx 0.25\)) with the y-intercept of the dotted black line (\(\approx 0.15\)). Looking at the x-axis, this works in favor of 2SLS as the corresponding inconsistency calculated by \(\frac{\theta^{B}\phi^{B}}{\gamma^{B}}<1\) unless the true degree of unobserved confounding is less than that of the benchmarked \(\gamma\).
Lastly, we will address the legend labels for which we have one for each line. "Benchmarked ER Violation" is the benchmarked exclusion restriction violation as quantified by \(R^{2}_{Y\sim Z|X,W,U}\) while "ER Violation Required" tells us the value of \(R^{2}_{Y\sim Z|X,W,U}\) required to return to ambivalence at the given multiplier. The comparison of the required numbers and the benchmarked violation gives analysts information to judge the plausibility of ambivalence. For example, on Figure 9, the degree violation required to return to ambivalence given \(M=1\) (0.061) is roughly 2.3 times the observed benchmark of 0.026.
### Independence Violation
**Proposition 4.2**.: \(\lambda_{4}=\left|\frac{c_{2}c_{I}}{c_{3}(1-c_{2}^{*})+c_{1}c_{2}}\right|= \left|\frac{\frac{c_{1}c_{2}(1-c_{2}^{*}-c_{2}^{*})}{1-c_{2}^{*}}}{1-(c_{5}+c_ {3}c_{7})^{2}-(1-c_{7}^{*})(c_{3}+\frac{14c_{7}}{1-c_{7}^{*}})^{2}}\right|= \lambda_{3}\) _can be re-written as \(\frac{\sqrt{R^{2}_{Z\sim U}}}{\sqrt{1-R^{2}_{Z\sim U|W}}}=\sqrt{R^{2}_{X\sim U |Z,W}}\Bigg{[}\frac{\sqrt{R^{2}_{X\sim Z|W}}s^{d(X\perp X,W)}s^{d(X\perp W)}s ^{d(Z\perp W)}}{1-R^{2}_{X\sim W}-(1-R^{2}_{Z\sim W})R^{2}_{X\sim Z|W}}\Bigg{]}\)__
Figure 9: A case where the inconsistency of OLS is more than that of 2SLS under an exclusion restriction violation
Once again, we partition the quantities in Proposition - as follows:
\[\phi =\frac{\sqrt{R_{X\sim Z|W}^{2}}sd(X^{\perp Z,W})sd(X^{\perp W})sd(Z^ {\perp W})}{1-R_{X\sim W}^{2}-(1-R_{Z\sim W}^{2})R_{X\sim Z|W}^{2}} \tag{32}\] \[\gamma =\sqrt{R_{X\sim U|Z,W}^{2}}\] (33) \[\theta =\frac{\sqrt{R_{Z\sim U}^{2}}}{\sqrt{1-R_{Z\sim U|W}^{2}}}\] (34) \[R_{Z\sim U}^{2} =\frac{\phi^{2}\gamma^{2}(1-R_{Z\sim W}^{2})^{2}}{1+\phi^{2} \gamma^{2}} \tag{35}\]
The IR in this case will be \(\frac{\phi\gamma}{\theta}\) with \(\theta\) being a monotonic function with the degree of the independence violation increasing. Though we may solve for \(R_{Z\sim U}^{2}\), as in line 35, we leave \(\theta\) in its current form to maintain the clean interpretation \(\gamma\). Then, given a value for \(\gamma\), \(\phi\), and the observed \(R_{Z\sim W}^{2}\), we calculate the value of \(R_{Z\sim U}^{2}\) in the legend entry for the corresponding line in the sensitivity plot. An example of the sensitivity plot for the independence violation is presented in Figure 10. The construction and interpretation of this figure is the same as in the exclusion restriction violation except for the fact that we now measure how large does an independence violation need to be to return to ambivalence. This is denoted by "U-Z B" and "U-Z Req." as in the \(R^{2}\) magnitude of the edge from \(U\) to \(Z\) as in the DAG in Figure 8. In this case, we simulated the data such that \(\lambda_{4}=\lambda_{3}\) so the purple line (multiplier of 1) and the required violation is close to the black anchor line and the observed benchmark. The corresponding benchmarks are detailed in Table 3.
### Treatment Effect Heterogeneity
**Proposition 4.3**.: \(\lambda_{2}=\beta_{1}+\frac{\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}}{1- \alpha_{4}^{2}-\frac{4\alpha_{5}^{2}+\alpha_{5}^{2}}{1+\alpha_{4}^{2}+\alpha_{5 }^{2}}}\) _and \(\lambda_{4}=\frac{\alpha_{1}\alpha_{3}\beta_{3}-\frac{2\alpha_{1}\alpha_{3} \alpha_{2}^{2}\beta_{3}}{\alpha_{4}^{2}+\alpha_{5}^{2}}}{\alpha_{1}^{2}+\alpha _{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{(\alpha_{1}^{2}+\alpha_{5}^{2})}}\) can be rewritten as_
\[\left|\frac{1/2-\frac{R_{X\sim ZW}^{2}}{Var(XW)}}{R_{X\sim W}^{2}+R _{X\sim ZW}^{2}-R_{XW\sim X}^{2}}\right|\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
Though there may be many combinations of what the signs of the individual coefficients may be, there are only three possible IRs
\[IR=\begin{cases}\frac{(\phi_{1}-1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&\text{ if }sign(\alpha_{2}\beta_{2})=sign(\alpha_{1}\alpha_{3}\beta_{3}),\\ \frac{(\phi_{1}+1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&\text{if }sign(\alpha_{2}\beta_{2}) \neq sign(\alpha_{1}\alpha_{3}\beta_{3})\text{ and }\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|}>|2 \alpha_{1}|,\\ -\frac{(\phi_{1}-1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&\text{if }sign(\alpha_{2}\beta_{2}) \neq sign(\alpha_{1}\alpha_{3}\beta_{3})\text{ and }\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|} \leq|2\alpha_{1}|.\end{cases}\]
Practically, there are only two IRs for us to consider because the first and third cases are opposite signs and we are concerned with the magnitude of the IR rather than the sign. Whether the condition of \(\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|}>|2\alpha_{1}|\) is met can be investigated through benchmarking. As an example of this condition, suppose \(\alpha=0.5\) - a fairly strong instrument in this context - then we require the treatment effect heterogeneity, \(\alpha_{3}\beta_{3}\), to be no stronger than the effect of \(U\) on \(X\) when \(Z=0\) (\(\alpha_{2}\)) as well as \(U\) on \(Y\) when \(X=0\) (\(\beta_{2}\)). Note that because \(X\) and \(Z\) are centered, \(\alpha_{2}\) and \(\beta_{3}\) are at the average value of these variables. We suggest that two graphs are computed for both cases and, following this, if the conclusions agree, then no further action should be taken. If they disagree, then one may benchmark the condition to give a sense of which condition is more likely or, alternatively, average the ICs assuming equal probability of each case.
\begin{table}
\begin{tabular}{|c|c|} \hline
**Quantity** & **Benchmark** \\ \hline \(R^{2}_{X\sim U}\) & \(\max_{j}R^{2}_{X\sim W_{j}}\) \\ \hline \(R^{2}_{Y\sim U|X,W,XU,XW}\) & \(\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},XW}\) \\ \hline \(R^{2}_{Y\sim U|X,W,XW}\) & \(\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},XW}\) \\ \hline \(R^{2}_{Y\sim XU|U,X,W,XW}\) & \(\max_{j}R^{2}_{Y\sim XW_{j}|X,\mathcal{W},X\mathcal{W}_{-j}}\) \\ \hline \(R^{2}_{Y\sim XU|W,XW}\) & \(\max_{j}R^{2}_{Y\sim XW_{j}|X,W,X\mathcal{W}_{-j}}\) \\ \hline \(R^{2}_{XU\sim U|W,XW}\) & \(\max_{j}R^{2}_{XW_{j}\sim W_{j}|\mathcal{W}_{-j},X\mathcal{W}_{-j}}\) \\ \hline \(R^{2}_{XW\sim XU}\) & \(\max_{i\neq j}R^{2}_{XW_{i}\sim XW_{j}}\) \\ \hline \(R^{2}_{X\sim ZU}\) & \(\max_{j}R^{2}_{X\sim ZW_{j}}\) \\ \hline \end{tabular}
\end{table}
Table 4: Benchmarking for \(\gamma\) and \(\theta\) Quantities for Treatment Effect Heterogeneity
Figure 11: A case where the inconsistency of OLS is less than that of 2SLS under a treatment effect heterogeneity violation
Simulation
In this section, we present results verifying the closed form derivations for all three assumption violation scenarios with and without covariates. Additionally, we further present the use of our sensitivity analysis procedure. All variables were generated via the structural equations provided in the earlier sections with the error terms following a normal distribution with mean zero. When the variables were independent from all other variables in the system, such as \(U\), they were standard normal. When the variable was determined by other variables in the system, such as \(X\), it was still normal but with the variance of the error term being equal to one minus the variance of the other variables in the structural equation. This is so that the total variance of terms like \(Var(X)\) will still equal one (see the "Notes about Simulations" section in the appendix). For example, in Eq 1, \(\sigma_{e_{1}}=1-c_{1}^{2}-c_{3}^{2}\) and, thus, \(\epsilon_{1}\sim N(0,1-c_{1}^{2}-c_{3}^{2})\).
To obtain the simulated numbers for OLS, we use the built-in 1m from R function while for 2SLS, we use the ivreg function from the ivreg R package. In the exclusion restriction and independence setting, we present the Monte Carlo averages over 500 simulations of 500 observations generated. For treatment effect heterogeneity because the empirical results take more samples to converge, we used 500 simulations of 3000 observations. Note that for all simulations, for demonstration purposes, we set the IV to be sufficiently strong such that the estimates would converge on the population value within a reasonable sample size. Nevertheless, our results otherwise hold for weak IVs if the number of observations in each simulation increased significantly.
To demonstrate the sensitivity analysis plots, we used one generation from the relevant set of structural equations with a sample size of 500 for the exclusion restriction and independence violations while we have a sample size of 1000 for the heterogeneity violation. To simulate a set of covariates that we may use to benchmark the unobserved quantities, we generated three independent covariates \((W_{1},W_{2},W_{3})\) from a standard normal distribution and constructed \(W\) via the first principal component.
### Exclusion Restriction
For the case with no covariates, we set the following structural parameters \(c_{0}=0.3\), \(c_{1}=0.5\), \(c_{2}=0.5\), \(c_{3}=0.5\), and \(c_{ER}=0.25\). For the case with covariates, we have \(c_{0}=0.25\), \(c_{1}=0.4\), \(c_{2}=0.4\), \(c_{3}=0.7\), \(c_{ER}=0.25\), \(c_{5}=0.4\), and \(c_{6}=0.4\). The results are presented in Table 5.
Using the values \(c_{1}=c_{2}=c_{5}=c_{6}=0.4\) and \(c_{3}=0.3\), an IV of relative moderate strength, we present how the sensitivity plots change in response to an increasing exclusion violation in Figure 12. We see that since the IV is not very strong, it will take only a small degree of the exclusion restriction violation to reach ambivalence and suggest that OLS is more appropriate. Note that for cases that there is no violation, benchmarking will pick up a small signal so the IC will not be 0 though small.
### Independence
For the case with no covariates, we set the following structural parameters \(c_{0}=0.3\), \(c_{1}=0.5\), \(c_{2}=0.5\), \(c_{3}=0.5\), \(c_{ER}=0.25\). For the case with covariates, we have \(c_{0}=0.3\), \(c_{1}=0.4\), \(c_{2}=0.4\), \(c_{3}=0.5\), \(c_{ER}=0.25\), \(c_{5}=0.4\), \(c_{6}=0.4\), and \(c_{7}=0.25\). The results are presented in Table 6.
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & & \multicolumn{3}{c}{Method} \\ \cline{3-5} & & OLS without Z & OLS with Z & 2SLS with Z \\ \hline \multirow{2}{*}{Without covariates} & Closed Form Result & 0.375 & 0.333 & 0.5 \\ & Simulated Result & 0.374 & 0.334 & 0.496 \\ \hline \multirow{2}{*}{With covariates} & Closed Form Result & 0.399 & 0.457 & 0.357 \\ & Simulated Result & 0.398 & 0.459 & 0.354 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Theoretical and Simulated Results for the Exclusion Restriction Violation
Figure 12: Increasing Exclusion Restriction Violation Sensitivity Analysis Results with Benchmarking
In Figure 13, we present the results of increasing the independence violation with \(c_{1}=c_{2}=c_{5}=c_{6}=0.4\) and \(c_{3}=0.4\), a relatively strong IV. As we vary \(c_{I}\), we set \(c_{7}\) equivalently under the assumption that our observed benchmark is roughly equal to that of unobserved violation. We see that a relatively large violation in the independence assumption is needed to render a strong IV more inconsistent than OLS with unobserved confounding.
### Treatment Effect Heterogeneity
For the case with no covariates, we set the following structural parameters \(\beta_{1}=0.1\), \(\beta_{2}=0.2\), \(\beta_{3}=0.1\), \(\alpha_{1}=0.45\), \(\alpha_{2}=0.15\), and \(\alpha_{3}=0.1\). For the case with covariates, we have \(\beta_{1}=0.1\), \(\beta_{2}=0.2\), \(\beta_{3}=0.1\), \(\beta_{4}\), \(\beta_{5}\), \(\alpha_{1}=0.45\), \(\alpha_{2}=0.15\), \(\alpha_{3}=0.1\), \(\alpha_{4}=0.15\), and \(\alpha_{5}=0.1\). The results are presented in Table 7.
Figure 14 reflects the how, with increasing treatment effect heterogeneity related to the IV, 2SLS could become more inconsistent than OLS. The structural parameters are the same as the above except that \(\alpha_{2}=\alpha_{3}=0.2\).
## 6 Applied Example
We demonstrate the application of our sensitivity analysis in studying neonatal outcomes as a function impacts to the mother during pregnancy. Specifically, we seek to measure the impact of perceived maternal stress during the pregnancy, as measured by the perceived stress scale (PSS),[19] on the birthweight of the neonate (in pounds). The PSS asks the participant several questions related to different symptoms of stress in her life where, in each item, they give a response from zero (almost never) to four (very often) on the Likert scale. Higher scores indicated a larger degree of stress and we treat the overall PSS score as continuous for the purposes of our analysis. From the literature, our hypothesis is that higher stress would lead to lower birthweights;[20] however, due to the observational nature of such data, we are concerned about unmeasured confounding. That is, there may be unobserved factors that affect both perceived maternal stress during pregnancy and birthweight. Therefore, we are motivated to use IVs to isolate the causal effect.
We study 147 mother-child dyads from ecological momentary assessment data collected at at University of California, Irvine. More details regarding the cohort inclusion-exclusion criteria and characteristics can be found in Lazarides et al.[21] For our purposes, at three points during their pregnancy (early, mid, and late),
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & & \multicolumn{3}{c}{Method} \\ \cline{3-5} & & OLS without Z & OLS with Z & 2SLS with Z \\ \hline \multirow{2}{*}{Without covariates} & Closed Form Result & 0.312 & 0.384 & 0.2 \\ & Simulated Result & 0.311 & 0.383 & 0.200 \\ \hline \multirow{2}{*}{With covariates} & Closed Form Result & 0.290 & 0.391 & 0.176 \\ & Simulated Result & 0.290 & 0.394 & 0.178 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Theoretical and Simulated Results for the Independence Violation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & & \multicolumn{3}{c}{Method} \\ \cline{3-5} & & OLS without Z & OLS with Z & 2SLS with Z \\ \hline \multirow{2}{*}{Without covariates} & Closed Form Result & 0.039 & 0.044 & 0.022 \\ & Simulated Result & 0.039 & 0.044 & 0.021 \\ \hline \multirow{2}{*}{With covariates} & Closed Form Result & 0.040 & 0.037 & 0.021 \\ & Simulated Result & 0.039 & 0.046 & 0.021 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Theoretical and Simulated Results for Treatment Effect Heterogeneity
Figure 13: Increasing Independence Violation Sensitivity Analysis Results with Benchmarking
Figure 14: Heterogeneity Violation Sensitivity Analysis Results with Benchmarking
the mothers were asked to fill out a questionnaire on their mobile devices about regarding their stress at ten times throughout the day. Furthermore, each mother repeated the questionnaire for four consecutive days, two of these days being weekends. For each trimester, day, and timepoint, we calculate the current PSS and then calculate the median PSS for that day, which is our exposure of interest for perceived stress. For example, a mother would fill out the questionnaire ten times during Thursday, Friday, Saturday, and Sunday. At each timepoint, we would calculate the PSS and subsequently have four median PSS measurements for each day the data was collected. For the purposes of the application of our sensitivity analysis tool, we only focus on data from the second collection point (i.e. mid-pregnancy).
A potential IV that we utilized was a binary indicator of whether the stress measurements were collected on a weekday or a weekend with the hypothesis being that stress is generally increased on weekdays compared to weekends. Because we had data for both weekdays and weekends, for each mother, we randomly sampled a single day to use as her stress measurement and recorded if this day was a weekday or weekend. This represents a realistic sampling scheme where the mothers may be asked only once to fill the questionnaire and they are randomized to a weekday or weekend. Regressing median PSS upon the weekday indicator IV corroborated our hypothesized relevance and found that weekdays were associated with 0.213 higher points, on average, compared to weekdays (95% CI: [0.038,0.388],\(p=0.017\)). Nevertheless, the IV was fairly weak with the first-stage \(F=5.775\).
For the confounders we adjusted for in both OLS and 2SLS, we utilized the neonate's sex, the mother's age, how many previous children the mother has had (i.e. parity), and an indicator for obstetric risk, which summarizes many factors that may result in adverse pregnancy outcomes such as low birth weight. OLS found an insignificant positive relationship where as median stress increased by one, on average, birthweight increased by 0.05 pounds (95% CI: [-0.212,0.313],\(p=0.702\)). 2SLS found an insignificant relationship but of the opposite sign where a one point increase in median PSS was associated with an 0.19 bound decrease in average birthweight (95% CI: [-2.587, 2.193], \(p=0.87\)).
The demonstration of our approach is presented in Figure 15. All variables were standardized before inputting it into our tool and, to form the general confounder \(W\), the confounders were combined using the first principal component. We may first look at the benchmarks of each assumption violation, which appear to be quite small. Even so, the benchmarked strength of unobserved confounding is relatively small, which overall, in conjunction with the weakness of the IV, leads us to empirically prefer using OLS rather than 2SLS.
Examining each assumption violation individually, if one believes the unobserved confounding is double the observed confounding (the blue line), we are in ambivalence with respect to the exclusion restriction violation but not the independence assumption. Theoretically, there may not be a basis for the weekday indicator to violate the IV assumptions and this is a factor the analyst should weigh against the observed evidence. If the analyst decides to use the IV, there may be concern that, due to exposure effect heterogeneity, the LACE will be far from the ACE. In this specific example, we would be concerned that there is an unobserved covariate that modifies the relationship between median PSS and birthweight in addition to modifying the effect of the weekday indicator on median PSS. Fortunately, the corresponding plot in Figure 15 shows that benchmarked heterogeneity related to the IV is low and will not cause large inconsistency in comparison to benchmarked unobserved confounding (unless we assign a multiplier of 0.5). Note that we only have one plot for treatment effect heterogeneity because both cases related to the signs yield similar plots and conclusions.
## 7 Discussion
In this paper, we have investigated two predominant ways that analysts may isolate causal effects: confounders, or the CAC, and via IV, or the IVAC. Furthermore, we have based our study on the notion that each approach may work imperfectly due to assumption violation as is most plausible in the vast majority of real world settings. Our closed-form results that capture interpretable rules of thumb based on DAG edge weights and coefficients, as well as our sensitivity analysis procedure, help guide analysts towards a practical philosophy on how one may execute observational studies. If the goal is to obtain an estimate of the ACE and there is a set of tools that one must choose from, such as the CAC and IVAC, one must ask: when is one tool more advantageous than the other? We have ultimately broken down this question by juxtaposing the degree
of unobserved confounding to the degree IV assumption violation, shifting away from perfectionism and providing results for analysts to offer evidence that the estimate computed was the least inconsistent it could have been given the scenario.
Our results show that properly defining confounders and IVs in the study paradigm is only a starting point. Whether a variable will be used in CAC or as an IV in IVAC is not necessarily congruent with its formal definition. In fact, we have presented analytically that, relative to OLS, there are scenarios where a variable should be used as an IV even though it does not meet the strict definition of an IV. For example, in an exclusion restriction violation the variable used as an IV is, by definition, a confounder and, although such a variable is not a "perfect IV," the relative performance of 2SLS would be better than that of OLS. Our sensitivity analysis procedure assists in detecting this by using the observed data.
Figure 15: Examining the Three Assumptions for the Weekday Indicator IV
Another relevant scenario lies in the fact that even though we may have a valid IV, the LACE estimand from 2SLS may be further away from the ACE than an estimand from OLS that is impacted by unobserved confounding. Through our closed form results and sensitivity analysis, we allow an analyst to judge how large heterogeneity would need to be in order for this to occur. Whether the resulting IV estimand remains scientifically useful is a subject of debate that we will not discuss here.[22] Rather, we are interested in directly targeting the ACE and will assume that treatment effect heterogeneity may provide a barrier to this goal. In this sense, the results from our paper may provide case-by-case evidence for and against those who may argue IV analyses may still be useful for the ACE.
Moving to the use of our sensitivity analysis tools, we essentially provide three separate plots for three distinct assumption violation scenarios. If all three plots all agree that either 2SLS is superior or OLS is superior then the suggested approach is clear to the analyst as was largely the case with the birthweight example. However, one may encounter a scenario that the graphs disagree: for example, the exclusion restriction and independence show 2SLS is superior while the treatment effect heterogeneity graph does not. In this case, one should rank the importance of each assumption and consider the observed benchmarks for each assumption violation. Using the multipliers provided or inputting a custom multiplier, one could consider the degree of unobserved confounding across the board in order for all three graphs to agree and evaluate whether these findings are reasonable in the particular study scenario. Similarly, one may further use the implied assumption violations in the legend relative to the benchmarked assumption violations for these purposes. Ultimately, sensitivity analyses are matters of judgement and we aim to provide quantitative tools for analysts to navigate these scenarios.
The results in this paper also have implications on more sophisticated, data-driven variable selection techniques. Users of CAC often will opt to capture confounding by adjusting for the propensity score (PS) using the fitted values of the exposure regressed on the confounders. In order to reduce Type I error, one would model the PS away from the outcome and, thus, it appears reasonable to use penalization or machine learning (ML) techniques to optimize prediction error. There are two main issues with this automated procedure: (1) strong though imperfect IVs would be selected leading to possible bias amplification and (2) omitted variable bias may occur due to shrinkage to zero of important confounders. These points have been mentioned by others[5, 3, 23] and our results further provide analysts information to act in the face of such issues. We may use our results to quantify how strong the confounder or IV should be in order to avoid adjustment altogether or, instead, use it in 2SLS. Future directions may include extending the framework developed in this paper to techniques such as augmented inverse probability of treatment weighting (AIPTW), post-LASSO (PL), targeted minimum loss estimation (TMLE), and double machine learning (DML) to weigh the approaches against one another.[24, 25, 26]
There are several avenues for future research. Firstly, in this paper, we focus only on consistency but in estimation, we may also want to know whether CAC may more efficient than IVAC or vice-versa. Furthermore, if the confidence intervals overlap on the point estimate, which would cause ambivalence. We may additionally wish to move beyond the setting with a continuous exposure and outcome. Nevertheless, if one believes linear probability models (LPMs) are appropriate for the study's context, then one may extend our results to a binary exposure and outcome. It has been shown that if the probabilities produced by the LPM are not outside of the range of \([0,1]\), or that the probabilities of exposure or outcome are not extreme in the population, then OLS and 2SLS may still give consistent results.[27, 28] For the sensitivity analysis procedure, the results are only as useful as variables available and chosen for benchmarking, which is partially mitigated by using the multipliers. Certainly, other benchmarking quantities for the unobserved quantities than the ones we chose could be used. The aggregation of the covariates into one general confounder \(W\) could be done via other methods besides PCA, which is limited when there are categorical variables. Using non-continuous variables also is limited when capturing associations using \(R^{2}\) due to the Frechet bounds on the correlation between non-continuous variables being potentially far narrower than \([-1,1]\).[29]
Isolating causal effects in observational data presents many challenges, which foremost include the effect of unobserved confounding. The CAC and IVAC offer potential avenues to mitigate this confounding. Even so, under untestable assumptions, choosing the optimal approach for the problem at hand involves much conjecture. Our closed-form findings and sensitivity analysis approach helps analysts quantitatively justify the approach that they ultimately believe produces an estimate closest to the ACE. The upshot is that, with
the information we provide, results from observational studies will both be more transparent and more useful in their interpretation.
## 8 Acknowledgements
RSZ and DLG are supported by NIH/NIA P30AG066519 and NIH/NIA 1RF1AG075107.
|
2307.08988 | EVIL: Evidential Inference Learning for Trustworthy Semi-supervised
Medical Image Segmentation | Recently, uncertainty-aware methods have attracted increasing attention in
semi-supervised medical image segmentation. However, current methods usually
suffer from the drawback that it is difficult to balance the computational
cost, estimation accuracy, and theoretical support in a unified framework. To
alleviate this problem, we introduce the Dempster-Shafer Theory of Evidence
(DST) into semi-supervised medical image segmentation, dubbed Evidential
Inference Learning (EVIL). EVIL provides a theoretically guaranteed solution to
infer accurate uncertainty quantification in a single forward pass. Trustworthy
pseudo labels on unlabeled data are generated after uncertainty estimation. The
recently proposed consistency regularization-based training paradigm is adopted
in our framework, which enforces the consistency on the perturbed predictions
to enhance the generalization with few labeled data. Experimental results show
that EVIL achieves competitive performance in comparison with several
state-of-the-art methods on the public dataset. | Yingyu Chen, Ziyuan Yang, Chenyu Shen, Zhiwen Wang, Yang Qin, Yi Zhang | 2023-07-18T05:59:27Z | http://arxiv.org/abs/2307.08988v1 | # Evil: Evidential Inference Learning for Trustworthy Semi-Supervised Medical Image Segmentation
###### Abstract
Recently, uncertainty-aware methods have attracted increasing attention in semi-supervised medical image segmentation. However, current methods usually suffer from the drawback that it is difficult to balance the computational cost, estimation accuracy, and theoretical support in a unified framework. To alleviate this problem, we introduce the Dempster-Shafer Theory of Evidence (DST) into semi-supervised medical image segmentation, dubbed _EVidential Inference_**Learning_**(_EVIL_)**. EVIL provides a theoretically guaranteed solution to infer accurate uncertainty quantification in a single forward pass. Trustworthy pseudo labels on unlabeled data are generated after uncertainty estimation. The recently proposed consistency regularization-based training paradigm is adopted in our framework, which enforces the consistency on the perturbed predictions to enhance the generalization with few labeled data. Experimental results show that EVIL achieves competitive performance in comparison with several state-of-the-art methods on the public dataset.
Yingyu Chen\({}^{1}\) Ziyuan Yang\({}^{1}\) Chenyu Shen\({}^{1}\) Zhiwen Wang\({}^{1}\) Yang Qin\({}^{1}\)
Yi Zhang\({}^{2,1}\), Senior Member, IEEE\({}^{1}\)College of Computer Science, Sichuan University, Chengdu, China
\({}^{2}\) School of Cyber Science and Engineering, Sichuan University, Chengdu, China
Medical Image Segmentation, Semi-Supervised Learning, Evidential Learning
## 1 Introduction
Medical image segmentation plays an essential role in subsequent clinical or computer-aided diagnosis and fully-supervised learning has achieved great success in the field of automatic image segmentation [1]. However, annotating medical images is laborious and requires rich professional knowledge [2].
Semi-supervised learning (SSL) has shown great potential to alleviate this problem by leveraging a large set of unlabeled data accompanied with a limited number of labeled data. These methods can be roughly categorized into two types: (1) pseudo-label retraining, which incorporates pseudo labels on unlabeled data for retraining [3, 4, 5]; and (2) consistency regularization, which enforces the prediction consistency to enhance generalization with various perturbations, such as input perturbation, feature perturbation, and network perturbation [6, 7, 8].
However, since these methods rely heavily on the prediction of pseudo label, false predictions will severely degrade the segmentation performance. To improve the quality of pseudo labels, some uncertainty-aware methods have been proposed, including Monte Carlo dropout (MC-dropout)-based [9], Information-Entropy-based [10], and Prediction Variance-based [11] methods. However, these methods suffer from some problems: (1) Although MC-dropout is mathematically guaranteed by Bayesian theory, its training process is costly due to the multiple sampling operations; (2) Due to the limited sampling times, MC-dropout can't obtain accurate uncertainty quantification; (3) Other two uncertainty estimation methods have advantages in computational cost, but they lack theoretical support, leading to unstable pseudo label generation.
To handle the above issues, we introduce the Dempster-Shafer Theory of Evidence (DST) into semi-supervised medical image segmentation, providing a theoretically guaranteed single-pass solution for uncertainty quantification inference, dubbed _EVidential Inference_**Learning_**(_EVIL_)**. Following the training paradigm proposed in [7], EVIL belongs to the consistency regularization method with network perturbation, which imposes the prediction consistency on two networks perturbed with different initialization. In particular, the two networks play different roles. One is a vanilla segmentation network (S-Net) which directly generates the segmentation result. The other network called evidential-network (E-Net) is built from the perspective of DST, which is theoretically guaranteed for reliable predictions. Different from S-Net, the output of E-Net is regarded as the evidence and parameterized into a Dirichlet distribution on segmentation probabilities. Subjective Logic theory (SL) [12] is employed to quantify the predictions and uncertainties of different categories with the Dirichlet distribution in a single inference, which significantly reduces the training time. Then, the trustworthy pseudo labels on unlabeled data are generated. In summary, there are three merits for our proposed EVIL: lower computation cost due to the single-pass
operation, accurate uncertainty estimation based on SL and theoretical guarantee based on DST.
The main contributions of this work are summarized as: 1) we introduce DST into SSL and provide a fast accurate uncertainty estimation with theoretical guarantee in a unified framework; 2) a novel network perturbation strategy is proposed, which allows different initialized network optimized with different objectives; and 3) extensive experiments are conducted to validate the effectiveness of our proposed EVIL.
## 2 Method
Given a labeled set \(D_{l}=\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{N_{l}}\) with \(N_{l}\) samples and an unlabeled set \(D_{u}=\{\mathbf{x}_{i}\}_{i=1}^{N_{u}}\) with \(N_{u}\) samples, where \(N_{u}\gg N_{l}\) in semi-supervised task.
As illustrated in Fig. 1, EVIL has two differently initialized networks, E-Net \(\mathcal{F}_{1}\) with parameter set \(\theta_{1}\) and S-Net \(\mathcal{F}_{2}\) with parameter set \(\theta_{2}\). For labeled data, S-Net is optimized with traditional joint cross-entropy loss and dice loss, while E-Net models a Dirichlet distribution and is optimized with evidential segmentation loss. \(P_{1}\), \(P_{2}\) are the segmentation predictions and \(Y_{1}\), \(Y_{2}\) are the corresponding pseudo labels generated by \(argmax\) function. For unlabeled data, E-Net generates pseudo labels \(Y_{1}\) and accurate uncertainty estimations \(\mathcal{M}\) simultaneously. Then, the trustworthy pseudo labels are calculated by \(\mathcal{M}\odot Y_{1}\) and used to guide the training of S-Net. Reversely, the pseudo labels \(Y_{2}\) generated by S-Net is leveraged for E-Net to explore more potential evidence to improve the generation of pseudo labels from unlabeled data.
### Uncertainty Modeling
In this section, we utilize DST to model the segmentation uncertainty and generate trustworthy prediction. For a \(K\)-class segmentation task, given an input \(\mathbf{x}_{i}\), the evidence vector \(\mathbf{e}_{i}\) is obtained with a transform function \(g\), which is defined in [13]:
\[\mathbf{e}_{i}=g(\mathcal{F}_{1}(\mathbf{x}_{i}))=exp^{(\tanh\mathcal{F}_{1}(\mathbf{x}_{i })/\tau)}, \tag{1}\]
where \(0<\tau<1\) is a scaling parameter set to \(1/K\). \(\mathcal{F}_{1}(\mathbf{x}_{i})\) is the output of E-Net with input \(\mathbf{x}_{i}\). Subjective Logic [12] computes the belief mass for category \(k\) and uncertainty as:
\[b_{i}^{k}=\frac{e_{i}^{k}}{S_{i}}=\frac{\alpha_{i}^{k}-1}{S_{i}}\quad\text{ and}\quad u_{i}=\frac{K}{S_{i}}, \tag{2}\]
where \(S_{i}=\sum_{k=1}^{K}(e_{i}^{k}+1)\), \(u_{i}+\sum_{k=1}^{K}b_{i}^{k}=1\) and \(\alpha_{i}^{k}=e_{i}^{k}+1\). \(\mathbf{\alpha}_{i}=\left[\alpha_{i}^{k},\ldots,\alpha_{i}^{K}\right]\) can be regarded as the parameters of Dirichlet distribution, which models the density of segmentation probability and uncertainty [14]. The density function is defined as:
\[D(\mathbf{p}_{i}\mid\mathbf{\alpha}_{i})=\left\{\begin{array}{ll}\frac{1}{B(\mathbf{ \alpha}_{i})}\prod_{k=1}^{K}p_{i}^{\alpha_{i}^{k}-1}&\text{for }\mathbf{p}_{i}\in\mathcal{S}_{K},\\ 0&\text{otherwise},\end{array}\right. \tag{3}\]
where \(\mathbf{p}_{i}\) is the segmentation probability, \(B(\mathbf{\alpha}_{i})\) is the \(K\)-dimensional multinomial beta function for parameter \(\mathbf{\alpha}_{i}\), and \(\mathcal{S}_{K}\) is the \(K\)-dimensional simplex.
### Evidential Net (E-Net)
We follow [14] and use cross-entropy loss to make the segmentation probabilities \(\mathbf{p}_{i}\) approach the ground-truth \(\mathbf{y}_{i}\). Notably, the density of \(\mathbf{p}_{i}\) follows the Dirichlet distribution parameterized with \(\mathbf{\alpha}_{i}\). The loss can be formulated as:
\[\begin{split}\mathcal{L}_{diag}&=\int\left[\sum_{k=1}^{K }-y_{i}^{k}\log\left(p_{i}^{k}\right)\right]D(\mathbf{p}_{i}\mid\mathbf{\alpha}_{i})d \mathbf{p}_{i}\\ &=\sum_{k=1}^{K}y_{i}^{k}\left(\psi\left(S_{i}\right)-\psi\left( \alpha_{i}^{k}\right)\right),\end{split} \tag{4}\]
where \(\psi(\cdot)\) is the _digamma_ function. By optimizing \(\mathcal{L}_{dig}\), the evidence of different classes for positive samples is generated. However, \(\mathcal{L}_{dig}\) cannot guarantee that negative samples generate evidence as close as zero. Therefore, Kullback-Leibler (KL) divergence is incorporated into our loss function to penalize the divergence from negative samples, which is defined as:
\[\begin{split}\mathcal{L}_{KL}&=KL\left[D\left(\mathbf{p }_{i}\mid\tilde{\mathbf{\alpha}}_{i}\right)\|D\left(\mathbf{p}_{i}\mid\mathbf{1}\right) \right]\\ &=\log\left(\frac{\Gamma\left(\sum_{k=1}^{K}\widetilde{\alpha}_{ i}^{k}\right)}{\Gamma(K)\sum_{k=1}^{K}\Gamma\left(\widetilde{\alpha}_{i}^{k} \right)}\right)\\ &+\sum_{k=1}^{K}\left(\widetilde{\alpha}_{i}^{k}-1\right)\left[ \psi\left(\widetilde{\alpha}_{i}^{k}\right)-\psi\left(\sum_{k=1}^{K} \widetilde{\alpha}_{i}^{k}\right)\right],\end{split} \tag{5}\]
where \(\Gamma(\cdot)\) is the _gamma_ function, \(D(\mathbf{p}_{i}|\mathbf{1})\) is the uniform Dirichlet distribution, and \(\widetilde{\mathbf{\alpha}}_{i}=\mathbf{y}_{i}+(1-\mathbf{y}_{i})\odot\mathbf{\alpha}_{i}\).
Figure 1: The overview of our EVidential Inference Learning framework (EVIL), where \(\mathcal{M}\) denotes uncertainty map estimated by E-Net and \(\odot\) denotes element-wise product. ‘\(\rightarrow\)’ presents forward operation, ‘\(\rightarrow\)’ presents supervision loss operation and ‘\(//\)’ on ‘\(\rightarrow\)’ presents stop-gradient.
For segmentation task, the evidence \(\mathbf{e}_{i}\) is obtained with \(\mathbf{x}_{i}\). Then, \(\mathbf{\alpha}_{i}=\mathbf{e}_{i}+1\) is parameterized into the corresponding Dirichlet distribution and the evidential loss is:
\[\mathcal{L}_{evi}=\mathcal{L}_{dig}+\beta\mathcal{L}_{KL}, \tag{6}\]
where \(\beta\) is a annealing coefficient and is set to \(\beta(t)=min(1.0,\frac{t}{0.5t_{max}})\). \(t\) is the current epoch and \(t_{max}\) is the total number of training epochs.
As shown in Fig. 2, the Subjective Logic model has two parts, the certain part called belief mass \(\mathbf{b}_{i}\) and the uncertain part \(u_{i}\). The evidential loss generates evidence to reduce the uncertainty. However, since the cross-entropy based evidential loss is based on pixel level, which ignores the relationships between pixels in segmentation task, we use the Dice loss on the certain part and the certain loss is defined as:
\[\mathcal{L}_{certain}=1-\frac{2\sum_{k=1}^{K}y_{i}^{k}\sum_{k=1}^{K}\hat{p}_{ i}^{k}}{\sum_{k=1}^{K}y_{i}^{k}+\sum_{k=1}^{K}\hat{p}_{i}^{k}}, \tag{7}\]
where \(\hat{\mathbf{p}}_{i}=softmax(\mathbf{b}_{i})\) presents a simplex transformed from the belief mass \(\mathbf{b}_{i}\) with a \(softmax\) function.
Then, our overall evidential segmentation loss is defined:
\[\mathcal{L}_{Eseg}=\mathcal{L}_{evi}+\gamma\mathcal{L}_{certain}, \tag{8}\]
where \(\mathcal{L}_{evi}\) and \(\mathcal{L}_{certain}\) denote the evidential loss and the certain loss, respectively. \(\gamma\) denotes the weighting parameter, which is set to \(1\). By optimizing \(\mathcal{L}_{evi}\), E-Net generates the evidence for positive samples, while reduces the evidence for negative samples. \(\mathcal{L}_{certain}\) is leveraged to constrain the relationship between different predicted pixels.
### EVIL Framework
The total loss \(\mathcal{L}\) for our whole framework contains two components: supervised loss \(\mathcal{L}_{sup}\) on labeled data and consistency loss \(\mathcal{L}_{con}\) on unlabeled data:
\[\mathcal{L}=\mathcal{L}_{sup}+\lambda\mathcal{L}_{con}, \tag{9}\]
where \(\lambda\) is the balancing parameter. We use Gaussian ramp-up function \(\lambda(t)=\lambda_{max}*e^{-5\left(1.0-\frac{t}{t_{max}}\right)^{2}}\) and \(\lambda_{max}=0.1\).
The supervision loss is formulated as:
\[\mathcal{L}_{sup}=\mathcal{L}_{Eseg}(\mathcal{F}_{1}(\mathbf{x}),\mathbf{y})+ \mathcal{L}_{Sseg}(\mathcal{F}_{2}(\mathbf{x}),\mathbf{y}), \tag{10}\]
where \(\mathcal{L}_{Sseg}=\frac{1}{2}(\mathcal{L}_{ce}+\mathcal{L}_{dice})\) denotes the loss component for S-Net. \(\mathcal{L}_{ce}\) and \(\mathcal{L}_{dice}\) are the cross-entropy loss and dice loss, respectively.
The pseudo label can be calculated as \(Y_{1}=argmax(\mathbf{b}_{i})\) for E-Net and \(Y_{2}=argmax(\mathcal{F}_{2}(\mathbf{x}))\) for S-Net. The consistency loss on the unlabeled data is written as:
\[\mathcal{L}_{con}=\mathcal{L}_{evi}(\mathcal{F}_{1}(\mathbf{x}),Y_{2})+\mathcal{L} _{ce}(\mathcal{F}_{2}(\mathbf{x}),\mathcal{M}\odot Y_{1}). \tag{11}\]
where \(\mathcal{M}=u<T\) is the mask to filter out high uncertain results with threshold \(T=0.2\). We only use the evidential and cross-entropy losses in consistency loss term due to the mask operation which preserves only the reliable pseudo pixel labels. The consistency loss encourages E-Net to generate potential evidence from S-Net using \(\mathcal{L}_{evi}\) and S-Net to learn the reliable pseudo labels using \(\mathcal{L}_{ce}\) from E-Net.
## 3 Experiment
### Experiment Setup
We evaluate our method on the Automated Cardiac Diagnosis Challenge (ACDC) [15] dataset which contains 200 annotated short-axis cardiac MR-cine images from 100 patients. We leverage 70 patients (140 scans) for training, 10 patients (20 scans) for validation and 20 patients (40 scans) for testing. All short-axis slices within 3D scans are resized to 256 \(\times\) 256 as 2D images. See SSL4MIS 1 for more details. For semi-supervised experiments, images from 7 patients, 14 patients and 21 patients are set as labeled ratio 10%, 20% and 30% in the training set, respectively. Standard data augmentation, including random cropping, random rotating, and random flipping, is used to enlarge the training set. Three widely
Figure 3: Visual comparison of segmentation results with different methods with 10% labeled images.
used metrics, Dice Coefficient (\(DSC\)), Hausdorff Distance 95 (\(HD_{95}\)) and Average Surface Distance (\(ASD\)) are employed to evaluate the performance of our method.
For the sake of fairness, Unet [1] is chosen as the backbone in all methods, and SGD is adopted as the optimizer. The initial learning rate is set to 0.01, and polynomial scheduler strategy is employed to update the learning rate. We implement the proposed framework with PyTorch, using a single NVIDIA GTX 1080Ti GPU. The batch size is set to 24, where 12 images are labeled. All methods perform 30000 iterations during training.
### Experimental Results
Several recently proposed semi-supervised segmentation methods are compared, including: Mean-Teacher (MT) [6], Uncertainty-Aware Mean Teacher (UA-MT) [9], Interpolation Consistency Training (ICT) [16], Cross Pseudo Supervision (CPS) [7], and Uncertainty Rectified Pyramid Consistency (URPC) [17]. For all competing methods, official parameter settings are adopted.
Tab. 1 illustrates the quantitative results on ACDC. The first and second rows list the quantitative results of supervised Unet and E-Net. In different labeled data ratio settings, EVIL outperforms all the other methods. When only 10% of data are labeled, our method improves \(DSC\) by more than 3% compared with other SOTA uncertainty-aware methods (UAMT and URPC). Moreover, we achieve 4 points improvement in \(HD_{95}\) and 1 point in \(ASD\) compared with CPS. Especially, we can see that the performance of EVIL using 20% labeled data has surpassed all compared methods using 30% labeled data.
Fig. 3 visualizes the segmentation results of two cases using different methods with 10% labeled data. It is easy to see that the compared methods mis-classify many pixels while EVIL obtains more accurate prediction. As shown in Fig. 4, sampling times affect the uncertainty estimation quality of MC-dropout and our E-Net has best accurate estimation.
Tab. 2 shows the training time with fixed batch size = 24, where 'Num', 'Uncertainty', 'Time', 'Cost' denotes the network number, uncertainty-based or not, time consuming, and the additional time consuming cost respectively. We treat Unet as the upper bound of single network method and MT as the baseline of the multi-network framework since it is the fastest method compared to others. Specially, we can see that the proposed method improve significantly without introducing too much computation overhead.
## 4 Conclusion
In this paper, we propose a novel uncertainty-aware semi-supervised medical image segmentation framework. The proposed EVIL introduces DST into the consistency regularization training paradigm and achieves fast accurate uncertainty estimation with solid theoretical guarantee. Extensive experiments demonstrate that EVIL achieves state-of-the-art performance on the widely used ACDC dataset.
## 5 Conflicts of Interest
The authors declare that they have no conflicts of interest.
\begin{table}
\begin{tabular}{c|c c c|c c|c c|c c} \hline \hline Method & \multicolumn{3}{c|}{10\%} & \multicolumn{3}{c|}{20\%} & \multicolumn{3}{c}{30\%} \\ & \(DSC\uparrow\) & \(HD_{95}\downarrow\) & \(ASD\downarrow\) & \(DSC\uparrow\) & \(HD_{95}\downarrow\) & \(ASD\downarrow\) & \(DSC\uparrow\) & \(HD_{95}\downarrow\) & \(ASD\downarrow\) \\ \hline Unet & 80.05 & 7.41 & 2.38 & 84.90 & 8.94 & 2.52 & 87.07 & 6.61 & 1.95 \\ E-Net (ours) & 81.05 & 11.17 & 3.26 & 85.68 & 7.39 & 2.12 & 87.45 & 8.12 & 2.23 \\ \hline MT & 81.06 & 10.17 & 2.64 & 86.01 & 8.13 & 2.40 & 87.37 & 4.81 & 1.49 \\ UA-MT & 80.81 & 11.73 & 3.52 & 85.38 & 7.77 & 2.70 & 87.53 & 6.32 & 2.05 \\ ICT & 83.54 & 8.42 & 2.46 & 85.28 & 5.65 & 1.64 & 87.49 & 8.25 & 2.23 \\ CPS & 84.70 & 8.25 & 2.35 & 87.47 & 5.98 & 1.74 & 88.21 & 6.49 & 1.90 \\ URPC & 82.07 & 5.62 & 1.88 & 85.13 & 5.71 & 1.75 & 86.99 & 4.43 & 1.31 \\ EVIL (ours) & **85.91** & **3.91** & **1.36** & **88.22** & **4.01** & **1.21** & **89.43** & **3.84** & **1.07** \\ \hline \hline \end{tabular}
\end{table}
Table 1: The comparison of different methods on ACDC dataset on different semi-supervised labeled data ratio settings.
\begin{table}
\begin{tabular}{c|c|c|c|c} \hline \hline Method & Num & Uncertainty & Time & Cost \\ \hline Unet & 1 & \(\times\) & 0.076 s & - \\ ICT & 1 & \(\times\) & 0.090 s & + 18.42 \% \\ URPC & 1 & \(\surd\) & 0.089 s & + 17.11 \% \\ E-Net (ours) & 1 & \(\surd\) & 0.085 s & + 11.84 \% \\ \hline MT & 2 & \(\times\) & 0.101 s & - \\ CPS & 2 & \(\times\) & 0.137 s & + 35.64 \% \\ UA-MT & 2 & \(\surd\) & 0.337 s & + 233.66\% \\ EVIL (ours) & 2 & \(\surd\) & 0.148 s & + 46.53 \% \\ \hline \hline \end{tabular}
\end{table}
Table 2: The comparison of training time.
Figure 4: Visualization of uncertainty estimation. ‘S’ denotes the MC-dropout sampling times.
## 6 Compliance with Ethical Standards
This research study was conducted retrospectively using real clinical exams acquired at the University Hospital of Dijon. Ethical approval was not required as confirmed by the license attached with the open access data.
## 7 Acknowledgement
This work was supported in part by the National Natural Science Foundation of China under Grant 62271335; in part by the Sichuan Science and Technology Program under Grant 2021JDJ00024; and in part by the Sichuan University "From 0 to 1" Innovative Research Program under Grant 2022SCUH0016.
|
2305.19024 | Unified triquark equations | We derive covariant equations describing the three-quark bound state in terms
of quark and diquark degrees of freedom. The equations are exact in the
approximation where three-body forces are neglected. A feature of these
equations is that they unify two often-used but seemingly unrelated approaches
that model baryons as quark-diquark systems; namely, (i) the approach using
Poincar\'{e} covariant quark+diquark Faddeev equations driven by a
one-quark-exchange kernel [pioneered by Cahill {\it et al.}, Austral.\ J.\
Phys.\ {\bf 42}, 129 (1989) and Reinhardt, Phys.\ Lett.\ B {\bf 244}, 316
(1990)], and (ii) the approach using the quasipotential quark-diquark
bound-state equation where the kernel consists of the lowest-order contribution
from an underlying quark-quark potential [pioneered by Ebert {\it et al.}, Z.\
Phys.\ C {\bf 76} 111 (1997)]. In particular, we show that each of these
approaches corresponds to the unified equations with its kernel taken in
different, non-overlapping, approximations. | A. N. Kvinikhidze, B. Blankleider | 2023-05-30T13:30:13Z | http://arxiv.org/abs/2305.19024v1 | # Unified triquark equations
###### Abstract
We derive covariant equations describing the three-quark bound state in terms of quark and diquark degrees of freedom. The equations are exact in the approximation where three-body forces are neglected. A feature of these equations is that they unify two often-used but seemingly unrelated approaches that model baryons as quark-diquark systems; namely, (i) the approach using Poincare covariant quark+diquark Faddeev equations driven by a one-quark-exchange kernel [pioneered by Cahill _et al._, Austral. J. Phys. **42**, 129 (1989) and Reinhardt, Phys. Lett. B **244**, 316 (1990)], and (ii) the approach using the quasipotential quark-diquark bound-state equation where the kernel consists of the lowest-order contribution from an underlying quark-quark potential [pioneered by Ebert _et al._, Z. Phys. C **76** 111 (1997)]. In particular, we show that each of these approaches corresponds to the unified equations with its kernel taken in different, non-overlapping, approximations.
Introduction
The use of diquarks as effective degrees of freedom in describing hadrons has a long history, as evidenced by a number of reviews over the last thirty years [1; 2; 3; 4]. Documented are different quark-diquark approaches for baryons, but to the best of our knowledge, no attempt has been made for their comparison on the basis of quantum field theory. In the present work, we would like to make such a comparison, demonstrating that two of the most often-used quark-diquark models of baryons, which have usually been considered as separate, unrelated models of baryons, are in fact two non-overlapping parts of the same quark-diquark model.
The first of these models, proposed more than thirty years ago [5; 6], is based on a description of three quarks using covariant Faddeev equations where the quark-quark t matrix is approximated by one or more diquark-pole terms (i.e., terms with a pole at the diquark mass, and with a residue that is expressed as an outer product of form factors \(\Gamma\) and \(\bar{\Gamma}\) for the transition between the diquark and two free quarks). The resulting coupled set of bound-state equations are illustrated in Fig. 1. Sometimes referred to as Poincare covariant quark+diquark Faddeev equations [4; 7], and sometimes as quark-diquark Bethe-Salpeter equations [8; 9], they have been used extensively over the years, see [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] for a representative selection of works.
The second model, proposed more that 25 years ago [19], is a relativistic description of the quark-diquark system using quasipotential equations (we will refer to it as the "quasipotential quark-diquark model"), which has likewise been often used over the years [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29].
Figure 1: Poincaré covariant quark+diquark Faddeev equations of Ref. [5; 6]. The amplitudes \(\Phi_{a}\) and \(\Phi_{c}\) are Faddeev components coupling the baryon to quark (single line) and diquark (double-line) states. The equation kernel corresponds to one-quark-exchange, with \(\Gamma_{c}\) and \(\bar{\Gamma}_{a}\) being vertex functions describing the disintegration and formation of the diquark.
In this model one first constructs a quark-quark potential of the form
\[V_{qq}=V_{\rm gluon}+V_{\rm conf} \tag{1}\]
where \(V_{\rm gluon}\) is the quark-quark (\(qq\)) one-gluon-exchange potential and \(V_{\rm conf}\) is a local confining potential, and then uses this to construct the quark-diquark potential which then forms the kernel of a relativistic quark-diquark quasipotential equation for the baryon. Illustrated in Fig. 2, this bound-state equation again has the form of a Faddeev equation, but with a kernel corresponding to a single rescattering of two quarks via potential \(V_{qq}\) (specified in the diagram as quarks a and c scattering via a potential \(K_{b}\) (\(=V_{qq}\)) with quark \(b\) being a spectator).
In the following, we derive covariant triquark bound-state equations that are exact for the case where three-body forces are neglected. These equations are illustrated in Fig. 3, and have the form of Faddeev equations where the kernel consists of an infinite series involving successive numbers of quark-exchanges between quark-diquark states. It is evident that the Poincare covariant quark+diquark Faddeev equations of Ref. [5; 6] correspond to keeping just the first term in the infinite series, and the quasipotential equations of Ref. [19] correspond to keeping just the second term in this series. As such, our triquark equations unify these two popular approaches for modeling baryons in terms of quark and diquark degrees of freedom. Moreover, it is evident that these two approaches should not be viewed as unrelated competing models of baryons, but rather, as different approximations of the same model. Indeed, any competition between these models at describing data, needs to be assessed by comparing their kernels, as these are non-overlapping terms appearing in the unified equations.
Figure 2: Equations corresponding to the quasipotential quark-diquark model of Ref. [19]. Similar to Fig. 1, amplitudes \(\Phi_{a}\) and \(\Phi_{c}\) are Faddeev components coupling the baryon to quark-diquark states, with \(\Gamma_{c}\) and \(\bar{\Gamma}_{a}\) being diquark vertex functions. However, the kernel of this equation involves a single scattering of two quarks (quarks a and c in this case) via a potential \(K_{b}\).
Ideally, the two approaches should be combined, with a kernel that is the sum of the first two terms of the infinite series illustrated in Fig. 3. Additionally, in light of the unification embodied in Fig. 3, all sorts of form-factors (electromagnetic, axial-vector, pseudoscalar, etc.) should also be unified correspondingly. This can be done by gauging the equation of Fig. 3[30], thereby obtaining contributions to the baryon form factors coming from both of the first two kernels in this figure. By contrast, the current situation is that the baryon form factors are being pursued intensively in each of the two approaches separately (just in the last few years, the Poincare covariant quark+diquark Faddeev equations have been used to calculate such form factors in Refs. [31; 32; 33; 34; 35; 36; 37; 38] and the quasipotential quark-diquark approach has been used to calculate them in Refs. [39; 40; 41; 42; 43; 44; 45; 46]).
It is worth noting that analogous unified equations were derived for the tetraquark [47].
## II Derivation
### Triquark equations for distinguishable quarks
For clarity of presentation, we first consider the case of three distinguishable quarks. To describe such a system where only pairwise interactions are taken into account, we follow the formulation of Faddeev [48]. Thus, assigning labels 1, 2 and 3 to the quarks, and using a notation where \((abc)\) is a cyclic permutation of (123), the three-body \((3q)\) kernel, \(K\), is written as
\[K=\sum_{a}K_{a} \tag{2}\]
where \(K_{a}\) is the kernel where quarks \(b\) and \(c\) are interacting while quark \(a\) is a spectator, as illustrated in Fig. 4.
Figure 3: Unified quark+diquark equations derived in this paper. The kernel of this equation is an infinite series whose first two terms, separately, correspond to the model of Ref. [5; 6] as illustrated in Fig. 1, and the model of Ref. [19] as illustrated in Fig. 2, respectively.
The \(3q\) bound-state wave function for distinguishable quarks is then
\[\Psi=G_{0}K\Psi \tag{3}\]
where \(G_{0}\) is the fully disconnected part of the full \(3q\) Green function \(G\). The three-body kernels \(K_{a}\) can be used to define the Faddeev components \(\Psi_{a}\) as
\[\Psi_{a}=G_{0}K_{a}\Psi, \tag{4}\]
so that
\[\Psi=\sum_{a}\Psi_{a}. \tag{5}\]
From Eq. (3) follow Faddeev's equations for the components,
\[\Psi_{a}=\sum_{b}G_{0}T_{a}\bar{\delta}_{ab}\Psi_{b} \tag{6}\]
where \(\bar{\delta}_{ab}=1-\delta_{ab}\) and \(T_{a}\) is the t matrix corresponding to kernel \(K_{a}\), so that
\[T_{a}=K_{a}+K_{a}G_{0}T_{a}. \tag{7}\]
Assuming that the \(qq\) interaction admits the creation of a diquark, the Green function \(G_{a}\) describing the scattering of quarks \(b\) and \(c\), will contain a corresponding pole at the diquark mass, so that one can write
\[G_{a}=G_{a}^{P}+G_{a}^{R} \tag{8}\]
where \(G_{a}^{P}\) is the Green function's pole term while \(G_{a}^{R}\) is its regular part. Then, because
\[T_{a}=K_{a}+K_{a}G_{a}K_{a}, \tag{9}\]
the t matrix \(T_{a}\) can be written as
\[T_{a}=K_{a}+T_{a}^{P}+T_{a}^{C} \tag{10}\]
Figure 4: Structure of the terms \(K_{a}\) (\(a=1,2,3\)) making up the three-body kernel \(K\) where only two-body forces are included. The coloured circles represent two-body kernels \(K_{bc}\) for the scattering of quarks \(b\) and \(c\), as indicated.
where \(T_{a}^{P}\) is \(T_{a}\)'s pole term, while the sum \(K_{a}+T_{a}^{C}\) constitutes its regular part. It is important to note that there is no overcounting in this decomposition; that is, the terms \(K_{a}\), \(T_{a}^{P}\) and \(T_{a}^{C}\) do not overlap. Note that in the case of unconfined quarks, the analytic structure of \(T_{a}\) would be represented by its pole part, \(T_{a}^{P}\), its part with the \(2q\) branch point, \(T_{a}^{C}\), and the part \(K_{a}\) again with a branch point, but above the \(2q\) mass.
We write Eq. (6) in matrix form as
\[\Psi={\cal T}\,\Psi \tag{11}\]
where \(\,\Psi\) is a column matrix of elements \(\Psi_{a}\), and \({\cal T}\) is a square matrix whose \((a,b)\)'th element is \({\cal T}_{ab}=G_{0}T_{a}\bar{\delta}_{ab}\). Similarly we write Eq. (10) in matrix form as
\[{\cal T}={\cal K}+{\cal T}^{P}+{\cal T}^{C} \tag{12}\]
where \({\cal K}_{ab}=G_{0}K_{a}\bar{\delta}_{ab}\), \({\cal T}_{ab}^{P}=G_{0}T_{a}^{P}\bar{\delta}_{ab}\), and \({\cal T}_{ab}^{C}=G_{0}T_{a}{}^{C}\bar{\delta}_{ab}\). Equation (11) can then be recast as
\[\Psi=(1-{\cal K}-{\cal T}^{C})^{-1}{\cal T}^{P}\,\Psi. \tag{13}\]
Using the separable form of the pole term,
\[T_{a}^{P}=\Gamma_{a}D_{a}\bar{\Gamma}_{a} \tag{14}\]
where \(\Gamma_{a}\) (similarly \(\bar{\Gamma}_{a}\)) and \(D_{a}\) are the diquark form factor and propagator, respectively, Eq. (13) implies that
\[\Phi_{a}=\sum_{bc}\bar{\Gamma}_{a}\bar{\delta}_{ab}\left[(1-{\cal K}-{\cal T} ^{C})^{-1}\right]_{bc}G_{0}\Gamma_{c}D_{c}\Phi_{c} \tag{15}\]
where
\[\Phi_{a}=\sum_{b}\bar{\Gamma}_{a}\bar{\delta}_{ab}\Psi_{b}. \tag{16}\]
Expanding the inverse term in Eq. (15) as
\[(1-{\cal K}-{\cal T}^{C})^{-1}=1+{\cal K}+{\cal T}^{C}+\ldots, \tag{17}\]
we obtain
\[\Phi_{a}=\sum_{bc}\bar{\Gamma}_{a}\bar{\delta}_{ab}(\delta_{bc}+G_{0}K_{b} \bar{\delta}_{bc}+\ldots)G_{0}\Gamma_{c}D_{c}\Phi_{c}, \tag{18}\]
which is illustrated in Fig. 3.
It is apparent that the first two terms of this series correspond to the models of Refs. [5; 6] and Ref. [19], respectively. Indeed, keeping just the first term in the series results in the bound-state equation
\[\Phi_{a}=\sum_{b}\bar{\Gamma}_{a}\bar{\delta}_{ab}G_{0}\Gamma_{b}D_{b}\Phi_{b} \tag{19}\]
which is illustrated in Fig. 1 and coincides with the Poincare covariant quark+diquark Faddeev equations of Ref. [5; 6], and keeping just the second term in the series results in the bound-state equation
\[\Phi_{a}=\sum_{bc}\bar{\Gamma}_{a}\bar{\delta}_{ab}G_{0}K_{b}\bar{\delta}_{bc}G _{0}\Gamma_{c}D_{c}\Phi_{c}, \tag{20}\]
which is illustrated in Fig. 2 and coincides with the quasipotential quark-diquark equations of Ref. [19].
Although each of the approaches of Refs. [5; 6] and Ref. [19], can be viewed as different approximations of the same unified equations, Eq. (18), the reality is that the quark-diquark picture of a baryon is described by a kernel that consists of at least the sum of the first two terms of the series in Eq. (18). This observation should clarify the true picture of quark-diquark dynamics in baryons.
### Triquark equations for indistinguishable quarks
To take into account the antisymmetry of identical quarks, we first note that the Faddeev equations for distinguishable particles, Eq. (6), possess fully antisymmetric solutions (as well as symmetric ones) where the component wave functions have the symmetry properties
\[P_{23}\Psi_{1}=-\Psi_{1}, P_{12}\Psi_{1}=-\Psi_{2}, P_{31}\Psi_{1}=-\Psi_{3},\] \[P_{31}\Psi_{2}=-\Psi_{2}, P_{23}\Psi_{2}=-\Psi_{3}, P_{12}\Psi_{2}=-\Psi_{1},\] \[P_{12}\Psi_{3}=-\Psi_{3}, P_{31}\Psi_{3}=-\Psi_{1}, P_{23}\Psi_{3}=-\Psi_{2}, \tag{21}\]
where \(P_{ab}\) is the operator that exchanges the quantum numbers of particles \(a\) and \(b\). Choosing a solution with these symmetry properties, Eq. (6) for \(\Psi_{1}\) reduces to
\[\Psi_{1}=-G_{0}T_{1}P_{12}\Psi_{1} \tag{22}\]
where \(T_{1}\) results from antisymmetrizing the t matrix for distinguishable particles, \(T_{1}^{d}\), using
\[T_{1}=(1-P_{23})T_{1}^{d}. \tag{23}\]
Equation (22) can be seen most easily by using Eqs. (21):
\[\Psi_{1} = G_{0}T_{1}^{d}(\Psi_{2}+\Psi_{3})=G_{0}T_{1}^{d}(1-P_{23})\Psi_{2} \tag{24}\] \[= G_{0}(1-P_{23})T_{1}^{d}\Psi_{2}\] \[= -G_{0}(1-P_{23})T_{1}^{d}P_{12}\Psi_{1}.\]
We can then again express \(T_{1}\) as
\[T_{1}=K_{1}+T_{1}^{P}+T_{1}^{C}, \tag{25}\]
where \(T_{1}^{P}\) and \(K_{1}+T_{1}^{C}\) are the pole and regular parts of \(T_{1}\), but this time with all quantities antisymmetric under the interchange of quark 2 and 3's quantum numbers. Equation (22) can then be recast as
\[\Psi_{1}=-\left[1+({\cal K}_{1}+{\cal T}_{1}^{C})P_{12}\right]^{-1}{\cal T}_{ 1}^{P}P_{12}\Psi_{1} \tag{26}\]
where \({\cal K}_{1}=G_{0}K_{1}\), \({\cal T}_{1}^{P}=G_{0}T_{1}^{P}\), and \({\cal T}_{1}^{C}=G_{0}T_{1}^{C}\). Using the separable form of the pole term,
\[T_{1}^{P}=\Gamma_{1}D_{1}\bar{\Gamma}_{1}, \tag{27}\]
where the diquark form factors are now antisymmetric, \(P_{23}\Gamma_{1}=-\Gamma_{1}\) and \(\bar{\Gamma}_{1}P_{23}=-\bar{\Gamma}_{1}\), we obtain the equation for the Faddeev component
\[\Phi_{1}=-\bar{\Gamma}_{1}P_{12}\left[1+G_{0}(K_{1}+T_{1}^{C})P_{12}\right]^{ -1}G_{0}\Gamma_{1}D_{1}\Phi_{1} \tag{28}\]
where
\[\Phi_{1}=\bar{\Gamma}_{1}P_{12}\Psi_{1}. \tag{29}\]
Expanding the inverse term in Eq. (28) as
\[\left[1+G_{0}(K_{1}+T_{1}^{C})P_{12}\right]^{-1}=1-G_{0}(K_{1}+T_{1}^{C})P_{12 }+\ldots, \tag{30}\]
leads to the final form of our unified equations for three identical quarks,
\[\Phi_{1}=-\bar{\Gamma}_{1}P_{12}\left[1-G_{0}(K_{1}+T_{1}^{C})P_{12}+\ldots \right]G_{0}\Gamma_{1}D_{1}\Phi_{1}. \tag{31}\]
Keeping only the first two terms of the series for the kernel, and making the further approximation, \(T_{1}^{C}=0\), leads to the equation
\[\Phi_{1}=\bar{\Gamma}_{1}P_{12}\left[1+K_{1}P_{12}\right]\Gamma_{1}d_{1}\Phi_ {1} \tag{32}\]
which covers both approaches of Refs. [5; 6] and Ref. [19].
## III Discussion
We have derived covariant equations that describe the bound state of the triquark in terms of quark and diquark degrees of freedom. These equations are illustrated in Fig. 3, with exact expressions given for distinguishable quarks in Eq. (18), and for indistinguishable quarks in Eq. (31). An essential aspect of these equations is that they are exact in the approximation where only two-body forces are retained. As a result, they are expected to encompass, and thus unify, all descriptions of triquarks that use quark and diquark degrees of freedom, and that assume two-body forces only.
It is worth noting that our procedure leading to Eq. (15), and hence to Eq. (18) and Eq. (31), is similar to the one used by Alt, Grassbeger, and Sandhas (AGS) to reduce three-particle Faddeev equations to that of coupled two-particle equations [49]; however, it differs from AGS in its details, and also in one essential way, namely, we have shown that the two-body matrix (in three-body space) \(T_{a}\), can be decomposed into three mutually exclusive parts, as in Eq. (10), where the two-body kernel \(K_{a}\) appears explicitly (AGS and related prior works, decomposed two-body t matrices into two parts, a separable one, and the rest). It is just this decomposition of \(T_{a}\) into three-parts involving \(K_{a}\), that has led to the unification of previous works, as outlined above.
This unification is demonstrated explicitly for two of the most prominent and longest-used approaches in the literature, namely the one using the covariant quark+diquark Faddeev equations of Refs. [5; 6], and the one using quasipotential quark-diquark equations of Ref. [19]. In particular, the covariant quark+diquark Faddeev equations correspond to keeping just the first term of the kernel in our equations [the one-quark-exchange diagram in Fig. 3], and the quasipotential quark-diquark equations correspond to keeping just the second term of the kernel in our equations [the \(qq\) rescattering diagram in Fig. 3]. It is noteworthy that our equations reveal that these two approaches, which have been pursued separately for more than 25 years in order to model not only bound states of baryons, but also various types of baryon form factors (electromagnetic, axial-vector, scalar, etc.), use equations with two, different, non-overlapping, kernels. Our equations indicate that it is the sum of the first two terms in the kernel (at least) that should have been used instead. Although this is not an issue for cases where only one pair of quarks (out of three possible pairs) can form a diquark, in which case only the kernel of the quasipotential quark-diquark equations
contributes [19; 22], it may be a serious problem for other case, like that of three identical quarks where both kernels contribute and therefore should be summed [29].
###### Acknowledgements.
A.N.K. was supported by the Shota Rustaveli National Science Foundation (Grant No. FR17-354).
|
2303.14856 | Automatic Number Plate Recognition using Random Forest Classifier | Automatic Number Plate Recognition System (ANPRS) is a mass surveillance
embedded system that recognizes the number plate of the vehicle. This system is
generally used for traffic management applications. It should be very efficient
in detecting the number plate in noisy as well as in low illumination and also
within required time frame. This paper proposes a number plate recognition
method by processing vehicle's rear or front image. After image is captured,
processing is divided into four steps which are Pre-Processing, Number plate
localization, Character segmentation and Character recognition. Pre-Processing
enhances the image for further processing, number plate localization extracts
the number plate region from the image, character segmentation separates the
individual characters from the extracted number plate and character recognition
identifies the optical characters by using random forest classification
algorithm. Experimental results reveal that the accuracy of this method is
90.9%. | Zuhaib Akhtar, Rashid Ali | 2023-03-26T23:49:43Z | http://arxiv.org/abs/2303.14856v1 | # Automatic Number Plate Recognition using Random Forest Classifier
###### Abstract
Automatic Number Plate Recognition System is a mass surveillance embedded system that recognizes the number plate of the vehicle. This system is generally used for traffic management applications. It should be very efficient in detecting the number plate in noisy as well as in low illumination and also within required time frame. This paper proposes a number plate recognition method by processing vehicle's rear or front image. After image is captured, processing is divided into four steps which are Pre-Processing, Number plate localization, Character segmentation and Character recognition. Pre-Processing enhances the image for further processing, number plate localization extracts the number plate region from the image, character segmentation separates the individual characters from the extracted number plate and character recognition identifies the optical characters by using random forest classification algorithm. Experimental results reveal that the accuracy of this method is 90.9 %.
Keywords:Automatic Number Plate Recognition System, Edge Detection, Character Recognition, Random Forest Classifier, Ensemble Learning
## 1 Introduction
As the number of vehicles has increased considerably during the recent years, more and more attention is required on advanced, efficient and accurate intelligent transportation system (ITSs). One of the important technique used in ITS is Automatic Number Plate Recognition (ANPR) System. It was invented way back in 1979 at the Police Scientific Development Branch in the UK. However, ANPR did not become widely used until new developments in software and hardware during the 1990s [1]. ANPR is a computer vision technology that recognizes the vehicle's number plate without direct human intervention. This system captures the image of the vehicle and extracts the characters of the number plate. These extracted characters then can be searched in the database to identify the owner of the vehicle.
Hence it can be used in traffic management applications like automatic gate control for authorized/non-authorized vehicles, entrance admissions in toll systems, monitoring of traffic violations, borders crossing control and premises where high security is needed, such as the Parliament building.
The ANPR system should work under noisy conditions and low contrast. There are various methodologies used in ANPR system. The proposed methodology consists of following steps: Pre-Processing, localization of number plate using edge detection, character segmentation and character recognition. Literature survey, proposed methodology and results are discussed in subsequent sections.
## 2 Literature Review
Number Plate recognition has garnered a lot of attention from the research community. One of the important characteristics of research in number plate recognition is that the research is restricted locally as number plate tends to be different for different regions. There is no standardization of number plate. For example, each state in US has a different number gate. License plate extraction is the most important step in the number plate recognition phase. Hontai and Koga proposed a method where no prior knowledge of position and shape is required to extract characters [2]. In this method Gaussian filters were used. In order to locate number plate, neural network based filters and post processor were used [3]. Neural networks analyzed a portion of image to decide whether that portion contains number plate or not. Gabour filters were used in this method. Becerikli et. al used colors of the number plate to extract the number plate from image. In this method neural network were used for obtaining the pixel values of number plate [4]. Saqib, Asad and Omer used Hough transform based technique to extract number plate [5]. Binarization with Sauvola method and moving window were used to detect the number plate by Chang et. al [6]. After obtaining the number plate, next step was to extract characters of the number plate. Panchal et. al, used Harris corner and connected component based method to segment the characters [7]. Haar-like features and AdaBoost algorithms were applied feature extraction was reported in another study [8]. Gabor wavelet transform and local binary operator were also used. Liu and Lin used both supervised k-means and support vector machine to classify characters [9]. In the proposed method edge detecting based on Sobel vertical edge detection is used to identify the potential number plate region. Vertical projection based technique is used to segment individual characters. Random Forest Classifier, which is based on ensemble of decision trees, is used to recognize characters of number plate.
## 3 Methodology
For the present study following methodology has been used as discussed in subsequent sections and is represented in Fig.1.
### Pre-Processing
Input image suffers from many factors like noise, distortion and lack of exposure. To minimize these factors Pre-Processing is required on the input image and hence processing of image becomes easy and computationally fast. Every stage in Pre-Processing is shown in Fig.2.
#### 3.1.1 Converting RGB (Red Green Blue) Image to Grayscale Image
RBG image is converted to grayscale image for two reasons as shown in Fig.2(b):
* RGB images are computationally intensive to process as compared to greyscale images simply because RGB images have three separate channels for red, blue and green values for a pixel whereas grayscale images have only single channel that represents the intensity of a pixel.
Figure 1: Methodology for Number Plate recognition
Figure 2: Pre-Processing steps for an image: (a) Input RGB image (b) Gray scale image (c) Noise removal (d) Contrast enhancement (e) Binary image (f) Dialated image
* Standard number plate has only two colors i.e., white and black. So there is no need to have the whole spectrum of colors in the image.
#### 3.1.2 Noise Removal
Bilateral filter is used for removing noise (Fig.2c), kernel size 5, from image as it is very effective in removing noise while keeping the edges of characters sharp. Filtering is the process in which each pixel value in the image is replaced by the average weighted sum of the pixels nearby. Number of pixels taking part in weighting average is decided by the size of kernel. Kernel size of 5 is efficient for removing noise as size greater than 5 are slow and size of kernel less than 5 in ineffective in removing noise in our case.
#### 3.1.3 Normalization
Increasing the contrast helps in cases where illumination is low. Increase in contrast increases the separation between colors which helps in separating the black characters from the white background on the number plate (Fig.2d). Contrast Limited Adaptive Histogram Equalization (CLASH) is used for increasing the contrast. This technique divides the image into small blocks called tiles. Tile of size \(8{\times}8\) was used and histogram equalization is applied on these tiles independently. This keeps the information in regions which is too much exposed to brightness.
Some examples which is significantly helped by contrast enhancement is shown in Fig. 3, that clearly shows it makes character on the number plate prominent.
#### 3.1.4 Converting to Binary Image
To make the color domain of image same as that of number plate, image is converted into binary image (Fig.2e), which reduces the pixel values to only two values i.e., black (0) and while (1). This is done by choosing a threshold (128 was chosen on the scale of 255). If the value of the pixel is less than threshold then it is converted to black color (value \(=0\)) and if the value is greater than threshold it is converted to white color (value \(=255\)).
#### 3.1.5 Dilation of Image
Dilation of image serves two purposes (Fig.2f):
* It makes the characters of the number plate bold which increases the area of the characters which in turn increases their edges. This is because of increase in area of an object that also increases the perimeter (edges) of the image. Increase in the edges of the character helps in localization of the number plate as the plate is localized by edge detection method.
* It removes the unnecessary edges around the number plate, hence prevents from capturing the "false" edges. This is because it reduces the perimeter of noise blot by combining noises nearby, hence reducing the edges of noise. This is shown in Fig. 4. Fig. 4 shows the number plate region without dilation and Fig. 4 shows its edges and vertical projection of the edges. Similarly, Fig. 4 shows the number plate region with dilation and Fig. 4 shows its edges and vertical projection of the edges. Fig. 4 clearly shows that peaks of noise is greater than peaks of noise in Fig. 4 which is present just below the characters of the number plate. Hence, dilation helps to reduce the edges of noise surrounding the number plate.
### Localization of Number Plate
To localize the number plate, edge detection is used, which gives the edges of the image. Most of the edges are localized on the number plate region which are actually the characters on the number plate. This is because standard number plate has black characters on the white number plate. Therefore, edges around the characters are distinctly separated from background. Also the characters are concentrated on the number plate which forms the region of maximum localized edges. Although, the edges are present throughout the image, but maximum edges are locally concentrated on the number plate. Pre-Processing makes these edges distinct. It's because of this property, bounding rectangle is taken, which represents the number plate. It is traversed on the image which counts the number of edges it bounds. Wherever it finds the maximum localized number of edges in the bounding rectangle, that portion of image is taken as the number plate.
Canny Edge detection method was tried to localize the number plate. Although Canny edge detection is more robust edge detection method, but it detects both horizontal and vertical edges. Generally in vehicle, just above the number plate there is a back windshield of car which has horizontal structure. Similarly, just below the number plate there is bumper which again has almost horizontal structure. The number plate itself and all the structures around the number plate gives strong horizontal edges as shown in Fig. 5(a). When Canny edge detection is applied it captures these edges and hence, only some portion or incorrect number plate region is captured, which is shown in Fig. 5(c) enclosed by green colored rectangle. Fig. 5(e) clearly shows partial number plate region is captured when canny edge detection is used.
Sobel vertical edge detection removes all the horizontal edges from the image. It only captures vertical edges which is clear in Fig. 5(b). Hence, all the horizontal edges in the image are removed. The edges of characters are still recognized as they are combination of both horizontal and vertical edges. Fig. 5(d) shows the localized number plate region is detected which is accurate and is shown by enclosed green colored rectangle. Fig 5(f) shows captured area of number plate when sobel vertical edge detection is used.
Green bounded rectangle shown in Fig.6(a) represents the maximum localized edges inside the bounding rectangle which is the number plate. That portion of image is cropped from dilated image (Fig.6b) and used for
Figure 5: Comparision between localization of number plate using Canny edge detection and Sobel vertical edge detection
### Character Segmentation
Before this stage, number plate has been localized. In this stage, localized number plate is taken as an input and the characters present in the number plate are extracted from it.
#### 3.3.1 Removal of Redundant Portion of the Number Plate
Before segmentation of characters is carried out, noise surrounding the characters is removed. If there is no noise around the characters then empty white spaces of the number plate should be removed to make computation faster. These portions are removed by first applying Sobel vertical edge detection on Fig. 7(b) which gives Fig 7(a). We are only interested in vertical edges because noises below and above the characters generally have greater horizontal component than vertical component. Hence, taking vertical edges remove all horizontal components of the noise. Characters are not affected by this because they are blend of horizontal and vertical components.
These edges (white pixels as shown in Fig. 7(a)) are projected on the vertical axis. Projection on the vertical axis is done by counting the number of white pixels (which represent edges) of each row and storing the value of count in an array. This array is converted into a histogram. The characters are generally represented by the highest and widest peaks in the histogram as shown in the Fig 8. If there is no noise then there is only single band of peaks representing characters.
First the numbers of bands are searched in the image. Then these bands are compared with each other on the basis of width and height of the bands.
As there is space between characters and noise, this space will separate the band of peaks of characters and noise into two separate bands. So, the band of peaks representing characters are cropped out, as they are highest and widest, hence, number plate free from noise is acquired as shown in Fig 9. After obtaining this image, it can now be sent to the stage of segmenting the individual characters.
There were some cases when the peaks of noise were higher than those of characters as shown in Fig 10. When Sobel vertical edge detection is applied on Fig. 10, peaks of noise becomes greater than that of characters as shown in Fig. 11. For such cases, these peaks of noise were eliminated due to the fact that characters represent wider band of peaks than noise and hence, number plate free from noise is obtained as shown in Fig. 12.
#### 3.3.2 Segmentation of Characters
The image shown in Fig.9 is free from noise. This image is projected on the horizontal axis. This is done by summing black pixels of each column and storing it in an array. This array is plotted in a histogram as shown in Fig 13. As there are gaps between each character so there are no black pixels in between and hence the value of histogram is zero in these gaps. Non-zero values in histogram represents characters. So to get the characters out of the number plate, these individual peaks are cropped out, from the position where the peak starts to the location where the peak ends, which represents a character.
This technique is applied throughout the image which gives all the characters. These individual characters are send to the next stage for recognition (Fig 14).
### 3.4 Character Recognition
For classifying the segmented characters into the respective digits and alphabets, various machine learning classification techniques were tried, such as neural networks, k-nearest neighbor (k-NN), support vector machines (SVM) and Random Forest Classifier (RF), as No Free Lunch theorem states that there will always be data sets where one classifier is better than another.
#### 3.4.1 k-NN Classification Algorithm
k-NN is a classification algorithm for supervised learning and it is non- parametric approach for classification. The value of 'k' was chosen as three, where k is number of nearest neighbor. The value of 'k' plays an important role in performance of k-NN. It is the tuning parameter of the model. k-NN groups the training images into classes, similar images are grouped together in the form of a class. When test image is provided, image is plotted on the same graph and 'k' closest training images are chosen from test image.
Figure 14: Separated characters from Number plate
Figure 13: After image becomes free from noise, it is projected on the horizontal axis. Peaks of histogram represents the number of black pixels in each column of number plate. Image between these peaks are cropped which gives the individual characters.
A major challenge was to classify special characters and symbols as shown in Fig 15 (encircled on number plate). These symbols appeared frequently in number plates throughout the dataset and needed to be eliminated from the output. To tackle this problem, these characters were classified into two different classes as shown in Fig. 16.
To disqualify images that belong to class A, number of black pixels were calculated. In actual characters there was healthy number of black pixels. In other words, lager area of the image was covered by the character itself. However, that is not the case with the images of Class A. Therefore a threshold (Tc) was set. Whenever total number of black pixels in an image was less than the set threshold, that image was discarded. This threshold was also used in subsequent classifiers.
For class B, this technique could not be used as they have large number of black pixels. Implementation of kNN in OpenCV returns the shortest distance between test characters and the characters that were used for training (neighbors). The more the neighbors are closer to test image, the closer the match. If this distance is very large, i.e., if the test image does not even remotely resemble any of the training images, then the image was unlikely to be a legitimate character. Again, a threshold value was set (Ts). If the distance between test image and nearest neighbors was greater than threshold, then that image was discarded.
#### 3.4.2 Multi-Layer Perceptron
Multi-Layer Perceptron or Neural Network (NN) is composed of artificial neurons which mimics human brain. Neural Network consists of layers and each layer has number of neurons. Layers are categorized into three types. Input layer, hidden layers and output layers. Input layer is equals to number of features or in this case, it is individual pixles. So, it is equals to 400 (size of image is 20\(\times\)20).
Number of neurons in output layer is equals to number of classes. We have 36 classes to represent digits and alphabet (10 digits and 26 alphabets). Hence, output layer has 36 neurons. In between input and output layer, neural network has hidden layers. For this problem, two hidden layers were chosen, first hidden layer has 270 neurons and second hidden layer has 150 neurons. Each neuron is generally connected by weights to all the previous layer neurons. Output of previous layer is multiplied and added to each other before it is fed to the neuron of next layer as an input.
Figure 16: Classification of special characters
Figure 15: Number plates with encircled special characters and symbols
At that neuron, output is calculated by activation function by using input. It maps the output of neuron between 0 and 1 or -1 and 1. There are several parameters that need to be set for neural network. One is activation function which is present in each neuron. Logistic function was chosen as an activation function which maps the values between 0 and 1. Learning rate was set to 0.001. Maximum iterations was set to 500 and training stopped at 278\({}^{\text{th}}\) iteration. To eliminate characters of class-A, count threshold 'Tc' was and to eliminate characters of class-B, threshold (Ns) was set at the output of neural network to ignore those special characters. All the important parameters are listed in Table 1.
#### 3.4.3 Support Vector Machine
SVM is a discriminative classifier. It categorizes the data by forming the hyperplane. It works best for binary class classification. Although, SVMs can be modified for multi-class problems. It usually consists of constructing binary classifiers which distinguish between one label and rest, called one-versus-all approach or between every pair of classes, called one-versus-one approach.
One-versus-all approach was chosen for SVM. There are three important parameters that needs to be set when SVM is applied which are: kernel, kernel width parameter (\(\gamma\)) and optimum cost parameter (C). Polynomial kernel of degree 3 was chosen for this case. Parameter 'C' decides the size of misclassification allowed, i.e., how much one wants to avoid misclassification of each training data. Large values of 'C' will choose smaller margin hyperplane. Conversely smaller values of 'C' will force classifier to look for larger margin, even if that hyperplane misclassifies the points. This is shown in Fig 17. Generally high value of parameter 'C' is desirable but it might lead to over fitting. Parameter '\(\gamma\)' affects the shape of the class dividing hyperplane. When '\(\gamma\)' is high, only points near the hyperplane is taken into consideration whereas, low value of '\(\gamma\)' will take far away points from
\begin{table}
\begin{tabular}{|l|c|} \hline Type of Classifier & Multi-Layer-Perceptron \\ \hline Total neurons in input layer & 400 \\ \hline Total neurons in first hidden layer & 270 \\ \hline Total neurons in second hidden layer & 150 \\ \hline Total neurons in output layer & 36 \\ \hline Learning rate & 0.0001 \\ \hline Activation function & Logistic function \\ \hline Maximum iterations for training set & 500 \\ \hline \end{tabular}
\end{table}
Table 1: Details of important neural network parameters
\begin{table}
\begin{tabular}{|l|c|} \hline Type of Classifier & Support Vector Machine \\ \hline Approach & One-vs-all \\ \hline optimum cost parameter (C) & 1 \\ \hline kernel width parameter (\(\gamma\)) & ‘auto deprecated’ \\ \hline degree of polynomial kernel & 3 \\ \hline \end{tabular}
\end{table}
Table 2: Details of important support vector machine parameters
hyperplane into consideration. Generally, low value of '\(\gamma\)' is desirable as larger value might lead to over fitting. High value of optimum cost parameter (C=1) was chosen with low value of kernel width parameter (a default value \(\gamma\) = 'auto deprecated'). To eliminate characters of class-A, count threshold 'Tc' was used and to eliminate characters of class-B, probability estimate was used by setting a threshold (Ps). All the important parameters are listed in Table 2.
#### 3.4.4 Random Forest Classifier
Random forest classification algorithm that is based on ensemble of decision trees where each of the tree is based on randomly selected subset of training set. Tree consists of nodes where the decision is taken on some parameter. This forest is time trained with a method which is based on bagging.
Random Forest uses slightly different kind of bagging approach where a subset of features is selected for the split at node, whereas, in bagging all features are used for node split. As the result of random forest is aggregation of trees, which reduces the affect of noise present in a single tree. Hence, bagging generally increases the overall result. Random forest are inherently multiclass which can be used in our problem case. There are two important parameters that needs to be set, one is number of features in each split (Fs) and number of decision trees (Nt) in the forest. The large value of parameter 'Nt' may be unnecessary but it does not harm the model. It will definitely make the predictions stronger but might make the model slower. Parameter 'Fs' is the number of features to consider while splitting a node. It is always subset of number of features. Value of 'Nt' was chosen as 100 and 'Fs' was set as square root of number of features for the model. To eliminate characters of class-A, count threshold 'Tc' was used and to eliminate characters of class-B, probability estimate was used by setting a threshold (Pe). All the important parameters are listed in Table 3.
\begin{table}
\begin{tabular}{|l|c|} \hline Type of Classifier & Random Forest \\ \hline Number of trees (Nt) & 100 \\ \hline Number of features in each split (Fs) & \(\sqrt{number\_of\_features}\) \\ \hline \end{tabular}
\end{table}
Table 3: Details of important random forest parameters
## 4 Results and Discussions
Automatic Number Plate Recognition System was written in Python (3.6.5). Libraries such as OpenCV (3.4.2), numpy (1.14.0), scikit-learn (0.19.1) and matplotlib (2.2.3) were used. Graphical User Interface was created using tkinter (8.6.8). Out of 350 images in the dataset (images of Croatian vehicles), 100 were used for testing, 220 for training and rest was used for validation to fine tune hyper parameters of the various models used. All the segmented characters used for training were resized to 20\(\times\)20. Different models were trained by training dataset. Characters were first extracted from the number plate by character segmentation and then these characters were used for training the various classification models. Final output of the Number Plate Recognition System is shown in Fig. 18 which shows recognized characters. Accuracy of various models is shown in Table 4.
### Testing
The final character accuracy for k-NN was found to be 83.40%. This classifier was found to be more prone to making mistakes between visually ambiguous characters, such as '8' and 'B','I' and '1','O' and 'D','G' and '6'. The results were near perfect for characters that are not optically ambiguous. Time taken by k-NN classifier was 0.3 seconds to give the output of an image.
\begin{table}
\begin{tabular}{|l|r|} \hline
**Classifier** & **Accuracy** \\ \hline k-NN & 83.4\% \\ \hline Neural Network & 89.47\% \\ \hline SVM & 87.5\% \\ \hline Random Forest & 90.9\% \\ \hline \end{tabular}
\end{table}
Table 4: Accuracy of various models tested
Figure 18: Final output of Number Plate Recognition System
In case of Neural Network, character accuracy was found to be 89.47%. This classifier was found to be prone to making mistakes between 'O' and 'G' and very rarely between '2' and 'Z'. Time taken by Neural Network was 0.23 seconds to give the output of an image.
In case of SVM classifier, character accuracy was found to be 87.50%. This classifier sometimes was found to be making mistakes between '8' and 'B', 'I' and '1', 'G' and '6', although it gave correct output for 'O' and 'D' most of the time. Time taken by SVM was 0.31 seconds to give the output of an image.
In case of Random forest classifier, character accuracy was found to be 90.9%. The classifier completely removed ambiguity between 'G' and '6' and reduced the error in detecting characters '8' and 'B', 'I' and '1' to a greater extent than SVM. Time taken by Random Forest was 0.35 seconds to give the output of an image.
### Discussion
There are various reasons why RF works better than SVM in this problem. First, Random forest is naturally a multi class classifier whereas SVM is binary classifier. For SVM to work in multi class problem, it is reduced to multiple binary class problem. Still, results show random forest outperforms SVM because it is intrinsically a multi class classifier. Second, images are inherently noisy even if Pre-Processing is performed on it. Noise resistant classier should perform better in this case. This claim is backed up by the results where RF performs better than SVM. RF gives the overall result by taking the consensus result of different trees present in the classifier. Hence even if some trees get trained on noisy input, overall result is expected to give the desired output. Also, RF actually don't take long to train, especially if you do so in parallel, something one cannot do with SVM. Neural networks are also noise tolerant; hence it gives decent accuracy. Still, it does not beat the accuracy of random forest which brings the robustness of ensemble of decision trees which is based on kind of bagging approach.
## 5 Conclusion
The algorithm for number plate recognition has been proposed. Pre-Processing, Number plate localization, Character segmentation and Character recognition are the steps used in the algorithm. Pre-Processing includes converting RGB image to grayscale image, removing noise by using Bilateral Filter, then increasing the contrast of image using CLASH, converting the image to a binary image and finally dilating the image. Number plate localization extracts the number plate region from the image using Sobel vertical edge detection. It is also shown in this proposed work that Canny edge detection does not work as effectively as Sobel vertical edge detection because of its intrinsic characteristics. Character segmentation first removes the redundant portion of number plate which might hinder the extraction
of characters and then segments the individual characters from the extracted number plate. Character recognition recognizes the individual optical characters by using Random Forest Classification Algorithm. For character recognition, Random Forest works best among several classification methods because it is based on ensemble of classifiers; precisely decision trees.
|
2304.01542 | Irregularity of polymer domain boundaries in two-dimensional polymer
solution | Polymer chains composing a polymer solution in strict two dimensions (2D) are
characterized with irregular domain boundaries, whose fractal dimension
($\mathcal{D}^{\partial}$) varies with the area fraction of the solution and
the solvent quality. {\color{black}Our analysis of numerical simulations of
polymer solutions finds} that $\mathcal{D}^{\partial}$ in good solvents changes
non-monotonically from $\mathcal{D}^{\partial}=4/3$ in dilute phase to
$\mathcal{D}^{\partial}=5/4$ in dense phase, maximizing to
$\mathcal{D}^{\partial}\approx 3/2$ at a crossover area fraction $\phi_{\rm
cr}\approx 0.2$, whereas for polymers in $\Theta$ solvents
$\mathcal{D}^{\partial}$ remains constant at $\mathcal{D}^{\partial}=4/3$ from
dilute to semi-dilute phase. Using polymer physics arguments, we rationalize
these values, and show that the maximum irregularity of
$\mathcal{D}^\partial\approx 3/2$ is due to "fjord"-like corrugations formed
along the domain boundaries which also maximize at the same crossover area
fraction. Our finding of $\mathcal{D}^\partial\approx 3/2$ is, in fact, in
perfect agreement with the upper bound for the fractal dimension of the
external perimeter of 2D random curves at scaling limit, which is predicted by
the Schramm-Loewner evolution (SLE). | Lei Liu, Changbong Hyeon | 2023-04-04T05:40:17Z | http://arxiv.org/abs/2304.01542v5 | # Irregularity of polymer domain boundaries in two dimensional polymer solution
###### Abstract
Polymer chains comprising a polymer solution in strict two dimensions (2D) are characterized with irregular domain boundaries, and their fractal dimension (\(\mathcal{D}^{\partial}\)) varies with the area fraction of the solution and the solvent quality. We find that \(\mathcal{D}^{\partial}\) in good solvents changes non-monotonically \(\mathcal{D}^{\partial}=4/3\) in dilute phase to \(\mathcal{D}^{\partial}=5/4\) in dense phase, maximizing to \(\mathcal{D}^{\partial}\approx 3/2\) at a critical area fraction, whereas for polymers in \(\Theta\) solvents \(\mathcal{D}^{\partial}\) remains constant at \(\mathcal{D}^{\partial}=4/3\) from dilute to semi-dilute phase. Using polymer physics arguments, we rationalize these values. We find that the maximum irregularity of \(\mathcal{D}^{\partial}\approx 3/2\) results from "fjord"-like corrugations formed along the domain boundaries which also maximize at the same critical area fraction. According to the Schramm-Loewner evolution (SLE), the outer-boundary of the \(SLE_{\kappa}\) curves with \(\kappa=4\), which lies at a crossover point between simple non-intersecting curves and those with self-intersection, has the fractal dimension of \(\mathcal{D}^{\partial}=3/2\), and this is in fact the upper bound of \(\mathcal{D}^{\partial}\) that \(SLE_{\kappa}\) can have.
_Introduction._ In polymer solution beyond the overlap concentration, adaptation of the polymer configurations in 2D is dramatically different from that in 3D. At thermodynamic equilibrium, polymer chains in strict 2D are bound to segregate and become territorial, forming entanglement-free polymer domains, whereas the chains in 3D tend to interpenetrate and are entangled to maximize the entropy of polymer solution [1]. Notably, the outer boundaries (external perimeters) of the domains in 2D are not smooth but irregular [2; 3; 4; 5], and the extent of the irregularity can be quantified using the fractal dimension [6]. Specifically, the "external perimeter" (\(E_{p}\)) increases with "the size of monomers constituting the perimeter," \(R_{p}^{2}=\frac{1}{2N_{p}^{2}}\sum_{i,j\in\partial}^{N_{p}}(\vec{r}_{i}-\vec{r }_{j})^{2}\) with \(\vec{r}_{i}\) the position of \(i\)-th monomer and \(N_{p}\) the number of monomers comprising the perimeter (\(\partial\)), obeying the relation
\[E_{p}\sim(R_{p})^{\mathcal{D}^{\partial}}\,, \tag{1}\]
where \(\mathcal{D}^{\partial}\in[1,2]\) quantifies the fractal (Hausdorff) dimension of the perimeter [2; 3; 4; 5; 6].
Calculating \(\mathcal{D}^{\partial}\) through the numerics of monodisperse polymer solutions with varying area fraction (\(\phi\)), we discover that \(\mathcal{D}^{\partial}\) exhibits qualitatively different \(\phi\)-dependences on the solvent quality. Remarkably, \(\mathcal{D}^{\partial}\) in good solvents (\(\mathcal{D}^{\partial}_{\rm SAW}\)) exhibits a non-monotonic variation with \(\phi\), whereas \(\mathcal{D}^{\partial}\) of \(\Theta\) chains (\(\mathcal{D}^{\partial}_{\Theta}\)) remains constant over the same range of \(\phi\). The constant \(\mathcal{D}^{\partial}_{\Theta}\) can be understood as an outcome of the compensation between attraction and repulsion that characterizes the nature of \(\Theta\) chain [7; 8; 9; 10; 11]. Despite a number of works that analyzed the irregularity of polymer domain boundary in 2D polymer solution [2; 3; 4; 5; 12; 13], our finding of the non-monotonic variation of \(\mathcal{D}^{\partial}_{\rm SAW}(\phi)\), especially over intermediate concentrations, has not been reported elsewhere.
In this Letter, we investigate the origin of \(\phi\)-dependent variation of \(\mathcal{D}^{\partial}\) by associating it with the fundamental scaling exponents of polymer in 2D. We also examine the problem under the hood of Schramm-Loewner evolution (SLE), a mathematical tool that offers quantitative description for the boundaries of 2D critical systems at their scaling limits [14; 15; 16; 17].
_Rationalizing the values of \(\mathcal{D}^{\partial}\) using polymer arguments._ Simulating 2D monodisperse polymer solution (Figs. 1 and S1), we study the configurations of polymer domains. To examine the \(\phi\)-dependent irregularity of the domain boundary (Fig.1A), we calculate \(E_{p}\) and analyze its variation against \(R_{p}\), and extracted \(\mathcal{D}^{\partial}\) as defined in Eq.1 (Fig. 1C). As shown in Fig.1D, in good solvents, \(\mathcal{D}^{\partial}\) of SAWs exhibits a non-monotonic variation with \(\phi\) (Fig.1D), starting from \(\mathcal{D}^{\partial}_{\rm SAW}=4/3(\approx 1.33)\) in dilute solution (\(\phi\approx 0\)), maximizing to \(\approx 3/2\) at a critical area fraction (\(\phi=\phi_{cr}\simeq 0.2\)), and reaching \(5/4(=1.25)\) in a dense phase (\(\phi\approx 1\)). On the other hand, \(\mathcal{D}^{\partial}\) of \(\Theta\) chains remains constant (\(\mathcal{D}^{\partial}_{\Theta}\simeq 4/3\)) over the range of \(\phi\) up to \(\phi\lesssim 0.4\), and also drops to \(5/4(=1.25)\) at \(\phi\approx 0.67\). The values of \(\mathcal{D}^{\partial}\) in Fig. 1D under the limiting and the crossover conditions are rationalized as follows.
(i) For 2D SAW chain in dilute phase (\(\phi<\phi^{*}\)), the majority of the monomers are exposed to the solvent (see the configurations of SAW in Fig. 1B). As a result, the perimeter of the polymer domain scales with the number of monomers, \(E_{p}(N)\sim N\). Just like the mean squared size of polymer and the mean squared end-to-end distance obey the same scaling relation \(R_{F}^{2}\sim\langle R_{ee}^{2}\rangle\sim N^{2\nu}\), \(R_{p}\) displays a scaling of \(R_{p}\sim N^{\nu}\) with \(\nu=\nu_{\rm SAW}=3/4\), and hence it is expected that \(E_{p}\sim N^{\nu\mathcal{D}^{\partial}}\sim N\); therefore, \(\mathcal{D}^{\partial}_{\rm SAW}=\nu_{\rm SAW}^{-1}=4/3\approx 1.33\)[5].
(ii) Configurations of \(\Theta\) chain differ from those of SAW (Fig. 1B) in that some monomers are buried inside the domain, whereas others are exposed to the periphery constituting the external perimeter. \(\Theta\) chains in dilute phase obey \(N\sim R_{F}^{\mathcal{D}\Theta}\), characterized by the fractal dimension of percolating clusters \(\mathcal{D}_{\Theta}=7/4\)[7; 9; 10; 12; 20; 21; 22; 23; 24]; however, we find that the perimeter of the \(\Theta\) chain is still self-avoiding, such that \(\mathcal{D}_{\Theta}^{\partial}=\mathcal{D}_{\rm SAW}=4/3\)[25; 26; 27].
(iii) In dense polymer solution, the external perimeter of a polymer domain is proportional to the number of interchain contacts [5].
\[E_{p}(N)\propto N\times f_{\rm inter}=N\frac{Z_{N,4}}{Z_{N,2}^{2}} \tag{2}\]
where \(f_{\rm inter}\), the fraction of such contacts per chain, can be associated with the partition sums, 4-arm (\(Z_{N,4}\)) and two 2-arm star polymers (\(Z_{N,2}^{2}\)), i.e., \(f_{\rm inter}=Z_{N,4}/Z_{N,2}^{2}\). Thus, the partition sum of an \(L\)-arm star polymer with each arm consisting of \(N\) segments, \(Z_{N,L}\sim\mu^{LN}N^{\gamma_{L}-1}\)[28; 29], leads to \(f_{\rm inter}=N^{\gamma_{4}-2\gamma_{2}+1}\) and \(E_{p}(N)\sim N^{\gamma_{4}-2\gamma_{2}+2}\)[2]. From \(E_{p}(N)\sim N^{\nu\mathcal{D}_{\rm D}^{\partial}}\), it follows that
\[\mathcal{D}_{\rm D}^{\partial}=\frac{1}{\nu}(\gamma_{4}-2\gamma_{2}+2). \tag{3}\]
Since \(\gamma_{L}=9/8+(3-L)L/32\) (see Eq. S24) [24; 29] and \(\nu=1/2\) for polymers in dense phases, we obtain \(\mathcal{D}_{\rm D}^{\partial}=5/4\).
(iv) The fractal dimension of the domain boundary maximizes to \(\mathcal{D}_{\rm SAW}^{\partial}\approx 1.5\) at \(\phi_{cr}\approx 0.2\) (Fig. 1D). The polymer configurations visualized at the five different values of \(\phi\) in Fig.1E indicate that the ruggedness of the domain boundary demarcated with orange lines also maximizes at an intermediate value of \(\phi\) (\(\phi=0.198\) and \(0.297\)), and it flattens out at the highest value (\(\phi=0.666\)).
Inspecting the configurations of polymers at varying \(\phi\) (Fig.1E), we notice that the outer-boundary of polymer domain features "fjord"-like configurations with
Figure 1: Fractal dimension of polymer domain boundary with varying area fraction \(\phi\). (A) Polymer solutions of SAW chains with \(N=640\) in 2D with increasing \(\phi\) (see Fig.S1 for polymer solutions of \(\Theta\) chains). Each panel is drawn in the 2D box of the identical dimension. (B) Configurations of SAW and \(\Theta\) chains with varying \(N\) in dilute solution (\(\phi<\phi^{*}\)). The overlap area fraction (\(\phi=\phi^{*}\)), in which the intra-monomer concentration (area fraction, \(\phi\)) is comparable to the inter-monomer concentration, i.e., \(\phi^{*}\sim Na^{d}/R_{F}^{d}\sim N^{1-\nu d}\), has previously been determined at \(\phi^{*}\approx 0.018\) and \(\phi^{*}\approx 0.266\) for the polymer solution with \(N=640\) under a good and \(\Theta\) solvent condition, respectively [11]. External perimeter (orange) of a polymer configuration (black) is calculated based on the turn-right tie-breaking rule [18; 19]. The interior of the domain enclosed by the perimeter is colored in pale blue. (C) Log-log plot of the external perimeter of the chain (\(E_{p}\)) versus the gyration radius of the perimeter (\(R_{p}\)) are produced using the chains with five different lengths (\(N=40\), \(80\), \(160\), \(320\), and \(640\)) for a given value of \(\phi\). (D) The fractal dimension \(\mathcal{D}^{\partial}\) was calculated from the data points in (C). The three characteristic values of \(\mathcal{D}^{\partial}=4/3\), \(3/2\), and \(5/4\), are marked in blue. (E) Typical configurations of SAW at varying \(\phi\)’s.
narrow "straits" (see Fig. 2A for illustration) [25; 26; 27], the number of which increases up to some \(\phi\), and that those fjords turn into lakes when \(\phi\) further increases reaching the dense phase, which merges the straits and smoothes out the domain boundary (orange line in Fig. 2A). More specifically, Fig.2B shows that the outer-boundary (orange line) of the polymer chain (black curve), drawn on a lattice, is characterized with closed loops or "fjord"-like configurations, the enclosed area of which is marked with blue dots. The total length of such fjords, contributing to the irregularity of the domain boundaries is small at \(\phi<\phi^{*}\), and it gradually increases up to \(\phi\approx(0.2-0.4)\) and decreases at higher \(\phi\). Explicit calculation of the average contour length of the fjords per chain also exhibits similar non-monotonic variations (Fig. 2). We surmise that the gradual increase of pressure [11] exerted by the neighboring chains facilitates the corrugation of intra-domain boundary to engender fjords until they merge into the interior of the domain at large \(\phi\).
_Schramm-Loewner evolution._ Assuming that the outer-boundary of polymer domain in 2D is a conformally invariant geometrical object, one can study it by means of the Schramm-Loewner evolution (SLE) [14; 15; 16; 17]. Specifically, a conformal map \(w=g_{t}(z)\) that satisfies the following differential equation
\[\partial_{t}g_{t}(z)=\frac{2}{g_{t}(z)-a_{t}} \tag{4}\]
with \(g_{0}(z)=z\) for \(z\to\infty\) uniformizes 2D curves, \(\gamma[0,t]\), in the upper half-plane (\(\mathbb{H}=\{\mathrm{Im}(z)>0;z\in\mathbb{C}\}\)) onto the real axis such that \(a_{t}=g_{t}(\gamma(t))\in\mathbb{R}\)[30]. If the driving function \(a_{t}\) is stochastic, satisfying \(\langle a_{t}\rangle=0\) and \(\langle a_{t}^{2}\rangle=\kappa t\) with \(\kappa\) corresponding to the diffusivity of the Brownian motion, the inverse mapping \(g_{t}^{-1}(\omega)\), namely \(\gamma(t)=g_{t}^{-1}(a_{t})\) can generate \(SLE_{\kappa}\), a 2D random curve whose behavior is decided solely by the value of a single parameter \(\kappa\).
The fractal (Hausdorff) dimension of the \(SLE_{\kappa}\) is given as [15; 31]
\[\mathcal{D}=\min{(2,1+\kappa/8)}. \tag{5}\]
Further, it has been conjectured that the outer boundary of \(SLE_{\kappa}\) for \(\kappa\geq 4\) corresponds to the curve of an \(SLE_{16/\kappa}\), which is known as "SLE duality" [31; 32]. Thus, for \(\kappa\geq 4\), the fractal dimension of the outer boundary is
\[\mathcal{D}^{\partial}=1+2/\kappa. \tag{6}\]
Eq. 6 can be used to validate our results in Fig. 1D.
(i) The fractal dimension of \(\Theta\) chain (\(\kappa=6\)) is \(\mathcal{D}_{\Theta}=7/4\) (Eq. 5), and that of its outer boundary is \(\mathcal{D}_{\Theta}^{\partial}=4/3\) (Eq. 6), which our numerics in Fig. 1D also confirms. Notably, despite the disparate polymer configurations in the two solvent conditions (see Fig. 1B), \(\mathcal{D}^{\partial}\) of \(\Theta\) chain is still identical to that of SAW. In fact, the external perimeter of the ideal random planar walk (Brownian motion) is also self-avoiding, which is the Mandelbrot's conjecture [33].
(ii) From the perspective SLE, polymer chain in dense solution is plane-filling with the corresponding value of \(\kappa\) being \(\kappa\geq 8\)[14; 15; 16]. In this case, we get \(\mathcal{D}=\mathcal{D}_{\mathrm{D}}=2\) (Eq.5) and \(\mathcal{D}^{\partial}=\mathcal{D}_{\mathrm{D}}^{\partial}=5/4\) (Eq.6).
(iii) Lastly, SLE can be used to account for the \(\phi\)-dependent non-monotonic variation of \(\mathcal{D}_{\mathrm{SAW}}^{\partial}\), the origin of which we have ascribed to the average length of fjords per chain that also exhibits a non-monotonic
Figure 2: Contribution of fjords to the perimeter ruggedness. (A) An illustration of fjord (top) and lake (bottom). When a narrow strait merges, the fjord turns into a lake, which smoothes out the perimeter of domain boundary. (B) A configuration of SAW chains with N = 320 at \(\phi=0.198\) (black solid line). The fjords, which are defined as the closed segments of the outer perimeter (orange), are demarcated using the blue dots. (C) The average length of fjords per chain (\(E_{f}\)), calculated on a square lattice with the lattice spacing \(a\), with increasing \(\phi\). The maximal \(E_{f}\) is identified at \(\phi=(0.2-0.3)\), gradually shifting towards the smaller \(\phi\) as \(N\) increases.
variation. A transformation \(h_{t}(z)=g_{t}(z)-a_{t}\) casts Eq.4 into \(\partial_{t}h_{t}(z)=2/h_{t}(z)+\xi_{t}\) where \(\xi_{t}\) is a white noise satisfying \(\langle\xi_{t}\rangle=0\) and \(\langle\xi_{t}\xi_{s}\rangle=\kappa\delta(t-s)\). If one considers the dynamics of SLE curves projected on the real axis \(\Re z=x\in\mathbb{R}\), \(h_{t}(x)=x_{t}\), the \(x_{t}\) is described by the 1D stochastic dynamics, known as _the Bessel process_[16; 34]
\[\frac{dx_{t}}{dt}=\frac{2}{x_{t}}+\xi_{t}. \tag{7}\]
Heuristically, the nature of the dynamics \(x_{t}\) is dictated by the value of \(\kappa\), and a crossover between a deterministic (\(x_{t}^{2}\sim 4t\)) and a stochastic growth (\(\langle x_{t}^{2}\rangle\sim\kappa t\)) occurs at \(\kappa=4\). This means that for \(\kappa<4\) the unprojected, original SLE curves, \(\gamma[0,t]\), are simple, and neither hit the real axis nor have self-intersections, whereas they are self-intersecting for \(\kappa>4\)[14; 15; 16]. The SLE curves at \(\kappa=4\) correspond to those lying precisely at the crossover point between the two contrasting behaviors. According to the SLE duality (Eq.6), the fractal dimension of the perimeter of SLE curves is upper bounded at \(\mathcal{D}^{\partial}_{\rm max}=3/2\) for \(\kappa=4\), and this number is consistent with the maximal value of \(\mathcal{D}^{\partial}\) calculated in Fig.1D, i.e., \(\mathcal{D}^{\partial}_{\rm max}\approx 1.53\pm 0.04\).
_Concluding Remarks._ Examining the \(\phi\)-dependent ruggedness of the polymer domain boundary in polymer solution (\(\mathcal{D}^{\partial}(\phi)\)), we discover the non-monotonic variation in the ruggedness for 2D polymer solution in good solvents. The values of \(\mathcal{D}^{\partial}\) at the crossover point as well as under the limiting conditions are rationalized using the fundamental critical exponents of polymer configurations in 2D (\(\nu\), \(\gamma_{2}\), and \(\gamma_{4}\)) and the idea of SLE. Among them, of particular note is the maximal ruggedness (\(\mathcal{D}^{\partial}_{\rm SAW}\))\({}_{\rm max}=3/2\), which is the universal upper bound of the fractal dimension for 2D interface, according to the SLE (Eq.6). Interestingly, similar values of maximal fractal dimension have been reported for the fractal interface in bacterial biofilms [35; 36]. We have associated (\(\mathcal{D}^{\partial}_{\rm SAW}\))\({}_{\rm max}\) with the maximal "fjord"-like corrugations that result from the marginal folding of domain boundary (see Fig. 2A). The adaptation of polymer configurations with \(\phi\), which gives rise to the non-monotonic variation of irregularity in the domain boundary, is unique and fundamentally differs from the \(\phi\)-dependent variation of osmotic pressure (\(\Pi\)) in that the latter is dictated by the exponent \(\nu\) alone and displays monotonic variation with \(\phi\)[37; 38; 11; 39].
Finally, our finding of the corrugations in polymer domain boundary is amenable to experimental verification which may require either a direct/indirect visualization of polymer configurations [40; 41; 42] or a careful investigation on rheological responses of ultrathin polymer films [43].
This work was in part supported by National Natural Science Foundation of China, ZSTU intramural grant (12104404, 20062226-Y to L.L.), and KIAS Individual Grant (CG035003 to C.H.) at Korea Institute for Advanced Study. We thank the Center for Advanced Computation in KIAS for providing computing resources.
|
2307.07463 | Structured quantum collision models: generating coherence with thermal
resources | Quantum collision models normally consist of a system interacting with a set
of ancillary units representing the environment. While these ancillary systems
are usually assumed to be either two level systems (TLS) or harmonic
oscillators, in this work we move further and represent each ancillary system
as a structured system, i.e., a system made out of two or more subsystems. We
show how this scenario modifies the kind of master equation that one can obtain
for the evolution of the open systems. Moreover, we are able to consider a
situation where the ancilla state is thermal yet has some coherence. This
allows the generation of coherence in the steady state of the open system and,
thanks to the simplicity of the collision model, this allows us to better
understand the thermodynamic cost of creating coherence in a system.
Specifically, we show that letting the system interact with the coherent
degrees of freedom requires a work cost, leading to the natural fulfillment of
the first and second law of thermodynamics without the necessity of {\it ad
hoc} formulations. | Stefano Cusumano, Gabriele De Chiara | 2023-07-14T16:43:46Z | http://arxiv.org/abs/2307.07463v2 | # Structured quantum collision models: generating coherence with thermal resources
###### Abstract
Quantum collision models normally consist of a system interacting with a set of ancillary units representing the environment. While these ancillary systems are usually assumed to be either two level systems (TLS) or harmonic oscillators, in this work we move further and represent each ancillary system as a structured system, i.e., a system made out of two or more subsystems. We show how this scenario modifies the kind of master equation that one can obtain for the evolution of the open systems. Moreover, we are able to consider a situation where the ancilla state is thermal yet has some coherence. This allows the generation of coherence in the steady state of the open system and, thanks to the simplicity of the collision model, this allows us to better understand the thermodynamic cost of creating coherence in a system. Specifically, we show that letting the system interact with the coherent degrees of freedom requires a work cost, leading to the natural fulfillment of the first and second law of thermodynamics without the necessity of _ad hoc_ formulations.
## I Introduction
Open quantum systems are those systems that do not evolve unitarily, i.e. according to the Schrodinger equation, due to the presence of a usually very large external system, dubbed the environment, which interacts with the system influencing its dynamical evolution [1; 2].
The main problem when studying such systems is the very large number of degrees of freedom of the environment, which make it practically impossible to compute the exact dynamics of the joint system. In order to overcome this problem, several techniques have been developed in order to take into account the effects of the environment, both analytical and numerical. Among the analytical methods we mention master equations methods [3], stochastic differential equations [4] and path integrals techniques, while among numerical methods one has the density matrix renormalization group [5; 6], matrix product states [7; 8] and other efficient truncation techniques to efficiently simulate the dynamics of the open system [9; 10].
Among these methods, growing attention has been given in recent years to collision models [11; 12; 13], which proved to be useful also for efficient quantum simulations [14]. In these models one depicts the environment as a collection of smaller units, called _ancillae_, which interact piecewise with the system giving rise to an open system dynamics. The main strength of collision models is their versatility in describing many different physical situations by acting only on few parameters of the model, like the state of the ancillae, the interaction time and the interaction Hamiltonian between the ancillae and the system or allowing the ancillae to interact more than once with the system. Acting on these few knobs it has been possible to examine different regimes of open system dynamics, ranging from Markovian [15; 16; 17] to non-Markovian dynamics [18; 19; 20; 21], and many different physical systems, ranging from optical systems [22; 23; 24] to spin chains [25]. Collision models have been used to study different phenomena such as dissipative dynamics, bath engineering, quantum darwinism, the difference between local and global Lindblad equations [26].
A very important field where collision models have found application is quantum thermodynamics [27; 28; 29; 30; 31; 32]. While in this context one is interested in the influence of quantum effects on work production and heat dissipation, collision models have proved useful in investigating the efficiency of quantum heat engines and the role of quantum coherence in thermodynamic phenomena [33; 34].
While in most scenarios the environment is represented as a collection of qubits and independent harmonic oscillators, in recent years some attention has been given to the case of structured environments, that is, environments which are not a simple collection of harmonic oscillators, but rather structured quantum systems, for instance by allowing the harmonic oscillators to have interaction among them or by considering lattice systems as environments [35; 36; 37; 38; 39; 40; 41]. Inspired by this, we introduce a new type of collision model in which the environment is depicted as a collection of ancillae which are intrinsically composite systems, that is, for instance, two or more interacting qubits.
These structured ancillae give rise to new terms in the master equation, providing a connection with the field of open quantum systems with structured environments. We will also use this model to introduce a situation where, in spite of the fact that the ancillae are in a thermal state, yet they possess coherence in specific degrees of freedom. This can be used to generate quantum coherence in the system's steady state, using solely thermal resources. Through this model, we are able to clarify the role of coherence in thermodynamics, highlighting how the interaction of the system with an ancilla with coherence requires work in order to be acti
vated. This represents an advancement with respect to previous works, where coherence in the ancillae was assumed a priori, thus hindering the work cost of accessing that coherence and leading to violations of the second law of thermodynamics and ad hoc formulations in order to preserve it.
The manuscript is organized as follows. In Sec. II we introduce our model and derive the most generic master equation for the case where the ancillae are structured instead of being simple TLSs or harmonic oscillators, highlighting the main differences between the two scenarios. Then in Sec. III we show some features of our model through some examples. In Sec. IV we use our model to explore the role of coherence in work production and heat dissipation. Finally in Sec. V we draw our conclusions and give some outlook for future works.
## II Collision models with structured ancillae
In what follows we want to analyze the dynamics of a quantum system \(S\) interacting with a structured environment \(\mathcal{E}\) by using a collision model. Here by structured environment we mean an environment composed of ancillae which are composite subsystems. As an example, one can think of each ancilla as being composed by two interacting TLSs or harmonic oscillators.
To represent the collision model, we assume to have an Hamiltonian of the form:
\[\hat{H}_{tot}=\hat{H}_{S}+\hat{H}_{\mathcal{E}}+\hat{H}_{S\mathcal{E}} \tag{1}\]
where the Hamiltonians \(\hat{H}_{\mathcal{E}},\hat{H}_{S\mathcal{E}}\) are of the form:
\[\hat{H}_{\mathcal{E}}=\sum_{i}\hat{H}_{E_{i}}\quad\hat{H}_{S\mathcal{E}}=\sum _{i}\hat{H}_{SE_{i}} \tag{2}\]
The Hamiltonians \(\{\hat{H}_{E_{i}}\}\) are the free Hamiltonians of the ancillae, while \(\hat{H}_{SE_{i}}\) are the interaction Hamiltonians between the system \(S\) and the ancillae. In order to compute the master equation describing the dynamics of \(S\), we start our analysis from the free Hamiltonian \(\hat{H}_{E_{i}}\) of the ancillae. Note that, as we are dealing with structured ancillae, each free Hamiltonian \(\hat{H}_{E_{i}}\) will be in general a many-body operator. Thus, we start our analysis by studying the most generic \(\hat{H}_{E_{i}}\) in the next subsection.
### Structured ancillae
In our model, we want to write the Hamiltonian of each ancilla in the form:
\[\hat{H}_{E_{i}} = \hat{H}_{E_{i}}^{\text{free}}+\hat{H}_{E_{i}}^{\text{int}}, \tag{3}\] \[\hat{H}_{E_{i}}^{\text{free}} = \sum_{k=1}^{K}\hbar\omega_{k}\hat{C}_{E,k}^{\dagger}\hat{C}_{E,k},\] (4) \[\hat{H}_{E_{i}}^{\text{int}} = \sum_{j\neq k=1}^{K}\kappa_{jk}\hat{D}_{E,ij}\hat{D}_{E,k}^{ \dagger}+h.c. \tag{5}\]
Here the operator \(\hat{H}_{E_{i}}^{\text{free}}\) includes the free Hamiltonians of each subsystem composing the ancilla, while the operator \(\hat{H}_{E_{i}}^{\text{int}}\) contains all the interaction terms between the subsystems of the ancilla.
Note that we are not making any assumption over the nature of the subsystems of the ancillae, as the operators \(\hat{C}_{k}\) can account for both the bosonic and fermionic case by imposing respectively the canonical commutation relations (CCR) or the canonical anti-commutation relations (CAR), that is:
\[\left[\hat{C}_{E_{i}k},\hat{C}_{E_{i}j}^{\dagger}\right]=\delta_{ jk},\;\;\left[\hat{C}_{E,k},\hat{C}_{E,j}\right]=\left[\hat{C}_{E_{i}k}^{ \dagger},\hat{C}_{E,j}^{\dagger}\right]=0, \tag{6}\]
for the case of bosons and
\[\left\{\hat{C}_{E_{i}k},\hat{C}_{E_{i}j}^{\dagger}\right\}= \delta_{jk},\;\;\left\{\hat{C}_{E,k},\hat{C}_{E,j}\right\}=\left\{\hat{C}_{E, k}^{\dagger},\hat{C}_{E,j}^{\dagger}\right\}=0. \tag{7}\]
for the case of fermions, where \([\cdot,\cdot]\) is the commutator between the operators, while \(\{\cdot,\cdot\}\) stands for the anticommutator.
Let us now consider the trivial case of unstructured ancillas, that is to say when \(\kappa_{jk}=0\;\forall j,k\). In this case we can easily define the energy eigenbasis of the ancilla as given by product states of each subsystems energy eigenstates:
\[\left|n_{1},n_{2},\cdots,n_{K}\right\rangle=\left|\vec{n}\right\rangle= \bigotimes_{k=1}^{K}\left|n_{k}\right\rangle. \tag{8}\]
where \(\left|n_{k}\right\rangle\) is the n-th energy eigenstate of the k-th subsystem.
However, as soon as interactions are present, i.e. for \(\kappa_{jk}\neq 0\), the Hamiltonian \(\hat{H}_{E_{i}}\) is not diagonal anymore in the basis \(\{\left|\vec{n}\right\rangle\}\). This implies that a thermal state of the form \(\exp\!\left[-\beta\hat{H}_{E_{i}}\right]\!\big{/}Z\) is non-diagonal in the basis \(\{\left|\vec{n}\right\rangle\}\) as well, so that if the interaction between the ancilla \(E_{i}\) and the system \(S\)\(\hat{H}_{SE_{i}}\) is defined in terms of eigenoperators of the Hamiltonian \(\hat{H}_{E_{i}}^{\text{free}}\), a unitary driving term may appear in the master equation for \(S\).
To continue setting up the notation, we define as \(\{\left|E_{j}^{\prime}\right\rangle\}\) the eigenbasis of the total Hamiltonian \(\hat{H}_{E_{i}}\), with
corresponding eigenvalues \(E_{j}^{\prime}\). The eigenbasis of \(\hat{H}_{E_{i}}\) and \(\hat{H}_{E_{i}}^{\text{free}}\) are related via a unitary transformation \(\hat{V}^{E_{i}}\):
\[|E_{k}^{\prime}\rangle=\sum_{\vec{n}}V_{k\vec{n}}^{E_{i}}\,|\vec{n}\rangle\,. \tag{9}\]
Practically, the columns of the matrix \(\hat{U}^{E_{i}}\) contain the coefficients of the expansion of the state \(|E_{k}^{\prime}\rangle\) in terms of the non-interacting basis \(\{|\vec{n}\rangle\}\).
Once we have completed our analysis of the Hamiltonian \(\hat{H}_{E_{i}}\) of the ancillae, we can move to the interaction between \(S\) and the ancillae.
### System-ancilla interaction and Markovian dynamics
We assume a system-ancilla Hamiltonian of the form:
\[\hat{H}_{SE_{i}}=\sum_{j,k}\hat{A}_{S}^{(k,j)}\otimes\hat{B}_{E_{i}k}^{(j)}, \tag{10}\]
where the only assumption is that the Hamiltonian \(\hat{H}_{SE_{i}}\) is Hermitian.
We also assume the initial states of \(S\) and the ancillae to be factorized:
\[\hat{R}_{S\mathcal{E}}(0)=\hat{\rho}_{S}(0)\bigotimes_{i=1}\hat{\eta}_{E_{i}}. \tag{11}\]
More generally, since after the interaction the ancilla is discarded, we indicate the joint state of \(S\) and \(\mathcal{E}\) after the collision as:
\[\hat{R}_{S\mathcal{E}}(i)=\hat{\rho}_{S}(i)\bigotimes_{j=i+1}\hat{\eta}_{E_{j }}. \tag{12}\]
The joint state of \(S\) and \(\mathcal{E}\) evolves unitarily according to:
\[\hat{R}_{S\mathcal{E}}(i)\rightarrow\hat{U}_{SE_{i+1}}\hat{R}_{S\mathcal{E}}( i)\hat{U}_{SE_{i+1}}^{\dagger}, \tag{13}\]
where the unitary operator \(\hat{U}_{SE_{i}}\) is defined as:
\[\hat{U}_{SE_{i}}=\exp\left[-\frac{i}{\hbar}\delta t\left(\hat{H}_{S}+\hat{H}_ {E_{i}}+\hat{H}_{SE_{i}}\right)\right]. \tag{14}\]
Here \(\delta t\) is the collision time during which \(S\) and \(E_{i}\) interact.
In order to connect the state of the system \(S\) before and after its interaction with an ancilla, we can expand the
Figure 1: In panel (a) one can see an example of structured ancilla made out of three subsystems each interacting with each other. Panels (b)-(d): pictorial representation of the Markovian collision model. In panel (b) the system interacts with the first ancilla, whose state is factorized from the one of all the other ancillas. After the system has interacted with the first ancilla, the latter is traced out and the system interacts with the second ancilla, as in panel (c). The dynamics, first interacting with an ancilla then tracing it out, as in panel (d).
unitary operator in Eq. (14) in powers of \(\delta t\), obtaining:
\[\hat{U}_{SE_{i}} \simeq \hat{1}-\frac{i}{\hbar}\delta t(\hat{H}_{S}+\hat{H}_{E_{i}}+\hat{H} _{SE_{i}}) \tag{15}\] \[-\frac{\delta t^{2}}{2\hbar^{2}}(\hat{H}_{S}+\hat{H}_{E_{i}}+\hat{ H}_{SE_{i}})^{2},\]
where \(\hat{1}\) is the identity operator. The dynamics now proceeds in discrete steps. First, the system \(S\) interacts with the first ancilla \(E_{1}\). After the interaction the degrees of freedom of \(E_{1}\) are traced away, leaving us with the state of the system after the first collision \(\hat{\rho}_{S}(1)\). Then the system interacts with the second ancilla \(E_{2}\), which is then traced away to leave us with the state \(\hat{\rho}_{S}(2)\) and so on. After the \(i\)-th collision one has:
\[\hat{\rho}_{S}(i+1)=\mathrm{Tr}_{E_{i+1}}\left[\hat{U}_{SE_{i+1}}\hat{R}_{SE}( i)\hat{U}_{SE_{i+1}}^{\dagger}\right]. \tag{16}\]
To obtain a master equation for \(\hat{\rho}_{S}\), we can use the power expansion in Eq. (15) and substitute it into Eq. (16). Retaining only terms up to \(\delta t^{2}\) one obtains:
\[\hat{U}_{SE_{i+1}}\hat{R}_{SE}(i)\hat{U}_{SE_{i+1}}^{\dagger}= \tag{17}\] \[\mathcal{U}_{0}\left(\hat{R}_{SE}(i)\right)+\mathcal{U}_{1}\left( \hat{R}_{SE}(i)\right)+\mathcal{U}_{2}\left(\hat{R}_{SE}(i)\right),\]
where the superoperators \(\mathcal{U}_{0},\mathcal{U}_{1},\mathcal{U}_{2}\) are the zero, first and second order contributions to the dynamics respectively. The explicit form of these contributions reads:
\[\mathcal{U}_{0}\left(\hat{R}_{SE}(i)\right)=\hat{R}_{SE}(i), \tag{18}\] \[\mathcal{U}_{1}(\hat{R}_{SE}(i))=\] \[\quad\quad\quad-\frac{i}{\hbar}\delta t\Big{[}\hat{H}_{S}+\hat{H }_{E_{i+1}}+\hat{H}_{SE_{i+1}},\hat{R}_{SE}(i)\Big{]},\] (19) \[\mathcal{U}_{2}(\hat{R}_{SE}(i))=\frac{\delta t^{2}}{2\hbar^{2}}\] \[\Big{(}2(\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}})\hat{R} _{SE}(i)(\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}})\] \[-\Big{\{}(\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}})^{2}, \hat{R}_{SE}(i)\Big{\}}\Big{)}. \tag{20}\]
It is immediate to observe that the zeroth order contribution corresponds simply to the identity superoperator, while the first order contribution has the form of a unitary contribution to the dynamics. Finally, the second order contribution is the one describing dissipative dynamics.
After computing all these contributions one can trace out the degrees of freedom of \(E_{i+1}\), obtaining:
\[\hat{\rho}_{S}(i+1)=\] \[\mathcal{L}_{0}(\hat{\rho}_{S}(i))+\mathcal{L}_{1}(\hat{\rho}_{S} (i))+\mathcal{L}_{2}(\hat{\rho}_{S}(i)), \tag{21}\]
where once again the superoperators \(\mathcal{L}_{0},\mathcal{L}_{1},\mathcal{L}_{2}\) represent the zero, first and second order contribution to the dynamics respectively. Using the expression of the interaction Hamiltonian in Eq. (10), we can write explicitly these contributions as:
\[\mathcal{L}_{0}(\hat{\rho}_{S}(i))=\hat{\rho}_{S}(i), \tag{22}\] \[\mathcal{L}_{1}(\hat{\rho}_{S}(i))=-\frac{i}{\hbar}\delta t\Bigg{[} \hat{H}_{S}+\sum_{k,j}g_{k}^{(j)}\hat{A}_{S}^{(k,j)},\hat{\rho}_{S}(i)\Bigg{]},\] (23) \[\mathcal{L}_{2}(\hat{\rho}_{S}(i))=\frac{\delta t^{2}}{2\hbar^{2}} \Bigg{[}\] \[\sum_{k,k^{\prime},j,j^{\prime}}g_{kk^{\prime}}^{(j^{\prime}j^{ \prime})}\left(2\hat{A}_{S}^{(k,j)}\hat{\rho}_{S}(i)\hat{A}_{S}^{(k^{\prime}, j^{\prime})\dagger}-\Big{\{}\hat{A}_{S}^{(k^{\prime},j^{\prime})}\hat{A}_{S}^{(k,j)}, \hat{\rho}_{S}(i)\Big{\}}\right)\] \[+\sum_{k,j}g_{k}^{(j)}\left(2\hat{H}_{S}\hat{\rho}_{S}\hat{A}_{S}^ {(k,j)\dagger}-\Big{\{}\hat{A}_{S}^{(k,j)\dagger}\hat{H}_{S},\hat{\rho}_{S}( i)\Big{\}}\right)\] \[+\sum_{k,j}g_{k}^{(j)}\left(2\hat{A}_{S}^{(j,k)}\hat{\rho}_{S}(i) \hat{H}_{S}-\Big{\{}\hat{H}_{S}\hat{A}_{S}^{(k,j)},\hat{\rho}_{S}(i)\Big{\}}\right)\] \[+\sum_{k,j}\Big{[}g_{kH_{E}}^{(j)}\hat{A}_{S}^{(k,j)}-g_{kH_{E}}^ {(j)*}\hat{A}_{S}^{(k,j)\dagger},\hat{\rho}_{S}(i)\Big{]}\] \[+\left(2\hat{H}_{S}\hat{\rho}_{S}(i)\hat{H}_{S}-\Big{\{}\hat{H}_{S} ^{2},\hat{\rho}_{S}(i)\Big{\}}\right)\Bigg{]}, \tag{24}\]
where the factors \(g_{k}^{(j)}\), \(g_{k}^{(j^{\prime}j^{\prime})}\) and \(g_{kH_{E}}^{(j)}\) are:
\[g_{k}^{(j)} = \mathrm{Tr}\left[\hat{B}_{E_{i}k}^{(j)}\hat{\eta}_{E_{i}k}\right], \tag{25}\] \[g_{kk^{\prime}}^{(j^{\prime}j^{\prime})} = \mathrm{Tr}\left[\hat{B}_{E_{i}k^{\prime}}^{(j^{\prime})\dagger} \hat{B}_{E_{k}k}^{(j)}\hat{\eta}_{E_{i}k}\right],\] (26) \[g_{kH_{E}}^{(j)} = \mathrm{Tr}\left[\hat{H}_{E_{i+1}}\hat{B}_{E_{i+1}k}^{(j)}\hat{\eta }_{E_{i+1}}\right]. \tag{27}\]
As expected, the term \(\mathcal{L}_{0}\) is simply the identity channel, mapping the density matrix \(\hat{\rho}_{S}\) onto itself. The term \(\mathcal{L}_{1}\) gives instead a unitary contribution, adding a driving term to the free Hamiltonian \(\hat{H}_{S}\). Finally, the term \(\mathcal{L}_{2}\) describes dissipation phenomena on \(S\). The effect of a structured environment can be observed by the presence of the double index \(kk^{\prime}\) in the correlation functions \(g_{kk^{\prime}}^{(j,j^{\prime})}\), which signal the presence of terms stemming from the common interaction of \(S\) with two different subsystems \(k,k^{\prime}\) of the ancilla \(E_{i}\).
Notice also that we did not make the usual assumption \(g_{k}^{(j)}=0\). This is usually a consequence of assuming the environment to be in a thermal state, for which first order correlation functions are identically null. However, since we are now considering a structured environment, even if it is in a thermal state the terms \(g_{k}^{(j)}\) could still be different from zero, due to the coherence between different components of the ancillae. This being said, let us now see two examples of a system interacting with a structured bath, in order to better understand when the presence of structured ancillae has effects on the dynamics, and which are those effects.
## III Examples of master equation with structured ancillae
### Example 1
As first example we want to consider the case where each ancilla is made out of two interacting TLS, so that the ancilla Hamiltonian reads:
\[\hat{H}_{E_{i}}=\frac{\omega_{1}}{2}\hat{\sigma}_{1}^{z}+\frac{\omega_{2}}{2} \hat{\sigma}_{2}^{z}+\kappa_{12}\left(\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}+ \hat{\sigma}_{1}^{-}\hat{\sigma}_{2}^{+}\right), \tag{28}\]
where from now on we set \(\hbar=1\) for ease of notation. Here \(\hat{\sigma}_{k}^{z}\) is the usual diagonal Pauli matrix, and \(\hat{\sigma}_{k}^{\pm}=\frac{1}{2}(\hat{\sigma}_{k}^{x}\pm i\hat{\sigma}_{k}^ {y})\). The bare energy levels of the two TLS are shown in the left panel of Fig. 2. To find the eigenenergies of \(\hat{H}_{E_{i}}\) shown in the right panel of Fig. 2 one needs to diagonalize the Hamiltonian in Eq. (28). To find the eigenenergies one has to solve the equation:
\[\lambda^{4}-\lambda^{2}(x^{2}+y^{2}+\kappa_{12}^{2})+x^{2}(y^{2}+ \kappa_{12}^{2})=0 \tag{29}\] \[x=\frac{\omega_{1}+\omega_{2}}{2},\quad y=\frac{\omega_{1}- \omega_{2}}{2} \tag{30}\]
whose solutions are:
\[E_{1}^{\prime}=-x=-\frac{\omega_{1}+\omega_{2}}{2}, \tag{31}\] \[E_{2}^{\prime}=-\sqrt{y^{2}+\kappa_{12}^{2}}=-\frac{1}{2}\sqrt{ (\omega_{1}-\omega_{2})^{2}+4\kappa_{12}^{2}},\] (32) \[E_{3}^{\prime}=\sqrt{y^{2}+\kappa_{12}^{2}}=\frac{1}{2}\sqrt{( \omega_{1}-\omega_{2})^{2}+4\kappa_{12}^{2}},\] (33) \[E_{4}^{\prime}=+x=\frac{\omega_{1}+\omega_{2}}{2}. \tag{34}\]
The corresponding eigenvectors are:
\[\ket{E_{1}^{\prime}} =\ket{E_{1}}, \tag{35}\] \[\ket{E_{2}^{\prime}} =\frac{E_{2}-y}{\sqrt{(E_{2}-y)^{2}+\kappa_{12}^{2}}}\left(\frac{ \kappa_{12}}{E_{2}-y}\ket{E_{2}}+\ket{E_{3}}\right),\] (36) \[\ket{E_{3}^{\prime}} =\frac{E_{3}-y}{\sqrt{(E_{3}-y)^{2}+\kappa_{12}^{2}}}\left(\frac{ \kappa_{12}}{E_{3}-y}\ket{E_{2}}+\ket{E_{3}}\right),\] (37) \[\ket{E_{4}^{\prime}} =\ket{E_{4}}. \tag{38}\]
Note that the weights in \(\ket{E_{2}^{\prime}},\ket{E_{3}^{\prime}}\) depend on the detuning \(y\). If the two TLS have the same frequency, i.e. \(\omega_{1}=\omega_{2}\), then \(\ket{E_{2}^{\prime}}\) and \(\ket{E_{3}^{\prime}}\) become a singlet and a triplet state with zero magnetization respectively.
We now assume that the open system \(S\) is also a TLS with free Hamiltonian:
\[\hat{H}_{S}=\frac{\omega_{S}}{2}\hat{\sigma}_{S}^{z}, \tag{39}\]
interacting with the ancillae through the interaction Hamiltonian:
\[\hat{H}_{SE_{i}}=\alpha_{1}(\hat{\sigma}_{S}^{+}\hat{\sigma}_{1}^{-}+\hat{ \sigma}_{S}^{-}\hat{\sigma}_{1}^{+})+\alpha_{2}(\hat{\sigma}_{S}^{+}\hat{ \sigma}_{2}^{-}+\hat{\sigma}_{S}^{-}\hat{\sigma}_{2}^{+}), \tag{40}\]
where in the right hand side (rhs) of the equation we have dropped for simplicity the index \(i\) and the operators \(\hat{\sigma}_{1,2}^{\pm}\) are the same appearing in Eq. (28). Moreover, we assume the ancillae to be initially in a thermal state at inverse temperature \(\beta\), that is:
\[\hat{\eta}_{E_{i}}=\frac{e^{-\beta\hat{H}_{E_{i}}}}{Z},\;Z=\text{Tr}\Big{[}e^{ -\beta\hat{H}_{E_{i}}}\Big{]}. \tag{41}\]
We first look at the possible unitary contributions, i.e. the ones stemming from the terms proportional to \(g_{k}^{(j)}\) or \(g_{kH_{E}}^{(j)}\). As shown in App. A all these terms are zero, and thus there is no unitary contribution to the dynamics in this case.
Next we look at the structure of the dissipative part of the dynamics, the one described by the term \(\mathcal{L}_{2}(\hat{\rho}_{S})\). To do this, we give a look at the correlation functions \(g_{kH}^{j,j^{\prime}}\) of the environment, starting with the ones where \(k=k^{\prime}\). There are four non-zero such correlation functions:
\[g_{11}^{(+,-)} =\text{Tr}\left[\hat{\sigma}_{1}^{-}\hat{\sigma}_{1}^{+}\hat{ \eta}_{E_{i}}\right],g_{11}^{(-,+)}=\text{Tr}\left[\hat{\sigma}_{1}^{+}\hat{ \sigma}_{1}^{-}\hat{\eta}_{E_{i}}\right], \tag{42}\] \[g_{22}^{(+,-)} =\text{Tr}\left[\hat{\sigma}_{2}^{-}\hat{\sigma}_{2}^{+}\hat{ \eta}_{E_{i}}\right],g_{22}^{(-,+)}=\text{Tr}\left[\hat{\sigma}_{2}^{+}\hat{ \sigma}_{2}^{-}\hat{\eta}_{E_{i}}\right], \tag{43}\]
which are computed explicitly in App. A. In addition to this, we also have cross correlation functions, i.e. those with \(k\neq k^{\prime}\). In this case we have eight possible terms, which are complex conjugates in couples:
\[g_{12}^{(+,-)} =g_{21}^{(+,-)*}=\text{Tr}\left[\hat{\sigma}_{1}^{-}\hat{\sigma} _{2}^{+}\hat{\eta}_{E_{i}}\right] \tag{44}\] \[g_{12}^{(-,+)} =g_{21}^{(-,+)*}=\text{Tr}\left[\hat{\sigma}_{1}^{+}\hat{\sigma}_ {2}^{-}\hat{\eta}_{E_{i}}\right]\] (45) \[g_{12}^{(-,-)} =g_{21}^{(-,-)*}=\text{Tr}\left[\hat{\sigma}_{1}^{+}\hat{\sigma} _{2}^{+}\hat{\eta}_{E_{i}}\right]\] (46) \[g_{12}^{(+,+)} =g_{21}^{(+,+)*}=\text{Tr}\left[\hat{\sigma}_{1}^{-}\hat{\sigma} _{2}^{-}\hat{\eta}_{E_{i}}\right]. \tag{47}\]
Figure 2: Panels (a): Energy levels of two non interacting TLS. Panel (b): Energy levels of two TLS interacting through the Hamiltonian in Eq. (28). It is immediate to see that the energy levels are changed by the interaction between the two TLS, and that the corresponding eigenstates are linear combinations of the non-interacting eigenstates.
As shown in App. A, the only non zero elements are the ones in Eqs.( 44, 45). Conversely, the terms with \(g_{12}^{(+,+)}\) and \(g_{12}^{(-,-)}\) are equal to zero. In the end, we can write the master equation for the dynamics of \(S\) in the interaction picture as follows:
\[\hat{\rho}_{S}(i+1)=\hat{\rho}_{S}(i)-i\delta t\Big{[}\hat{H}_{S},\hat{\rho}_{S}(i)\Big{]}\] \[+\frac{\delta t^{2}}{2}\Bigg{[}\sum_{i,j=1,2}g_{ij}^{(+,-)} \mathcal{D}[\hat{\sigma}_{S}^{+}](\hat{\rho}_{S}(i))\] \[\sum_{i,j=1,2}g_{ij}^{(-,+)}\mathcal{D}[\hat{\sigma}_{S}^{-}]( \hat{\rho}_{S}(i))+\frac{\omega_{S}^{2}}{4}\mathcal{D}[\hat{\sigma}_{S}^{z}]( \hat{\rho}_{S}(i))\Bigg{]}, \tag{48}\]
where we have defined:
\[\mathcal{D}[\hat{A}](\cdots)=2\hat{A}\cdots\hat{A}^{\dagger}- \Big{\{}\hat{A}^{\dagger}\hat{A},\cdots\Big{\}}, \tag{49}\] \[\mathcal{D}^{\prime}[\hat{A},\hat{B}](\cdots)=2\hat{A}\cdots\hat {B}-\Big{\{}\hat{B}\hat{A},\cdots\Big{\}}. \tag{50}\]
This is the usual discrete Markovian master equation for a system interacting with a thermal bath. The presence of coherence in the thermal bath has no effect on the dynamics and thus the system will simply reach thermal equilibrium at the steady state, without the formation of any coherence. One can in fact compute explicitly the steady state satisfying \(\hat{\rho}_{S}(i+1)=\hat{\rho}_{S}(i)\) to obtain:
\[\hat{\rho}_{S}^{steady}=\frac{1}{\sum_{i,j,\pm}g_{ij}^{(\pm,+)}} \begin{bmatrix}\sum_{i,j}g_{ij}^{(-,+)}&0\\ 0&\sum_{i,j}g_{ij}^{(+,-)}\end{bmatrix}. \tag{51}\]
With this example we can see that when system \(S\) interacts with each qubit of the ancilla independently, then the presence of coherence in the bath degrees of freedom has no effects on the dynamics. In the next example we are going to see how, when the interaction between the system and the ancilla involves joint degrees of freedom of the ancilla, i.e. degrees of freedom that have coherence, then the presence of coherence becomes significant for the dynamics.
### Example 2
As a second example we consider the same system we examined in Sec. III.1, but this time with an interaction Hamiltonian between the system and the ancillae of the form:
\[\hat{H}_{SE_{i}}=\alpha\hat{\sigma}_{S}^{+}\hat{\sigma}_{E_{i}}^{ -}+\alpha^{*}\hat{\sigma}_{S}^{-}\hat{\sigma}_{E_{i}}^{+} \tag{52}\]
where the operators \(\hat{\sigma}_{E_{i}}^{\pm}\) are defined as:
\[\hat{\sigma}_{E_{i}}^{+}=|E_{3}\rangle\!\langle E_{2}|\quad\hat{ \sigma}_{E_{i}}^{-}=|E_{2}\rangle\!\langle E_{3}|\,, \tag{53}\]
and the states \(|E_{2}\rangle\,,|E_{3}\rangle\) are the bare levels of the ancilla Hamiltonian. Thanks to the results of the previous section we can immediately write the Hamiltonian and dissipative parts of the dynamics. We have:
\[\mathcal{L}_{1} = -i\Big{[}\Big{(}g_{E_{i}}^{(+)}\Big{)}\hat{\sigma}_{S}^{-}+ \Big{(}g_{E_{i}}^{(-)}\Big{)}\hat{\sigma}_{S}^{+},\hat{\rho}_{S}\Big{]} \tag{54}\] \[= -i\Big{[}\text{Re}\Big{\{}g_{E_{i}}^{(+)}\Big{\}}\hat{\sigma}_{S} ^{x}+\text{Im}\Big{\{}g_{E_{i}}^{(+)}\Big{\}}\hat{\sigma}_{S}^{y},\hat{\rho}_{ S}\Big{]},\]
where the constants \(g_{E_{i}}^{(\pm)}\) are:
\[g_{E_{i}}^{(+)}=\text{Tr}\left[\hat{\sigma}_{E_{i}}^{+}\hat{\eta }_{E_{i}}\right]=g_{E_{i}}^{(-)*}. \tag{55}\]
Note that for an unstructured environment these terms would be identically zero, as they are proportional to the amount of coherence between the levels \(|E_{2}\rangle\) and \(|E_{3}\rangle\). In practice, the term \(\mathcal{L}_{1}(\hat{\rho}_{S})\) describe the unitary evolution due to an effective Hamiltonian of the form:
\[\hat{H}_{S}^{\text{eff}}=\text{Re}\Big{\{}g_{E_{i}}^{(+)}\Big{\}} \hat{\sigma}_{S}^{x}+\text{Im}\Big{\{}g_{E_{i}}^{(+)}\Big{\}}\hat{\sigma}_{S}^ {y}.\]
While we leave the explicit calculations of the coefficients to App. B, we can immediately write down the master equation as:
\[\hat{\rho}_{S}(i+1)=\hat{\rho}_{S}(i)-i\delta t\Big{[}\hat{H}_{S} +g_{E_{i}}^{(-)}\hat{\sigma}_{S}^{+}+g_{E_{i}}^{(+)}\hat{\sigma}_{S}^{-},\hat{ \rho}_{S}(i)\Big{]}\] \[+\frac{\delta t^{2}}{2}\Bigg{[}\frac{\omega_{S}^{2}}{4}\mathcal{ D}[\hat{\sigma}_{S}^{z}](\hat{\rho}_{S}(i))+g_{E_{i}}^{(+,-)}\mathcal{D}[\hat{ \sigma}_{S}^{-}](\hat{\rho}_{S}(i))\] \[+g_{E_{i}}^{(-,+)}\mathcal{D}[\hat{\sigma}_{S}^{-}](\hat{\rho}_{S} (i))+\frac{g_{E_{i}}^{(+)}\omega_{S}}{2}\mathcal{D}^{\prime}[\hat{\sigma}_{S}^{- },\hat{\sigma}_{S}^{z}](\hat{\rho}_{S}(i))\] \[+\frac{g_{E_{i}}^{(+)}\omega_{S}}{2}\mathcal{D}^{\prime}[\hat{ \sigma}_{S}^{z},\hat{\sigma}_{S}^{-}](\hat{\rho}_{S}(i))+\frac{g_{E_{i}}^{(-)} \omega_{S}}{2}\mathcal{D}^{\prime}[\hat{\sigma}_{S}^{+},\hat{\sigma}_{S}^{z}]( \hat{\rho}_{S}(i))\] \[+\frac{g_{E_{i}}^{(-)}\omega_{S}}{2}\mathcal{D}^{\prime}[\hat{ \sigma}_{S}^{z},\hat{\sigma}_{S}^{+}](\hat{\rho}_{S}(i))\Bigg{]}. \tag{56}\]
The dynamics described by this master equation consists in the usual dissipative contribution and dephasing term driving the system towards the thermal equilibrium, plus a unitary contribution which creates coherence in the system. Specifically, the matrix elements are mapped according to:
\[\rho_{11}\rightarrow\rho_{11}+\delta t^{2}\left(g_{E_{i}}^{(-,+)} \rho_{22}-g_{E_{i}}^{(+,-)}\rho_{11}\right)+\frac{\delta t^{2}}{2}\omega_{S} \left(g_{E_{i}}^{(+)}\rho_{12}+g_{E_{i}}^{(-)}\rho_{21}\right)+i\delta t\left(g_ {E_{i}}^{(+)}\rho_{12}-g_{E_{i}}^{(-)}\rho_{21}\right), \tag{57}\] \[\rho_{22}\rightarrow\rho_{22}-\delta t^{2}\left(g_{E_{i}}^{(-,+)} \rho_{22}-g_{E_{i}}^{(+,-)}\rho_{11}\right)-\frac{\delta t^{2}}{2}\omega_{S} \left(g_{E_{i}}^{(+)}\rho_{12}+g_{E_{i}}^{(-)}\rho_{21}\right)-i\delta t\left(g _{E_{i}}^{(+)}\rho_{12}-g_{E_{i}}^{(-)}\rho_{21}\right),\] (58) \[\rho_{12}\rightarrow\rho_{12}+\frac{\delta t^{2}}{2}\omega_{S}g_ {E_{i}}^{(-)}(\rho_{11}-\rho_{22})+i\delta tg_{E_{i}}^{(-)}(\rho_{11}-\rho_{22 })-i\delta t\omega_{S}\rho_{12}-\frac{\delta t^{2}}{2}\rho_{12}\left(g_{E_{i}} ^{(+,-)}+g_{E_{i}}^{(-,+)}+\omega_{S}^{2}\right),\] (59) \[\rho_{21}\rightarrow\rho_{21}+\frac{\delta t^{2}}{2}\omega_{S}g_ {E_{i}}^{(+)}(\rho_{11}-\rho_{22})-i\delta tg_{E_{i}}^{(+)}(\rho_{11}-\rho_{2 2})+i\delta t\omega_{S}\rho_{21}-\frac{\delta t^{2}}{2}\rho_{21}\left(g_{E_{i} }^{(+,-)}+g_{E_{i}}^{(-,+)}+\omega_{S}^{2}\right). \tag{60}\]
On one hand one can see that the populations, i.e. \(\rho_{11}\) and \(\rho_{22}\), evolve according to the usual term due to dissipation, plus another term due to the presence of the unitary driving. On the other hand, the coherences \(\rho_{12}\) and \(\rho_{21}\), besides the usual decay term due to the presence of the environment, evolves also according to terms due to the presence of the unitary driving, thus leading to a non-zero value in the steady state.
In fact it is possible to compute the steady state of system \(S\) by imposing \(\hat{\rho}(i+1)=\hat{\rho}(i)\), obtaining the following values for \(\langle\hat{\sigma}_{z}\rangle,\langle\hat{\sigma}_{x}\rangle,\langle\hat{ \sigma}_{y}\rangle\):
Figure 3: Plots of the evolution of populations and coherence. In all the plots \(\delta t=0.1\), \(\omega_{S}=1\), \(\omega_{1}=0.5\), \(\omega_{2}=1.5\), \(\kappa_{12}=0.3\), \(\alpha=0.1\). The initial state is a thermal state of the form \(\hat{\rho}_{S}(0)=\exp\!\left[-\beta_{S}\hat{H}_{S}\right]\!/Z_{S}\), \(Z_{S}\) being the partition function and \(\beta_{S}=1\). Panel (a): plot of the expectation value \(\langle\hat{\sigma}_{z}\rangle\) as a function of the number of collisions. As one can see, the population tends to equilibrate towards the steady state value, which is not simply the thermal population, as the steady state also has coherence. This observation in particular justifies the behavior of the population for the case \(\beta=\beta_{S}\), which without the creation of coherence should just be a line. In panels (b) and (c) the expectation values \(\langle\hat{\sigma}_{x}\rangle\) and \(\langle\hat{\sigma}_{y}\rangle\) are plotted. One can observe how the absolute amount of coherence is maximal when the system and the environment have the same temperature. In panel (d), (e), (f) the same quantities are plotted, this time for different values of \(\kappa_{12}\), the environmental temperature being \(\beta=\beta_{S}=1\). As one can observe, there is a small difference in the equilibrium populations, due to the fact that the amount of coherence created is different. Moreover, one can observe that the stronger the interaction between the two TLSs composing the ancilla, the greater the amount of coherence created.
\[\langle\hat{\sigma}_{z}\rangle =\frac{\left(g_{E}^{(-,+)}-g_{E}^{(+,-)}\right)\left[4\omega_{S}^{2} +\delta t^{2}\left(g_{E}^{(-,+)}+g_{E}^{(+,-)}\omega_{S}^{2}\right)^{2}\right]}{K}, \tag{61}\] \[\langle\hat{\sigma}_{x}\rangle =\frac{\left(g_{E}^{(-,+)}-g_{E}^{(+,-)}\right)2\operatorname{Re} \left[g_{E}^{(-)}\left(4\omega_{S}+2i\delta t\left(g_{E}^{(-,+)}+g_{E}^{(+,-)} \right)+\delta t^{2}\omega_{S}\left(g_{E}^{(-,+)}+g_{E}^{(+,-)}+\omega_{S}^{2} \right)\right)\right]}{K},\] (62) \[\langle\hat{\sigma}_{y}\rangle =-\frac{\left(g_{E}^{(-,+)}-g_{E}^{(+,-)}\right)2\operatorname{Im }\left[g_{E}^{(-)}\left(4\omega_{S}+2i\delta t\left(g_{E}^{(-,+)}+g_{E}^{(+,-) }\right)+\delta t^{2}\omega_{S}\left(g_{E}^{(-,+)}+g_{E}^{(+,-)}+\omega_{S}^{2 }\right)\right)\right]}{K},\] (63) \[K =8g_{E}^{(-)}g_{E}^{(+)}\left(g_{E}^{(-,+)}+g_{E}^{(-,+)}-\omega_ {S}^{2}\right)+4\omega_{S}^{2}\left(g_{E}^{(-,+)}+g_{E}^{(+,-)}\right)\] \[+\delta t^{2}\left(g_{E}^{(-,+)}+g_{E}^{(+,-)}\right)\left[\left( g_{E}^{(-,+)}+g_{E}^{(+,-)}\right)^{2}+2\omega_{S}^{2}\left(g_{E}^{(-,+)}+g_{E}^{(+,-) }-g_{E}^{(-)}g_{E}^{(+)}\right)+\omega_{S}^{4}\left(1-2g_{E}^{(-)}g_{E}^{(+)} \right)\right]. \tag{64}\]
One can see immediately that the coherences are non null in the steady state, as both \(\langle\hat{\sigma}_{x}\rangle,\langle\hat{\sigma}_{y}\rangle\) are non-zero. Moreover, it is worth noticing that the equilibrium populations will not simply be the thermal populations, due to the fact that coherence is created in the system \(S\).
We now want to analyze with more detail the dynamics of \(S\), in particular checking how the populations and the coherence evolve, that is, we want to look at the transient dynamics before reaching the steady state. In Fig. 3 some plots for the populations and the coherence of system \(S\) are shown. In the upper plots the evolution of these quantities is shown for different values of the environmental temperature \(\beta\), while in the bottom plots the same quantities are plotted for different values of \(\kappa_{12}\).
Some observations are to be made. First, one can notice that the equilibrium populations are not the thermal ones, consistently with the fact that coherence is created in system \(S\). Furthermore, one can observe oscillations of the value of both \(\langle\hat{\sigma}_{x}\rangle\) and \(\langle\hat{\sigma}_{y}\rangle\). These oscillations are slowly suppressed as the system reaches its steady state. Finally, one can clearly observe that the total amount of coherence available to system \(S\) is strongly dependent on the amount of coherence in the ancillae, ultimately depending on the value of \(\kappa_{12}\).
Moreover, in Fig. 4 we have plotted a detail of the evolution of the populations for various temperatures \(\beta\) and different \(\kappa_{12}\). It can be observed, consistently with the presence of the unitary driving term, that while going to their steady state value, the populations also oscillates. One can also observe that these oscillations become wider for larger values of \(\kappa_{12}\), as the strength of the unitary driving becomes larger.
In Fig. 5 we have plotted the steady state value of \(\langle\hat{\sigma}_{x}\rangle\) as a function of \(\kappa_{12}\) for several values of the environmental temperature \(\beta\) and as a function of \(\beta\) for different values of \(\kappa_{12}\). As one can see in Fig. 5(a), on one hand for all values of \(\beta\), as \(\kappa_{12}\) grows large the steady state coherence converges towards an asymptotic value. On the other hand, one can observe a peak in \(\langle\hat{\sigma}_{x}\rangle\) for specific values of \(\kappa_{12}\) and \(\beta\). In Fig. 5(b) one can instead observe a different behavior as a function of the temperature for different values of \(\kappa_{12}\). In fact, for \(\kappa_{12}\lesssim 1\) it can be observed that the steady state coherence goes to zero as \(\beta\) grows large. Conversely, for values of \(\kappa_{12}\gtrsim 1\), as \(\beta\) grows
large one can observe that the steady state coherence goes towards a non zero value, the largest value being obtained for \(\kappa_{12}\simeq 1\). In conclusion, the dependence of \(\langle\hat{\sigma}_{x}\rangle\) on \(\kappa_{12}\) and \(\beta\) is in general non trivial, though its main characteristics can be extrapolated. These plots demonstrate the result we anticipated and one of the main contributions of this paper: the generation of steady-state coherence using only thermal resources.
## IV Thermodynamics
In this section we want to analyze the thermodynamics of a system when interacting with a bath made out of structured ancillae. We will first define the relevant thermodynamic quantities, such as internal energy, work and heat. Then we will make some general considerations based on the general form of the master equation before moving to a practical example based on the system described in Sec. III.2. We will show that the creation of coherence in the steady state requires work to be injected into the system, thus recovering the first principle of thermodynamics in its standard form and fulfilling the second principle of thermodynamics.
### Defining the thermodynamic quantities
We define the thermodynamic quantities starting from the internal energy of system \(S\), which is simply:
\[U=\mathrm{Tr}\left[\hat{H}_{S}\hat{\rho}_{S}\right], \tag{65}\]
so that at each step the internal energy variation can be defined as:
\[\Delta U(i+1) = \mathrm{Tr}\left[\hat{H}_{S}\left(\hat{\rho}_{S}(i+1)-\hat{\rho} _{S}(i)\right)\right] \tag{66}\] \[= \mathrm{Tr}\left[\hat{H}_{S}\left(\hat{U}_{SE_{i+1}}\hat{R}_{S \mathcal{E}}(i)\hat{U}_{SE_{i+1}}-\hat{R}_{S\mathcal{E}}(i)\right)\right].\]
By inserting the expressions in Eqs. ( 19, 20) into Eq. (66) we obtain:
\[\Delta U(i+1)=\mathrm{Tr}\left[\hat{H}_{S}\mathcal{U}_{1}(\hat{R }_{S\mathcal{E}}(i))\right]+\mathrm{Tr}\left[\hat{H}_{S}\mathcal{U}_{2}(\hat{ R}_{S\mathcal{E}}(i))\right]\] \[=-i\delta t\,\mathrm{Tr}\left[\left[\hat{H}_{S},\hat{H}_{SE_{i+1 }}\right]\hat{R}_{S\mathcal{E}}(i)\right]+\frac{\delta t^{2}}{2}\,\mathrm{Tr} \left[\left[\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}},\left[\hat{H}_{S },\hat{H}_{SE_{i+1}}\right]\right]\hat{R}_{S\mathcal{E}}(i)\right]. \tag{67}\]
We now define the heat exchanged by the system, assumed to be positive when it is absorbed by the system. At each interaction between the system and an ancilla, we thus define the heat exchanged as the opposite of the energy variation of the ancilla:
\[\delta Q(i+1)=-\,\mathrm{Tr}\left[\hat{H}_{E_{i}}(\hat{\eta}_{E_{ i+1}}^{out}-\hat{\eta}_{E_{i+1}}^{in})\right]\] \[=-\,\mathrm{Tr}\left[\hat{H}_{E_{i+1}}\left(\hat{U}_{SE_{i+1}} \hat{R}_{S\mathcal{E}}(i)\hat{U}_{SE_{i+1}}^{\dagger}-\hat{R}_{S\mathcal{E}}( i)\right)\right], \tag{68}\]
Figure 5: Plots of \(\langle\hat{\sigma}_{x}\rangle\) at the steady state. The values of the parameters are \(\delta t=0.1\), \(\omega_{S}=1\), \(\omega_{1}=0.5\), \(\omega_{2}=1.5\), \(\alpha=0.2\) Panel (a): plot of \(\langle\hat{\sigma}_{x}\rangle\) as a function of \(\kappa_{12}\) for different values of \(\beta\). In this case it can be noticed that in general the amount of coherence generated in the steady state is larger for lower temperatures. However the behavior of the coherence for large values of \(\kappa_{12}\) shows that the amount of coherence reaches a steady value independently of the temperature. Moreover, one can see that for specific values of \(\kappa_{12}\) the amount of coherence is larger. Panel (b): plot of \(\langle\hat{\sigma}_{x}\rangle\) as a function of \(\beta\) for different values of \(\kappa_{12}\). On one hand, it can be immediately noticed that, for values of \(\kappa_{12}\lesssim 1\) one has larger values of \(\langle\hat{\sigma}_{x}\rangle\) for larger values of \(\kappa_{12}\). On the other hand, for values of \(\kappa_{12}\gtrsim 1\), in the limit of very large \(\beta\), the coherence reaches always the same steady value. The case \(\kappa_{12}\simeq 1\) represents the border between the two scenarios, where the coherence, in the limit of large \(\beta\) reaches the higher value.
where we have defined:
\[\tilde{\eta}^{out}_{E_{i+1}}=\mathrm{Tr}_{S}\left[\hat{U}_{SE_{i+1}}\hat{R}_{S \mathcal{E}}(i)\hat{U}^{\dagger}_{SE_{i+1}}\right]. \tag{69}\]
Definition (68) is justified since the initial state of the environment is thermal. Once again we can insert the expressions in Eqs. ( 19, 20) to obtain:
\[\delta Q(i+1)=-\operatorname{Tr}\left[\hat{H}_{E_{i+1}}\mathcal{U }_{1}(\hat{R}_{S\mathcal{E}}(i))\right]+\operatorname{Tr}\left[\hat{H}_{E_{i+ 1}}\mathcal{U}_{2}(\hat{R}_{S\mathcal{E}})\right]\] \[=i\delta t\operatorname{Tr}\left[\left[\hat{H}_{E_{i+1}},\hat{H} _{SE_{i+1}}\right]\hat{R}_{S\mathcal{E}}(i)\right]-\frac{\delta t^{2}}{2} \operatorname{Tr}\left[\left[\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}},\left[\hat{H}_{E_{i+1}},\hat{H}_{SE_{i+1}}\right]\right]\hat{R}_{S\mathcal{E} }(i)\right],\]
Moreover, any variation of the system Hamiltonian will contribute to work, so that we define:
\[\delta W=\operatorname{Tr}\Bigl{[}\Delta\hat{H}_{S}\hat{\rho}_{S}\Bigr{]}. \tag{70}\]
As a consequence, we also have to take into consideration the switching work contribution [27], namely the one stemming from turning on and off of the interaction between the system and the ancilla, which reads:
\[W_{sw}(i+1) = \operatorname{Tr}\left[\hat{H}_{SE_{i}}\left(\hat{R}_{S\mathcal{E }}(i)-\hat{U}_{SE_{i+1}}\hat{R}_{S\mathcal{E}}(i)\hat{U}^{\dagger}_{SE_{i+1}} \right)\right], \tag{71}\] \[=i\delta t\operatorname{Tr}\left[\left[\hat{H}_{SE_{i+1}},\hat{H }_{SE_{i+1}}\right]\hat{R}_{S\mathcal{E}}(i)\right]-\frac{\delta t^{2}}{2} \operatorname{Tr}\left[\left[\hat{H}_{S}+\hat{H}_{E_{i+1}}+\hat{H}_{SE_{i+1}},\left[\hat{H}_{SE_{i+1}},\hat{H}_{E_{i+1}}\right]\right]\hat{R}_{S\mathcal{E} }(i)\right].\]
\[\Delta U=-i\delta t\sum_{j,k}g_{k}^{(j)}\,{\rm Tr}\left[\left[\hat{H} _{S},\hat{A}_{S}^{(k,j)}\right]\hat{\rho}_{S}(i)\right]\frac{\delta t^{2}}{2} \sum_{k,j}g_{k}^{(j)}\,{\rm Tr}\left[\left[\hat{A}_{S}^{(k,j)\dagger},\hat{H}_{ S}\right]\hat{H}_{S}\hat{\rho}_{S}(i)\right]\] \[+\frac{\delta t^{2}}{2}\sum_{k,j}g_{k}^{(j)}\,{\rm Tr}\left[\left[ \hat{H}_{S},\hat{A}_{S}^{(k,j)}\right]\hat{\rho}_{S}(i)\hat{H}_{S}\right]+ \frac{g\delta t^{2}}{2}\sum_{k,k^{\prime},j,j^{\prime}}g_{kk^{\prime}}^{(j,j^{ \prime})}\,{\rm Tr}\left[\left(2\hat{A}_{S}^{(k,j)}\hat{H}_{S}\hat{A}_{S}^{(k,j)\dagger}-\left\{\hat{A}_{S}^{(k,j)\dagger}\hat{A}_{S}^{(k,j)},\hat{H}_{S} \right\}\right)\hat{\rho}_{S}(i)\right]\] \[+\frac{\delta t^{2}}{2}\sum_{j,k}{\rm Tr}\left[\left[\hat{H}_{S},g_{kHz}^{(j)}\hat{A}_{S}^{(k,j)}-g_{kHz}^{(j)*}\hat{A}_{S}^{(k,j)\dagger} \right]\hat{\rho}_{S}(i)\right], \tag{74}\]
for the variation of internal energy, while the heat exchanged reads:
\[\delta Q=i\delta t\sum_{k,j}(g_{kHz}^{(j)}-g_{kHz}^{(j)*})\,{\rm Tr }\left[\hat{A}_{S}^{(k,j)}\hat{\rho}_{S}(i)\right]-\frac{\delta t^{2}}{2}\sum _{j,k}{\rm Tr}\left[\left[\hat{H}_{S},g_{kHz}^{(j)}\hat{A}_{S}^{(k,j)}-g_{kHz }^{(j)*}\hat{A}_{S}^{(k,j)\dagger}\right]\hat{\rho}_{S}(i)\right]\] \[+\frac{\delta t^{2}}{2}\sum_{j,k}{\rm Tr}\left[\hat{A}_{S}^{(j,k) }\hat{\rho}_{S}(i)\right]{\rm Tr}\left[(2\hat{H}_{E}\hat{B}_{Ek}^{(j)}\hat{H}_ {E}-\left\{\hat{H}_{E}^{2},\hat{B}_{Ek}^{(j)}\right\})\hat{\eta}_{E_{i+1}}\right]\] \[-\frac{\delta t^{2}}{2}\sum_{k,k^{\prime},j,j^{\prime}}{\rm Tr} \left[\hat{A}_{S}^{(k,j)\dagger}\hat{A}_{S}^{(k,j)}\hat{\rho}_{S}\right]{\rm Tr }\left[\left[\hat{H}_{E},\hat{B}_{Ek}^{(j)}\right]\hat{\eta}_{E_{i+1}}\hat{B}_ {Ek}^{(j)\dagger}\right]+\frac{\delta t^{2}}{2}\sum_{k,k^{\prime},j,j^{\prime} }{\rm Tr}\left[\hat{A}_{S}^{(k,j)}\hat{A}_{S}^{(k,j)\dagger}\hat{\rho}_{S} \right]{\rm Tr}\left[\left[\hat{H}_{E},\hat{B}_{Ek}^{(j)}\right]\hat{B}_{Ek}^ {(j)\dagger}\hat{\eta}_{E_{i+1}}\right].\]
Finally, we will not write explicitly the switching work, since one can easily derive it from:
\[W_{sw}=\Delta U-\delta Q. \tag{76}\]
Let us now turn our attention to the entropic quantities. The von Neumann entropy of a quantum state is defined as:
\[S(\hat{\rho}_{S})=-\,{\rm Tr}\left[\hat{\rho}_{S}\log\hat{\rho}_{S}\right]. \tag{77}\]
We thus define the system \(S\) entropy variation during one collision as:
\[\Delta S_{S}(i+1)=S(\hat{\rho}_{S}(i+1))-S(\hat{\rho}_{S}(i)). \tag{78}\]
The entropy production, describing the total amount of entropy created during one collision, including the contributions due to the change of the ancilla state and the one due to the loss of information, can be defined as [27]:
\[\Sigma(i)={\cal I}(\hat{\rho}_{S}(i);\hat{\eta}_{E_{i}}^{out})+S(\hat{\eta}_{ E_{i}}^{out}||\hat{\eta}_{E_{i}}^{in}), \tag{79}\]
where
\[{\cal I}(\hat{\rho}_{S}(i);\hat{\eta}_{E_{i}}^{out})= \tag{80}\] \[=S(\hat{\rho}_{S}(i))+S(\hat{\eta}_{E_{i}}^{out})-S(\hat{U}_{SE_{i }}R_{S\mathcal{E}}(i)\hat{U}_{SE_{i}})\] \[=S(\hat{\rho}_{S}(i))+S(\hat{\eta}_{E_{i}}^{out})-S(\hat{\rho}_{ S}(i-1))-S(\hat{\eta}_{E_{i}}^{in}),\]
is the mutual information between system \(S\) and the ancilla \(E_{i}\) after their interaction, while:
\[S(\hat{\eta}_{E_{i}}^{out}||\hat{\eta}_{E_{i}}^{in})={\rm Tr}\left[\hat{\eta}_ {E_{i}}^{out}\log\hat{\eta}_{E_{i}}^{out}\right]-{\rm Tr}\left[\hat{\eta}_{E_ {i}}^{out}\log\hat{\eta}_{E_{i}}^{in}\right],\]
is the relative entropy between the state of the ancilla after the interaction and the one before the interaction. Summing the two quantities together one obtains:
\[\Sigma(i)=\Delta S_{S}(i)-\beta\,\delta Q(i), \tag{81}\]
where we have exploited the fact that the initial ancilla state \(\hat{\eta}_{E_{i}}^{in}\) is thermal. The second principle of thermodynamics can thus be expressed as:
\[\Sigma(i)\geq 0\quad\forall i. \tag{82}\]
### Thermodynamics of the system in Example 2
We now want to apply the formulas derived in the previous section to the system in Sec. III.2. By substituting the Hamiltonians in Eqs.( 28, 39, 52) we obtain for the internal energy variation:
\[\Delta U=-\frac{\delta t^{2}}{2}\omega_{S}\] \[\left[\left\langle\hat{\sigma}_{S}^{z}\right\rangle\left(g_{E_{i} }^{(+,-)}+g_{E_{i}}^{(-,+)}\right)+\left(g_{E_{i}}^{(+,-)}-g_{E_{i}}^{(-,+)} \right)\right]\] \[+\frac{\delta t^{2}}{2}\omega_{S}^{2}\left(g_{E_{i}}^{(+)}\rho_{ 12}+g_{E_{i}}^{(-)}\rho_{21}\right)\] \[+i\delta t\omega_{S}\left(g_{E_{i}}^{(+)}\rho_{12}-g_{E_{i}}^{(-)} \rho_{21}\right). \tag{83}\]
Using the same method, it is also possible to derive the expressions for the heat exchanged and the switching work, the former reading:
\[\delta Q=-\frac{\delta t^{2}}{2}(E_{3}-E_{2})\] \[\left[\left\langle\hat{\sigma}_{S}^{z}\right\rangle\left(g_{E_{i} }^{(+,-)}+g_{E_{i}}^{(-,+)}\right)+\left(g_{E_{i}}^{(+,-)}-g_{E_{i}}^{(-,+)} \right)\right]\] \[+\frac{\delta t^{2}}{2}\kappa_{12}|\alpha|^{2}\left(\eta_{23}+ \eta_{32}\right), \tag{84}\]
while the switching work reads:
\[W_{sw}=\frac{\delta t^{2}}{2}(E_{3}-E_{2}-\omega_{S})\] \[\left[\langle\hat{\sigma}_{S}^{z}\rangle\left(g_{E_{i}}^{(+,-)}+g_ {E_{i}}^{(-,+)}\right)+\left(g_{E_{i}}^{(+,-)}-g_{E_{i}}^{(-,+)}\right)\right]\] \[-\frac{\delta t^{2}}{2}\kappa_{12}|\alpha|^{2}\left(\eta_{23}+ \eta_{32}\right)\] \[+\frac{\delta t^{2}}{2}\omega_{S}^{2}\left(g_{E_{i}}^{(+)}\rho_{1 2}+g_{E_{i}}^{(-)}\rho_{21}\right)\] \[+i\delta t\omega_{S}t\left(g_{E_{i}}^{(+)}\rho_{12}-g_{E_{i}}^{(- )}\rho_{21}\right). \tag{85}\]
The first term in the switching work is simply due to the mismatch of the energy difference \(E_{3}-E_{2}\) in the structured ancilla and the system's energy separation \(\omega_{S}\) and is common to all collision models. The last two terms arise because of the specific form of the interaction Hamiltonian of the system with the structured ancillae.
In Fig. 6 we plot \(\Delta U\), \(\delta Q\) and \(W_{sw}\) for the initial part of the dynamics. One can observe, consistently with Eqs. ( 83, 85), that since \(\Delta U\) and \(W_{sw}\) depend on the coherences in system \(S\), both quantities presents oscillations before reaching their steady state values. Conversely, as the dissipated heat only depends on \(\langle\hat{\sigma}_{z}\rangle\) and on the ancillae coherences, no oscillations are observed in this case.
After reaching the steady state, the variation of internal energy \(\Delta U\) becomes identically zero. In the plots in Fig. 7 one can observe that in all cases, at the steady state, a positive non-null amount of work is injected into the system in order to keep the steady state coherence. This energy is then exactly dissipated as heat into the thermal bath. This is perfectly consistent with the first principle of thermodynamics, which after the reaching of the steady state simply becomes \(W_{sw}=-\delta Q\). This is also consistent with the second law of thermodynamics: no work can be created with a device operating with only one thermal bath. Moreover, in both plots one can observe that the amount of work is proportional to the amount of coherence created in the steady state, i.e. the larger the coherence the larger the work needed.
Furthermore, in Fig. 8 we have plotted the step-by-step entropy production \(\Sigma(i)\). It is immediate to note that this quantity is always positive, in fulfillment of the second principle of thermodynamics, and that it does never get to zero, since even at the steady state work has to be injected into the system and then dissipated as heat in order to retain coherence.
## V Conclusions
In this work we have introduced a new kind of collision model by allowing the ancillae to be composite systems, rather than simple TLS or harmonic oscillators. We have seen how to compute the discrete master equation for an open quantum system interacting with a structured environment using our model. We have shown how the presence of structured ancillae has consequences on the dynamics of \(S\), most noticeably the possibility of creating coherence in the system in the steady state even if the environment is in a thermal state. We have seen practically through a couple of examples involving qubits how the presence of the structured environment can lead to additional terms in the master equation, in particular the unitary driving terms leading to the steady state coherence, and how this is possible only when the system interacts with the degrees of freedom of the environment that possess coherence.
Furthermore, we have shown how the interaction between the system and the degrees of freedom of the environment possessing coherence requires a work cost to be activated, thus leading to the natural fulfillment of the
Figure 7: Plot of \(W_{sw}\) and \(\delta Q\) at the steady state. In both plots \(\omega_{S}=1\), \(\omega_{1}=0.5\), \(\omega_{2}=1.5\), \(\alpha=0.2\). In panel (a) the quantities are plotted as a function of \(\kappa_{12}\) for different values of the environmental temperature. It can be noticed that the work cost for having coherence in the steady state is always equal to the heat dissipated into the environment. Moreover, the amount of work is proportional to the amount of coherence created in system \(S\). In panel (b) the same quantities are plotted as a function of the environmental temperature \(\beta\) for different values of \(\kappa_{12}\). Also in this case it can be noticed that the work cost is higher for lower temperatures, that is, for larger amounts of coherence.
first and second principle of thermodynamics without invoking any further assumption. We have also been able to compute analytically all the relevant thermodynamic quantities, and shown that one needs to keep injecting work into the system in order to retain the coherence in the steady state.
In future works we plan to use the model presented here to study the efficiency of heat engines and refrigerators exploiting coherence, and how including the cost of creating coherence in the system influences quantities such as the efficiency and the efficiency at maximum power. Finally, the present model might be useful also in different context than thermodynamics, for instance for investigating whether the presence of coherence in the environment can influence non Markovian dynamics or whether more complicated structured ancillae can be exploited to further engineer the steady state of the open quantum systems in order to obtain useful resources for quantum technologies.
## Acknowledgements
SC thanks D. Tamascelli for a useful comment during IQIS 2022 conference. SC acknowledges support by the Foundation for Polish Science (IRAP project, ICTQT, Contract No. 2018/MAB/5, cofinanced by the EU within the Smart Growth Operational Programme). G.D.C. acknowledges the support by the UK EPSRC EP/S02994X/1, the Royal Society IEC\(\backslash\)R2\(\backslash\)222003.
|
2306.08479 | A mathematical model of delay discounting with trait and state
impulsivity | Existing mathematical models of delay discounting consider impulsivity as a
mono-dimensional entity. However, the present article derives a novel
mathematical model of delay discounting involving its bi-dimensional
characteristic. This new mathematical model, named the Extended Effective
Exponential Model or E$^3$M, considers impulsivity as a variable represented by
two positive and fluctuating quantities: trait and state impulsivity. To derive
the model, the superstatistics method, which has been used to describe
fluctuating physical systems like a thermal plasma, has been adapted. The model
successfully describes the results of the existing studies used in the paper
and indicates a dominance of the state dimension among the participants of the
studies. | Shanu Shukla, Trambak Bhattacharyya | 2023-06-14T12:49:32Z | http://arxiv.org/abs/2306.08479v1 | # A mathematical model of delay discounting with trait and state impulsivity
###### Abstract
Existing mathematical models of delay discounting consider impulsivity as a mono-dimensional entity. However, the present article derives a novel mathematical model of delay discounting involving its bi-dimensional characteristic. This new mathematical model, named the Extended Effective Exponential Model or E\({}^{3}\)M, considers impulsivity as a variable represented by two positive and fluctuating quantities: trait and state impulsivity. To derive the model, the superstatistics method, which has been used to describe fluctuating physical systems like a thermal plasma, has been adapted. The model successfully describes the results of the existing studies used in the paper and indicates a dominance of the state dimension among the participants of the studies.
Delay discounting model, trait impulsivity, state impulsivity
## I Introduction
Delay discounting, a well-explored phenomenon in various fields including behavioural science, economics, and finance, describes the tendency for the subjective value of a reward to decrease as its delivery is postponed. This can be illustrated by a hypothetical scenario where an individual is presented with a choice between receiving $50 immediately or $70 after a month. Such a decision involves selecting between a smaller-sooner reward and a larger-later reward, influenced by personal factors like impulsivity and situational factors such as trust. Some individuals may opt for the immediate reward, exhibiting impulsive behaviour according to behavioural analysts [22]. This behaviour suggests that a distant reward is perceived as less valuable than an immediate one. Delay discounting has been observed across various species, reward types, and sample populations, as evidenced in studies [9; 18; 19; 24; 37]. Researchers have proposed several mathematical models to comprehend this phenomenon, including the exponential, hyperbolic, and \(q\)-exponential models of delay discounting. These models commonly incorporate impulsivity as a parameter. However, these models often treat impulsivity as a singular entity, while numerous studies acknowledge impulsivity as a multifaceted variable [12].
To account for this complexity, we introduce a mathematical model that considers impulsivity as having two dimensions, as examined in Refs. [5; 39]. These dimensions are trait and state impulsivity. In this paper, we utilize this model to analyze existing datasets obtained from previous experiments.
The structure of this paper is as follows: the subsequent section provides a brief overview of existing mathematical models of delay discounting. Section III presents a comprehensive derivation of our proposed model. In Section IV, we analyze available datasets and present the findings. Finally, we summarize and conclude our study in Section V.
## II Existing mathematical models of delay discounting
### Exponential model
In the experiments studying delay discounting, the subjective values of a reward are established by finding indifference points [10]. In the case of rational agents, indifference points follow an exponential decay with delay. This model is called the exponential model [26] (EM) and considers an exponential dependence of the subjective value of a reward on delay given by,
\[V(D)=V(0)\exp(-\kappa_{0}D),\qquad\text{(--EM-)} \tag{1}\]
where \(V(0)\) is the undiscounted value of a reward, and \(V(D)<V(0)\) is the discounted value of the reward after a delay \(D\). The quantity \(\kappa_{0}>0\) is described as the impulsivity parameter.
### Hyperbolic model
The rationality of agents has long been a matter of discussion and debate [35] in behavioural science. Studies in delay discounting recognize inter-temporal consistency as a characterization of rationality. The exponential model describes such agents well who display a constant discounting over time. However, for non-rational agents, one needs to look beyond the exponential model and the hyperbolic model is the first step. One of the earliest one-parameter versions of this model can be found in e.g., Refs. [19; 24]
\[V(D)=\frac{V(0)}{(1+\kappa_{\rm h}D)}. \tag{2}\]
There are other two-parameter variants of the models. The first one, described in Ref. [16], is given by,
\[V(D)=\frac{V(0)}{(1+\kappa_{\rm h}D^{s})}. \tag{3}\]
The second variant, given below, is proposed in Ref. [23].
\[V(D)=\frac{V(0)}{(1+\kappa_{\rm h}D)^{s}}, \tag{4}\]
For a comprehensive comparison of the models given by Eqs. (1)-(4), please see Ref. [20].
### The \(q\)-exponential models
#### ii.3.1 Takahashi model
The hyperbolic model is not able to distinguish between inter-temporal inconsistency and impulsivity, and hence Takahashi [30] proposed a model containing the \(q\)-exponential function, also utilized in Ref. [9].
\[V(D)\ =\ \frac{V(0)}{(1+(1-q)\kappa_{1}D)^{\frac{1}{1-q}}}. \tag{5}\]
The \(q\)-exponential model has been used in many behavioral studies [14; 15; 18; 28; 31; 32].
#### ii.3.2 Effective Exponential Model
Models given by Eqs. (1)-(5) follow a phenomenological approach. However, we are interested in another formulation, called superstatistics, whose application in social systems was proposed recently in Ref. [29]. Interestingly, superstatistics was first introduced to the study of physical systems [6; 38]. But as shown in Ref. [29], superstatistics provides an ab initio derivation of the effective exponential model, that is dual to the Takahashi \(q\)-exponential model given by Eq. (5).
Superstatistics assumes that impulsivity, which has long been used in various models of delay discounting as a parameter, is a positive and fluctuating quantity. This comes as no surprise as within a social group impulsivity does vary. With this observation, the effective exponential model, which contains a power-law function resembling the one discussed by Constantino Tsallis [34], was formulated in Ref. [29]. It was shown that the \(q\)-exponential model of delay discounting proposed by Takahashi in Ref. [30] is dual to the effective exponential model in Ref. [29] by a \(q\leftrightarrow 2-q\) transformation.
Hence, the effective exponential and the Takahashi model are similar, but not exactly the same in appearance. Notwithstanding, Takahashi's model was not derived from superstatistics, and hence the approach followed in Ref. [29] (and to be adapted in the present article) differs.
At this point, it will be worthwhile to briefly recapitulate some key details of the EEM. It was motivated that in a social system, impulsivity as a positive random variable should have a distribution. In the absence of any more knowledge of how the impulsivity distribution may look like, we may choose one of the 'least informative options' - the gamma distribution [8; 25].
-dimensional impulsivityity variable from scratch.
By weighting impulsivity \(\kappa\) in the exponential model (EM) of delay discounting in Eq. (1) with the gamma distribution, the effective exponential factor, that looks like the Tsallis-like \(q\)-exponential power-law function, was obtained. The EEM proposed in Ref. [29] is given by,
\[V(D) = V(0)\exp_{q}(-\kappa_{1}D) \tag{6}\] \[= \frac{V(0)}{\left(1+(q-1)\kappa_{1}D\right)^{\frac{1}{q-1}}}. \qquad\text{(--EEM--)}\]
In the above equation, \(q\) is may be called the 'noextensivity parameter' and \(\kappa_{1}\) is impulsivity. In Ref. [29], they were shown to be related to the relative variance and the mean of impulsivity, respectively. It is also straightforward to verify the \(q\leftrightarrow 2-q\) duality between Eqs. (5) and (6).
_Statement of the problem_: This article aims to extend the Effective Exponential Model by incorporating impulsivity as a two-dimensional construct, characterized by trait and state dimensions. The resulting model is referred to as the Extended Effective Exponential Model or E\({}^{3}\)M. To the best of our knowledge, this is the first attempt to derive a mathematical model of delay discounting that incorporates a two-dimensional impulsivity variable from scratch. Furthermore, the novelty of this paper lies in the utilization of the superstatistics approach. Similar to the EEM, the E\({}^{3}\)M identifies an effective exponential factor that can represent temporal discounting. In the following section, we will delve into the details of the steps to obtain such a factor.
## III A new mathematical model of delay discounting
### Step 1: finding a joint distribution
Just like the EEM, one begins by assuming that both trait and state impulsivity, represented by \(k_{\rm T}\) and \(k_{\rm S}\), are positive random variables. In the present article, they are also considered to be uncorrelated. In the absence of any further knowledge of how they are distributed in a (social) system, one of the least informative options, _i.e._, a joint gamma distribution of \(k_{\rm T}\) and \(k_{\rm S}\) given by [8; 25],
\[f(k_{\rm T},k_{\rm S})=\frac{k_{\rm T}^{\frac{1}{q-1}-1}k_{\rm S}^{\frac{1}{q -1}-1}\exp(-\frac{k_{\rm T}+k_{\rm S}}{\theta})}{\theta^{\frac{1}{q-1}}\Gamma \left(\frac{1}{q-1}\right)\Gamma\left(\frac{1}{q_{\rm S}-1}\right)}, \tag{7}\]
for a single scale parameter \(\theta>0\) and shape parameters \(1<q_{\rm T}<\infty\) and \(1<q_{\rm S}<\infty\), may be considered.
Now, we aim to define another impulsivity variable \(k_{\rm R}\) that is formed by combining \(k_{\rm T}\) and \(k_{\rm S}\). This step is reminiscent of the consideration in the previous paragraph that impulsivity is bi-faceted. Hence, we have to find out a gamma distribution involving \(k_{\rm R}\), and not \(k_{\rm T}\) and \(k_{\rm S}\). Fortunately, such a ritual is clearly described in textbooks of statistics [25] and involves introducing another dummy variable \(k_{\rm A}=k_{\rm T}+k_{\rm S}\). So, the variable transformation we are looking for is \(\{k_{\rm T},k_{\rm S}\}\rightarrow\{k_{\rm R},k_{\rm A}\}\). The distribution in terms of \(k_{\rm R}\) and \(k_{\rm A}\) is related to that of \(k_{\rm T}\) and \(k_{\rm S}\) (Eq. 7) by the following relation.
\[f(k_{\rm R},k_{\rm A})=f[k_{\rm T}(k_{\rm R},k_{\rm A}),k_{\rm S}(k_{\rm R},k_ {\rm A})]\left|\det\,J\right|, \tag{8}\]
where \(\left|\det\,J\right|\) is the absolute value of the determinant of the Jacobian variable transformation matrix defined by,
\[J=\begin{pmatrix}\frac{\partial k_{\rm S}}{\partial k_{\rm R}}&\frac{\partial k _{\rm S}}{\partial k_{\rm A}}\\ \frac{\partial k_{\rm T}}{\partial k_{\rm R}}&\frac{\partial k_{\rm T}}{ \partial k_{\rm A}}\end{pmatrix} \tag{9}\]
It is apparent from Eq. (9) that to find the joint distribution we must know what \(k_{\rm R}\) is. So far, we have just mentioned that \(k_{\rm R}\) is formed out of combining \(k_{\rm T}\) and \(k_{\rm S}\), but did not explore what exactly that combination may be. We address this question in the next section.
Figure 1: Fitting of observed data obtained from Ref. [33] using the EM and EEM. EM: \(\kappa_{0}=0.0034\pm 0.0008\). EEM: \(q=7.157\pm 0.244,\ \kappa_{1}=0.016\pm 0.006\).
### Step 2: combining \(k_{\rm T}\) and \(k_{\rm S}\)
Following Ref. [8], we may consider the two random variables, \(k_{\rm T}\) and \(k_{\rm S}\), to be two impulsed moments induced by fluctuations in a medium. In terms of financial markets, these two random variables may be similar to predicted increase or decrease of stock prices. So, fluctuation in price is given by \(k_{\rm T}-k_{\rm S}\). Now, such a system's response will be dependent on \(k_{\rm T}\) and \(k_{\rm S}\) and can be represented by some kind of mean (\({\cal M}\)) between the two variables. In such a scenario, \(k_{\rm R}\) can be treated as weighted fluctuation given by,
\[k_{\rm R}={\cal M}^{-1}(k_{\rm T},k_{\rm S})(k_{\rm T}-k_{\rm S}). \tag{10}\]
But such an ansatz has a problem. In Eq. (10), \(k_{\rm R}\) can be negative. Now, according to our consideration, impulsivity is a (a) positive and (b) fluctuating variable (\(0<k_{\rm R}<\infty\)). So, although the ansatz in Eq. (10) satisfies condition (b), condition (a) is not satisfied. Author in Ref. [8] also explores the combination \(k_{\rm R}=k_{\rm T}(k_{\rm T}+k_{\rm S})^{-1}\) that may be obtained from Eq. (10). But this ansatz for \(k_{\rm R}\), although yields a positive fluctuating variable, makes it bounded between 0 and 1. So, \(k_{\rm R}=k_{\rm T}(k_{\rm T}+k_{\rm S})^{-1}\) may be a possible candidate modulo the boundedness restriction. Further exploration brings us to the last option suggested in Ref. [8] given by \(k_{\rm R}=k_{\rm T}k_{\rm S}^{-1}\). As one notices, the last ansatz yields a positive, random variable that is not bounded, and it fulfils our criteria. Hence, this is the option we choose for our analysis.
### Step 3: finding a joint distribution in terms of \(k_{\rm R}\)
Now, with \(k_{\rm R}=k_{\rm T}k_{\rm S}^{-1}\) and \(k_{\rm A}=k_{\rm T}+k_{\rm S}\), we find that,
\[k_{\rm T}=k_{\rm A}k_{\rm R}(k_{\rm R}+1)^{-1};\ k_{\rm S} = k_{\rm A}(k_{\rm R}+1)^{-1}; \tag{11}\] \[\frac{\partial k_{\rm S}}{\partial k_{\rm R}}=-k_{\rm A}(k_{\rm R} +1)^{-2};\ \frac{\partial k_{\rm S}}{\partial k_{\rm A}}=(k_{\rm R}+1)^{-1};\ \frac{\partial k_{\rm T}}{ \partial k_{\rm R}} = k_{\rm A}(k_{\rm R}+1)^{-1}-k_{\rm A}k_{\rm R}(k_{\rm R}+1)^{-2};\ \frac{\partial k_{\rm T}}{ \partial k_{\rm A}}=k_{\rm R}(k_{\rm R}+1)^{-1};\] (12) \[\mbox{and hence, }|\det J| = k_{\rm A}(k_{\rm R}+1)^{-2}.\]
Putting Eqs. (11) and (12) in Eq. (8), we obtain the following joint distribution,
\[f(k_{\rm R},k_{\rm A}) = \frac{k_{\rm A}}{(k_{\rm R}+1)^{2}}\left(\frac{k_{\rm A}k_{\rm R} }{k_{\rm R}+1}\right)^{\alpha_{\rm T}-1}\left(\frac{k_{\rm A}}{k_{\rm R}+1} \right)^{\alpha_{\rm S}-1}\frac{\exp\left(-\frac{k_{\rm A}}{\theta}\right)}{ \theta^{\alpha_{\rm T}+\alpha_{\rm S}}\Gamma\left(\alpha_{\rm T}\right)\Gamma \left(\alpha_{\rm S}\right)} \tag{13}\] \[= \frac{k_{\rm R}^{\alpha_{\rm T}-1}k_{\rm A}^{\alpha_{\rm T}+ \alpha_{\rm S}-1}}{(k_{\rm R}+1)^{\alpha_{\rm T}+\alpha_{\rm S}}}\frac{\exp \left(-\frac{k_{\rm A}}{\theta}\right)}{\theta^{\alpha_{\rm T}+\alpha_{\rm S} }\Gamma\left(\alpha_{\rm T}\right)\Gamma\left(\alpha_{\rm S}\right)},\]
where we have defined two positive quantities \(\alpha_{\rm S}=(q_{\rm S}-1)^{-1}\) and \(\alpha_{\rm T}=(q_{\rm T}-1)^{-1}\).
### Step 4: Finding the extended effective exponential factor
Similar to the effective exponential model caculations [29] we can define an 'extended effective exponential factor' (essentially an average) defined by,
\[\epsilon_{\rm eff}^{\rm ext} = \frac{\int_{0}^{\infty}dk_{\rm R}\exp(-k_{\rm R}D)f(k_{\rm R},k_{ \rm A})}{\int_{0}^{\infty}dk_{\rm R}f(k_{\rm R},k_{\rm A})} \tag{14}\] \[= \frac{\int_{0}^{\infty}dk_{\rm R}\exp(-k_{\rm R}D)k_{\rm R}^{\alpha _{\rm T}-1}(k_{\rm R}+1)^{-\alpha_{\rm T}-\alpha_{\rm S}}}{\int_{0}^{\infty}dk _{\rm R}k_{\rm R}^{\alpha_{\rm T}-1}(k_{\rm R}+1)^{-\alpha_{\rm T}-\alpha_{ \rm S}}}\] \[= \frac{\Gamma(\alpha_{\rm S}+\alpha_{\rm T})}{\Gamma(\alpha_{\rm S })}\ {\cal U}(\alpha_{\rm T},1-\alpha_{\rm S},D), \tag{15}\]
and replace the exponential factor of the exponential model with \(\epsilon_{\rm eff}^{\rm ext}\). Hence the E\({}^{3}\)M can be written as,
\[V(D) = V(0)\times\epsilon_{\rm eff}^{\rm ext}=V(0)\ {\cal U}(\alpha_{\rm T},1- \alpha_{\rm S},D)\ \frac{\Gamma(\alpha_{\rm S}+\alpha_{\rm T})}{\Gamma(\alpha_{\rm S})} \tag{16}\] \[= V(0)\ {\cal U}\left(\frac{1}{q_{\rm T}-1},\frac{q_{\rm S}-2}{q_{ \rm S}-1},D\right)\ \frac{\Gamma\left(\frac{1}{q_{\rm S}-1}+\frac{1}{q_{\rm T}-1}\right)}{\Gamma \left(\frac{1}{q_{\rm S}-1}\right)}\qquad\mbox{(--E${}^{3}$M$--)}\]
In the above equation, \(\Gamma\) is the gamma function represented by the integral [3],
\[\Gamma(n)=\int_{0}^{\infty}dx\exp(-x)x^{n-1};\ \forall\ {\cal R}(n)>0, \tag{17}\]
and \({\cal U}\) is the confluent hypergeometric function of the second kind represented by the integral [2],
\[{\cal U}(a,b,z)=\frac{1}{\Gamma(a)}\int_{0}^{\infty}dt\exp(-zt)t^{a-1}(1+t)^{b- a-1};\ \forall\ {\cal R}(a),{\cal R}(z)>0. \tag{18}\]
Hence, the numerator of Eq. (14) yields the confluent hypergeometric function \({\cal U}\). On the other hand, the denominator of Eq. (14) can be calculated in terms of the Beta function represented by [1],
\[{\cal B}(m+1,n+1)=\int_{0}^{\infty}\frac{u^{m}du}{(1+u)^{m+n+2}}=\frac{\Gamma( m+1)\Gamma(n+1)}{\Gamma(m+n+2)}. \tag{19}\]
Eq. (16) is the first main result of our paper. To further explore the result, variation of the factor \(\epsilon_{\rm eff}^{\rm ext}\) with delay is shown in Figs. (2) and (3). We can see that the factor monotonically decreases with delay and hence is suitable to describe the gradual decrease of indifference points in experimental observables. In section IV, we will consider a set of observed data to investigate the applicability of the model. However, before that we explore the mathematical implications of the model further.
### Some comments about the model parameters
In this subsection, we aim to find out the significance of the parameters \(q_{\rm S}\) and \(q_{\rm T}\) that are to be determined by fitting experimental data. For this, we compute the average value of the variable \(k_{\rm R}\) that is given by,
\[\langle k_{\rm R}\rangle=\left\langle\frac{k_{\rm T}}{k_{\rm S}}\right\rangle = \frac{\int_{0}^{\infty}dk_{\rm R}k_{\rm R}f(k_{\rm R},k_{\rm A}) }{\int_{0}^{\infty}dk_{\rm R}f(k_{\rm R},k_{\rm A})} \tag{20}\] \[= \frac{\int_{0}^{\infty}dk_{\rm R}k_{\rm R}^{\alpha_{\rm T}}(k_{ \rm R}+1)^{-\alpha_{\rm T}-\alpha_{\rm S}}}{\int_{0}^{\infty}dk_{\rm R}k_{\rm R }^{\alpha_{\rm T}-1}(k_{\rm R}+1)^{-\alpha_{\rm T}-\alpha_{\rm S}}}\] \[= \frac{\alpha_{\rm T}}{\alpha_{\rm S}-1}=\frac{q_{\rm S}-1}{(q_{ \rm T}-1)(2-q_{\rm S})}\ \ \ \ \ ({\rm using\ Eq.\ 19}).\]
Since \(k_{\rm R}\) is a positive quantity, its average should be positive. This imposes a new constraint \(q_{\rm S}<2\) for Eq. (20) to be satisfied (since \(q_{\rm S},q_{\rm T}>1\)).
Eq. (20) is another important result obtained in this paper. As far as the experimental data are concerned, the ratio of model parameters obtained from the fitting of observed data will give us an idea of a sample's average ratio of trait to state impulsivity. If the ratio \(\alpha_{\rm T}/(\alpha_{\rm S}-1)\) exceeds 1, the average ratio of trait to state impulsivity of the participants is greater than 1. On the other hand, a sub-unity value of the ratio will imply a dominance of the state impulsivity. In our opinion, this ratio should be considered an important quantity in the study of mathematical models of delay discounting. We name this ratio the 'inherence ratio' \(r_{\rm inh}\) that is related to the relative value of inherent (trait) impulsivity with respect to the temporary (state) one. So, the inherence ratio is given by,
\[r_{\rm inh}=\frac{\alpha_{\rm T}}{\alpha_{\rm S}-1}=\frac{q_{\rm S}-1}{(q_{ \rm T}-1)(2-q_{\rm S})}. \tag{21}\]
Since we have \(1<q_{\rm S}<2\) and \(q_{\rm T}>1\), \(r_{\rm inh}\) can be any positive real number. Consequently, our model can not predict if the trait characteristic will overshadow state or vice-versa and the values should depend on the sample being studied. This aspect will be explored in the next section with the help of observed datasets.
## IV Data analysis using the E\({}^{3}\)M
_Methodology_: We utilize our model in Eq. (16) to explain existing observed data for indifference points in delay discounting tasks. We consider eight datasets from three different studies (two longitudinal and one cross-sectional). Datasets are analyzed with the help of C++ codes utilizing the MINUIT package of the ROOT program [7]. For visualization purposes and additional verification of results, the Mathematica software [40] has been used.
_Findings_: Considering the longitudinal data set associated with Refs. [4] (Study 1) and [11] (Study 2), we obtain Figs. (4)-(10). Additionally, we utilize the cross-sectional data set associated with Ref. [33] (Study 3) and obtain Fig. (11). The indifference points obtained from the experiments have been averaged over all the participants. Details of the studies yielding the datasets utilized in this paper are provided in Table 1. We observe that our model closely follows the data points in all these plots. Table 2 summarizes the results obtained from the description of the datasets using the E\({}^{3}\)M.
Figs. (12) and (13) graphically represent the parameter values obtained from the studies and the inherence ratio values. In these figures, Studies 1, 2, and 3 have been depicted by different shaded regions of red, green, and blue respectively. Interestingly, for these datasets, the additional constraint \(q_{\rm S}<2\), obtained from Eq. (20), is followed and \(r_{\rm inh}<1\). According to the E\({}^{3}\)M, this implies that the average relative value of state impulsivity with respect to trait impulsivity exceeds 1 for the analyzed samples, implying a dominance of the state characteristics.
From Fig. (12), it seems that the \(q_{\rm S}\) values (with error bars almost coinciding with the diameter of the data points used to represent the central values) are almost constant, but on closer inspection, we notice that the trend followed by \(q_{\rm S}\) is similar to that of \(q_{\rm T}\). Hence, the inherence ratios calculated from these parameter values also follow the same trend. In both longitudinal studies (1 and 2), a dip in the inherence ratio is observed. For Study 1, the dip is observed for the dizyotic twins at the age 16 (D16) and for Study 2, there is a dip in \(r_{\rm inh}\) value at time point 2 (TP2).
The dip in Study 1 is particularly of interest as the point is distinctly'separated' from the other three considering even the error bars. As the inherence ratio signifies the average ratio of the trait to state impulsivity, the dip suggests a sudden increase in state impulsivity. Interestingly, in both studies, the \(r_{\rm inh}\) value recovers afterward. It will be interesting to investigate data from other longitudinal studies to find whether this trend is found.
\begin{table}
\begin{tabular}{c c c c c c} \hline Study & Reference & Sample size & Age (years) & Gender & Other criteria \\ \hline \hline
1 & [4] & 560 (USA) & 16-18 & Female = 50.7\% & 1. No head trauma 2. No health conditions that restrict physical movement \\ \hline
2 & [11] & 23 (Mexico) & 18-22 & Female = 65.2\% & 1. No substance use problems 2. No psychiatric diagnosis 3. No psychiatric medication \\ \hline
3 & [33] & 33 (USA) & 19-48 & Female = 57.6\% & 1. No smokers 2. No pregnant/breastfeeding participants 3. No medication that affects appetite 4. No allergies to study foods 5. No active effort to lose weight \\ \hline \end{tabular}
\end{table}
Table 1: Descriptive statistics of the studies.
We also observe that the \(k_{1}\) parameter values obtained from the fitting of the same set of data (using Eq. (6)) is very close to (or overlapping with) the \(r_{\rm inh}\) values. Since the \(k_{1}\) parameter can be explained as the average impulsivity of the sample [29] and \(r_{\rm inh}\) is related to the average ratio of trait to state impulsivity, the closeness of these two sets of values increases the reliability of our choice of the variable \(k_{\rm R}\) in terms of \(k_{\rm T}\) and \(k_{\rm S}\).
## V Summary, conclusion and outlook
In this article, we propose an extension of the effective exponential model forwarded in Ref. [29]. The present model, named the extended effective exponential model or E\({}^{3}\)M, considers bi-faceted impulsivity composed of trait and state parts. We assumed that trait and state impulsivity are represented by two positive, random variables. This is a sensible consideration given the fact that impulsivity does fluctuate in a social system from person to person. We find that with this consideration, the subjective value of a reward follows a power-law decay. However, unlike the EEM, this power-law decay is now dictated by a special function called the confluent hypergeometric function of the second kind.
The novelty of this article lies (a) in the superstatistics approach it takes and (b) in the fact that trait and state impulsivity are treated as two distinct variables. To our knowledge, this is the first such attempt to derive a mathematical model of delay discounting involving two distinct facets of impulsivity.
As we proceed with the mathematical analysis, an additional constraint like \(q_{\rm S}<2\) is discovered and a ratio called the inference ratio represented by \(r_{\rm inh}\), which is related to the average ratio of the trait to state impulsivity of a sample has been proposed. Given the constraint \(q_{\rm S}<2\), we argue that the inherence ratio can be any positive number for observed datasets. Hence, no conclusions from the mathematical set-up could be drawn about the relative values of trait and state impulsivity. From the datasets analyzed in this paper, we observe that \(r_{\rm inh}<1\), implying a dominance of the state characteristics of the samples.
However, we consider only three different studies in this work, and more datasets need to be analyzed with the help of the E\({}^{3}\)M. It is also worth mentioning that the present article only studies temporal discounting. Similar studies may be carried out when other types of behaviour in effort discounting [21], probability discounting [13] and social discounting [36] are taken into account.
Another interesting formal development constitutes extending the E\({}^{3}\)M, considering more dimensions of impulsivity. There are some behavioural studies relying on a three-factor [17], or even the four-factor UPPS model [27] of impulsivity. The three-factor model posits cognitive impulsivity, behavioural impulsivity, and impatience/restlessness as the three facets, whereas the four-factor UPPS model considers the UPPS Impulsivity Scale comprising of four sub-scales: Urgency, (lack of) Premeditation, (lack of) Perseverance, and Sensation Seeking. We think that present method adapted in Section III may very well be extended to models accommodating multiple (\(>\)2) variables, but we reserve that work for the future.
To conclude, the approach adapted in the article is a generalized one involving random variables. It is, by no means, limited only to the studies of impulsivity and delay discounting. In fact, any system represented by random variables has benefited (e.g., physical systems like a thermal plasma) and will benefit from these lines of argument. So, we hope that more and more investigations in this direction will throw light on the applicability of the work in
\begin{table}
\begin{tabular}{c c c c c c} \hline Study & Reference & Dataset & \(q_{\rm T}\) & \(q_{\rm S}\) & \(r_{\rm inh}\) \\ \hline \hline
1 & [4] & Monozygotic, 16 yrs (M16) & \(6.463\pm 0.816\) & \(1.113\pm 0.040\) & \(0.023\pm 0.003\) \\ & & Monozygotic, 18 yrs (M18) & \(7.137\pm 0.705\) & \(1.134\pm 0.038\) & \(0.025\pm 0.003\) \\ & & Dizygotic, 16 yrs (D16) & \(5.414\pm 0.800\) & \(1.057\pm 0.021\) & \(0.014\pm 0.002\) \\ & & Dizygotic, 18 yrs (D18) & \(7.011\pm 0.624\) & \(1.120\pm 0.030\) & \(0.023\pm 0.002\) \\ \hline
2 & [11] & Time point 1 (TP1) & \(3.525\pm 1.006\) & \(1.026\pm 0.024\) & \(0.011\pm 0.003\) \\ & & Time point 2 (TP2) & \(3.113\pm 0.698\) & \(1.015\pm 0.009\) & \(0.007\pm 0.002\) \\ & & Time point 3 (TP3) & \(4.044\pm 1.086\) & \(1.039\pm 0.038\) & \(0.013\pm 0.004\) \\ \hline
3 & [33] & - & \(7.036\pm 0.261\) & \(1.091\pm 0.022\) & \(0.017\pm 0.001\) \\ \hline \end{tabular}
\end{table}
Table 2: A summary of the results obtained from description of existing datasets using the E\({}^{3}\)M.
explaining systems represented by multi-faceted random variables.
|
2306.10236 | Beat Pilot Tone (BPT): Simultaneous MR Imaging and RF Motion Sensing at
Arbitrary Frequencies | Purpose: To introduce a simple system exploitation with the potential to turn
MRI scanners into general-purpose RF motion monitoring systems.
Methods: Inspired by Pilot Tone (PT), this work proposes Beat Pilot Tone
(BPT), in which two or more RF tones at arbitrary frequencies are transmitted
continuously during the scan. These tones create motion-modulated standing wave
patterns that are sensed by the receiver coil array, incidentally mixed by
intermodulation in the receiver chain, and digitized simultaneously with the
MRI data. BPT can operate at almost any frequency as long as the
intermodulation products lie within the bandwidth of the receivers. BPT's
mechanism is explained in electromagnetic simulations and validated
experimentally.
Results: Phantom and volunteer experiments over a range of transmit
frequencies suggest that BPT may offer frequency-dependent sensitivity to
motion. Using a semi-flexible body receiver array, BPT appears to sense
cardiac-induced body vibrations at microwave frequencies (1.2 GHz and greater).
At lower frequencies, it exhibits a similar cardiac signal shape to PT, likely
due to blood volume changes.
Other volunteer experiments with respiratory, bulk, and head motion show that
BPT can achieve greater sensitivity to motion than PT and greater separability
between motion types. Basic multiple-input multiple-output (4x22 MIMO)
operation with simultaneous PT and BPT in head motion is demonstrated using two
transmit antennas and a 22-channel head-neck coil.
Conclusion: BPT may offer a rich source of motion information that is
frequency-dependent, simultaneous, and complementary to PT and the MRI exam. | Suma Anand, Michael Lustig | 2023-06-17T02:33:52Z | http://arxiv.org/abs/2306.10236v2 | # Beat Pilot Tone: Versatile, Contact-Free Motion Sensing in MRI with Radio Frequency Intermodulation
###### Abstract
Motion in Magnetic Resonance Imaging (MRI) scans results in image corruption and remains a barrier to clinical imaging. Motion correction algorithms require accurate sensing, but existing sensors are limited in sensitivity, comfort, or general usability. We propose Beat Pilot Tone (BPT), a radio frequency (RF) motion sensing system that is sensitive, comfortable, versatile, and scalable. BPT operates by a novel mechanism: two or more transmitted RF tones form standing wave patterns that are modulated by motion and sensed by the same receiver coil arrays used for MR imaging. By serendipity, the tones are mixed through nonlinear intermodulation in the receiver chain and digitized simultaneously with the MRI data. We demonstrate BPT's mechanism in simulations and experiments. Furthermore, we show in healthy volunteers that BPT can sense head, bulk, respiratory, and cardiac motion, including small vibrations such as displacement ballistocardiograms. BPT can distinguish between different motion types, achieve greater sensitivity than other methods, and operate as a multiple-input multiple-output (MIMO) system. Thus, BPT can enable motion-robust MRI scans at high spatiotemporal resolution in many applications.
radio frequency, magnetic resonance imaging, motion sensing, microwave
## 1 Introduction
Magnetic Resonance Imaging (MRI) is a premier noninvasive imaging modality, prized for its excellent soft tissue contrast and ability to produce high resolution images of the body in deep tissue structures. It remains an essential medical technology for clinicians [1; 2]. MRI data is typically acquired by sequentially sampling the spatial Fourier transform of the object (\(k\)-space) along straight lines or smooth trajectories. The MR signal is sensed by Faraday induction using one or more resonant RF coils that are tuned to the center ("Larmor") frequency (\(21-300\)MHz for \(0.5-7\)T systems, respectively). Modern MRI systems can have up to 128 receivers, with 32 being very common. Because the data acquisition is long (seconds to minutes, depending on the type of scan), physiological motion of the patient can easily corrupt the scan data. Motion is therefore the most common unanticipated event in a clinical MRI examination [3]. Even when anticipated, such as with respiratory and cardiac motion, motion during the examination degrades image quality, resulting in repeated exams and increased costs [4]. Moreover, as spatial resolution continues to
improve, motion is accentuated, causing greater image corruption [5; 6; 7].
The simplest technique to mitigate motion is to suppress it; for example, by restricting the imaged body part or breath-holding. However, this is often not robust enough. Other approaches require measuring the motion during the exam and correcting artifacts either prospectively, retrospectively, or both. For instance, "gating" is a commonly used method that synchronizes the MR acquisition to measured cardiac or respiratory cycles. While it is possible to use motion correction algorithms that rely only on the MR data itself [9; 10; 11], external motion monitoring signals offer a rich source of extra information that may improve the quality and speed of the correction [12; 13; 14; 15; 16; 17; 18; 19].
Ideally, a general-purpose motion sensing system for MRI should have three main characteristics (Fig. 1a): easy to implement within the MRI scanner, sensitive to the motion being measured, and comfortable for the patient or subject [8]. State-of-the-art methods fulfill some but not all of these criteria. Some of the most popular modalities listed in Fig. 1a include optical sensors (e.g., cameras), radio frequency (RF) sensors, and NMR/MRI signals ("navigators") [7; 8; 9; 15; 16; 19]. Cameras [20; 21] are accurate in marker-based and markerless tracking; however, they require a line-of-sight path that is often blocked by the MRI coils [12]. NMR or MRI-based navigators [7; 8; 9; 15; 16; 19] require changes to the MRI acquisition and thus have found widespread adoption in a limited number of applications [22]. Recently, an RF-based sensor known as Pilot Tone (PT) [23; 24] was introduced, which involves transmitting an RF tone near the Larmor frequency that interacts with the body and is detected by the MRI receiver coils. PT is simple yet powerful; it is able to sense many motion types with minimal hardware additions. Consequently, it has been integrated into commercial systems and shown to provide useful respiratory, cardiac, and head motion information in a number of studies [25; 26; 27; 28; 29]. However, PT is limited in sensitivity to motion because it is tied to the Larmor frequency (21/64/127/300MHz for 0.5/1.5T/3/7T systems, respectively). This limitation holds for similar RF-based motion sensing methods which also use existing MRI receiver or transmitter hardware [30; 31; 32; 33]. On the other hand, ultra-high frequency (UHF) RF sensors such as radars [34; 35; 36] may offer greater sensitivity to motion but require significant engineering efforts to be integrated with the MRI scanner. Moreover, they may have fewer receivers than the MRI receiver array and thus fewer observations of the underlying motion.
In this paper, we introduce an RF-based general-purpose motion monitoring system called Beat Pilot Tone (BPT) that overcomes the limitations of other motion sensors by offering easy implementation, sensitivity to many motion types, and non-contact hardware (Fig. 1a-c). By transmitting two microwave RF tones, BPT combines the ease of implementation of PT with the sensitivity of UHF radar: like PT, BPT is acquired alongside the MR image using the receiver array and can be used in most scans; however, unlike PT, BPT creates standing wave patterns in the bore, which are modulated by subtle physiological changes in the body such as motion (Fig. 1d). Motion sensitivity to the surface or the inside of the body can be tuned based on the choice of the BPT transmit frequencies. Higher frequencies can yield greater sensitivity to surface motions.
BPT's standing wave mechanism gives rise to interesting phenomena in simulations and experiments. In electromagnetic (EM) field simulations, we show that BPT samples standing wave patterns, which increase in complexity with frequency, leading to greater sensitivity to motion at higher frequencies. This increased sensitivity allows BPT to sense small vibrations of the receiver coil at microwave frequencies (\(\geq\) 1.8GHz). Moreover, in volunteer experiments, we show that BPT can sense large motions, such as respiratory and bulk motion, and very small motions, such as a displacement ballistocardiogram (dBCG), which measures vibrations of the body due to pulsing of blood. BPT can easily be scaled to operate as a MIMO system, and thereby separate two different head motions (nodding "yes" and shaking "no") using two transmitters placed at different locations. Ultimately, BPT opens up new possibilities both for exploring a whole spectrum of RF motion sensing and developing novel motion correction techniques that build on the rich BPT dataset.
## 2 Results
### Signal Reception
Unlike PT and other methods that use a fixed transmit frequency [23; 24; 25; 26; 27; 30; 31; 32; 33], BPT can choose this parameter freely, which may enable tunable sensitivity to motion. In BPT, we leverage the inherent non-linearity in MRI receiver chain. We transmit a minimum of two RF tones at frequencies spaced such that an intermodulation product will fall close to the Larmor frequency. The tones create EM fields that interact with the
Figure 1: **BPT concept**. a) Motion sensors for MRI have three main desired characteristics [8]. Current motion sensors, color-coded by modality, fulfill some of these characteristics; however, our proposed method, Beat Pilot Tone (BPT), has all three built in. b) Conventional MRI suite sensors (left) are bulky and require contact with the patient, while our method (right) utilizes MRI coil arrays and requires only a transmit (tx) antenna in or near the bore to provide motion estimates (c). Multiple antennas can be used for MIMO operation. d) Simulated magnitudes of the H-field generated by a tx dipole antenna at 127.8MHz (left) and 2.4GHz (right), windowed to the same relative levels. At 2.4GHz, standing wave patterns emerge.
bore and the body and are then sensed by the MRI receiver coil elements. Once passed through the preamplifier chain, these fields are serendipitously mixed via nonlinear intermodulation; the resulting intermodulation product is within the bandwidth (BW) of the receiver chain and is digitized along with the MRI data (Fig. 2b). For example, two tones transmitted at \(f_{1}=2.4\)GHz, \(f_{2}=2.5278\)GHz will have an intermodulation product at frequency
\[f_{BPT}=f_{2}-f_{1}=127.8\ \text{MHz}, \tag{1}\]
which falls within our GE MR750W (GE Healthcare; Waukesha, WI) 3T scanner bandwidth. This example is a second-order intermodulation; however, it is possible to use any order, as long as
\[f_{BPT}=mf_{2}+nf_{1}, \tag{2}\]
where \(m\) and \(n\) are signed integers. Because the MR coils and receiver chain are not originally designed for this purpose, the conversion is inefficient. Fortunately, the MR signal is small (\(<-30\)dBm) [39], so even little transmit power (\(<20\)dBm) is sufficient to induce an intermodulation product at a similar amplitude to the MR signal. Both BPT and PT appear as a peak in the Fourier transform of each _k_-space line that is easy to separate from the MR data (Fig. 2a).
### A Complex Electro-Magnetic Field Distribution
We show in simulation and experiment that the BPT signal is fundamentally different in origin from the PT and other near-field RF methods; thus, BPT can achieve greater sensitivity to motion. We simulated the EM field at multiple frequencies inside an MRI bore, idealized as a cylindrical perfect conductor. The field was transmitted by an antenna outside the bore (Fig. 3a) and sensed by three coils. The coils were moved 10cm back and forth along the bore (longitudinal, or z-direction), as shown in Fig. 3a.
The main differences between BPT and PT are due to the nonlinear reception of BPT and the frequency-dependent characteristics of the MRI bore, which acts as a cylindrical waveguide [40]. Fig. 3c shows the simulated H-field at 127.8MHz. Because this frequency is below the waveguide cutoff frequency, it produces a standing wave that decays uniformly with increasing distance (Fig. 3c, left) [40]. However, at frequencies greater than the cutoff, such as 2.4GHz, the standing wave begins to form patterns that vary in space (Fig. 3c, right) [40]. As a result of these field patterns, the received BPT signals show spatial variation and are altered by small changes to the boundary conditions (Fig. 3d). We hypothesize that this mechanism enables BPT to obtain greater signal modulation for the same motion compared to PT.
In order to validate this simulation, we measured PT and BPT at the same frequencies and displacements as the simulated values. We moved a posterior receiver array back and forth along the z-axis using Rocker, an application that moves the scanner bed using the built-in motor (Fig. 3b). In Fig. 3d and 3e, respectively, we compare the simulation results to measurements from three of the posterior coils. All signals are displayed in units of percent modulation relative to the mean. The first subplot is PT at 127.8MHz; the remaining
Figure 2: **Pipeline for BPT extraction**. a) BPT appears as a peak in the Fourier transform of each _k_-space acquisition. The beat frequency \(f_{BPT}\) falls within the BW of the receiver, but outside the BW of the MR image data. b) Model of the receiver chain: the electromagnetic fields at frequencies \(f_{1}\) (red) and \(f_{2}\) (blue) are sensed by MRI receiver coils, and mixed by the preamplifiers to generate a beat frequency [37; 38]. c) Example of the magnitude of BPT signal from two receiver coils, corresponding to the time-frequency plot in a), and showing amplitude modulation due to cardiac motion in a breath-holding volunteer.
plots are BPT, labeled by the average of the two transmit frequencies.
In the simulated and measured signals, the number of peaks and level of modulation appear to
Figure 3: **BPT finite-element EM field simulations**. a) Simulation setup for an idealized bore. Flux was computed through rectangular coils moving 10cm back and forth along the z-axis. b) The experimental measurement setup matched the simulation. c) The magnitude of the H-field at the central coronal slice of the bore at (left) 127.8MHz and (right) 2463.9MHz. The colored rectangles show the physical coil locations, while the white dashed lines indicate the start and end of the motion. The right plot shows the product of the H-field magnitudes at 2400 and 2527.8MHz. While the magnitude of the field at 127.8MHz decays with increasing distance from the transmitter, the field at 2463.9MHz shows standing wave patterns. d) The simulated magnitude of BPT per coil for each frequency in units of percent modulation relative to the mean. The maximum theoretical number of peaks is indicated on the bottom right. e) The experimental magnitude of the BPT for three coils in the posterior array. The number of peaks and level of modulation appear to match between simulation and experiment.
increase with frequency due to the standing waves and nonlinear sensing, suggesting that higher frequencies may have greater sensitivity to motion. Qualitatively, the signal shapes appear similar between simulation and experiment. Quantitatively, the number of theoretically expected peaks in the signal appears to match the measured value across frequencies (see Section 4.8), as indicated in the bottom right corner of each subplot in Fig. 3d and 3e. The simulated and experimental results differ slightly, and limitations of the experiment are discussed in Section 3. Nevertheless, the measured levels of modulation are comparable to the simulated values. At 127.8MHz, the modulation is within +/- 20%, while for 2463.9MHz it ranges from \(-50\) to 100%. This suggests that higher BPT transmit frequencies can result in greater signal modulation.
The simulation and experiment also suggest that the coil signals vary significantly from one another with increasing frequency, which can lead to greater spatial sensitivity. For 127.8MHz, the three signals are very similar; however, at 2463.9MHz, the locations of the peaks in each signal are not aligned in space. The coil signals reflect the underlying standing wave patterns. Because each BPT coil samples the local magnetic field, BPT can capture greater spatial information compared to PT or other near-field sensing methods. Moreover, the spatial sensitivity is tunable based on the choice of transmit frequencies.
### Motion Sensitivity
We performed a second experiment to demonstrate that BPT is more sensitive to small motions at higher frequencies than at lower frequencies. Using the same antenna and coil setup, we caused the coils to vibrate by placing the coil array on a plastic structure. We moved the scanner bed at the maximum speed (100mm/s) and a small displacement (1cm). We stopped the bed for 5 seconds after each movement to enable vibrations to dissipate and measured the PT and BPT sequentially at the same set of frequencies as the simulation. Fig. 4b shows the results of this experiment, with setup in 4a and physical coil arrangement in Fig. 4c. At the two highest frequencies (1.839GHz and 2.4639GHz), the coil signals show small fluctuations, indicated by arrows in Fig. 4b. We hypothesized that these are vibrations of the coil when the bed suddenly pushes the plastic support underneath it.
We validated the origin of these fluctuations by comparing BPT at the two highest frequencies to displacement calculated from accelerometer measurements. Fig. 4e suggests that the displacement (purple) qualitatively matches BPT. Both signals appear to start and end at the same times and exhibit similar decaying amplitudes. The correspondence is not exact due to limitations of the measurement setup (See Section 3). Nevertheless, the correlation is high. We hypothesize that the BPT is able to sense such small motions because the complex EM field patterns change rapidly over space and are more modulated by motion as frequency increases. In volunteer experiments, we show that this sensitivity enables us to sense a displacement BCG (dBCG) -- a signal that arises due to small vibrations of the body.
Figure 4: **BPT vibration measurement and validation.** a) The scanner bed was moved at maximum speed and stopped for 5 seconds after each displacement. b) The first period of the PT (top left) and BPT (remaining plots) is displayed after low-pass filtering. At frequencies \(>\) 1.2GHz, the BPT senses the vibration of the coils due to the motor (black arrows). c) The physical arrangement of the coils. d) The physical setup of the accelerometer experiment. e) The measured BPT signal at the two highest frequencies vs. displacement (\(\Delta\)d) measured simultaneously with an accelerometer (accelerometer (accel). The signals appear to match qualitatively.
### Volunteer Experiments
To demonstrate the broad applicability of the BPT for motion sensing in various MRI settings, we performed three volunteer experiments with different motion types: respiratory, bulk, cardiac, and head motion. It is important to note that for all of these experiments, the BPT was simultaneous to the MR data acquisition and non-contact. The BPT transmit antenna(s) were placed away from the subject at the top and/or side of the bore, while many of the sensors used for comparison (PT antenna, PPG, ECG, and accelerometer) were placed on the subject's chest or abdomen.
#### Respiratory Motion Experiment
The respiratory motion experiment highlights the sensitivity of the BPT to changes in breathing and bulk motion. A healthy volunteer was asked to perform different breathing types: chest breathing, stomach breathing, rapid-shallow breathing, and breathing with simultaneous bulk motion of the chest (i.e., shaking the torso). The PT and BPT were transmitted simultaneously.
Fig. 5 shows the results of this experiment. Fig. 5 displays the two most modulated coils from BPT and PT on the same y-axis scale (excluding coils with low means, for which the percent modulation is artificially large). It suggests that the BPT magnitude signal during breathing is more than seven times as modulated as the PT. Similarly, the BPT phase signal appears 10 times as modulated as the PT. The numbered coils are physically arranged as in Fig. 5. When overlaid on a patch of the image over time, both the BPT (top) and PT (bottom) breathing signals appear comparable (Fig. 5).
The BPT is also more modulated than the PT in the bulk motion portion of the experiment (Fig. 5). While the BPT magnitude shows much sharper peaks for bulk motion compared to breathing, the PT magnitude is much smoother and appears similar to the breathing signal. This suggests that it may be more difficult to distinguish among different motion types using the PT.
We further demonstrate the BPT's ability to separate between motion types by applying Principal Component Analysis (PCA) to the different receiver signals. We compared the time evolution of the three main PCs of the PT and BPT for a portion of the data with breathing and bulk motion. Fig. 5 suggests that breathing is much better separated from bulk motion in the BPT (right) compared to the PT (left), with the green dashed line indicating a separating plane. Separation in the PC feature space may be a good metric to distinguish between motion types. For instance, after a training period, one could use this information to reject outliers or prospectively acquire data with only respiratory motion.
#### Cardiac Motion Experiment
To demonstrate cardiac sensing with the BPT, we performed a series of breath-held cardiac scans of a healthy volunteer at frequencies ranging from 127.8MHz to 2.4GHz, the same as the simulated frequency range. We acquired the BPT and MR images along with simultaneous physiological monitoring (electrocardiogram [ECG] and photoplethysmogram [PPG]).
Cardiac dynamics are visible in the raw magnitude of the BPT signal. Fig. 6 shows the unfiltered PT and BPT magnitude from the two coils with the greatest energy in the cardiac frequency range. While the cardiac signal is barely visible in the raw coil data for frequencies \(<800\)MHz, they become obvious for BPT \(\geq 1.2\)GHz. This finding suggests that BPT cardiac signal can be extracted from the raw data with minimal processing, which makes BPT attractive for real-time cardiac gating applications.
With more processing, PT and lower frequency BPT signals do reveal cardiac information (Fig. 6). PT/BPT magnitude signals were median-filtered, followed by PCA, ICA, and filtering [25]. The ICA component with the greatest energy in the cardiac frequency range is plotted in Fig. 6 along with the PPG (orange) and ECG (blue). At lower frequencies (\(<1.2\)GHz), BPT changes smoothly and is qualitatively more similar to the PPG. This result is consistent with other PT studies [24]. At higher frequencies (\(\geq 1.2\)GHz), the BPT signal contains much sharper peaks which appear between the R peak of the ECG and the main peak of the PPG in time. We hypothesize that BPT is capturing a displacement ballistocardiogram (dBCG).
Figure 5: **Respiratory motion sensing with BPT and PT**. a) BPT and PT signals during chest, stomach, and rapid breathing, chosen from the two most modulated coils and displayed in percent modulation units. The BPT and PT magnitudes (first and second rows) are on the same y-axis scale, as are the BPT and PT phases (third and fourth rows). The BPT modulation is much greater than PT. b) Coil arrangement, with feet-first orientation and colors corresponding to a), c), and d). c) BPT magnitude and phase modulation (first and third row) and PT magnitude and phase modulation (second and last row) during bulk motion. BPT modulation is larger and sharper than the PT. d) BPT (top) and PT (bottom) magnitude overlaid on a patch of the image (orange box) for coils 9 and 3, respectively. Both appear to match the displacement of the patch qualitatively. e) The time evolution of the three main PCs of the PT and BPT during breathing and bulk motion. The green arrow and line indicate a separating plane between breathing and bulk motion.
#### dBCG Validation
We show that the cardiac signals acquired by BPT highly correlate with dBCG, which measures the recoil of the body due to the ballistic forces of blood [44] (Fig. 7a); therefore, BPT could offer information about the cardiac cycle that is complementary to the MR images. dBCG can be measured in many ways, including pneumatic sensors [44], cameras [45], doppler radar [46], or accelerometers [45]. We compared BPT in a healthy volunteer to a conventional dBCG acquired with a tri-axial accelerometer placed on the chest (Fig. 7b), as well as to PT, ECG, and PPG [41].
We performed a least squares fit to compare dBCG and BPT quantitatively by regressing the multi-coil BPT signals to the dBCG. We denote the resulting signal as BPT-dBCG. Fig. 7c demonstrates the timing of the raw BPT signal from the coil in Fig. 7d to the dBCG, PPG, and ECG signals. The timing and peaks of the BPT signal qualitatively match dBCG. Quantitatively, BPT-dBCG has a strong correlation of 0.81 with dBCG (Fig. 7e). We hypothesize that the BPT senses mechanical vibrations; therefore, BPT-dBCG may be enhanced due to the stiffness of the receiver array. It is possible to identify individual features of the dBCG from a single heartbeat, as indicated on the right half of Fig. 7e [42]. These features contain important information about cardiac function. For example, the time interval between the I and J waves may represent aortic pulse transit
Figure 6: **Cardiac motion sensing with PT and BPT. a) Cardiac PT and BPT signals were obtained from a healthy volunteer during breath-held scans across various frequencies. Unfiltered magnitude signals from two coils (red and cyan) are displayed. Initially, the cardiac signal is not visible in a single coil, but as the transmit frequency increases, it becomes clearly observable. This suggests that the BPT cardiac signal can be extracted from the raw data with minimal processing. b) The PT and BPT are displayed after median-filtering, PCA, ICA, and band-pass filtering. The extracted cardiac signal from PT/BPT is compared to PPG (orange) and ECG (blue), which were acquired simultaneously. The cardiac PT/BPT signal appears smooth at frequencies \(<1.2\)GHz but contains sharp peaks for higher frequencies.**
time, which is a predictor of cardiovascular risk [43].
#### Head Motion Experiment
In the head motion experiment, we demonstrated the ability of BPT to act as a MIMO system and distinguish between two motion types. Two antennas were placed orthogonally to one another at the top and side of the bore (Fig. 8a). BPTs were acquired at 2.4/2.5278GHz and 2.4/2.5276GHz. For comparison, we performed a separate scan with two PTs at 127.6MHz and 127.8MHz. For each condition (multi-BPT and multi-PT), we performed 2-D scans in two planes: axial, in which the volunteer shook their head "no" (Fig. 8b, left), and sagittal, in which the volunteer nodded "yes" (Fig. 8b, right) using a head coil.
Figure 7: **dBCG measurement and validation with BPT.** a) Displacement BCG (dBCG) measures the recoil of the body due to the ballistic forces of blood through the aorta. It was measured with a tri-axial accelerometer, then integrated twice to obtain displacement. b) Placement of the BPT tx antenna, PT tx antenna, and the accelerometer in the bore [41]. c) Raw BPT signal from a single coil vs. computed dBCG, PPG, and ECG. The first wave of BPT and dBCG appear later than ECG, but earlier than PPG. d) Physical location of the BPT coil corresponding to c). e) BPT signals were linearly combined by a least-squares fit to dBCG, then low-pass filtered; this is denoted as BPT-dBCG. BPT-dBCG correlates strongly with dBCG (correlation \(=0.81\)). On the right, a single heartbeat identifies particular features of the dBCG, denoted by H, I, J, and K [42]. These features correlate with cardiac function and could be a predictor of cardiovascular risk [43].
To evaluate BPT and PT's ability to separate these two motions ("no" and "yes"), we computed PCA of the signals from both antennas and plotted the top three PCs versus time (Fig. 8c) and in the PC feature space (Fig. 8d). The two head motions are well separated for BPT, suggesting that it is possible to learn these features easily. As with breathing and bulk motion, after a short training period, the learned motion could be corrected prospectively or retrospectively [47].
## 3 Discussion
In this paper, we present a new method for RF-based motion sensing that is simultaneous to the MRI acquisition. BPT uses standing waves to obtain increased motion sensitivity at RF and microwave frequencies compared to a single RF wave. It is applicable to many different types of motion, including respiratory, bulk, cardiac, and head motion. Moreover, it is independent of the MRI sequence, contact-free, and simple to implement.
Simulations suggest that BPT operates by sampling standing wave patterns in the bore that evolve with motion (Fig. 3). These simulations agree well with experimental results. Differences between simulation and experiment could be due to antenna characteristics and additional structures in the bore, such as the cradle, cables, and additional embedded coils in the bed, which are not modeled in the simulation. RF fields could radiate not only from the antenna but also from the long cable used to transmit the BPT or PT, leading to additional reflections from other surfaces in the room. These reflections could create maxima in the middle of the bore that are not simulated, thus resulting in measured signal shapes with opposite polarity to the simulated signal shapes in Fig. 3. Further exploration with accurate modeling of the human body may be necessary to simulate the behavior of BPT on a human subject.
BPT offers tunable sensitivity to different motion types by choice of transmit frequency and antenna placement. We have shown that BPT is highly sensitive to small motions such as vibration of the coil at frequencies \(>1.2\)GHz (Fig. 4). In volunteers, BPT appears to be more sensitive to surface motions, such as dBCG, at higher frequencies compared to lower frequencies (Fig. 6). BPT correlates strongly with dBCG acquired simultaneously with an accelerometer (Fig. 7). We have demonstrated BPT's ability to capture motion at different spatial and temporal scales by using the same acquisition setup for respiratory (Fig. 5), cardiac (Fig. 6), and head motion (Fig. 8). BPT can easily be scaled to multiple antennas (Fig. 8), allowing the possibility to separate and learn different motion trajectories.
While BPT appears to sense small vibrations, the correspondence shown in Fig. 4 is not perfectly exact. The measured displacement of the phantom appears to be at a lower frequency than BPT. This could be due to the accelerometer placement: the accelerometer had to be placed at the back of the array because of the constrained MRI environment and RF interference (Fig. 4c). Therefore, the accelerometer did not sense the exact position of the BPT coils. In addition, a USB cable
Figure 8: **MIMO head motion sensing with BPT**. a) Two antennas were placed at the top and side of the bore, each transmitting a BPT or PT. b) The volunteer was instructed to shake “no” (left-right) and nod “yes” (updown). c) The three main principal components (PCs) of PT and BPT plotted on the same scale versus time and d) in the PC feature space. The two head motions (shake “no”, nod “yes”) are well separated for BPT, suggesting that it is possible to learn these features easily.
connected to the accelerometer may have had a damping effect.
An important parameter in BPT is the order of intermodulation. In this work, we focused on second order intermodulation for the volunteer experiments (i.e., \(f_{BPT}=f_{2}-f_{1}\)). It may be desirable to choose a higher order to allow for a larger bandwidth of intermodulation; however, this may require greater transmit power to overcome attenuation by the receiver chain.
BPT offers a rich dataset of motion information. Obtaining dBCG measurements with BPT could characterize cardiac function simultaneously to the MRI exam [42; 43]. Using BPT with the MRI system has many other potential applications, including cardiac and respiratory gating [38]; motion sensing for low-field systems, in which body impedance changes are minimal [48; 49]; sensing motion deep in the body such as fetal motion [50]; or using multiple transmitters to reconstruct microwave images [51]. The simplicity of implementation allows for easy, widespread adoption in many MRI systems.
## 4 Methods
### BPT Acquisition Setup
BPT is implemented with consumer-grade hardware using the block diagram in Supp. Fig. 1a and with the hardware components listed in Supp. Tables 1-4. We used two software-defined radios (SDRs) to produce the two tones (Ettus Research B200; National Instruments, TX, USA), synchronized to the system 10MHz clock. The tones were combined, amplified, filtered and transmitted using an antenna placed inside or outside the bore. The filter ensured that no intermodulation from the transmit amplifier was present in the transmitted tones. The experiments had slight differences in the particular components used (e.g., the model of the filter or the combiner); however, the setup was largely the same. The antennas and placements used for each experiment are listed in Supp. Table 1.
For all experiments except the cardiac experiment, we adjusted the SDR transmit power to ensure that all frequencies were at approximately the same received level. It was not possible to do this for the cardiac experiment because the cardiac signal was not visible at lower frequencies without boosting the power above the 2.4GHz BPT signal level.
The respiratory motion experiment used the hardware in Supp. Table 2, which details the part numbers of the transmitters, combiner, amplifier, filter, and antenna. The measured output power of the BPT at 2.4GHz was +13dBm at the antenna, while the PT was at \(-\)36dBm.
The cardiac and dBCG experiments used the same antenna setup for BPT as in the respiratory motion experiment (Supp. Table 2); however, different amplifiers and filters were used in the cardiac experiment depending on the BPT transmit frequencies. The hardware used for each frequency is summarized in Supp. Table 3.
The head motion experiment used two sets of the hardware listed in Supp. Table 2 (i.e., four USRP SDRs and 2 4G LTE antennas) to transmit BPT. PT used the same antennas, but only needed two of the SDRs without the combiner, amplifier, or filter.
For the experiments with Rocker, we used a wideband amplifier for BPT frequencies 363.9MHz and greater (ZX60-83MP-S+; Minicircuits, Brooklyn, NY, USA) in order to obtain greater SNR. We replaced the high-pass filter with a notch filter (ZX75BS-125-S+; Minicircuits) in order to use the same filter for all frequencies.
### Volunteer Experiments
All experiments were performed on a GE 3T 750W MRI system (GE Healthcare; Waukesha, WI). For the phantom, head, respiratory, and cardiac scans, a 2D Balanced SSFP sequence was used with with multiple imaging frames and a frame rate of 1.1s (BW=250kHz, TR=4.4ms, FA=35, FOV=50cm, resolution=2mm). The dBCG validation experiments (Fig. 7) used a 2D SPGR sequence with TR=8.7ms and BW=62.5kHz in order to allow time for data transfer via SPI and to increase SNR of the BPT. The phantom, cardiac, and dBCG validation experiments were acquired with gradients and RF excitation off; the respiratory and head experiments had gradients and RF on.
The head scans were acquired with a 22-channel GEM head-neck coil (GE Healthcare; Waukesha, WI), and the abdominal free-breathing scans were acquired with a GEM 16-channel anterior array (GE Healthcare; Waukesha, WI). The abdominal breath-held scans were acquired using
the same GEM 16-channel anterior array, but with 16 additional posterior coils (GE Healthcare; Waukesha, WI). The Rocker scans used the same 32-channel anterior and posterior array coil as the breath-held scans (GE Healthcare; Waukesha, WI).
All scans were acquired on healthy volunteers. The same volunteer was used for the head scan and the breath-held cardiac scans, in different scan sessions. Two different volunteers were scanned for the respiratory and dBCG experiments. In scans with ECG and PPG, the ECG leads were placed directly on the chest, while PPG was placed on the subject's index finger.
### BPT Magnitude Processing
To process the data, we extracted the complex signal at the BPT frequency (Supp. Fig 1c). The BPT appears as a line in the Fourier domain (Fig. 1c), which does not overlap with the MR image and can easily be separated. The magnitude is straightforward to extract (Supp. Fig. 1c). All processing except for the SNR calculation (Section 4.10) was implemented in Python. The SNR code was implemented in MATLAB (Mathworks; Natick, MA). We will make the code and data freely available.
The low-pass filter cutoffs for each experiment are detailed in Supp. Table 4 and were chosen for best visualization without loss of signal features. The cardiac and head experiment included additional processing. In the cardiac experiment, the magnitude BPT/PT data were median-filtered with a kernel size of 3, followed by PCA, ICA, and low-pass filtering [25] (Fig. 6b). The scikit-learn implementation of the Fast ICA algorithm in Python was used with three components. In the head motion experiment, the data was corrected for vibration-induced artifacts before filtering (described further in Section 4.9).
### BPT Phase Processing
The phase was wrapped due to scanner-specific phase variations such as \(B_{0}\) eddy current compensation. We estimated the unwrapped phase by applying the pseudo-inverse of the relative phase operator \(\mathbf{P}\) to the data. We derive the pseudoinverse for a single timepoint below; the same operations can be applied to all timepoints independently.
The BPT signal \(\mathbf{X}\) at a given time point is an array of size \(c\times 1\), where \(c\) is the number of coils. Let \(x_{1},x_{1},....,x_{c}\) be the complex-valued BPT sample for each coil, respectively, at said time point. We can then write \(\mathbf{X}\) as follows:
\[\mathbf{X} =\begin{bmatrix}x_{1}\\ \vdots\\ x_{c}\end{bmatrix} \tag{3}\] \[=\begin{bmatrix}a_{1}e^{j\theta_{1}}\\ \vdots\\ a_{c}e^{j\theta_{c}}\end{bmatrix} \tag{4}\]
The goal is to recover the underlying unwrapped phase \(\boldsymbol{\theta}\), where
\[\boldsymbol{\theta}=\begin{bmatrix}\theta_{1}\\ \vdots\\ \theta_{c}\end{bmatrix} \tag{5}\]
Due to phase wrapping, \(\boldsymbol{\theta}\neq\angle\mathbf{X}\). We can try to unwrap the data by applying an unwrapping operator \(\mathbf{U}\) to \(\angle\mathbf{X}\). \(\mathbf{U}\) attempts to remove large discontinuities of more than \(\pi\). However, this fails when there are many phase wraps. Instead, we can first compute the phase relative to a chosen reference coil, then unwrap it. We compute the unwrapped relative phase vector \(\mathbf{X_{ph,r}}\) as:
\[\mathbf{X_{ph,r}} =\mathbf{U}\angle\left[x_{r}^{*}\cdot\mathbf{X}\right] \tag{6}\] \[=\mathbf{U}\angle\begin{bmatrix}a_{r}e^{-j\theta_{r}}a_{1}e^{j \theta_{1}}\\ \vdots\\ a_{r}e^{-j\theta_{r}}a_{c}e^{j\theta_{c}}\end{bmatrix}\] (7) \[=\mathbf{U}\angle\begin{bmatrix}a_{r}a_{1}e^{j(\theta_{1}-\theta_ {r})}\\ \vdots\\ a_{r}a_{c}e^{j(\theta_{c}-\theta_{r})}\end{bmatrix}\] (8) \[=\begin{bmatrix}\theta_{1}-\theta_{r}\\ \vdots\\ \theta_{c}-\theta_{r}\end{bmatrix}\] (9) \[=\boldsymbol{\theta}+\begin{bmatrix}0&-1&\ldots&0\\ \vdots&\vdots&\ddots\\ 0&-1&\ldots&0\end{bmatrix}\boldsymbol{\theta}\] (10) \[=(\mathbf{I}+\mathbf{S_{r}})\boldsymbol{\theta}, \tag{11}\]
where \(*\) denotes the complex conjugate and \(\mathbf{S_{r}}\) consists of -1 in the \(r\)th column and 0s everywhere else. We can compute \(\mathbf{X_{ph,r}}\) for all values of \(r\) and stack them into a matrix \(\mathbf{X_{ph}}\):
\[\mathbf{X_{ph}}=\begin{bmatrix}\mathbf{X_{ph,1}}\\ \vdots\\ \mathbf{X_{ph,c}}\end{bmatrix} \tag{12}\]
\[=\begin{bmatrix}\mathbf{I}+\mathbf{S_{1}}\\ \vdots\\ \mathbf{I}+\mathbf{S_{c}}\end{bmatrix}\boldsymbol{\theta} \tag{13}\] \[=\mathbf{P}\boldsymbol{\theta} \tag{14}\]
\(\boldsymbol{\theta}\) can then be obtained by applying the pseudoinverse of \(\mathbf{P}\) to \(\mathbf{X_{ph}}\):
\[\boldsymbol{\theta}=\mathbf{P}^{\dagger}\mathbf{X_{ph}} \tag{15}\]
### HFSS Simulations
We used the finite element solver High Frequency Structure Simulator (HFSS; Ansys, Canonsburg, PA, USA) to perform EM field simulations. The bore was simulated as a perfect electrical conductor (PEC) containing vacuum. The bore was 70cm in diameter and 135cm in length, similar to the size of the GE 3T 750W system used for all experiments. The calculated cutoff frequency for an air-filled bore of this size is 251.1MHz [52].
The simulated antenna was a wideband log periodic antenna (300MHz - 3GHz) from the HFSS component library. The EM field was simulated as a Multi-Frequency simulation with frequencies 127.8, 300, 427.8, 800, 927.8, 1200, 1327.8, 1800, 1927.8, 2400, and 2527.8 MHz.
We simulated the reception of the BPT by computing the product of the complex flux through three rectangular receiver coils for each BPT frequency pair. The coils were modelled as rectangular surfaces that were matched in size (\(10\times 19\)cm) and position to the posterior coils used in experiments (Fig. 3b). The flux \(\Phi\) was computed through each of the simulated coils as \(\Phi=\int\vec{H}\cdot\vec{dA}\). Though the voltage is the time derivative of the flux, because the fields are complex exponentials in time, the voltage is directly proportional to the flux, i.e. \(V(t)=e^{j\omega}\Phi(t)\). The scaling does not affect the magnitude of the signal. Therefore, we omitted it.
For the simulation results shown in Figure 1d, we simulated a vacuum-filled bore as a PEC with a diameter of 70cm and length of 67.5cm in order to reduce the mesh size. A dipole antenna from the HFSS component library was placed at the top of the bore, and the EM field was simulated as a Multi-Frequency simulation with frequencies 127.8, 300, 427.8, 800, 927.8, 1200, 1327.8, 1800, 1927.8, 2400, and 2527.8 MHz. Figure 1d shows the magnitude of the complex H-field vector at 127.8MHz and 2.4GHz, each windowed between the 0th and 90th percentiles.
### Phase Contributions
We chose to compare only the magnitude of the computed flux between simulation and experiment because there are possible contributions to the phase that were not part of the simulation, such as AM-PM modulation [53]. Due to these effects, the level of modulation in the measured phase is much greater than expected in all experiments.
### Accelerometer Experiments
For the phantom and volunteer accelerometer comparisons, a tri-axial SCL3300 accelerometer (Murata Electronics; Kyoto, Japan) was controlled by an Arduino Pro Mini (Arduino; Somerville, MA, USA). The accelerometer was synchronized to the BPT and PT via a Transistor-Transistor Logic (TTL) signal from the scanner. The displacement \(\Delta d\) was computed from raw acceleration by high-pass filtering and integrating the signal twice. In the phantom rocker experiment, the accelerometer was placed away from the coils that were chosen for comparison in order to reduce electronic noise (Fig. 4c-d). Due to the slow motion of the phantom, we chose a high pass filter cutoff of 2Hz for the phantom experiment, whereas for the volunteer experiment, we used a cutoff of 4Hz.
### Standing Wave Calculations
The standing wave patterns can be idealized as two sinusoids in space with amplitudes \(A_{1}\) and \(A_{2}\):
\[A_{1}=a_{1}cos(2\pi f_{1}d+\phi_{1}) \tag{16}\] \[A_{2}=a_{2}cos(2\pi f_{2}d+\phi_{2}) \tag{17}\]
where \(a_{1}\) and \(a_{2}\) are constants, \(f_{1}\) and \(f_{2}\) are the two transmit frequencies, \(d\) is the longitudinal distance vector over the waveguide, and \(\phi_{1}\) and \(\phi_{2}\) are the phases for each sinusoid.
The amplitude of the BPT field over space can be modelled as the product of the two sinusoids:
\[A_{BPT}= A_{1}A_{2} \tag{18}\] \[= \frac{a_{1}a_{2}}{2}\left[cos(2\pi(f_{1}+f_{2})d+\phi_{1}+\phi_{2})\right.\] \[\left.+cos(2\pi(f_{1}-f_{2})d+\phi_{1}-\phi_{2})\right]\]
As Equation 18 suggests, the amplitude of the BPT can be written as the sum of a fast sinusoid with frequency \(f_{1}+f_{2}\) in space and a slow sinusoid with frequency \(f_{1}-f_{2}\). The fast sinusoid is amplitude-modulated by the slow sinusoid; therefore, the number of peaks over a distance \(\Delta d\) is governed only by the fast sinusoid:
\[N_{peaks}\leq\left[\Delta d\left(\frac{1}{D_{1}}+\frac{1}{D_{2}}\right)\right], \tag{19}\]
where \(D_{1}\) and \(D_{2}\) are the wavelengths corresponding to frequencies \(f_{1}\) and \(f_{2}\), i.e., \(D_{1}=300/f_{1}\) for \(f_{1}\) in MHz, and \(\left\lceil\right\rceil\) denotes the ceiling operation.
### Vibration artifacts
While BPT does not affect the signal-to-noise ratio (SNR) of the MR acquisition (Supp. Fig. 2b-c), some aspects of the MR acquisition environment corrupt the BPT signal; namely, we hypothesize that the antenna and receiver coils are caused to vibrate by mechanical and electrical coupling to the scanner. Mechanically, the gradient coils and other structures in the scanner vibrate due to Lorentz forces on the coil windings from the main magnetic field. If the antenna or receiver array is placed on any vibrating surface, it will also vibrate. Electrically, the antenna and coils may vibrate due to induced eddy currents from the switching gradient fields, which experience their own Lorentz forces. The electrical coupling occurs only if the antenna and coils are placed inside the MRI bore.
This artifact can be mitigated by filtering, placing the antenna outside the bore, or dampening its vibration with heavy materials. For the head scans, we approximated the artifact using a linear fit per-coil and subtracted it from the data. For a single coil, the multi-frame BPT data \(\mathbf{b}\) is of size \([N_{f}\times N_{pe}]\), where \(N_{f}\) is the number of frames, \(N_{pe}\) is the number of k-space lines. The artifact \(\mathbf{l}\) can be modeled as:
\[\mathbf{l}=\alpha_{0}\mathbf{c}+\alpha_{1}\mathbf{y_{pe}} \tag{20}\]
where \(\mathbf{y_{pe}}\) is a vector of phase encode indices, which ranges linearly from -1 to 1 over a single frame and repeats over frames, and \(\mathbf{c}\) is a constant-valued vector. The linear fit is approximated by solving a linear system of equations \(\mathbf{A}\boldsymbol{\alpha}=\mathbf{b_{f}}\) for \(\boldsymbol{\alpha}\), where:
\[\mathbf{A}=\left[\begin{array}{c|c}&\\ \mathbf{c}&\mathbf{y_{pe}}\\ \left\lvert\mathbf{l}\right\rvert&\end{array}\right], \tag{21}\]
\[\boldsymbol{\alpha}=\begin{bmatrix}\alpha_{0}\\ \alpha_{1}\end{bmatrix}, \tag{22}\]
and \(\mathbf{b_{f}}\) is the flattened BPT data of size \([(N_{pe}\times N_{f})\times 1]\). The corrected BPT signal is then \(\mathbf{b_{f}}-\mathbf{l}\).
Supp. Fig. 2a shows an example of one of the BPT signals in the head motion experiment with and without artifact correction. Both uncorrected and corrected signals were low-pass filtered with a cutoff of 5Hz. For the respiratory data, low-pass filtering was sufficient to correct for the artifact.
### Snr
To ensure that the nonlinear nature of the BPT reception does not adversely impact image SNR, we performed an image SNR comparison. A uniform phantom was scanned with BPT, PT, and no BPT/PT using a 2D SPGR sequence (TR=34ms, FA=30, resolution=0.9mm). The BPT and PT were cropped out of the image. An SNR map was computed using the method developed by Kellman et al [54]. Supp. Fig. 2b shows the computed SNR maps. The SNR maps (Supp. Fig. 2b) and line profiles (Supp. Fig. 2c) are nearly identical between conditions. This suggests that PT and BPT have no adverse impact on SNR. For the transmit power levels used in the volunteer experiments and SNR maps, there is no visible gain compression.
We would like to thank Katie Lamar-Bruno for her feedback on the writing and help with various experimental setups, including the log-periodic antenna apparatus and the cabling for the accelerometer validation. We
Additionally thank Efrat Shimron for editing the paper and Alan Dong for interesting discussions. We acknowledge Karthik Gopalan, Jason Stockmann, and Jonathan Polimeni for their code to compute image SNR. We acknowledge support from GE Healthcare, the NSF GRFP fellowship, and NIH grants U01EB025162, R01HLL136965, and U01EB029427.
# Beat Pilot Tone Supplementary Information
Suma Anand
1Electrical Engineering and Computer Sciences, University of California, Berkeley, 253 Cory Hall, Berkeley, 94720, CA, USA. 1
Michael Lustig
1Electrical Engineering and Computer Sciences, University of California, Berkeley, 253 Cory Hall, Berkeley, 94720, CA, USA. 1
###### Abstract
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**Supplementary Figure 2** The BPT may be sensitive to certain artifacts. a) If the tx antenna is placed inside the bore, switching gradient fields cause eddy currents to be induced on it, which then vibrate in the scanner due to the Lorentz force from the main magnetic field. For some sequences, this artifact can be approximated from a linear fit and subtracted from the data. An example BPT signal from the head motion experiment is shown here without correction (blue) and with correction using the linear fit approximation (orange). Both signals were low-pass filtered with a cutoff of 5Hz. b) Image SNR was compared in three acquisitions with no PT, BPT, and PT using a uniform phantom with a 32-channel body coil. SNR maps across the three acquisitions are shown. c) Line plots through the center of the maps, indicated by the dashed line.
**Supplementary Table 1**: Type and placement of BPT and PT antennas
\begin{tabular}{l c c c c} \hline Experiment & BPT Type & PT Type & BPT Placement & PT Placement \\ \hline Rocker & Log periodic & " & Outside & " \\ Respiratory & 4G LTE & Loop & Top of bore & Top of bore \\ Cardiac & 4G LTE & Loop & Top of bore & Top of bore \\ dBCG & Bluetooth & Loop & Top of bore & On chest \\ Head & 2 \(\times\) 4G LTE & " & Top and side of bore & " \\ \hline \end{tabular}
This table shows the type and placement of antennas for BPT and PT. Ditto marks (") indicate that the antenna or placement was the same, e.g. the same log periodic antenna was used to transmit both BPT and PT.
**Supplementary Table 2**: 2.4GHz BPT Hardware
\begin{tabular}{l c c c} \hline Hardware element & Model & Company \\ \hline Transmitter & USRP B200 & Ettus Research \\ Power combiner & ZN2PD-63+ & Minicircuits \\ Amplifier & Sunhans 2.4GHz 34dBm WiFi Signal Booster & Sunhans \\ High Pass Filter & VHF-440+ & Minicircuits \\ \hline \end{tabular}
The hardware setup for transmitting the BPT at 2.4GHz.
**Supplementary Table 3**: Cardiac BPT hardware
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz ; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz ; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz ; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0.5 & Linear fit \\ \hline \end{tabular}
Filtering and artifact correction details for each experiment, with corresponding Figure number in the second column.
\begin{tabular}{l c c c} \hline Experiment & Figure & Filter cutoff (Hz) & Artifact correction \\ \hline Rocker & 3 & 5 & None \\ Vibration & 4 & 1 for \(<863.9\)MHz ; 5 for \(\geq 863.9\)MHz & None \\ Respiratory & 5 & 2 & None \\ Cardiac & 6 & 3 for \(<863.9\)MHz ; 15 for \(\geq 863.9\)MHz & None \\ dBCG & 7 & 15 & None \\ Head & 8 & 0. |
2304.00884 | Dialog-to-Actions: Building Task-Oriented Dialogue System via
Action-Level Generation | End-to-end generation-based approaches have been investigated and applied in
task-oriented dialogue systems. However, in industrial scenarios, existing
methods face the bottlenecks of controllability (e.g., domain-inconsistent
responses, repetition problem, etc) and efficiency (e.g., long computation
time, etc). In this paper, we propose a task-oriented dialogue system via
action-level generation. Specifically, we first construct dialogue actions from
large-scale dialogues and represent each natural language (NL) response as a
sequence of dialogue actions. Further, we train a Sequence-to-Sequence model
which takes the dialogue history as input and outputs sequence of dialogue
actions. The generated dialogue actions are transformed into verbal responses.
Experimental results show that our light-weighted method achieves competitive
performance, and has the advantage of controllability and efficiency. | Yuncheng Hua, Xiangyu Xi, Zheng Jiang, Guanwei Zhang, Chaobo Sun, Guanglu Wan, Wei Ye | 2023-04-03T11:09:20Z | http://arxiv.org/abs/2304.00884v1 | # Dialog-to-Actions: Building Task-Oriented Dialogue System via Action-Level Generation
###### Abstract.
End-to-end generation-based approaches have been investigated and applied in task-oriented dialogue systems. However, in industrial scenarios, existing methods face the bottlenecks of reliability (e.g., domain-inconsistent responses, repetition problem, etc) and efficiency (e.g., long computation time, etc). In this paper, we propose a task-oriented dialogue system via action-level generation. Specifically, we first construct dialogue actions from large-scale dialogues and represent each natural language (NL) response as a sequence of dialogue actions. Further, we train a Sequence-to-Sequence model which takes the dialogue history as the input and outputs a sequence of dialogue actions. The generated dialogue actions are transformed into verbal responses. Experimental results show that our light-weighted method achieves competitive performance, and has the advantage of reliability and efficiency.
Task-Oriented Dialogue System, Action-Level Generation, Dialog-to-Actions 2021
Yuncheng Hua and Xiangyu Xi contributed equally to this research.
2021Task-Oriented Dialogue System, Action-Level Generation, Dialog-to-Actions 2021
Yuncheng Hua, Xiangyu Xi, Zheng Jiang, Guanwei Zhang, Chaobo Sun, Guanglu Wan, and Wei Ye. 2023Dialog-to-Actions: Building Task-Oriented Dialogue System via Action-Level Generation. In _Proceedings of ACM Conference'17_. ACM, New York, NY, USA, 5 pages. [https://doi.org/10.1145/mnmnmn.mnmnn](https://doi.org/10.1145/mnmnmn.mnmnn)
## 1. Introduction
Recently, the end-to-end generation-based methods that directly output appropriate NL responses or API calls have been deeply investigated in task-oriented chatbots [(2; 5; 9; 16; 22)], and have been proven valuable for real-world business, especially after-sale customer services [(1; 1; 7; 8; 13; 14; 19; 21; 24; 25)]. Based on the large-scale pre-trained language models [(10; 11)], generation-based methods have the advantage of simpler architecture and anthropomorphic interaction. Despite the significant progress, we find these token-level generation methods suffer from the following two limitations in practical scenarios.
**1. The token-level generation methods have limited reliability, which is essential for industrial task-oriented dialogue systems.** Due to the pre-trained language models' characteristics, the models may generate responses that are learned from the pre-training corpus. In certain cases, such responses are meaningless and not semantically incoherent with the current business domain, interrupting online interaction. Worse still, the models occasionally get stuck in generating repetitive responses across multiple turns (e.g., repeatedly enquiring the users for the same information). Above issues are also widely observed by other researchers [(4; 12)] and practitioners.1
Footnote 1: [https://github.com/microsoft/DialoGPT/issues/45](https://github.com/microsoft/DialoGPT/issues/45)
**2. The token-level generation methods may fail to meet the efficiency requirement of the industrial systems, especially with large decoding steps.** The long computation time of the token-level generation models leads to unacceptable response latency of online dialogue systems, especially when the model generates a sentence of length that exceeds a threshold (e.g., 1,544 ms of T5 for a sentence of 30 words, as Figure 3 shows). Owing to the latency problem, a large number of service requests may be suspended or blocked during the peak period. Also, the computation resources (e.g., GPUs) required by the aforementioned systems might be unaffordable for small companies.
To address the above two problems, in this paper, we propose a task-oriented dialogue system based on the action-level generation method. Inspired by Xi et al. [(19)], we represent responses with **Dialogue Actions**, i.e., a class of the responses with unique and identical semantic meaning that can be automatically obtained by clustering. While Xi et al. [(19)] directly treats a whole response as a specific dialogue action, we split one response into multiple
segments (Bordner et al., 2017) and each segment can be mapped to a dialogue action. In this way, each response is represented as a sequence of dialogue actions. Given the dialogue context, a Seq2Seq model with an action-level recurrent decoder is used to generate the sequence of dialogue actions. Further, a frequency-based sampling method is used to compose the final response, based on the generated sequence of dialogue actions. Since the core component of our approach is the generation model which takes the _dialogue context_ as the inputs and outputs _actions_, our method is named as DialogTo-Actions (abbr. **DTA**). Compared with existing token-level generation-based systems, our DTA has the advantage of 1) reliability, since the generated natural language responses derive from the predefined dialogue actions; 2) efficiency, since the decoding space (i.e., dialogue actions) and the decoding steps are much smaller.
## 2. Framework Description
### Overview
We follow the workflow employed in the previous end-to-end task-oriented dialogue systems (Han et al., 2017; Chen et al., 2018), where the system takes the dialogue history as input, and generates a text string that either serves as a verbal staff response to the user or API calls (e.g., information inquiry, action execution, etc). When an API is invoked, the information returned from the API will be incorporated into the system's next response. A dialogue sample following such system interaction life cycle can be found in Figure 1 (b).
The key idea of our work is to generate dialogue actions and then compose a verbal response. To do so, we first construct dialogue actions from large-scale dialogues (Step 1) and represent each response as a sequence of dialogue actions (Step 2), as Figure 1 (a) shows. A Seq2Seq model with an action-level recurrent decoder is utilized to generate dialogue actions (Step 3), and the generated actions are further used to compose the verbal response (Step 4). We exemplify using the after-sale customer service of electric bike rental business, where users and staffs communicate online through text messages. The technical details are introduced as follows.
### Step 1: Dialogue Action Construction
A dialogue action refers to a cluster of utterances or utterance fragments that share identical semantic meaning and represent a common communicative intention, for instance making a request or querying information. Xi et al. (2018) views a group of utterances with identical semantic information as dialogue action and selects a response corresponding to a specific staff action. However, the oversimplified setting, i.e., abstracting a whole utterance into an action, leads to relatively limited expressiveness and scalability. To make the responses more targeted and flexible, we construct dialogue actions based on utterance segments (of staff) rather than utterances. Specifically, each utterance is divided into multiple segments by a rule-based approach (Bordner et al., 2017). Further, following Xi et al. (2018), we exploit a two-stage method to cluster the segments. Specifically, ConSERT (Xie et al., 2018) is utilized to generate representations for each utterance segment, and K-means is then applied to cluster the segments. We choose the number of clusters \(K\) empirically to balance the purity and the number of the clusters, and treat each cluster of segments as a dialogue action (e.g., \(A_{1}\) and \(A_{2}\) in Figure 1 (a)).
### Step 2: Response Standardization
Response standardization aims to standardize the responses (from the large-scale dialogues) by mapping each response to a sequence of dialogue actions. Following Yu et al. (2019), we exploit a retrieval-based method, which retrieves clustered segments that are most similar to the given input utterance segment and label the input based on the corresponding clusters. As Figure 2 shows, given an input segment \(x\), we use BM25 to recall top \(k\) segments \(\{u_{1},...,u_{k}\}\) from all clustered segments. Further, we exploit a BERT-based text similarity computation model \(S\) to rerank the \(k\) segments and select
Figure 1. The system architecture and dialogue sample. In (b), the dialogue action and corresponding utterance segment are marked by the same color (e.g., “A226” and “1 am really sorry”).
the segment \(\hat{u}\) with the highest similarity to \(x\), denoting as:
\[\hat{u}=\operatorname*{argmax}_{u_{i}\in\{u_{1},...,u_{R}\}}S(x,u_{i}) \tag{1}\]
where \(S(x,u_{i})\) refers to the similarity between \(x\) and \(u_{i}\). \(x\) is then annotated with the dialogue action \(A_{i}\) that \(\hat{u}\) belongs to.
Furthermore, we record the correspondence between the dialogue actions and utterance segments, as well as the frequencies of utterance segments in the dialogues. Specifically, we employ a key-value dictionary \(\mathfrak{D}\), in which a key refers to a dialogue action \(A_{i}\) while its value is a nested dictionary where the mapping relationship between the unique segments \(\{x_{1},...,x_{n}\}\) and \(A_{i}\) and the segments' occurrence frequencies \(\{f_{i},...,f_{n}\}\) are recorded. The dictionary \(\mathfrak{D}\) is used for composing the verbal response (Step 4).
### Step 3: Action Sequence Prediction
Given a dialogue \(\mathcal{D}=\{U_{1},S_{1},...,U_{T},S_{T}\}\) as a set of utterances exchanged between user (\(U_{i}\)) and staff (\(S_{i}\)) alternatively, through above carefully-designed steps, each staff utterance \(S_{i}\) is represented as an action sequence \(\mathcal{A}_{S_{i}}\). At the \(m\)-th turn, given a dialogue history \(\mathcal{H}_{m}=\{U_{m-w},S_{m-w},...,S_{m-1},U_{m}\}\), we propose a Seq2Seq model to produce staff responses \(S_{m}\). We first resort to the Seq2Seq model to output an action sequence of \(\mathcal{A}_{S_{m}}=(A_{1}^{S_{m}},A_{2}^{S_{m}},...,A_{k}^{S_{m}})\), where \(k\) denotes the length of the action sequence, and then use the action sequence to form verbal response (in SS 2.5).
**Encoder** Given the dialogue history \(\mathcal{H}_{m}\), we sequentially concatenate all the utterances in \(\mathcal{H}_{m}\) and each staff utterance's corresponding action sequence \(\mathcal{A}_{S_{i}}\), forming a token sequence \((w_{1},...,w_{n})\). The Bi-LSTM model is used to encode the token sequence into a sequence of continuous representations \(\mathbf{H}\):
\[\mathbf{H}=(\mathbf{h}_{1},...,\mathbf{h}_{n})=\text{BiLSTM}(w_{1},...,w_{n}) \tag{2}\]
**Decoder** Considering the efficiency requirement and small decoding space, we use the Luong attention method and employ LSTM as the decoder to calculate hidden state \(\mathbf{s_{t}}\) at time-step \(t\) as follows:
\[\mathbf{s_{t}}=\text{LSTM}(\mathbf{s_{t-1}},\mathbf{g}(\mathbf{y_{t-1}}),\mathbf{c_{t-1}}) \tag{3}\]
where \(\mathbf{y_{t-1}}\) denotes the probability distribution over dialogue action space at step \(t\)-1 and \(\mathbf{g}(\mathbf{y_{t-1}})\) denotes the action has the highest probability. After obtaining hidden state \(\mathbf{s_{t}}\) and context vector \(\mathbf{c_{t}}\), we generate probability distribution at time-step \(t\) as follows:
\[\mathbf{y_{t}}=\text{Softmax}(\mathbf{W_{d}}[\mathbf{s_{t}};\mathbf{c_{t}}]) \tag{4}\]
where \(\mathbf{W_{d}}\) is weight parameter. Given the ground-truth label \(y_{t}\) at time-step \(t\), we use \(p(y_{t}|y_{<t},\mathcal{H}_{m})\) to denote the cross-entropy loss at step \(t\) where \(y_{<t}\) denotes the previously-generated actions.
The optimization objective is defined as:
\[L_{\textit{Gen}}=-\sum_{\mathcal{D}\in\mathcal{C}}\sum_{\mathcal{H}_{m}\in \mathcal{D}}\sum_{t=1}^{l_{m}}p(y_{t}|y_{<t},\mathcal{H}_{m}) \tag{5}\]
where \(\mathcal{C}\) denotes the set of dialogues and \(l_{m}\) denotes the length of the dialogue action sequence at the \(m\)-th turn of dialogue \(\mathcal{D}\).
### Step 4: Response Generation
Based on the action sequence generated in SS 2.4, we compose the verbal response by selecting an utterance segment for each action and combining the segments sequentially. Considering the segments with higher frequencies are more likely to be the formal utterance that staff commonly use, we sample the segments from \(\mathfrak{D}\) (built in SS 2.3) following the principle that the higher the frequency, the more likely the segment to be selected. By doing this, we ensure the quality as well as the diversity of the verbal responses.
## 3. Experiments
### Experimental Settings
#### 3.1.1. Dataset
We perform an experiment with a Chinese online after-sale customer service of electric bike rental business. In this scenario, the users may finish riding but forgot to lock the bike, and thus request the staff to remotely lock the bike and reduce the fees. The staff is required to judge whether the fee can be reduced by checking the status of the order via the back-end APIs. We collect the user-staff dialogues from the logs of online services for a week. The data statistics are shown in Table 1. The dialogues are randomly split into train, dev, and test sets with a ratio of 8:1:1. We construct 1,420 dialogue actions (SS 2.2), and each API call is treated as a dialogue action. The dataset will be released online.
#### 3.1.2. Baselines
To evaluate the effectiveness and efficiency of our method, we compare it with the following state-of-the-art baselines: (1) **LSTM** which exploits a classical LSTM-based sequence-to-sequence architecture (He et al., 2017); (2) **Transformer** which uses Transformer (He et al., 2017) as encoder and decoder; (3) **CDiaGPT** which is a GPT model pre-trained on a large-scale Chinese conversation dataset (He et al., 2018); (4) **T5** which is a Text-to-Text Transfer Transformer (T5) model (He et al., 2017) pretrained with the CLUE Corpus.
#### 3.1.3. Evaluation Metrics
To comprehensively evaluate the effectiveness of different models, we perform both offline evaluation (i.e., on the dataset) and online evaluation (i.e., online A/B testing).
**Offline Evaluation** The models take a specific ground-truth conversation history (i.e., context) as input and generate a response. Following Byrne et al. (2018), we report the BLEU-4 score of each model in the test set. Considering the API calls are highly important, we observe the generated API calls in each turn and report the macro Precision (P), Recall (R), and F1-Score (F1) of API calls.
\begin{table}
\begin{tabular}{l c} \hline \hline
**STAT TYPE** & **VALUE** \\ \hline Dialogs & 8,363 \\ Total turns & 55,576 \\ Avg. turns per dialog & 6.65 \\ Dialogue Actions & 1,420 \\ \hline \hline \end{tabular}
\end{table}
Table 1. Data statistics.
Figure 2. The workflow of response standardization.
**Online Evaluation** Following Xi et al. (2018), we deploy the models online and perform A/B testing. For each model, 120 dialogues are randomly sampled. The annotators who possess domain knowledge are required to perform the satisfaction assessment by grading each dialogue as "Low", "Medium", or "High" satisfaction degree. 2
Footnote 2: The grading criteria can be summarized as follows: (i) Score “Low” denotes that Chabot can not handle user requirements correctly. (ii) Score “Middle” denotes that Chabot can handle user requirements correctly, but may generate a disfluent or incomplete response. (iii) Score “High” denotes that Chabot can handle user requirements correctly and complete the conversation perfectly.
### Main Results
The offline evaluation and online evaluation are shown in Table 3 and 2 respectively, from which we have the following observations: (1) Large-scale pre-trained language models significantly improve the performance of token-level generation models. For example, compared with the plain LSTM model, T5 achieves an absolute improvement of 26.20% for the BLEU-4 score and 4.98% for F1. (2) Compared with the CDiaGPT and the T5 models, our light-weighted DTA achieves competitive performance in the offline evaluation, and earns the highest satisfaction rating in the online evaluation, verifying the effectiveness of our proposed method.
### In-Depth Analysis
#### 3.3.1. Effect of Efficiency Issue
To investigate the effect of efficiency issue, we collect each model's computation time for processing the test samples under the same infrastructure (i.e., Tesla v100, 32GB RAM size, etc). Considering the computation time is highly correlated with the decoding steps, we first divide the generated responses into 10 subsets based on the response length, and then calculate the average computation time of each subset, as Figure 3 shows. We can observe that: (1) Despite the comparable performance, DTA has a significant advantage in computation efficiency over other models (e.g., 3.37 ms of DTA v.s. 1265.56 ms of CDiaGPT v.s. 2470.69 ms of T5 for subset "[(50, 59)]"). (2) DTA outperforms other models more significantly with longer responses. The reason is that the decoding steps of the token-level generation models are identical to the response length, while DTA performs action-level decoding. The above observation verifies that our system provides an effective solution to build online dialogue services with limited computation resources.
#### 3.3.2. Effect of Reliability Issue
We investigate the effect of reliability issue by quantitatively inspecting the repetition problem. Specifically, we calculate the Jaccard index of responses of each turn and the previous turns (Beng et al., 2017). The average Jaccard index of each model is shown in Table 4, from which we can observe that: (1) The Jaccard index of human response is the smallest, indicating that existing models have room for improvement in terms of the repetition problem. (2) Our method has a much smaller Jaccard index than CDiaGPT and T5. The action sequence generation, together with the sampling strategy, can effectively alleviate the repetition problem.
A concrete online dialogue is shown in Figure 1 (b), where the CDiaGPT generates exactly the same responses. Though DTA generates similar action sequences (e.g., combinations of Actions A226, A372, A249 and A109), there are much fewer cases where the action sequence exactly repeats previous turns. The small differences in action sequences can lead to large changes in verbal responses. Besides, the sampling mechanism ensures that the same action in different turns corresponds to different segments (e.g., A226 in three turns), which further enables DTA with better diversity.
## 4. Conclusion
In this paper, we propose a task-oriented dialogue system via action-level generation. An effective framework is proposed to build the generation model from the large-scale dialogues with minimum manual effort. The experimental analyses demonstrate our system's capability of tackling the reliability and the efficiency problems encountered with the existing end-to-end generation methods. In the future, we are interested in exploring an integrated system that unifies the discrete modules in DTA in an end-to-end architecture.
\begin{table}
\begin{tabular}{l l} \hline \hline Model & Jaccard Index \\ \hline Human Response & 0.129 \\ \hline CDiaGPT & 0.214 \\ TS & 0.207 \\ DTA & 0.142 \\ \hline \hline \end{tabular}
\end{table}
Table 4. Jaccard Index of different models.
\begin{table}
\begin{tabular}{l c c c} \hline \hline Model & Low & Medium & High \\ \hline LSTM & 30.00 & 29.17 & 40.83 \\ Transformer & 27.50 & 28.33 & 44.17 \\ CDiaGPT & 12.50 & 13.33 & 74.17 \\ T5 & 5.83 & 15.83 & 78.33 \\ DTA & 8.33 & 12.50 & 79.17 \\ \hline \hline \end{tabular}
\end{table}
Table 2. Statistical Results of Online Evaluation (%)
Figure 3. The computation time of different models.
\begin{table}
\begin{tabular}{l c c c} \hline \hline Model & BLEU-4 & P & R & F1 \\ \hline LSTM & 20.62 & 66.16 & 78.84 & 71.72 \\ Transformer & 26.21 & 62.59 & 75.90 & 64.74 \\ CDiaGPT & 42.54 & **71.18** & 86.43 & 77.11 \\ T5 & **46.83** & 70.91 & 87.25 & 76.70 \\ DTA & 44.82 & 68.03 & **90.80** & **77.74** \\ \hline \hline \end{tabular}
\end{table}
Table 3. Statistical Results of Offline Evaluation (%).
## 5. Presenter Biography
Presenter: Yuncheng Hua. He is an algorithm engineer at Meituan, focusing on researching and building dialogue systems.
## 6. Company Portrait
Meituan is China's leading shopping platform for locally found consumer products and retail services including entertainment, dining, delivery, travel and other services.
|
2305.09551 | Interactive and Incremental Learning of Spatial Object Relations from
Human Demonstrations | Humans use semantic concepts such as spatial relations between objects to
describe scenes and communicate tasks such as "Put the tea to the right of the
cup" or "Move the plate between the fork and the spoon." Just as children,
assistive robots must be able to learn the sub-symbolic meaning of such
concepts from human demonstrations and instructions. We address the problem of
incrementally learning geometric models of spatial relations from few
demonstrations collected online during interaction with a human. Such models
enable a robot to manipulate objects in order to fulfill desired spatial
relations specified by verbal instructions. At the start, we assume the robot
has no geometric model of spatial relations. Given a task as above, the robot
requests the user to demonstrate the task once in order to create a model from
a single demonstration, leveraging cylindrical probability distribution as
generative representation of spatial relations. We show how this model can be
updated incrementally with each new demonstration without access to past
examples in a sample-efficient way using incremental maximum likelihood
estimation, and demonstrate the approach on a real humanoid robot. | Rainer Kartmann, Tamim Asfour | 2023-05-16T15:51:14Z | http://arxiv.org/abs/2305.09551v1 | # Interactive and Incremental Learning of Spatial Object Relations from Human Demonstrations
###### Abstract
Humans use semantic concepts such as spatial relations between objects to describe scenes and communicate tasks such as "Put the tea to the right of the cup" or "Move the plate between the fork and the spoon." Just as children, assistive robots must be able to learn the sub-symbolic meaning of such concepts from human demonstrations and instructions. We address the problem of incrementally learning geometric models of spatial relations from few demonstrations collected online during interaction with a human. Such models enable a robot to manipulate objects in order to fulfill desired spatial relations specified by verbal instructions. At the start, we assume the robot has no geometric model of spatial relations. Given a task as above, the robot requests the user to demonstrate the task once in order to create a model from a single demonstration, leveraging cylindrical probability distribution as generative representation of spatial relations. We show how this model can be updated incrementally with each new demonstration without access to past examples in a sample-efficient way using incremental maximum likelihood estimation, and demonstrate the approach on a real humanoid robot.
Cognitive Robotics, Learning Spatial Object Relations, Semantic Scene Manipulation, Incremental Learning, Interactive Learning
## 1 Introduction
While growing up, humans show impressive capabilities to continually learn intuitive models of the physical world as well as concepts which are essential to communicate and interact with others. While an understanding of the physical world can be created through exploration, concepts such as the meaning of words and gestures are learned by observing and imitating others. If necessary, humans give each other explicit explanations and demonstrations to purposefully help the learner improve their understanding of a specific concept. These can be requested by the learner after acknowledging their incomplete understanding, or by the teacher when observing a behavior that does not match their internal
model (Grusec, 1994). Assistive robots that naturally interact with humans and support them in their daily lives should be equipped with such continual and interactive learning abilities, allowing them to improve their current models and learn new concepts from their users interactively and incrementally.
One important class of concepts children need to learn are the meanings of spatial prepositions such as _right of, above_ or _close to_. Such prepositions define geometrical relationships between spatial entities (O'Keefe, 2003), such as objects, living beings or conceptual areas, which are referred to as _spatial relations_(Stopp et al., 1994; Aksoy et al., 2011; Rosman and Ramamoorthy, 2011). Spatial relations play an important role in communicating manipulation tasks in natural language, e. g., in "Set the table by placing a plate _on_ the table, the fork to _the left_ of the plate, and the knife to the _right of_ the plate." By abstracting from precise metric coordinates and the involved entities' shapes, spatial relations allow the expression of tasks on a semantic, symbolic level. However, a robot performing such a task must be able to derive subsymbolic placing positions that are needed to parameterize actions. Such _signal-to-symbol gap_ remains a grand challenge in cognitive robotics (Kruger et al., 2011). Just like a child, a robot should be able to learn such mapping of spatial object relations from demonstrations provided by humans.
In this work, we consider a robot that has no prior knowledge about the geometric meaning of any spatial relations yet. When given the task to manipulate a scene to fulfill a desired spatial relation between two or more objects, such as a cup _in front of_ a bottle, the robot should request a demonstration from the user if it has no model of the spatial relation or if its current model is insufficient (Fig. 1). Similarly, the robot should be able to receive corrections from the human after executing the task. Finally, having received a new demonstration, the robot should be able to derive a model of the spatial relation from the very first sample and subsequently update its model incrementally with each new demonstration, i. e., without the need to retrain the model with all previously observed demonstrations (Losing et al., 2018).
These goals pose hard requirements for the underlying representation of spatial relations and the cognitive system as a whole. The robot needs to inform the user in case it cannot perform the task by asking for help while maintaining an internal state of the interaction. In addition, the robot will only receive very sparse demonstrations - every single demonstration should be used to update the robot's model of the spatial relation at hand. As a consequence, we require a very sample-efficient representation that can be constructed from few demonstrations and incrementally updated with new ones.
Obtaining a sample-efficient representation can be achieved by introducing bias about the hypothesis space (Mitchell, 1982). Bias reduces the model's capacity, and thereby its potential variance, but more
Figure 1: Incremental learning of spatial relations from human demonstrations.
importantly, also reduces the amount of data required to train the model. This effect is also known as the bias-variance tradeoff. Compared to a partially model-driven approach with a stronger bias, a purely data-driven black-box model can offer superfluous capacity, slowing down training. Bias can be introduced by choosing a model whose structure matches that of the problem at hand. For the problem of placing objects according to desired spatial relations, we proposed to represent spatial relations as parametric probability distributions defined in cylindrical coordinates in our previous work (Kartmann et al., 2020, 2021), observing that capturing relative positions in terms of horizontal distance, horizontal direction, and vertical distance closely matches the notions of common spatial prepositions.
An interesting consideration arises when learning a model from a single demonstration. A single example does not hold any variance; consequently, the learner's only option is to reproduce this demonstration as closely as possible. When receiving more examples, the learner can add variance to its model, thus increasing its ability to adapt to more difficult scenarios. This principle is leveraged by version space algorithms (Mitchell, 1982), which have been used in robotics to incrementally learn task precedence (Pardowitz et al., 2005). While our approach is not a version space algorithm per se, it behaves similarly with respect to incremental learning: The model starts with no variance and acquires more variance with new demonstrations, which increases its ability to handle more difficult scenarios such as cluttered scenes.
We summarize our contributions as follows:
1. We present an approach for a robot interacting with a human that allows the robot to (a) request demonstrations from a human for how to manipulate the scene according to desired spatial relations specified by a language instruction if the robot has not sufficient knowledge about how to perform such a task,, as well as (b) to use corrections after each execution to continually improve its internal model about spatial relations.
2. We show how our representation of spatial relations based on cylindrical distributions proposed in (Kartmann et al., 2021) can be incrementally learned from few demonstrations based only on the current model and a new demonstration using incremental maximum likelihood estimation.
We evaluate our approach in simulation and real world experiments on the humanoid robot ARMAR-6 (Asfour et al., 2019)1. Footnote 1: [https://youtu.be/x6KKUzd_SeE](https://youtu.be/x6KKUzd_SeE)
## 2 Related Work
In this section, we discuss related work in the areas of using spatial relations in the context of human-robot interaction and the incremental learning of spatial relation models.
### Spatial Relations in Human-Robot Interaction and Learning from Demonstrations
Spatial relations have been used to enrich language-based human-robot interaction. Many works focus on using spatial relations to resolve referring expressions identifying objects (Bao et al., 2016; Tan et al., 2014) as well as locations (Fasola and Mataric, 2013; Tellex et al., 2011) for manipulation and navigation. In these works, the robot passively tries to parse the given command or description without querying the user for more information if necessary. As a consequence, if the sentence is not understood correctly, the robot is unable to perform the task. Other works use spatial relations in dialog to resolve such ambiguities. Hatori et al. (2018) query for additional expressions if the resolution did not score a single object higher
by a margin than all other objects. Shridhar and Hsu (2018); Shridhar et al. (2020) and Dogan et al. (2022) formulate clarification questions describing candidate objects using, among others, spatial relations between them. These works use spatial relations in dialog to identify objects which the robot should interact with. However, our goal is to perform a manipulation task defined by desired spatial relations between objects.
A special form of human-robot interaction arises in the context of robot learning from human demonstrations (Ravichandar et al., 2020). There, the goal is to teach the robot a new skill or task instead of performing a command. Spatial relations have been used in such settings to specify parameters of taught actions. Similar to the works above, given the language command and current context, Forbes et al. (2015) resolve referring expressions to identify objects using language generation to find the most suitable parameters for a set of primitive actions. Prepositions from natural language commands are incorporated as action parameters in a task representation based on part-of-speech tagging by Nicolescu et al. (2019). These works focus on learning the structure of a task including multiple actions and their parameters. However, action parameters are limited to a finite set of values, and spatial relations are implemented as fixed position offsets. In contrast, our goal is learning continuous, geometric models of the spatial relations themselves.
### Learning Spatial Relation Models
Many works have introduced models to classify existing spatial relations between objects to improve scene understanding (Rosman and Ramamoorthy, 2011; Sjoo and Jensfelt, 2011; Fichtl et al., 2014; Yan et al., 2020) and human activity recognition (Lee et al., 2020; Zampogiannis et al., 2015; Dreher et al., 2020). These models are either hand-crafted or are not learned incrementally. In contrast, our models are learned incrementally from demonstrations collected during interaction. Few works consider models for classifying spatial relations which could be incrementally updated with new data. Mees et al. (2020) train a neural network model to predict a pixel-wise probability map of placement positions given a camera image of the scene and an object to be placed according to a spatial relation. However, their models are not trained incrementally, and training neural networks incrementally is, in general, not trivial (Losing et al., 2018). In an earlier work, Mees et al. (2017) propose a metric learning approach to model spatial relations between two objects represented by point clouds. The authors learn a distance metric measuring how different the realized spatial relations in two scenes are. Recognizing the spatial relation in a given scene is then reduced to a search of known examples that are similar to the given scene according to the learned metric. Once the metric is learned and kept fixed, this approach inherently allows adding new samples to the knowledge base, which potentially changes the classification of new, similar scenes. However, their method requires storing all encountered samples to keep a notion of known spatial relations, while our models can be updated incrementally with a limited budget of stored examples (Losing et al., 2018). Mota and Sridharan (2018) follow a related idea to learn classification models of spatial relations incrementally. They encode spatial relations as 1D and 2D histograms over the relative distances or directions (encoded as azimuth and elevation angles), of points in two point clouds representing two objects. These histograms can be incrementally updated by merely adding a new observation to the current frequency bins.
However, all of these models are _discriminative_, i. e., they determine the existing relations between two objects in the current scene. In contrast, our goal is to _generate_ a new target scene given desired spatial relations. While discriminative models can still be applied by exhaustively sampling the solution space (e. g., possible locations of manipulated objects), classifying the relations in these candidates and choosing one that contains the desired relation, we believe that it is more effective to directly learn and
apply generative geometric models of spatial relations. In our previous works, we introduced generative representations of 3D spatial relations in the form of parametric probability distributions over placing positions (Kartmann et al., 2021). These probabilistic models can be sampled to obtain suitable placing positions for an object to fulfill a desired spatial relation to one or multiple reference objects. We have shown how these models can be learned from human demonstrations which were collected offline. In this work, we show how demonstrations can be given interactively and how the models can be updated in a fully incremental manner, i. e., relying solely on the current model and a new demonstration.
## 3 Problem Formulation and Concept
In the following, we formulate the problem of interactive and incremental learning of spatial relation models from human demonstrations and introduce the general concept of the work. In Section 3.1, we summarize the actual task of semantic scene manipulation which the robot has to solve. In Section 3.2, we describe the semantic memory system as part of the entire cognitive control architecture used on the robot. In Section 3.3, we formulate the problem of incremental learning of spatial relation models. In Section 3.4, we describe the envisioned human-robot interaction task and explain how we approached each subproblem in Section 4.
### Semantic Scene Manipulation
We consider the following problem: Given a scene with a set of objects and a language command specifying spatial relations between these objects, the robot must transfer the initial scene to a new scene fulfilling the specified relations by executing an action of a set of actions. We denote points in time as \(t_{0},\ldots,t_{k},\ldots\ (t_{k}\in\mathbb{R})\) that can be viewed as events where the robot is given a command or makes an observation. The scene model is part of the robot's working memory and contains the current configuration
\[\mathbf{P}^{(t_{k})}=\left\{\mathbf{P}_{1}^{(t_{k})},\ldots,\mathbf{P}_{n}^{(t_ {k})}\right\} \tag{1}\]
of \(n^{(t_{k})}\) objects at time \(t_{k}\), where \(\mathbf{P}_{i}^{(t_{k})}\in\mathrm{SE}(3)\) is the pose of object \(i\) with position \(\mathbf{p}_{i}^{(t_{k})}\in\mathbb{R}^{3}\) and orientation \(\mathbf{Q}_{i}^{(t_{k})}\in\mathrm{SO}(3)\). \(\mathrm{SE}(3)\) and \(\mathrm{SO}(3)\) denote the special Euclidean group and special orthogonal group, respectively. The desired relation
\[R^{*}=(s^{*},u^{*},\mathcal{V}^{*})\leftarrow\mathrm{ground}(C) \tag{2}\]
is obtained by parsing and grounding the natural language command \(C\), i. e., extracting the phrases referring to objects and relations and mapping them to the respective entities in the robot's working and long-term memories2. This desired relation consists of a symbol \(s^{*}\in\mathcal{S}\) describing the identity of the relation, a _target_ object \(u^{*}\in\left\{1,\ldots,n^{(t_{k})}\right\}\) and a set of _reference_ objects \(\mathcal{V}^{*}\subseteq\left\{1,\ldots,n^{(t_{k})}\right\}\setminus\left\{u^ {*}\right\}\). \(\mathcal{S}\) is the set of known spatial relation symbols. The robot's task is to place the target object \(u^{*}\) at an appropriate pose \(\mathbf{P}_{u^{*}}^{(t_{k+1})}\) which fulfills the desired spatial relation \(R^{*}\). We aim at keeping the object's original orientation, therefore this task is reduced to finding a suitable position \(\mathbf{p}_{u^{*}}^{(t_{k+1})}\).
Footnote 2: Please refer to Kartmann et al. (2021) for more details on the natural language understanding part.
Our approach to finding suitable placing positions is based on a generative model \(G\) of spatial relations. This model is able to generate suitable target object positions fulfilling a relation \(R=(s,u,\mathcal{V})\) based on
the current scene \(\mathbf{P}^{(t_{k})}\) and semantic object information \(\mathcal{O}\) (object names and geometry, see (5) below), that is formally
\[\mathbf{p}_{u}^{(t_{k+1})}\sim G_{s}\Big{(}u,\mathcal{V},\mathcal{O},\mathbf{P} ^{(t_{k})}\Big{)}\,. \tag{3}\]
The generative models \(G_{s}\) of spatial relations \(s\) can be learned from human demonstrations. In our previous work, we recorded human demonstrations for each spatial relation using real objects and learned the generative models \(G_{s}\) offline. Each demonstration consisted of the initial scene \(\mathbf{P}^{(t_{k})}\), a desired relation \(R^{*}=(s^{*},u^{*},\mathcal{V}^{*})\) verbalized as a language command for the human demonstrator, and the resulting scene \(\mathbf{P}^{(t_{k+1})}\) created by the demonstrator by manipulating the initial scene. Therefore, each demonstration has the form
\[D=\left(\mathbf{P}^{(t_{k})},R,\mathbf{P}^{(t_{k+1})}\right),\quad R=(s,u, \mathcal{V}) \tag{4}\]
and can be used to learn the generative model \(G_{s}\) of the relation \(s\). In contrast to the previous work, in this work we consider the problem of interactively collecting samples by querying demonstrations from the user and incrementally updating the generative models of the spatial relations in the robot's memory with each newly collected sample.
### Robot Semantic Memory
The robot's semantic memory consists of two parts: the _prior knowledge_, i. e., information defined a-priori by a developer, and the _long-term memory_, i. e., experience gathered by the robot itself (Asfour et al., 2017). In our scenario, the prior knowledge contains semantic information about \(N\) known objects,
\[\mathcal{O}=\{O_{i}\}_{i=1}^{N}=\{(g_{i},\eta_{i})\}_{i=1}^{N}\,, \tag{5}\]
including object names \(\eta_{i}\) and 3D models \(g_{i}\), as well as names of spatial relations, so that language phrases referring to both objects and relations can be grounded in entities stored in the robot's memory by natural language understanding as indicated in (2). We assume no prior knowledge about the geometric meaning of the relations. Instead, this geometric meaning is learned by the robot during interaction in a continual manner, and is thus part of the long-term memory. In other words, the long-term memory contains generative models \(G_{s}^{(t_{k})}\) of spatial relations \(s\in\mathcal{S}^{(t_{k})}\) representing the robot's current understanding of spatial relations at time \(t_{k}\),
\[\mathcal{G}^{(t_{k})}=\left\{G_{s}^{(t_{k})}\Big{|}\,s\in\mathcal{S}^{(t_{k}) }\right\}, \tag{6}\]
where \(\mathcal{S}^{(t_{k})}\subseteq\mathcal{S}\) is the set of spatial relations for which the robot has learned a generative model at \(t_{k}\). These models are based on samples collected from human demonstrations \(\mathcal{D}\), which are contained in the long-term memory as well:
\[\begin{split}\mathcal{D}^{(t_{k})}&=\left\{ \mathcal{D}_{s}^{(t_{k})}\,\Big{|}\,s\in\mathcal{S}^{(t_{k})}\right\},\\ \mathcal{D}_{s}^{(t_{k})}&=\{D_{sj}\}_{j=1}^{m_{s}^ {(t_{k})}}=\left\{\mathbf{P}_{sj}^{\binom{t_{k_{sj}}}{g_{j}}},R_{sj},\mathbf{P }_{sj}^{\binom{t_{k_{sj}+1}}{g_{j}}}\right\}_{j=1}^{m_{s}^{(t_{k})}},\end{split} \tag{7}\]
where \(\mathcal{D}_{s}^{(t_{k})}\) is the set of \(m_{s}^{(t_{k})}\) collected samples of spatial relation \(s\) at time \(t_{k}\), and \(t_{k_{sj}},t_{k_{sj}+1}\) refer to the time a sample was collected. At the beginning, the robot's long-term memory is empty, therefore
\[\mathcal{G}^{(t_{0})}=\mathcal{D}^{(t_{0})}=\mathcal{S}^{(t_{0})}=\emptyset. \tag{8}\]
### Learning Spatial Relations: Batch vs Incremental
During interactions, the robot will collect new samples from human demonstrations. When the robot receives a new demonstration for relation \(s\) at time \(t_{k}\)\((k>0)\), it first stores the new sample \(D_{s}\) in its long-term memory:
\[\mathcal{D}_{s}^{(t_{k})}\leftarrow\{D_{s}\}\cup\begin{cases}\mathcal{D}_{s}^ {(t_{k-1})},&s\in\mathcal{S}^{(t_{k-1})}\\ \emptyset,&\text{else}\end{cases} \tag{9}\]
Afterwards, the robot may query its long-term memory for the relevant samples \(\mathcal{D}_{s}^{(t_{k})}\) collected to date and use them to update its model \(G_{s}\) of \(s\):
\[G_{s}^{(t_{k})}\leftarrow\mathrm{update}\Big{(}G_{s}^{(t_{k-1})},\mathcal{D}_ {s}^{(t_{k})}\Big{)} \tag{10}\]
Note that such an incremental model update with each new sample requires a very sample-efficient representation. We refer to (10) as the _batch update problem_. The model is expected to adapt meaningfully to each new sample, but all previous samples are needed for the model update. In the machine learning community, incremental learning can be defined as proposed by Losing et al. (2018):
"We define an incremental learning algorithm as one that generates on a given stream of training data \(s_{1},s_{2},\ldots,s_{t}\) a sequence of models \(h_{1},h_{2},\ldots,h_{t}\). In our case [...] \(h_{i}:\mathbb{R}^{n}\rightarrow\{1,\ldots,C\}\) is a model function solely depending on \(h_{i-1}\) and the recent \(p\) examples \(s_{i},\ldots,s_{i-p}\), with \(p\) being strictly limited."
Comparing this definition to (10), it becomes clear that it is not an instance of incremental learning in this sense, as the number of samples \(|\mathcal{D}_{s}|\) in the memory is not bounded by a constant. This raises the question of whether representations of spatial relations in the form of cylindrical distributions as proposed in our previous work can be learned in a truly incremental way using a limited budget of stored samples. In this work, we further investigate the question of incremental learning of spatial relations without even storing _any_ samples, i. e., solely based on the current model \(G_{s}^{(t_{k-1})}\) and the latest sample \(D_{s}\), forming the _incremental update problem_:
\[G_{s}^{(t_{k})}\leftarrow\mathrm{update}\Big{(}G_{s}^{(t_{k-1})},D_{s}\Big{)} \tag{11}\]
### Interactive Learning of Spatial Relations
We now describe a scenario of a robot interacting with a human where the human gives a semantic manipulation command to the robot while the robot can gather new samples from human demonstrations. The scheme is illustrated in Fig. 2. The procedure can be described as follows:
1. The human gives a command to the robot at \(t_{k}\), specifying a spatial relation \(R^{*}=(s^{*},u^{*},\mathcal{V}^{*})\) as in (2).
2. The robot observes the current scene \(\mathbf{P}^{(t_{k})}\) and plans the execution of the given task using its current model \(G_{s^{*}}^{(t_{k})}\). Planning may be successful or fail due to different reasons. Assuming the task given by the user is solvable, the failure can be attributed to an insufficient model.
3. Depending on the outcome of 2.: 1. [label=0.,leftmargin=*]
4. _Planning is successful:_ The robot found a suitable placing position and executes the task by manipulating the scene. If the execution was successful and the user is satisfied, the interaction is finished. 2. _Planning fails:_ The robot's current model is insufficient, thus, it queries the human for a demonstration of the task.
5. The human was queried for a demonstration (3b.) or wants to correct the robot's execution (3a.). In both cases, the human performs the task by manipulating the scene.
6. The human signals that the demonstration is complete by giving a speech cue (e. g., "Put it here") to the robot.
7. When receiving the speech cue, the robot observes the changed scene \(\mathbf{P}^{(t_{k+1})}\), creates a new sample \(D=\left(\mathbf{P}^{(t_{k})},R^{*},\mathbf{P}^{(t_{k+1})}\right)\), stores the sample in the long-term memory and updates its model to obtain the new model \(G_{s^{*}}^{(t_{k+1})}\) as described in Section 3.3.
## 4 Methods and Implementation
To solve the underlying task of semantic scene manipulation, we rely on our previous work which is briefly described in Section 4.1. We outline the implementation of the robot's semantic memory in Section 4.2. We describe how each new sample is used to update a spatial relation's model in Section 4.3. Finally, we explain how we implement the defined interaction scenario in Section 4.4.
Figure 2: Scheme for a robot interacting with a human to learn geometric models of spatial relations.
### 3D Spatial Relations as Cylindrical Distributions
In (Kartmann et al., 2021), we proposed a model of spatial relations based on a cylindrical distribution \(\mathcal{C}\)
\[(r,\phi,h)\sim\mathcal{C}(\theta)\,,\quad\theta=\big{(}\theta_{rh},\theta_{\phi} \big{)} \tag{12}\]
over the cylindrical coordinates radius \(r\in\mathbb{R}_{\geq 0}\), azimuth \(\phi\in[-\pi,\pi]\) and height \(h\in\mathbb{R}\). The radius and height follow a joint Gaussian distribution \(\mathcal{N}\), while the azimuth, as an angle, follows a von Mises distribution \(\mathcal{M}\) (which behaves similar to a Gaussian distribution but is defined on the unit circle),
\[(r,h)\sim\mathcal{N}(\theta_{rh})\,,\quad\theta_{rh}=(\boldsymbol{\mu}_{rh}, \boldsymbol{\Sigma}_{rh})\,,\qquad\phi\sim\mathcal{M}\big{(}\theta_{\phi} \big{)}\,,\quad\theta_{\phi}=\big{(}\mu_{\phi},\kappa_{\phi}\big{)} \tag{13}\]
so the joint probability density function of \(\mathcal{C}\) is given by
\[p_{\theta}(r,\phi,h)=p_{\theta_{rh}}(r,h)\cdot p_{\theta_{\phi}}(\phi) \tag{14}\]
with \(p_{\theta_{\phi}}(\phi)\) and \(p_{\theta_{rh}}(r,h)\) being the respective probability density functions of the distributions in (13). We claim that cylindrical coordinates are a suitable space for representing spatial relations as they inherently encode horizontal distance (_close to, far from_), direction (_left of, behind_) and vertical distance (_above_, _below_), which are the qualities many spatial relations used by humans are defined by. Therefore, we leverage a cylindrical distribution as a distribution over suitable placing positions of a target object relative to one or more reference objects. The corresponding cylindrical coordinate system is centered at the bottom-projected centroid of the axis-aligned bounding box enclosing all reference objects and scales with the size of that bounding box. By defining the cylindrical coordinate system based on the reference objects' joint bounding box, we can apply the same spatial relation models to single and multiple reference objects. Especially, this allows considering spatial relations that inherently involve multiple reference objects (e. g., _between_, _among_).
### Initialization of Robot's Semantic Memory
We build our implementation of the cognitive architecture of our robot in ArmarX (Vahrenkamp et al., 2015; Asfour et al., 2017). The architecture consists of three-layer for 1) low-level sensorimotor control, 2) high-level for semantic reasoning and task planning and 3) mid-level as a memory system and mediator between the symbolic high-level and subsymbolic low-level. The memory system contains segments for different modalities, such as object instances or text recognized from speech. A segment contains any number of entities that can receive new observations and thus evolve over time. Here, we define three new segments:
* The _sample segment_ implements the set of human demonstrations \(\mathcal{D}\) in (7). It contains one entity for each \(s\in\mathcal{S}^{(t_{k})}\), each one representing "all collected samples of relation \(s\)." New samples are added as new observations to the respective entity. Thus, \(\mathcal{D}_{s}^{(t_{k})}\) in (7) is obtained by querying the sample segment for all observations of entity \(s\).
* The _spatial relation segment_ contains the robot's knowledge about spatial relations. There is one entity per \(s\in\mathcal{S}\), which holds the information from prior knowledge such as the relations' names for language grounding and verbalization. In addition, each entity contains the current geometric model \(G_{s}^{(t_{k})}=\theta\) with \(\theta\) as in (12).
* The _relational command_ segment contains the semantic manipulation tasks \(R^{*}=(s^{*},u^{*},\mathcal{V}^{*})\) extracted from language commands (2). The latest observation is the current or previous task.
In the beginning, the sample segment and the relational command segment are initialized to be empty, i. e., there are no collected samples yet and no command has been given yet. The spatial relation segment is partially initialized from prior knowledge. However, in accordance with the sample segment, the entities contain no geometric model \(G_{s}\) yet, i. e. \(G_{s}^{(t_{0})}=\emptyset\).
### Incremental Learning of Spatial Relations
Now, we describe how the batch update problem can be solved using cylindrical distributions. Then, we show how the same mathematical operations can be performed incrementally, i. e., without accessing past samples.
Batch UpdatesDue to the simplicity of our representation of spatial relations, updating the geometric model of a spatial relation, i. e., its cylindrical distribution, is relatively straightforward. To implement the batch update (10), we query all samples of the relation of interest collected so far, perform Maximum Likelihood Estimation (MLE) to obtain the cylindrical distribution's parameters,
\[G_{s}^{(t_{k})}=\theta_{s}^{(t_{k})}\leftarrow\mathrm{MLE}\!\left(\mathcal{D} _{s}^{(t_{k})}\right), \tag{15}\]
and update the spatial relation segment.
A cylindrical distribution is a combination of a bivariate Gaussian distribution over \((r,h)\) and a von Mises distribution over \(\phi\), see (13). Hence, to perform the MLE of a cylindrical distribution in (15), two independent MLEs are performed for the Gaussian distribution \(\mathcal{N}(\theta_{rh})\) and the von Mises distribution \(\mathcal{M}\!\left(\theta_{\phi}\right)\), respectively,
\[\theta_{rh}^{*}\leftarrow\mathrm{MLE}_{\mathcal{N}}\!\left(\{r_{j},h_{j}\}_{ j=1}^{m_{s}^{(t_{k})}}\right),\qquad\theta_{\phi}^{*}\leftarrow\mathrm{MLE}_{ \mathcal{M}}\!\left(\{\phi_{j}\}_{j=1}^{m_{s}^{(t_{k})}}\right), \tag{16}\]
where \((r_{j},\phi_{j},h_{j})\) are the cylindrical of sample \(D_{j}\in\mathcal{D}_{s}^{(t_{k})}\) (see (Kartmann et al., 2021) for more details).
Updating the model with each new sample requires a representation that can be generated from few examples, including just one, which is the case for cylindrical distributions. However, special attention has to be paid to the case of encountering the first sample (\(|\mathcal{D}_{s}|=1\)). As a single sample holds no variance, an estimated distribution would collapse in a single point. Note that the model's expected behavior is well-defined: It should reproduce the single sample when generating placing positions, which often is a valid candidate. Because cylindrical distributions generalize to objects of different sizes, the model can be directly applied to new scenes. The caveat is that the model is not able to provide alternatives if this placing candidate is, e. g., in collision with other objects. If more samples are added (\(|\mathcal{D}_{s}|\geq 2\)), the model is generated using standard MLE. The resulting distribution will have a variance according to the samples, which allows the robot to generate different candidates for placing positions.
Technically, in order to allow deriving a generative model from a single demonstration while avoiding special cases in the mathematical formulation, we perform a small data augmentation step: After transforming the sample to its corresponding cylindrical coordinates \(\mathbf{c}=(r,\phi,h)^{\top}\), we create \(n\) copies \(\mathbf{c}_{1}^{\prime},\ldots,\mathbf{c}_{n}^{\prime}\) of the sample to which we add small Gaussian noise
\[\mathbf{c}_{i}^{\prime}\leftarrow\mathbf{c}+\left(\varepsilon_{r},\varepsilon _{\phi},\varepsilon_{h}\right)^{\top},\quad\varepsilon_{r},\varepsilon_{\phi },\varepsilon_{h}\sim\mathcal{N}(0,\sigma_{\varepsilon}) \tag{17}\]
for \(i\in\{1,\ldots,n\}\), and perform MLE on \(\{\mathbf{c},\mathbf{c}^{\prime}_{1},\ldots,\mathbf{c}^{\prime}_{n}\}\). This keeps the distribution concentrated on the original sample while allowing small variance parameters to keep the mathematical formulation simple and computations numerically stable. In our experiments, we used \(n=2\) and \(\sigma_{\varepsilon}=10^{-3}\). Note that the variance created by this augmentation step is usually negligible compared to the variance created by additional demonstrations.
Incremental UpdatesNow, we present how the model update (15) can be performed in an incremental manner, i. e., with access only to the current model and the new sample. We follow the method by Welford (1962) for the incremental calculation of the parameters of a univariate Gaussian distribution. By applying this method to the multivariate Gaussian and using a similar method for von Mises distributions to implement the MLEs (16), we can incrementally estimate cylindrical distributions.
Welford (1962) proved that the mean \(\mu\) and standard deviation \(\sigma\) of a one-dimensional series \((x_{i}),\)\(x_{i}\in\mathbb{R},\) can be incrementally calculated as follows. Let \(n\) be the current time step, \(\mu_{1}=x_{1}\) and \(\sigma=\frac{1}{n}\ \tilde{\sigma}\),
\[\mu_{n} =\left(\frac{n-1}{n}\right)\cdot\mu_{n-1}+\frac{1}{n}\cdot x_{n} (n>1)\, \tag{18}\] \[\tilde{\sigma}_{n} =\tilde{\sigma}_{n-1}+\left(\frac{n-1}{n}\right)\cdot\left(x_{n} -\mu_{n-1}\right)^{2} (n\geq 1). \tag{19}\]
This method can be extended to a multivariate Gaussian \(\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})\) over a vector-valued series \((\mathbf{x}_{i})\) with \(\mathbf{x}\in\mathbb{R}^{d}\). With analogous conditions as above,
\[\boldsymbol{\mu}_{n} =\left(\frac{n-1}{n}\right)\cdot\boldsymbol{\mu}_{n-1}+\frac{1}{ n}\cdot\mathbf{x}_{n} (n>1)\, \tag{20}\] \[\tilde{\boldsymbol{\Sigma}}_{n} =\tilde{\boldsymbol{\Sigma}}_{n-1}+\left(\frac{n-1}{n}\right) \cdot\left(\mathbf{x}_{n}-\boldsymbol{\mu}_{n-1}\right)\left(\mathbf{x}_{n} -\boldsymbol{\mu}_{n-1}\right)^{\top} (n\geq 1). \tag{21}\]
A von Mises distribution \(\mathcal{M}(\mu,\kappa)\) over angles \(\phi_{i}\) is estimated as follows (Kasarapu and Allison, 2015). Let \(\mathbf{x}_{i}\) be the directional vectors in the 2D plane corresponding to the directions \(\phi_{i}\), and \(\bar{r}\) the normalized length of their sum,
\[\mathbf{x}_{i}\coloneqq\begin{pmatrix}\cos(\phi_{i})\\ \sin(\phi_{i})\end{pmatrix}\in\mathbb{R}^{2}\quad(i=1,\ldots,n),\qquad\bar{r} \coloneqq\frac{\left\|\sum_{i=1}^{n}\mathbf{x}_{i}\right\|}{n}. \tag{22}\]
The mean angle \(\mu\) is the angle corresponding to the mean direction \(\tilde{\boldsymbol{\mu}}\in\mathbb{R}^{2}\),
\[\tilde{\boldsymbol{\mu}}=\begin{pmatrix}\tilde{\mu}_{x}\\ \tilde{\mu}_{y}\end{pmatrix}=\frac{1}{n\cdot\bar{r}}\sum_{i=1}^{n}\mathbf{x}_{ i},\qquad\mu=\tan^{-1}\biggl{(}\frac{\tilde{\mu}_{y}}{\tilde{\mu}_{x}}\biggr{)}. \tag{23}\]
The concentration \(\kappa\) is computed as the solution of
\[A_{d}(\kappa)=\bar{r}\,\qquad\text{where }A_{d}(\kappa)\coloneqq\frac{I_{d/2}( \kappa)}{I_{d/2-1}(\kappa)}\, \tag{24}\]
\(I_{s}(\kappa)\) denotes the modified Bessel function of the first kind and order \(s\), and \(d\) denotes the dimension of vectors \(\mathbf{x}\in\mathbb{R}^{d}\) on the \((d-1)\)-dimensional sphere \(\mathbb{S}^{d-1}\) (since we consider the circle embedded in the
2D plane, \(d=2\) in our case). Equation (24) is usually solved using closed-form approximations (Sra, 2012; Kasarapu and Allison, 2015). However, the important insight here is that (24) does not depend on the values \(\mathbf{x}_{i}\), but only on the normalized length \(\bar{r}\) of their _sum_. This means that the MLE for both \(\mu\) and \(\kappa\) of a von Mises distribution only depends on the sum \(\mathbf{r}:=\sum_{i=1}^{n}\mathbf{x}_{i}\) of the directional vectors \(\mathbf{x}_{i}\), which can easily be computed incrementally. With \(\mathbf{r}_{1}=\mathbf{x}_{1}\),
\[\mathbf{r}_{n} =\mathbf{r}_{n-1}+\mathbf{x}_{n} (n>1), \tag{25}\] \[\bar{r} =\frac{\|\mathbf{r}_{n}\|}{n}\,\quad\tilde{\boldsymbol{\mu}}= \frac{1}{n}\;\mathbf{r}_{n} (n\geq 1), \tag{26}\]
and the remaining terms as in (23) and (24). Overall, this allows to estimate cylindrical distributions fully incrementally. Note that the batch and incremental updates are mathematically equivalent and thus yield the same results.
### Interactive Teaching of Spatial Relations
Finally, we explain how we implement the different steps of the interaction scenario sketched in Section 3.4 and Fig. 2. An example with a real robot is shown in Fig. 3.
_1. Command:_ Following (Kartmann et al., 2021), we use a Named Entity Recognition (NER) model to parse object and relation phrases from a language command and ground them to objects and relations in the robot's memory using substring matching. In addition, the resulting task \(R^{*}=(s^{*},u^{*},\mathcal{V}^{*})\) is stored in the relational command segment of the memory system. This is required to later construct the sample from a demonstration.
_2. Plan:_ We query the current model \(G_{s^{*}}^{(t_{k})}\) from the spatial relation memory segment. If \(G_{s^{*}}^{(t_{k})}=\emptyset\), the geometric meaning of the spatial relation is unknown, and the robot cannot solve the task. In this case, the robot verbally expresses its lack of knowledge and requests a demonstration of what to do from the human. Otherwise, \(G_{s^{*}}^{(t_{k})}=\theta\) defines a cylindrical distribution, which is used to sample a given number of candidates (\(50\) in our experiments). As in (Kartmann et al., 2021), the candidates are filtered for feasibility, ranked according to the distribution's probability density function \(p_{\theta}(r,\phi,h)\), and the best candidate is selected for execution. We consider a candidate to be feasible if it is free of collisions, reachable and stable. To decide whether a candidate is free of collisions, the target object is placed virtually at the candidate position and tested for collisions with other objects within a margin of \(25\,\mathrm{mm}\) at \(8\) different rotations around the gravity vector in order to cope with imprecision of action execution, e. g., the object rotating inside the robot's hand while grasping or placing (note that our method aims to keep the object's orientation; the different rotations are only used for collision checking). If a feasible candidate is found, planning is successful. However, after filtering, it is possible that none of the sampled candidates is feasible. This is especially likely when only few samples have been collected so far and, thus, the model's variance is low. Before, this event marked a failure, but in this work, the robot is able to recover by expressing its inability to solve the task and ask the human for a demonstration. Note that the query mechanism of both failure cases, i. e. a missing model and an insufficient one, are structurally identical. In both cases, the human is asked to perform the task it originally gave to the robot, i. e., manipulate the scene to fulfill the relation \(R^{*}\).
_3a. Execution by Robot and 3b. Query:_ If planning was successful (3a.), the robot executes the task by grasping the target object and executing a placing action parameterized based on the selected placing position. If planning failed (3b.), the robot verbally requests a demonstration of the task from the human
using sentences generated from templates and its speech-to-text system. Examples of generated verbal queries are "I am sorry, I don't know what 'right' means yet, can you show me what to do?" (no model) and "Sorry, I cannot do it with my current knowledge. Can you show me what I should do?" (insufficient model). Then, the robot waits for the speech cue signaling the finished demonstration.
_4. Execution by Human and 5. Speech Cue:_ After being queried for a demonstration (3b.), the human relocates the target object to fulfill the requested spatial relation. To signal the completed demonstration, the human gives a speech cue such as "Place it here" to the robot, that is detected using simple keyword spotting and triggers the recording of a new sample. This represents a third case where demonstrations can be triggered: The robot may have successfully executed the task in a qualitative sense (3a.), but the human may not be satisfied with the chosen placing position. In this case, the human can simply change the scene in order to demonstrate what the robot _should have done_, and give a similar speech cue "No, put it here." Again, note that this case is inherently handled by the framework without a special case: When the speech cue is received, the robot has all the knowledge it requires to create a new sample, independently of whether it executed the task before or explicitly asked for a demonstration.
_6. Observation and Model Update:_ When the robot receives the speech cue, it assembles a new sample by querying its memory for the relevant information. It first queries its relational command segment for the latest command, which specifies the requested relation \(R^{*}\) that was just fulfilled by the demonstration. It then queries its object instance segment for the state of the scene \(\mathbf{P}^{(t_{k})}\) when the command was given and the current state \(\mathbf{P}^{(t_{k+1})}\). Combined, this information forms a new sample \(D=\left(\mathbf{P}^{(t_{k})},R^{*},\mathbf{P}^{(t_{k+1})}\right)\). Afterwards, the robot stores the new sample in its long-term memory and updates \(G_{s^{*}}\) as described in Section 3.4. Finally, the robot thanks the human demonstrator for the help, e. g., "Thanks, I think I now know the meaning of 'right' a bit better."
## 5 Results and Discussion
We evaluate our method quantitatively by simulating the interaction scheme with a virtual robot and (Asfour et al. (2019)).
### Quantitative Evaluation
With our experiments, we aim at investigating two questions: (1) How many demonstrations are necessary to obtain a useful model for a given spatial relation? And (2) how does our model perform compared to baseline models using fixed offsets instead of learned cylindrical distributions? To this end, we implement the proposed human-robot interaction scheme based on human demonstrations collected in simulation.
#### 5.1.1 Experimental Setup
For the design of our experiments, a human instructs the robot to manipulate an object according to a spatial relation to other objects. The robot tries to perform the task, and if it is successful, the interaction is finished. If the robot is unable to solve the task, it requests the user to demonstrate the task and updates its model of the spatial relation from the given demonstration. We call this procedure one _interaction_. We are interested in how the robot's model of a spatial relation develops over the course of multiple interactions, where it only receives a new demonstration if it fails to perform the task at hand.
In our experiments, we consider the \(12\) spatial relations \(\mathcal{S}\) listed in Table 1. Consider one relation \(s\in\mathcal{S}\), and assume we have \(10\) demonstrations of this relation. As defined in (4), each demonstration
\((i=1,\ldots,10)\) has the form \(D_{i}=\left(\mathbf{P}_{i}^{(t_{k})},R_{i},\mathbf{P}_{i}^{(t_{k+1})}\right)\), with initial object poses \(\mathbf{P}_{i}^{(t_{k})}\), desired spatial relation \(R_{i}\), and object poses after the demonstration \(\mathbf{P}_{i}^{(t_{k+1})}\). Each demonstration \(D_{i}\) also implicitly defines a _task_\(T_{i}=\left(\mathbf{P}_{i}^{(t_{k})},R_{i}\right)\) consisting of only the initial scene and the desired relation. Given this setup, we define a _learning scenario_ as follows: First, we initialize a virtual robot that has no geometric model of \(s\) as it has not received any demonstrations yet. Then, we consecutively perform one interaction for each task \(T_{1},\ldots,T_{10}\). After each interaction, we evaluate the robot's current model of \(s\) on all tasks.
More precisely, in the beginning we ask the robot to perform the first task \(T_{1}\). As it has no model of \(s\) at this point, it will always be unable to solve the task. Following our interaction scheme, the robot requests the corresponding demonstration and is given the target scene \(\mathbf{P}_{i}^{(t_{k+1})}\). The robot learns its first model of \(s\) from this demonstration, which concludes the first interaction. We then evaluate the learned model on all tasks \(T_{1},\ldots,T_{10}\), i. e., for each task we test whether the model can generate a feasible
Figure 3: Interactively teaching the humanoid robot ARMAR-6 to place objects _to the right of_ other objects. After the first command by the human (A), the robot queries a demonstration (B). The human demonstrates the task (C) which is used by the robot to create or update its model (D). When given the next command (E), the robot can successfully perfom the task (F).
placing position for the target object. Here, we consider \(T_{1}\) as _seen_, and the remaining tasks \(T_{2},\ldots,T_{10}\) as _unseen_. Accordingly, we report the success ratios, i. e., the proportion of solved tasks, among the seen task, the unseen tasks and all tasks. We then proceed with the second interaction, where the robot is asked to perform \(T_{2}\). This time, the model learned from the previous interaction may already be sufficient to solve the task. If this is the case, the robot does _not_ receive the corresponding demonstration and thus keeps its current model. Otherwise, it is given the new demonstration \(D_{2}\) and can incrementally update its model of \(s\). Again, we test the new model on all tasks, where \(T_{1}\) and \(T_{2}\) are now considered as seen and \(T_{3},\ldots,T_{10}\) as unseen. We continue in this manner for all tasks left. After the last interaction, all tasks \(T_{1},\ldots,T_{10}\) are seen and there are no unseen tasks. For completeness, we also perform an evaluation on all tasks before the first interaction; here, the robot trivially fails on all tasks since it initially has no model of \(s\).
Overall, for one learning scenario of relation \(s\), we report the proportions of solved tasks among all tasks, the seen tasks and the unseen tasks after each interaction (and before the first interaction). Note that, as explained above, the number of seen and unseen tasks change over the course of a learning scenario. In addition, we report the number of demonstrations the robot has been given after each interaction. Note that this number may be smaller than the number of performed interactions, as a demonstration is only given during an interaction if the robot fails to perform the task at hand. As the results depend on the order of the tasks and demonstrations, we run \(10\) repetitions of each learning scenario with the tasks randomly shuffled for each repetition, and report all metrics' means and standard deviations over the repetitions. Finally, to obtain an overall result over all spatial relations, we aggregate the results of all repetitions of learning scenarios of all relations \(s\in\mathcal{S}\), and report the resulting means and standard deviations of success ratios and number of demonstrations with respect to the number of performed interactions.
We compare our method with baseline models for all relations which place the target object at a fixed offset from the reference objects. Table 1 gives their descriptions and the definitions of the placing position \(\mathbf{P}_{u}^{(t_{k+1})}\) they generate. The baseline models of direction-based static relations (_right of_, _left of_, _behind_, _in front of_, _on top of_) yield a candidate at a fixed distance towards the respective direction. Distance-based static relations (_close to_, _far from_, _among_) place the target object at a fixed distance towards its initial direction. The baseline models of dynamic spatial relations (_closer_, _farther from_, _on the other side_) are similar, but the distance of the placing position to the reference object is defined relative to the current distance between the objects. We conduct the experiments on the baseline models in the same way as described above, except that (1) the robot has access to the models from the beginning, and (2) we only report the success ratio among all tasks over one repetition, as the baseline models are constant and do not change during a learning scenario or depending on the order of tasks.
For collecting the required demonstrations for each relation, we use a setup with a human, several objects on a table in a simulated environment and a command generation based on sentence templates. Given the objects present in the initial scene and the set of spatial relation symbols \(\mathcal{S}\), we generate a verbal command specifying a desired spatial relation \(R=(s,u,\mathcal{V})\) such as "Place the cup between the milk and the plate." The human is given the generated command and performs the task by moving the objects in the scene. We record the initial scene \(\mathbf{P}^{(t_{k})}\), the desired spatial relation \(R\), and the final scene \(\mathbf{P}^{(t_{k+1})}\), describing a full demonstration \(D=\left(\mathbf{P}^{(t_{k})},R,\mathbf{P}^{(t_{k+1})}\right)\) as required for the experiment. Aside from the objects involved in the spatial relation, the scenes also contain inactive objects which can lead to collisions when placing the object at a generated position, thus rendering some placements infeasible.
#### 5.1.2 Results
Fig. 4 shows the means and standard deviations of the percentage of solved tasks among all tasks, the seen tasks and the unseen tasks after each interaction aggregated over all relations and repetitions as described above. Furthermore, it shows the averages of total number of demonstrations the robot has received after each interaction. As explained above, note that not all interactions result in a demonstration. Also, note that the number of seen and unseen tasks change with the number of interactions; especially, there are no seen tasks before the first interaction and there are no unseen tasks after the last one. For comparison, the success ratio of the baseline models averaged over all relations are shown as well.
In addition, Fig. 5 gives examples of success ratios and number of received demonstrations aggregated only over the repetitions of learning scenarios of single spatial relations. Note that here, the baseline's performance is constant over all repetitions as the number of tasks it can solve is independent of the order of tasks in the learning scenario. Finally, Fig. 6 presents concrete examples of solved and failed tasks involving the spatial relations in Fig. 5 demonstrating the behavior of our method and the baseline during the experiment. Note that the cases represented by the rows in Fig. 6 are not equally frequent; our method more often outperforms the baseline models than vice-versa as indicated by the success ratios in Figs. 4 and 5.
#### 5.1.3 Discussion
The baseline models achieved only \(52.5\pm 24.9\,\%\) success rate over all relations, showing that many tasks in our demonstrations were not trivially solvable using fixed offsets. Also, the standard deviation of 52.5 \(\pm\) 24.9 %. The models were successful if their candidate placing position was free and on the tables.
\begin{table}
\begin{tabular}{c|l c} \hline \hline
**Spatial Relations** & \multicolumn{2}{c}{**Baseline Model**} \\ \((s\in\mathcal{S})\) & **Description** & **Placing Position**\(\mathbf{p}_{u}^{(t_{k+1})}\) \\ \hline _right of_ & \(20\,\mathrm{cm}\) in \(+x\) direction & \(\mathbf{p}_{v}+20\,\mathrm{cm}\cdot\mathbf{x}\) \\ _left of_ & \(20\,\mathrm{cm}\) in \(-x\) direction & \(\mathbf{p}_{v}-20\,\mathrm{cm}\cdot\mathbf{x}\) \\ _behind_ & \(20\,\mathrm{cm}\) in \(+y\) direction & \(\mathbf{p}_{v}+20\,\mathrm{cm}\cdot\mathbf{y}\) \\ _in front of_ & \(20\,\mathrm{cm}\) in \(-y\) direction & \(\mathbf{p}_{v}-20\,\mathrm{cm}\cdot\mathbf{y}\) \\ _on top of_ & \(10\,\mathrm{cm}\) in \(+z\) direction & \(\mathbf{p}_{v}+10\,\mathrm{cm}\cdot\mathbf{z}\) \\ _close to_ & \(20\,\mathrm{cm}\) towards initial position of \(u\) & \(\mathbf{p}_{v}+10\,\mathrm{cm}\cdot\mathrm{dir}(\mathbf{p}_{u}-\mathbf{p}_{v})\) \\ _far from_ & \(60\,\mathrm{cm}\) towards initial position of \(u\) & \(\mathbf{p}_{v}+50\,\mathrm{cm}\cdot\mathrm{dir}(\mathbf{p}_{u}-\mathbf{p}_{v})\) \\ _between_ & Mid point of reference objects & \(\nicefrac{{1}}{{n}}\cdot\sum_{i=1}^{n}\mathbf{p}_{v_{i}}\) \\ _among_ & \(20\,\mathrm{cm}\) from reference objects’ & \\ _mid point towards initial position of \(u\) & \(\mathbf{p}_{v}+10\,\mathrm{cm}\cdot\mathrm{dir}\big{(}\mathbf{p}_{u}-\nicefrac{{ 1}}{{n}}\cdot\sum_{i=1}^{n}\mathbf{p}_{v_{i}}\big{)}\) \\ _closer_ & Half the distance from reference to \(u\) & \(\mathbf{p}_{v}+\nicefrac{{1}}{{2}}\cdot(\mathbf{p}_{u}-\mathbf{p}_{v})\) \\ _farther from_ & Twice the distance from reference to \(u\) & \(\mathbf{p}_{v}+2\cdot(\mathbf{p}_{u}-\mathbf{p}_{v})\) \\ _on the other side of_ & Distance between reference and target in opposite direction of target & \(\mathbf{p}_{v}-(\mathbf{p}_{u}-\mathbf{p}_{v})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Left column: Spatial relation symbols \(s\in\mathcal{S}\) used in the experiments. Right columns: Their corresponding baseline model, where \(\mathbf{p}_{v},\mathbf{p}_{v_{i}}\in\mathbb{R}^{3}\) refer to the positions of reference objects, \(\mathbf{p}_{u}\) refers to the initial position of target object \(u\), \(\mathbf{x},\mathbf{y},\mathbf{z}\) are the unit vectors in the robot’s coordinate system, and \(\mathrm{dir}(\mathbf{v})=\nicefrac{{\mathsf{v}}}{{\lVert\mathbf{v}\rVert}}\) is the direction of a vector \(\mathbf{v}\in\mathbb{R}^{3}\).
However, if the single candidate was, e. g., obstructed by other objects, the models could not fall back to other options. This was especially frequent among relations such as _close to, between_ and _among_ which tend to bring the target object close to other objects. Other collisions were caused because the sizes of the involved objects were not taken into account, especially among relations with fixed distances.
As expected, using our method the robot could never solve a task before the first interaction. However, after the first interaction (which always results in a demonstration), the robot could already solve about \(83.3\pm 13.1\,\mathrm{\char 37}\) of seen and about \(51.9\pm 18.3\,\mathrm{\char 37}\) of unseen tasks on average, almost equaling the baseline models on the unseen tasks. After just two interactions, the mean success ratios on seen tasks reaches a plateau of at \(89\,\mathrm{\char 37}\) to \(92\,\mathrm{\char 37}\), with \(1.53\pm 0.19\) demonstrations on average. Importantly, the success ratios among the seen tasks stays high and does not decrease after more interactions, which indicates that our method generally does not "forget" what it has learned from past demonstrations. The success ratios among unseen tasks rise consistently with the number of interactions, although more slowly than among seen tasks. Nonetheless, after five interactions, the robot could solve \(85.3\pm 9.9\,\mathrm{\char 37}\) of unseen tasks after having received \(2.13\pm 0.41\) demonstrations on average, which shows that our method can generalize from few demonstrations. After completing each learning scenario, i. e., after all interactions
Figure 4: Means and standard deviations of percentage of tasks solved by our method among the seen (green), unseen (orange) and all (blue) tasks, by the baseline models using fixed offsets (gray) as well as number of demonstrations (purple) the robot has been given after a given number of interactions. All metrics are aggregated over multiple repetitions and all spatial relations.
Figure 5: Examples of success ratios and number of demonstrations aggregated over the repetitions of single relations. Colors are as in Fig. 4. Note that for single relations, the success ratios of the baseline models are constant and, thus, have no variance.
have been performed, the robot could solve \(93.0\pm 9.1\,\%\) of all tasks while having received \(2.81\pm 0.87\) demonstrations on average.
One might wonder why the robot is not always able to successfully reproduce the first demonstration on the single seen task after the first interaction. This can be explained in two ways: First, the human is free to change the target object's orientation during the demonstration, which can allow the human to place it closer to other objects than without rotating it. However, as our method only generates new positions while trying to keep the original object orientation, the robot cannot reproduce this demonstration as it would lead to a collision. Second, we use a rather conservative strategy for collision detection in order to anticipate inaccuracies in action execution (e. g., the object rotating in the robot's hand during grasping or placing). More precisely, we use a collision margin of \(25\,\mathrm{mm}\) to check for collisions at different hypothetical rotations of the target object (see Section 4.4). Therefore, if the human placed the object very closely to another in the demonstration, the robot might discard this solution to avoid collisions. These are situations that can only be solved after increasing the model's variance through multiple demonstrations.
We can observe the behavior of our method in more detail by focusing on the results of single relations shown in Fig. 5 and the examples in Fig. 6. First, the standard deviation over the success ratios among
Figure 6: Examples of solved and failed tasks from our experiment. In all images, the reference objects are highlighted in yellow, the target object is highlighted in light blue, and the generated placing position is visualized in green if successful and red if no feasible candidate was found. The orange arrow indicates the mean of a cylindrical distribution (Ours) and the placement according to the baseline model, respectively. The cylindrical distribution’s p.d.f. is visualized around the reference objects (low values in purple, high values in yellow). Sampled positions are shown as grey marks. The left column of each relations shows our model, while the right column shows the result of the baseline model in the same task. For our model, the current cylindrical distribution is visualized as well. For good qualitative coverage, each row show another case with respect to the models’ success: (1) both succeed, (2) ours succeeds, baseline fails, (3) ours fails, baseline succeeds, (4) both fail.
the unseen tasks tends to increase towards the end of the learning scenarios. This is likely due to the decreasing number of unseen tasks towards the end: Before the final interaction, there is only one unseen task left, so the success ratio is either \(0\,\%\) or \(100\,\%\) in each repetition, leading to a higher variance than when the success ratio is averaged over more tasks. As for the relation _right of_, common failure cases were caused by the conservative collision checking in combination with a finite number of sampled candidates (Fig. 6, third row), or the mean distance of the learned distribution being too small for larger objects such as the plate (Fig. 6, fourth row). With the relation _further_, failures were often caused by the candidate positions being partly off the table in combination with the distance variance being too small to generate alternatives (Fig. 6, third row), or scenes where the sampled area was either blocked by other objects or off the table (Fig. 6, fourth row).
The relation _between_ was one of the relations that were more difficult to learn, with a success ratio of only \(67.0\pm 4.6\,\%\) among all tasks after receiving an average of \(4.90\pm 0.83\) demonstrations at the end of the learning scenario (the baseline model achieved only \(10\,\%\)). In the demonstrations, the area between the two reference objects was often cluttered, which prevented our method from finding a collision free placing location. The success ratio among the seen tasks starts at \(70.0\pm 45.8\,\%\) after one interaction. The large variance indicates that, compared to other relations, there were many demonstrations that could not be reproduced after the first interaction, with the success ratio among seen tasks being either \(0\,\%\) or \(100\,\%\) leading to a high variance, similar to the success ratios among unseen tasks towards the end of the learning scenarios. Moreover, the success ratio among the seen tasks decreases to \(56.7\pm 26.0\,\%\) after the third interaction, although it slightly increases again afterwards. In this case, the \(1.50\pm 0.50\) additional demonstrations caused the model to "unlearn" to solve the first tasks in some cases. However, after the third interaction, the success ratios among seen and unseen tasks stabilize and do not change significantly with more demonstrations. Apparently, the models reached their maximum variance after a few interactions, with new demonstrations not changing the model significantly; however our conservative collision detection often caused all candidates to be considered infeasible. One especially difficult task is shown in Fig. 6 (fourth row), where the two reference objects were standing very close to each other, leaving little space for the target object. Finally, note that the tasks for the _between_ relation shown in the first and the third row of Fig. 6 are the same. This is because the baseline could only solve this single task. The corresponding failure example of our model (third row) was shows a model learned from only one demonstration. Indeed, with two or more demonstrations, our method was always able to solve the this task (example in first row).
To summarize, while some aspects can still be improved, the overall results demonstrate that our generative models of spatial relations can be effectively learned in an incremental manner from few demonstrations, while our interaction scheme allows the robot to obtain new demonstrations to improve its models if they prove insufficient for a task at hand.
### Validation Experiments on Real Robot
We performed validation experiments on the real humanoid robot (Asfour et al. (2019)) which are shown in the video3. An example is shown in Fig. 3 and described in more detail here: In the beginning, the robot has no geometric model of the _right of_ relation. First, the user commands the robot to place an object _on the right side of_ a second object (step 1. in Section 3.4). The robot grounds the relation phrase "on the right side of" to the respective entry in its memory (2.), and responds that it has "not learned what _right_ means yet," and requests the user to show it what to do (3a.). Consequently, the user gives a demonstration
by performing the requested task (4.) and gives the speech cue "Put it here" (5.). The robot observes the change in the scene and creates a model of the spatial relation _right of_ (6.). Afterwards, the user starts a second interaction by instructing the robot to put the object _on the right side of_ a third one (1.). This time, the robot has geometric model of _right of_ in its memory (2.) and is able to perform the task (3b.). Beyond that, we show examples of demonstrating and manipulation the scene according to the relations _in front of_, _on top of_, _on the opposite side of_, and _between_.
## 6 Conclusion and Future Work
In this work, we described how a learning humanoid robot which has the task to manipulate the scene based on desired spatial object relations can query and use demonstrations from a human during interaction to incrementally learn generative models of spatial relations. We demonstrated how the robot can communicate its inability to solve a task in order to collect more demonstrations in a continual manner. In addition, we showed how a parametric representation of object spatial relations can be learned incrementally from few demonstrations. In future work, we would like to make the human-robot interaction even more natural by detecting when a demonstration is finished, thus releasing the requirement of a speech cue indicating this event. Furthermore, we want to explore how knowledge about different spatial relations can be transferred between them and leveraged for learning new ones.
## Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
## Author Contributions
RK developed the methods and their implementation and performed the evaluation experiments. The entire work was conceptualized and supervised by TA. The initial draft of the manuscript was written by RK and revised jointly by RK and TA.
## Funding
This work has been supported by the Carl Zeiss Foundation through the JuBot project and by the German Federal Ministry of Education and Research (BMBF) through the OML project (01IS18040A).
## Supplemental Material
This paper is supplemented by a video giving an overview of our approach and showing the validation experiments described in Section 5.2.
## Data Availability Statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation. |
2304.09678 | Column Subset Selection and Nyström Approximation via Continuous
Optimization | We propose a continuous optimization algorithm for the Column Subset
Selection Problem (CSSP) and Nystr\"om approximation. The CSSP and Nystr\"om
method construct low-rank approximations of matrices based on a predetermined
subset of columns. It is well known that choosing the best column subset of
size $k$ is a difficult combinatorial problem. In this work, we show how one
can approximate the optimal solution by defining a penalized continuous loss
function which is minimized via stochastic gradient descent. We show that the
gradients of this loss function can be estimated efficiently using
matrix-vector products with a data matrix $X$ in the case of the CSSP or a
kernel matrix $K$ in the case of the Nystr\"om approximation. We provide
numerical results for a number of real datasets showing that this continuous
optimization is competitive against existing methods. | Anant Mathur, Sarat Moka, Zdravko Botev | 2023-04-19T14:12:21Z | http://arxiv.org/abs/2304.09678v1 | # Column Subset Selection and Nystrom Approximation via Continuous Optimization
###### Abstract
We propose a continuous optimization algorithm for the Column Subset Selection Problem (CSSP) and Nystrom approximation. The CSSP and Nystrom method construct low-rank approximations of matrices based on a predetermined subset of columns. It is well known that choosing the best column subset of size \(k\) is a difficult combinatorial problem. In this work, we show how one can approximate the optimal solution by defining a penalized continuous loss function which is minimized via stochastic gradient descent. We show that the gradients of this loss function can be estimated efficiently using matrix-vector products with a data matrix \(\mathbf{X}\) in the case of the CSSP or a kernel matrix \(\mathbf{K}\) in the case of the Nystrom approximation. We provide numerical results for a number of real datasets showing that this continuous optimization is competitive against existing methods.
Machine Learning, ICML, ICML
## 1 Introduction
Recent advances in the technological ability to capture and collect data have meant that high-dimensional datasets are now ubiquitous in the fields of engineering, economics, finance, biology, and health sciences to name a few. In the case where the data collected is not labeled it is often desirable to obtain an accurate low-rank approximation for the data which is relatively low-cost to obtain and memory efficient. Such an approximation is useful to speed up downstream matrix computations that are often required in large-scale learning algorithms. The Column Subset Selection Problem (CSSP) and Nystrom method are two such tools that generate low-rank approximations based on a subset of data instances or features from the dataset. The chosen subset of instances or features are commonly referred to as "landmark" points. The choice of landmark points determines how accurate the low-rank approximation is.
The challenge in the CSSP is to select the best \(k\) columns of a data matrix \(\mathbf{X}\in\mathbb{R}^{m\times n}\) that span its column space. That is, for any binary vector \(\boldsymbol{s}\in\{0,1\}^{n}\), compute
\[\operatorname*{argmin}_{\boldsymbol{s}\in\{0,1\}^{n}}\|\mathbf{X}-\mathbf{P}_ {s}\mathbf{X}\|_{F}^{2},\quad\text{subject to }\|\boldsymbol{s}\|_{0}\leq k, \tag{1}\]
where \(\|\cdot\|_{F}\) is the Frobenius matrix norm, \(\|\boldsymbol{s}\|_{0}=\sum_{j=1}^{n}I(s_{j}=1)\) and \(\mathbf{P}_{s}\) is the projection matrix onto \(\operatorname*{span}\{\boldsymbol{x}_{j}:s_{j}=1,j=1,\ldots,n\}\) (\(\boldsymbol{x}_{j}\) being the \(j\)-th column of \(\mathbf{X}\)).
Solving this combinatorial problem exactly is known to be NP-complete (Shitov, 2021), and is practically infeasible even when \(k\) is of moderate size. We propose a novel continuous optimization algorithm to approximate the exact solution to this problem. While an optimization approach via Group Lasso (Yuan & Lin, 2006) exists for the convex relaxation of this problem (Bien et al., 2010), to the best of our knowledge, no continuous optimization method has been developed to solve the highly non-convex combinatorial problem (1). To introduce our approach for the CSSP, instead of searching over binary vectors \(\boldsymbol{s}\in\{0,1\}^{n}\), we consider the hyper-cube \([0,1]^{n}\) and define for each \(\boldsymbol{t}\in[0,1]^{n}\) a matrix \(\widetilde{\mathbf{P}}(\boldsymbol{t})\) which allows the following well-defined penalized continuous extension of the exact problem,
\[\operatorname*{argmin}_{\boldsymbol{t}\in[0,1]^{n}}\|\mathbf{X}-\widetilde{ \mathbf{P}}(\boldsymbol{t})\mathbf{X}\|_{F}^{2}+\lambda\sum_{j=1}^{n}t_{j}.\]
The parameter \(\lambda>0\) plays an analogous role to that of the regularization parameter in regularized linear regression methods (Tibshirani, 1996) and controls the sparsity of the solution, that is, the size of \(k\). Two aspects of this continuous extension make it useful for approximating the exact solution. Firstly, the continuous loss agrees with the discrete loss at every corner point \(\boldsymbol{s}\in\{0,1\}^{n}\) of the hypercube \([0,1]^{n}\), and secondly, for large datasets the gradient can be estimated via an unbiased stochastic estimate. To obtain an approximate solution to the exact problem, _stochastic gradient descent_ (SGD) is implemented on the penalized loss. After starting at an interior point of the hyper-cube, under SGD, the vector \(\boldsymbol{t}\) moves towards a corner point, and some of the \(\boldsymbol{t}_{j}\)'s exhibit shrinkage to zero. It is these values that indicate which columns in \(\mathbf{X}\) should not be selected as landmark points.
The Nystrom approximation (Williams & Seeger, 2000; Drineas et al., 2005) is a popular variant of the CSSP for
positive semi-definite kernel matrices. The Nystrom method also constructs a low-rank approximation \(\widehat{\mathbf{K}}\in\mathbb{R}^{n\times n}\) to the true kernel matrix \(\mathbf{K}\in\mathbb{R}^{n\times n}\) using a subset of columns. Once the \(k\) columns are selected, \(\widehat{\mathbf{K}}\) (in factored form) takes \(O(k^{3})\) additional time to compute, requires \(O(nk)\) space to store, and can be manipulated quickly in downstream applications, e.g., inverting \(\widehat{\mathbf{K}}\) takes \(O(nk^{2})\) time. In addition to the continuous extension for the CSSP, in this paper, we provide a continuous optimization algorithm that can approximate the best \(k\) columns to be used to construct \(\widehat{\mathbf{K}}\) (Section 2.2).
The continuous algorithm for the CSSP formulated in this paper utilizes SGD where at each iteration one can estimate the gradient with a cost of \(O(mn)\). We show that the gradients of the penalized continuous loss can be estimated via linear solves with random vectors that are approximated with the conjugate gradient algorithm (CG) (Golub & Van Loan, 1996), which itself is an iterative algorithm that only requires matrix-vector multiplications (MVMs) with the \(m\times n\) matrix \(\mathbf{X}\). Similarly, for the Nystrom method we show that at each step of the gradient descent, the gradient can be estimated in \(O(n^{2})\) time requiring only matrix-vector multiplications with the kernel matrix \(\mathbf{K}\). This is especially useful in cases where we only have access to a black-box MVM function. The fact that both these algorithms require only matrix-vector multiplications to estimate the gradients lends itself to utilizing GPU hardware acceleration. Moreover, the computations in the proposed algorithm can exploit the sparsity that is achieved by working only with the columns of \(\mathbf{X}\) that are selected by the algorithm at any given iteration.
### Related Work
There exists extensive literature on random sampling methods for the approximation of the exact CSSP and Nystrom problem. Sampling techniques such as adaptive sampling (Deshpande & Vempala, 2006), ridge leverage scores (Gittens & Mahoney, 2013; Musco & Musco, 2017; Alaoui & Mahoney, 2015) attempt to sample "important" and "diverse" columns. In particular, recent attention has been paid to Determinantal Point Processes (DPPs) (Hough et al., 2006; Derezinski & Mahoney, 2021). DPPs provide strong theoretical guarantees (Derezinski et al., 2020) for the CSSP and Nystrom approximation and are amenable to efficient numerical implementation (Li et al., 2016; Derezinski et al., 2019; Calandriello et al., 2020; Derezinski, 2019). Outside of sampling methods, iterative methods such as Greedy selection (Farahat et al., 2011, 2013) have been shown to perform well in practice and exhibit provable guarantees (Altschuler et al., 2016).
Column selection has been extensively studied in the supervised context of linear regression (more commonly referred to as feature or variable selection). Penalized regression methods such as the Lasso (Tibshirani, 1996) have been widely applied to select columns of a predictor matrix that best explain a response vector. The canonical \(k\)-best subset or \(l_{0}\)-penalized regression problem is another penalized regression method, where the goal is to find the best subset of \(k\) predictors that best fit a response \(\mathbf{y}\)(Beale et al., 1967; Hocking & Leslie, 1967). The recently proposed _Continuous Optimization Method Towards Best Subset Selection_ (COMBSS) algorithm (Moka et al., 2022) attempts to solve the \(l_{0}\)-penalized regression problem by minimizing a continuous loss that approximates the exact solution. The algorithm we propose for the CSSP in this paper can be viewed as an adaptation of COMBSS to the unsupervised setting. In this setting, the goal is to find the best subset of size \(k\) for a multiple multivariate regression model where both the response and predictor matrix are \(\mathbf{X}\). Interestingly, this framework can be extended to include a continuous selection loss for the Nystrom approximation.
The rest of the paper is structured as follows. In Section 2 we describe the continuous extension for the CSSP and the Nystrom method. In Section 3 we provide steps for the efficient implementation of our proposed continuous algorithm on large matrices and in Section 4 we provide numerical results on a variety of real datasets.
## 2 Continuous Loss for Landmark Selection
In this section, we formally define the CSSP and the best size \(k\)-Nystrom approximation. Then, we provide the mathematical setup for the continuous extension of the exact problem.
### Column Subset Selection
Let \(\mathbf{X}\in\mathbb{R}^{m\times n}\) and for any binary vector \(\mathbf{s}=(s_{1},\dots,s_{n})^{\top}\in\{0,1\}^{n}\), let \(\mathbf{X}_{[\mathbf{s}]}\) denote the matrix of size \(m\times\|s\|_{0}\) keeping only columns \(j\) of \(\mathbf{X}\) where \(s_{j}=1\), for \(j=1,\dots,n\). Then for every integer \(k\leq n\) the CSSP finds
\[\operatorname*{argmin}_{\mathbf{s}\in\{0,1\}^{n}}\|\mathbf{X}-\mathbf{P}_{s} \mathbf{X}\|_{F}^{2},\quad\text{subject to }\|\mathbf{s}\|_{0}\leq k, \tag{2}\]
where \(\mathbf{P}_{s}:=\mathbf{X}_{[\mathbf{s}]}\mathbf{X}_{[\mathbf{s}]}^{\dagger}\) (\(\dagger\) denotes Moore-Penrose inverse) is the projection matrix onto \(\text{span}\{\mathbf{x}_{j}:s_{j}=1\}\) and \(\mathbf{x}_{j}\) is the \(j\)-th column of \(\mathbf{X}\). By expanding the Frobenius norm it is easy to see that the discrete problem (2) can be reformulated as,
\[\operatorname*{argmin}_{\mathbf{s}\in\{0,1\}^{n}}-\operatorname{tr}\left[\mathbf{X} ^{\top}\mathbf{P}_{\mathbf{s}}\mathbf{X}\right],\quad\text{subject to }\|\mathbf{s}\|_{0}\leq k.\]
We now define a new matrix function on \(\mathbf{t}\in[0,1]^{n}\) which acts as a continuous generalization of \(\mathbf{P}_{\mathbf{s}}\).
**Definition 2.1**.: For \(\mathbf{t}=(t_{1},\dots,t_{n})^{\top}\in[0,1]^{n}\), define \(\mathbf{T}:=\operatorname{Diag}(\mathbf{t})\) as the diagonal matrix with diagonal elements \(t_{1},\dots,t_{n}\) and
\[\widetilde{\mathbf{P}}(\mathbf{t}):=\mathbf{X}\mathbf{T}\left[\mathbf{T}\mathbf{X} ^{\top}\mathbf{X}\mathbf{T}+\delta(\mathbf{I}-\mathbf{T}^{2})\right]^{\dagger }\mathbf{T}\mathbf{X}^{\top},\]
where \(\delta>0\) is a fixed constant.
Although not explicitly stated in (Moka et al., 2022), \(\widetilde{\mathbf{P}}(\mathbf{t})\) is used as the continuous generalization for the hat matrix \(\mathbf{P}_{\mathbf{s}}\) to solve the \(l_{0}\)-penalized regression problem.
The main difference between this definition and traditional sampling methods is that instead of multiplying \(\mathbf{X}\) by a sampling matrix to obtain \(\mathbf{X}_{[\mathbf{s}]}\) we compute the matrix \(\mathbf{X}\mathbf{T}\) which weights column \(j\) of \(\mathbf{X}\) by the parameter \(t_{j}\in[0,1]\). Intuitively, the matrix \(\mathbf{T}\mathbf{X}^{\top}\mathbf{X}\mathbf{T}+\delta(\mathbf{I}-\mathbf{T}^ {2})\) can be viewed as a convex combination of the matrices \(\mathbf{X}^{\top}\mathbf{X}\) and \(\delta\mathbf{I}\).
From an evaluation standpoint, the pseudo-inverse need not be evaluated for any interior point in this newly defined function. We remark that for any \(\mathbf{t}\in[0,1)^{n}\) the matrix inverse in Definition 2.1 exists and therefore,
\[\widetilde{\mathbf{P}}(\mathbf{t})=\mathbf{X}\mathbf{T}\left[\mathbf{T}\mathbf{X} ^{\top}\mathbf{X}\mathbf{T}+\delta(\mathbf{I}-\mathbf{T}^{2})\right]^{-1} \mathbf{T}\mathbf{X}^{\top}.\]
We now state two results for the function \(\widetilde{\mathbf{P}}(\mathbf{t})\) and its relationship with the projection matrix \(\mathbf{P}_{s}\). The following Lemmas (2.2 and 2.3) are extensions of the results stated in (Moka et al., 2022).
**Lemma 2.2**.: _For any binary vector \(\mathbf{s}\in\{0,1\}^{n}\), \(\widetilde{\mathbf{P}}(\mathbf{s})\) exists and_
\[\widetilde{\mathbf{P}}(\mathbf{s})=\mathbf{P}_{\mathbf{s}}=\mathbf{X}_{[\mathbf{s}]} \mathbf{X}_{[\mathbf{s}]}^{\dagger}.\]
**Lemma 2.3**.: \(\widetilde{\mathbf{P}}(\mathbf{t})\) _is continuous element-wise over \([0,1]^{n}\). Moreover, for any sequence \(\mathbf{t}^{(1)},\mathbf{t}^{(2)}\cdots\in[0,1)^{n}\) converging to \(\mathbf{t}\in[0,1]^{n}\), the limit \(\lim_{l\to\infty}\widetilde{\mathbf{P}}(\mathbf{t}^{(l)})\) exists and_
\[\lim_{l\to\infty}\widetilde{\mathbf{P}}(\mathbf{t}^{(l)})=\widetilde{\mathbf{P}}( \mathbf{t}).\]
We note that the proof of Lemma 2.3 follows identically to the proof of _Theorem 3_ in (Moka et al., 2022) where it is stated that the function \(\|\mathbf{y}-\widetilde{\mathbf{P}}(\mathbf{t})\mathbf{y}\|_{2}^{2}\) is continuous over \([0,1]^{n}\) for any fixed vector \(\mathbf{y}\in\mathbb{R}^{n}\).
Given \(\widetilde{\mathbf{P}}(\mathbf{t})\) is continuous on \([0,1]^{n}\) and agrees with \(\mathbf{P}_{\mathbf{s}}\) at every corner point we can define the continuous generalization of the exact problem (2),
\[\operatorname*{argmin}_{\mathbf{t}\in[0,1]^{n}}-\operatorname{tr}\left[\mathbf{X} ^{\top}\widetilde{\mathbf{P}}(\mathbf{t})\mathbf{X}\right],\quad\text{subject to }\sum_{j=1}^{n}t_{j}\leq k.\]
Instead of solving this constrained problem, for a tunable parameter \(\lambda\), we consider minimizing the Lagrangian function,
\[\operatorname*{argmin}_{\mathbf{t}\in[0,1]^{n}}f_{\lambda}(\mathbf{t}),\quad f_{ \lambda}(\mathbf{t}):=-\operatorname{tr}\left[\mathbf{X}^{\top}\widetilde{\mathbf{ P}}(\mathbf{t})\mathbf{X}\right]+\lambda\sum_{j=1}^{n}t_{j}.\]
In Section 3 we reformulate this box-constrained problem into an equivalent unconstrained problem via a nonlinear mapping \(\mathbf{t}=\mathbf{t}(\mathbf{w})\) for \(\mathbf{w}\in\mathbb{R}^{n}\) that forces \(\mathbf{t}\) to be in the hypercube \([0,1]^{n}\). We solve this optimization via continuous gradient descent. To this end, we need to evaluate the gradient \(\nabla f_{\lambda}(\mathbf{t})\) for any interior point.
**Lemma 2.4**.: _Let \(\mathbf{K}=\mathbf{X}^{\top}\mathbf{X}\), \(\mathbf{Z}=\mathbf{K}-\delta\mathbf{I}\) and \(\mathbf{L}_{\mathbf{t}}=\mathbf{T}\mathbf{Z}\mathbf{T}+\delta\mathbf{I}\). Then, for \(\mathbf{t}\in(0,1)^{n}\),_
\[\nabla f_{\lambda}(\mathbf{t})=2\operatorname{Diag}\left[\mathbf{L}_{\mathbf{t}}^{-1} \mathbf{T}\mathbf{K}^{2}\left(\mathbf{T}\mathbf{L}_{\mathbf{t}}^{-1}\mathbf{T} \mathbf{Z}-\mathbf{I}\right)\right]+\lambda\mathbf{1}.\]
Evaluating \(\nabla f_{\lambda}(\mathbf{t})\) has a computational complexity of \(O(n^{3})\) due to the required inversion of \(\mathbf{L}_{t}\). In Section 3 we detail an unbiased estimate for \(\nabla f_{\lambda}(\mathbf{t})\) which utilizes the CG algorithm, where the most expensive operations involved are matrix-vector multiplications with \(\mathbf{X}\) and \(\mathbf{X}^{\top}\), which reduces the computational complexity to \(O(mn)\).
### Nystrom Method
We now turn our attention to defining a continuous objective for the landmark points in the Nystrom approximation. We consider optimizing the landmark points first with respect to the trace matrix norm and then to the Frobenius matrix norm.
In many applications, we are interested in obtaining a low-rank approximation to a _kernel_ matrix \(\mathbf{K}\in\mathbb{R}^{n\times n}\). Consider an input space \(\mathcal{X}\) and a positive semi-definite kernel function \(h:\mathcal{X}\times\mathcal{X}\to\mathbb{R}\). Given a set of \(n\) input points \(\mathbf{x}^{\prime}_{1},...,\mathbf{x}^{\prime}_{n}\in\mathcal{X}\), the kernel matrix \(\mathbf{K}\in\mathbb{R}^{n\times n}\) is defined by \(\mathbf{K}_{i,j}=h(\mathbf{x}^{\prime}_{i},\mathbf{x}^{\prime}_{j})\) and is positive semi-definite.
For any binary vector \(\mathbf{s}\in\{0,1\}^{n}\) let \(\mathbf{K}_{[\mathbf{s}]}\) be the \(n\times\|\mathbf{s}\|_{0}\) matrix with columns indexed by \(\{j:s_{j}=1\}\) and \(\mathbf{K}_{[\mathbf{s},\mathbf{s}]}\) be the \(\|\mathbf{s}\|_{0}\times\|\mathbf{s}\|_{0}\) principal sub-matrix indexed by \(\{j:s_{j}=1\}\). The Nystrom low-rank approximation for \(\mathbf{K}\) is given by,
\[\widehat{\mathbf{K}}_{\mathbf{s}}:=\mathbf{K}_{[\mathbf{s}]}\mathbf{K}_{[\mathbf{s},\mathbf{s} ]}^{\dagger}\mathbf{K}_{[\mathbf{s}]}^{\top}.\]
The following observation appearing in (Derezinski et al., 2020) connects the CSSP and the Nystrom approximation with respect to the trace matrix norm.
Suppose we have the decomposition of the kernel matrix \(\mathbf{K}=\mathbf{X}^{\top}\mathbf{X}\) where \(\mathbf{X}\in\mathbb{R}^{m\times n}\). Then, the Nystrom approximation is given by \(\widehat{\mathbf{K}}_{\mathbf{s}}=\left(\mathbf{P}_{\mathbf{s}}\mathbf{X}\right)^{\top} \mathbf{P}_{\mathbf{s}}\mathbf{X}\) and
\[\|\mathbf{K}-\widehat{\mathbf{K}}_{\mathbf{s}}\|_{*}=\|\mathbf{X}-\mathbf{P}_{\mathbf{s }}\mathbf{X}\|_{F}^{2}.\]
where \(\|\mathbf{A}\|_{*}=\sum_{i=1}^{\min\{m,n\}}\sigma_{i}(\mathbf{A})\) for \(\mathbf{A}\in\mathbb{R}^{m\times n}\) is the trace matrix norm. This connection is used in (Derezinski et al., 2020) to provide shared approximation bounds for both the CSSP and Nystrom approximation. Given that the kernel matrix is always positive semi-definite, the decomposition \(\mathbf{K}=\mathbf{X}^{\top}\mathbf{X}\) always exists and one can solve the
CSSP for \(\mathbf{X}\) to obtain the best \(k\)-landmark Nystrom approximation with respect to the trace norm. We note that such a decomposition is not unique, e.g., it can be the Cholesky decomposition or the symmetric square-root decomposition.
The matrix \(\mathbf{X}\) does not need explicit evaluation in order to perform CSSP as one can attain \(\nabla f_{\lambda}(\mathbf{t})\) with the matrix \(\mathbf{K}\) instead (see, Lemma 2.4). Therefore, finding the decomposition \(\mathbf{K}=\mathbf{X}^{\top}\mathbf{X}\) is not required, and one can approximately solve the CSSP by minimizing \(\nabla f_{\lambda}(\mathbf{t})\) with the kernel matrix \(\mathbf{K}\).
Suppose instead we want to use the Frobenius matrix norm to find the best choice of columns of the matrix \(\mathbf{K}\) to construct the Nystrom approximation. This problem is formulated as
\[\operatorname*{argmin}_{\mathbf{s}\in\{0,1\}^{n}}\|\mathbf{K}-\widehat{\mathbf{K} }_{\mathbf{s}}\|_{F}^{2},\quad\text{subject to }\|\mathbf{s}\|_{0}\leq k. \tag{3}\]
Similar to \(\widetilde{\mathbf{P}}(\mathbf{t})\) we can weight each column \(j\) of \(\mathbf{K}\) by \(t_{j}\in[0,1]\) instead of sampling the columns \(\mathbf{K}_{[\mathbf{s}]}\) for the Nystrom approximation. We define continuous generalization for the Nystrom approximation,
**Definition 2.5**.: For \(\mathbf{t}=(t_{1},\ldots,t_{n})^{\top}\in[0,1]^{n}\) let \(\mathbf{T}:=\operatorname{Diag}(\mathbf{t})\) and
\[\widetilde{\mathbf{K}}(\mathbf{t}):=\mathbf{K}\mathbf{T}\left[\mathbf{T}\mathbf{K} \mathbf{T}+\delta(\mathbf{I}-\mathbf{T}^{2})\right]^{\dagger}\mathbf{T}\mathbf{ K},\]
where \(\delta>0\) is a fixed constant. Similar to \(\widetilde{\mathbf{P}}(\mathbf{t})\), for any \(\mathbf{t}\in[0,1)^{n}\) the matrix \(\mathbf{T}\mathbf{K}\mathbf{T}+\delta(\mathbf{I}-\mathbf{T}^{2})\) is invertible.
In the following two results, we state that \(\widetilde{\mathbf{K}}(\mathbf{t})\) is a continuous function on \([0,1]^{n}\) and agrees with the exact Nystrom approximation at every corner point.
**Lemma 2.6**.: _For any corner point \(\mathbf{s}\in\{0,1\}^{n}\), \(\widetilde{\mathbf{K}}(\mathbf{s})\) exists and_
\[\widetilde{\mathbf{K}}(\mathbf{s})=\widehat{\mathbf{K}}_{s}=\mathbf{K}_{[\mathbf{s}]} \mathbf{K}_{[\mathbf{s},\mathbf{s}]}^{\dagger}\mathbf{K}_{[\mathbf{s}]}^{\top}.\]
**Lemma 2.7**.: \(\widetilde{\mathbf{K}}(\mathbf{t})\) _is continuous element-wise over \([0,1]^{n}\). Moreover, for any sequence \(\mathbf{t}^{(1)},\mathbf{t}^{(2)}\ldots\in[0,1)^{n}\) converging to \(\mathbf{t}\in[0,1]^{n}\), the limit \(\lim_{l\to\infty}\widetilde{\mathbf{K}}(\mathbf{t}^{(l)})\) exists and_
\[\lim_{l\to\infty}\widetilde{\mathbf{K}}(\mathbf{t}^{(l)})=\widetilde{\mathbf{K}}( \mathbf{t}).\]
We therefore have the continuous generalization of the exact problem (3),
\[\operatorname*{argmin}_{\mathbf{t}\in[0,1]^{n}}\|\mathbf{K}-\widetilde{\mathbf{K} }(\mathbf{t})\|_{F}^{2},\quad\text{subject to }\sum_{j=1}^{n}t_{j}\leq k.\]
Instead of solving this constrained problem, for a tunable parameter \(\lambda\), we consider minimizing the Lagrangian function,
\[\operatorname*{argmin}_{\mathbf{t}\in[0,1]^{n}}g_{\lambda}(\mathbf{t}),\quad g_{ \lambda}(\mathbf{t}):=\|\mathbf{K}-\widetilde{\mathbf{K}}(\mathbf{t})\|_{F}^{2}+ \lambda\sum_{j=1}^{n}t_{j}.\]
As with the continuous extension for CSSP we use a gradient descent method to solve the above problem. The following result provides an expression for \(\nabla g_{\lambda}(\mathbf{t})\) for \(\mathbf{t}\in(0,1)^{n}\).
**Lemma 2.8**.: _Let \(\mathbf{Z}=\mathbf{K}-\delta\mathbf{I}\), \(\mathbf{L}_{\mathbf{t}}=\mathbf{T}\mathbf{Z}\mathbf{T}+\delta\mathbf{I}\) and \(\mathbf{D}=\widetilde{\mathbf{K}}(\mathbf{t})-\mathbf{K}\). Then, for \(\mathbf{t}\in(0,1)^{n}\),_
\[\nabla g_{\lambda}(\mathbf{t})=4\operatorname{Diag}\left[\mathbf{L}_{\mathbf{t}}^{-1} \mathbf{T}\mathbf{K}\mathbf{D}\mathbf{K}\left(\mathbf{I}-\mathbf{T}\mathbf{L} _{\mathbf{t}}^{-1}\mathbf{T}\mathbf{Z}\right)\right]+\lambda\mathbf{1}.\]
Evaluating \(\nabla g_{\lambda}(\mathbf{t})\) has a computational complexity of \(O(n^{3})\) due to the required inversion of \(\mathbf{L}\) and evaluation of \(\mathbf{K}(\mathbf{t})\). As with \(\nabla f_{\lambda}(\mathbf{t})\) we detail an unbiased estimate for \(\nabla g_{\lambda}(\mathbf{t})\) in Section 3 which utilizes matrix-vector multiplications with \(\mathbf{K}\) and that helps in reducing the computational cost.
## 3 Implementation
In this section, we detail how to efficiently solve the continuous problems posed in Section 2. In particular, we detail a non-linear transformation that was also used in (Moka et al., 2022) to make both the CSSP and Nystrom approximation optimization problems unconstrained. We then show how one can estimate the gradients using MVMs with \(\mathbf{X}\) and \(\mathbf{K}\).
### Handling Box Constraints (Moka et al., 2022)
The continuous extension of the CSSP and Nystrom approximation requires minimizing the functions \(f_{\lambda}(\mathbf{t})\) and \(g_{\lambda}(\mathbf{t})\) over \(\mathbf{t}\in[0,1]^{n}\). We now consider a non-linear transformation to make both optimization problems unconstrained. Consider the mapping \(\mathbf{t}=\mathbf{t}(\mathbf{w})\) given by,
\[t_{j}(w_{j})=1-\exp(-w_{j}^{2}),\quad j=1,\ldots,n,\]
then if we consider the optimization of continuous CSSP,
\[\mathbf{w}^{*}=\operatorname*{argmin}_{\mathbf{w}\in\mathbb{R}^{p}}f_{\lambda}(\mathbf{t}( \mathbf{w})),\]
we attain the solution to (2.1) by evaluating \(\mathbf{t}(\mathbf{w}^{*})\). This is true because for any \(a,b\in\mathbb{R}\),
\[1-\exp(-a^{2})<1-\exp(-b^{2})\quad\text{if and only if }a^{2}<b^{2}.\]
In vector form the transformation is \(\mathbf{t}(\mathbf{w})=\mathbf{1}-\exp(-\mathbf{w}\odot\mathbf{w})\) (here \(\odot\) denotes element-wise multiplication) and using the chain rule we obtain for \(\mathbf{w}\in\mathbb{R}^{p}\),
\[\frac{\partial f_{\lambda}(\mathbf{t}(\mathbf{w}))}{\partial\mathbf{w}}=\frac{\partial f_{ \lambda}(\mathbf{t}(\mathbf{w}))}{\partial\mathbf{t}}\odot(2\mathbf{w}\odot\exp(-\mathbf{w}\odot \mathbf{w})).\]
We can now implement a gradient descent algorithm to approximately obtain \(\mathbf{t}(\mathbf{w}^{*})\). Using this approximation we can select an appropriate binary vector as a solution to the exact problem (2). The same transformation can be applied to solve \(g_{\lambda}(\mathbf{t}(\mathbf{w}))\) over \(\mathbf{w}\in\mathbb{R}^{n}\).
### Stochastic Estimate for the Gradient
As discussed in Section 2, \(\nabla f_{\lambda}(\mathbf{t})\) and \(\nabla g_{\lambda}(\mathbf{t})\) are problematic to compute for large \(n\) due to the \(O(n^{3})\) complexity of inverting a matrix. Here we show that we can implement a stochastic gradient descent (SGD) which has strong theoretical guarantees (Robbins and Monro, 1951) by using an unbiased estimate for \(\nabla f_{\lambda}(\mathbf{t})\) and \(\nabla g_{\lambda}(\mathbf{t})\).
The method we employ is a factorized estimator \(\hat{\ell}\) for the diagonal of a square matrix. Suppose we wish to estimate the diagonal of the matrix \(\mathbf{A}=\mathbf{B}\mathbf{C}^{\top}\) where \(\mathbf{A},\mathbf{B},\mathbf{C}\in\mathbb{R}^{n\times n}\). Let \(\mathbf{z}\in\mathbb{R}^{n}\) be a random vector sampled from the Rademacher distribution, whose entries are either \(-1\) or \(1\), each with probability \(1/2\). Then an unbiased estimate for \(\mathrm{Diag}\left(\mathbf{A}\right)\) is \(\hat{\ell}=\mathbf{B}\mathbf{z}\odot\mathbf{C}\mathbf{z}\), see (Martens et al., 2012). Further analysis of its properties including its variance can be found in (Mathur et al., 2021). We note that when \(\mathbf{B}=\mathbf{A}\) and \(\mathbf{C}=\mathbf{I}\), this estimator reduces to the well-known (Bekas et al., 2007) estimator for the diagonal.
The two following results provide an unbiased estimate for \(\nabla f_{\lambda}(\mathbf{t})\) and \(\nabla g_{\lambda}(\mathbf{t})\) using the factorized estimator for the diagonal of a matrix.
**Lemma 3.1**.: _Recall that in the continuous CSSP optimization for \(\mathbf{X}\), we have the definitions \(\mathbf{T}=\mathrm{Diag}(\mathbf{t})\) for \(t\in[0,1]^{n}\), \(\mathbf{K}=\mathbf{X}^{\top}\mathbf{X}\), \(\mathbf{Z}=\mathbf{K}-\delta\mathbf{I}_{n}\) and \(\mathbf{L}_{\mathbf{t}}=\mathbf{T}\mathbf{Z}\mathbf{T}+\delta\mathbf{I}_{n}\). Suppose \(\mathbf{z}\in\mathbb{R}^{n}\) follows a Rademacher distribution and let:_
\[(1)\,\mathbf{a}=\mathbf{K}\mathbf{z},\,(2)\,\mathbf{b}=\mathbf{L}_{\mathbf{t}}^{-1}(\mathbf{t} \odot\mathbf{a})\text{ and }\]
\[\mathbf{\phi}=\mathbf{b}\odot\mathbf{Z}(\mathbf{t}\odot\mathbf{b})-\mathbf{a}\odot\mathbf{b}.\]
_Then for \(\mathbf{t}\in(0,1)^{n}\),_
\[\nabla f_{\lambda}(\mathbf{t})=2\mathbb{E}\left[\mathbf{\phi}\right]+\lambda\mathbf{1}.\]
**Lemma 3.2**.: _Recall that in the continuous Nystrom optimization for a kernel matrix \(\mathbf{K}\), we have the definitions \(\mathbf{T}=\mathrm{Diag}(\mathbf{t})\) for \(t\in[0,1]^{n}\), \(\mathbf{Z}=\mathbf{K}-\delta\mathbf{I}\) and \(\mathbf{L}_{\mathbf{t}}=\mathbf{T}\mathbf{Z}\mathbf{T}+\delta\mathbf{I}\) Suppose \(\mathbf{z}\in\mathbb{R}^{n}\) follows a Rademacher distribution and let:_
\[(1)\,\mathbf{a}=\mathbf{K}\mathbf{z},\,(2)\,\mathbf{b}=\mathbf{L}_{\mathbf{t}}^{-1}(\mathbf{t} \odot\mathbf{a}),\,(3)\,\mathbf{c}=\mathbf{K}(\mathbf{t}\odot\mathbf{b})-\mathbf{a},\,(4)\,\mathbf{d} =\mathbf{K}\mathbf{c},\,(5)\,\mathbf{e}=\mathbf{L}_{\mathbf{t}}^{-1}(\mathbf{t}\odot\mathbf{d}) \text{ and }\]
\[\mathbf{\psi}=\mathbf{b}\odot\mathbf{d}+\mathbf{a}\odot\mathbf{e}-\mathbf{e}\odot\mathbf{Z}(\mathbf{t} \odot\mathbf{b})-\mathbf{b}\odot\mathbf{Z}(\mathbf{t}\odot\mathbf{e}).\]
_Then for \(\mathbf{t}\in(0,1)^{n}\),_
\[\nabla g_{\lambda}(\mathbf{t})=2\mathbb{E}\left[\mathbf{\psi}\right]+\lambda\mathbf{1}.\]
Using these results, we can obtain for a Monte-Carlo size \(M\), the approximations \(\nabla f_{\lambda}(\mathbf{t})\approx 2\left(\frac{1}{M}\sum_{i=1}^{M}\mathbf{\phi}^{(i)} \right)+\lambda\mathbf{1}\) and \(\nabla g_{\lambda}(\mathbf{t})\approx 2\left(\frac{1}{M}\sum_{i=1}^{M}\mathbf{\psi}^{(i)} \right)+\lambda\mathbf{1}\), where \(\mathbf{\phi}^{(i)}\) and \(\mathbf{\psi}^{(i)}\) are evaluated using a sample \(\mathbf{z}^{(i)}\) drawn from the Rademacher distribution.
These results show that to evaluate stochastic gradients one needs to solve linear systems efficiently with the matrix \(\mathbf{L}_{\mathbf{t}}\). These systems can be iteratively solved using the conjugate gradient (CG) algorithm (Golub and Van Loan, 1996) which uses a sequence of MVMs with \(\mathbf{L}_{\mathbf{t}}\). Multiplying a vector with \(\mathbf{L}_{\mathbf{t}}\) can be reduced to a single MVM with the matrix \(\mathbf{K}\) and a sequence of element-wise vector multiplications and additions.
### Obtaining a Solution
While we have re-framed both the CSSP and the Nystrom problem as an optimization over \(\mathbf{t}\in[0,1]^{n}\), the priority remains to obtain an approximate solution \(\mathbf{s}\in\{0,1\}^{n}\) to (2) and (3). To obtain such a binary vector, we first initialize SGD from a starting point \(\mathbf{t}^{(0)}\) and return the final value \(\mathbf{t}^{*}\) after a termination condition for SGD has been satisfied. Under SGD the iterative sequence \(\{\mathbf{t}^{(i)}\}_{i\geq 0}\) moves towards a corner point of the hypercube. To obtain the closest corner point \(\mathbf{s}\in\{0,1\}^{n}\), we map the insignificant \(t_{j}^{*}\)'s to \(0\) and all the other \(t_{j}^{*}\)'s to \(1\) for some tolerance parameter \(\tau\in(0,1)\). This implementation is shown Algorithm 1. In Figure 1 we provide example solution paths \(\{\mathbf{t}^{(i)}\}_{i\geq 0}\) under both batch gradient descent and SGD.
When choosing the value for \(\mathbf{t}^{(0)}\) it is important to consider the following true statements: \(t_{j}=0\) if and only if \(w_{j}=0\) and
\[\lim_{w_{j}\to 0}\frac{\partial f_{\lambda}(\mathbf{t}(\mathbf{w}))}{\partial w_{j}}=\lim_{ w_{j}\to 0}\frac{\partial g_{\lambda}(\mathbf{t}(\mathbf{w}))}{\partial w_{j}}=0.\]
These facts imply that if \(t_{j}\) is set to zero during the course of the optimization it will remain unchanged thereafter. Therefore, it is important to choose \(\mathbf{t}^{(0)}\) that is away from any corner point. It is for this reason, we set \(\mathbf{t}^{(0)}=(1/2,\ldots,1/2)^{\top}\) in all our experiments.
### Dimensionality Reduction
In Section 3.3 we stated that if \(t_{j}\) is set to zero during the course of the SGD then it will remain unchanged thereafter. This opens the possibility to reduce the computational cost of estimating \(\nabla f_{\lambda}(\mathbf{t}(\mathbf{w}))\) and \(\nabla g_{\lambda}(\mathbf{t}(\mathbf{w}))\) by only focusing on terms where \(t_{j}\neq 0\).
Let \(\mathcal{N}=\{1,\dots,n\}\) and for any \(\mathbf{t}\in[0,1)^{n}\) let \(\mathcal{I}_{\mathbf{t}}=\{j:t_{j}=0\}\). For a vector \(\mathbf{a}\in\mathbb{R}^{n}\), denote the vector \((\mathbf{a})_{+}\) of dimension \(n-|\mathcal{I}_{\mathbf{t}}|\) (respectively \(|\mathcal{I}_{\mathbf{t}}|\)) constructed from \(\mathbf{a}\) by removing the elements with indices that are in \(\mathcal{I}_{\mathbf{t}}\) (respectively, in \(\mathcal{N}\setminus\mathcal{I}_{\mathbf{t}}\)). Likewise, for a matrix \(\mathbf{A}\in\mathbb{R}^{n\times n}\), denote the principal sub-matrix \((\mathbf{A})_{+}\) (respectively, \((\mathbf{A})_{0}\)) that is constructed by removing the rows and columns with indices that are in \(|\mathcal{I}_{\mathbf{t}}|\) (respectively, in \(\mathcal{N}\setminus\mathcal{I}_{\mathbf{t}}\)). Then, we have the following result.
**Lemma 3.3**.: _For any expression \(\mathbf{q}=\mathbf{L}_{\mathbf{t}}^{-1}(\mathbf{t}\odot\mathbf{r})\) where \(\mathbf{r}\in\mathbb{R}^{n}\) and \(\mathbf{t}\in[0,1)^{n}\),_
\[(\mathbf{q})_{0}=\mathbf{0}\quad\text{and}\quad(\mathbf{q})_{+}=\left((\mathbf{L}_{\mathbf{t} })_{+}\right)^{-1}\left((\mathbf{t})_{+}\odot(\mathbf{r})_{+}\right),\]
_where,_
\[(\mathbf{L}_{\mathbf{t}})_{+}=(\mathbf{T})_{+}(\mathbf{K})_{+}(\mathbf{T})_{+}+ \delta(\mathbf{I}-(\mathbf{T})_{+}^{2}).\]
To incorporate this result in our algorithm, we set a small constant \(\epsilon\) and during the course of SGD if \(\mathbf{t}(\mathbf{w})_{j}<\epsilon\), we set its value to zero. Thereafter, when solving \((2)\) and \((5)\) in either Lemma 3.1 (CSSP) or Lemma 3.2 (Nystrom) the dimension of the linear system is \(n-|\mathcal{I}_{\mathbf{t}}|<n\).
### Complexity Analysis
The main computational cost of our algorithm is the complexity attributed to estimating the gradients at each iteration of SGD. For simplicity of analysis, we assume the dimensionality reduction described in Section 3.4 is not carried out. The cost to solve \((2)\) and \((5)\) in either Lemma 3.1 or 3.2 via CG is \(O(T_{mult}M\ell)\) flops where \(\ell\) is the number of CG iterations and \(T_{mult}\) is the cost of computing a matrix-vector product with either \(\mathbf{X}^{\top}\mathbf{X}\) (CSSP) or kernel matrix \(\mathbf{K}\) (Nystrom). Generally, only \(\ell\ll n\) iterations of CG are required to obtain an accurate solution to the linear system.
The cost \(T_{mult}\) is \(O(mn)\) and \(O(n^{2})\) via direct computation for \(\mathbf{X}^{\top}\mathbf{X}\) and \(\mathbf{K}\) respectively. For kernel matrices with specific structure, this cost can be reduced. For example, for Toeplitz matrices or for matrices constructed from a kernel function that is analytic and isotropic, the cost can be reduced to quasi-linear complexity (Dietrich and Newsam, 1997; Gardner et al., 2018; Ryan et al., 2022). Utilizing GPU hardware for accelerating matrix computations has gained significant recent attention and numerous software regimes (Charlier et al., 2021; Hu et al., 2022) have been proposed to accelerate kernel MVMs. These methods can be implemented out-of-the-box and allow MVMs to be feasible on very large datasets (\(n\sim 10^{8}\)). Another advantage of these algorithms is that, as long as the kernel function \(h(\mathbf{x}_{i}^{\prime},\mathbf{x}_{j}^{\prime})\) is given, MVMs can be computed directly without ever storing the kernel matrix \(\mathbf{K}\). This is an advantage of our method when compared to other methods such as the greedy selection method for the Nystrom approximation in (Farahat et al., 2011), which has a cost of \(O(n^{2}k)\) and requires the full explicit matrix to be stored in memory.
### Role of parameters \(\delta\) and \(\lambda\)
The tuning parameter \(\lambda\) controls the size of the penalty \(\|\mathbf{t}\|_{1}\) which is added to the Frobenius matrix loss. It is intuitive then that for a larger value of \(\lambda\) a stronger shrinkage is applied to \(\mathbf{t}\) during the course of the continuous optimization. In terms of curvature, as \(\lambda\) increases so does the directional slope of \(f_{\lambda}(\mathbf{t}(\mathbf{w}))\) and \(g_{\lambda}(\mathbf{t}(\mathbf{w}))\) in the region around \(w_{j}=0\). For this reason, it is likelier that more \(w_{j}\)'s will be pushed
Figure 1: Convergence of \(\mathbf{t}\) for continuous Column Subset Selection using the MNIST dataset. Blue trajectories correspond to selected columns. Only a subset of 300 randomly chosen column trajectories (out of 784) are displayed. For both (a) and (b), \(\lambda=10\) and \(\delta=10\). In (b) the Monte-Carlo size is \(M=5\).
towards zero when the value for \(\lambda\) is large. This behavior is similar to that of the parameter \(\lambda\) in the COMBSS method (Moka et al., 2022) where a more formal analysis can be found. We note that the relationship between \(\lambda\) and \(k\) is data dependent and it is suggested that the user apply an efficient grid search regime to obtain an appropriate \(\lambda\) for their use.
With respect to the parameter \(\delta\) we first note that Lemma 2.2 and Lemma 2.6 remain true regardless of the choice of \(\delta\). Therefore, the value of \(\delta\) affects the behavior of the penalized loss only at the interior points \(\mathbf{t}\in(0,1)^{n}\). We would like a choice of \(\delta\) such that for all the interior points \(\mathbf{t}\in(0,1)^{n}\) the functions \(f_{\lambda}(\mathbf{t})\) and \(g_{\lambda}(\mathbf{t})\) are well-behaved. When \(\delta\) is very small the linear systems that require solving at \(\mathbf{t}\in(0,1)^{n}\) may be close to singular and numerical issues can arise more frequently. Moreover, when \(\delta\) is large we observe large shifts in the value of the objective approaching a corner point. Our simulations indicate that \(\delta=1\) produces a well-behaved function.
## 4 Numerical Experiments and Results
In this section, we provide numerical examples with real data designed to demonstrate that our proposed continuous optimization method outperforms well-known sampling-based methods for small and large datasets. Moreover, we demonstrate that when it is feasible to run greedy selection, our continuous method exhibits very similar performance.
Numerical experiments were conducted on the small to medium-sized datasets: Residential and Building dataset (\(m=372\), \(n=109\)), MNIST1K (\(m=1000\), \(n=784\))1, Arrhythmia dataset (\(m=452\), \(n=279\)), SECOM (\(m=1567\), \(n=591\)). Numerical experiments for Nystrom landmark selection were also conducted on the larger datasets: Power Plant dataset (\(m=4\), \(n=9568\)), HTRU2 dataset (\(m=8\), \(n=17898\)) and Protein dataset (\(m=9\), \(n=45730\)). All datasets except MNIST are downloaded from UCI ML Repository (Asuncion and Newman, 2007). All datasets were standardized such that all columns had mean zero and variance equal to one.
Footnote 1: [https://yann.lecun.com/exdb/mnist/](https://yann.lecun.com/exdb/mnist/)
For the small to medium-sized datasets, we use the best rank-k approximation factor to compare our method to existing methods (see Figure 2 and Figure 3). The best rank-k approximation factor is given by
\[\text{Approximation Factor}:=\frac{\|\mathbf{A}-\widehat{\mathbf{A}}_{s}\|_{F}^ {2}}{\|\mathbf{A}-\widehat{\mathbf{G}}\|_{F}^{2}}.\]
where \(\widehat{\mathbf{A}}_{s}\) is either the Nystrom or CSSP low-rank matrix and \(\widehat{\mathbf{G}}\) is the best rank-k approximation computed using the Singular Value Decomposition (SVD) of \(\mathbf{A}\).
In these experiments, we compare the proposed continuous landmark selection method executed with SGD (\(M=10\)) with the following four well-known methods: Uniform Sam
Figure 3: The mean CSSP empirical approximation factor over 50 trials for the MNIST dataset and three UCI datasets for different methods. Approximation factor is plotted on a logarithmic scale.
Figure 2: The mean Nyström empirical approximation factor over 50 trials for the UCI Residential Building and MNIST dataset where \(\mathbf{K}\) is constructed using the Gaussian Radial Basis Function (RBF) kernel: \(\mathbf{K}_{i,j}=h(\mathbf{x}_{i}^{\prime},\mathbf{x}_{j}^{\prime})=\exp\left(-\|\mathbf{x }_{i}^{\prime}-\mathbf{x}_{j}^{\prime}\|^{2}\right)/\sigma^{2}\). Approximation factor is plotted on a logarithmic scale.
pling (Williams and Seeger, 2000), Recursive RLS (Ridge Leverage Scores) - Nystrom sampling (Musco and Musco, 2017), k-DPP sampling (Derezinski and Mahoney, 2021) and Greedy selection (Farahat et al., 2011; 2013).
For the experiments conducted on the larger datasets (see Figure 4) we exclude the k-DPP sampling and greedy methods as it is either too costly to compute the choice of landmark points or too costly to store the full kernel matrix on a GPU. In our implementation of continuous Nystrom landmark selection, we use the KeOps library (Charlier et al., 2021) to efficiently compute MVMs and linear solves on a GPU without ever storing the matrix \(\mathbf{K}\), thus negating the need to store any \(O(n^{2})\) objects. These experiments were run using an NVIDIA Tesla T4 GPU with 16GB memory.
In Figure 2 and Figure 3 we observe the approximation factor for Nystrom and CSSP landmark selection with different subset sizes \(k\). A lower approximation factor indicates a better approximation and an approximation factor close to one implies near-best-case performance for the given subset size \(k\). The results indicate that the continuous optimization method is better than every tested sampling method and is very similar to greedy selection in performance (whenever the greedy selection is feasible). In most cases, for the CSSP, as the proportion of selected columns increases the continuous method starts to marginally outperform the greedy method.
In Figure 4, we observe for all three datasets (Power Plant, HTRU2 and Protein) that the continuous landmark selection achieves better accuracy than the Recursive RLS (Ridge Leverage Scores) - Nystrom sampling and Uniform sampling methods. While Recursive RLS sampling (complexity: \(O(nk^{2})\)) and uniform sampling are faster at selecting landmark points, for a fixed \(k\) the continuous method obtains a more accurate Nystrom approximation. Thus, if a memory budget for the size of the Nystrom approximation is given, as is often the case, the continuous method will compute a superior approximation.
## 5 Conclusion
In this paper, we have introduced a novel algorithm that exploits unconstrained continuous optimization to select columns for both the CSSP and Nystrom approximation. The algorithm selects columns by minimizing an extended objective which is defined over the hypercube \([0,1]^{n}\) rather than iterating over the corner points of the hypercube which correspond to all of the \(\binom{n}{k}\) subsets. The extended objective for both the CSSP and Nystrom approximation can be minimized via SGD where the gradients are estimated with an unbiased estimator which requires only MVMs with either \(\mathbf{X}\) (CSSP) or \(\mathbf{K}\) (Nystrom). On the real-world examples that we considered in this article, the proposed method has proven to be more accurate without incurring higher computational cost.
|
2308.07580 | AutoLTS: Automating Cycling Stress Assessment via Contrastive Learning
and Spatial Post-processing | Cycling stress assessment, which quantifies cyclists' perceived stress
imposed by the built environment and motor traffics, increasingly informs
cycling infrastructure planning and cycling route recommendation. However,
currently calculating cycling stress is slow and data-intensive, which hinders
its broader application. In this paper, We propose a deep learning framework to
support accurate, fast, and large-scale cycling stress assessments for urban
road networks based on street-view images. Our framework features i) a
contrastive learning approach that leverages the ordinal relationship among
cycling stress labels, and ii) a post-processing technique that enforces
spatial smoothness into our predictions. On a dataset of 39,153 road segments
collected in Toronto, Canada, our results demonstrate the effectiveness of our
deep learning framework and the value of using image data for cycling stress
assessment in the absence of high-quality road geometry and motor traffic data. | Bo Lin, Shoshanna Saxe, Timothy C. Y. Chan | 2023-08-15T05:51:25Z | http://arxiv.org/abs/2308.07580v1 | # AutoLTS: Automating Cycling Stress Assessment via Contrastive Learning and Spatial Post-processing
###### Abstract
Cycling stress assessment, which quantifies cyclists' perceived stress imposed by the built environment and motor traffics, increasingly informs cycling infrastructure planning and cycling route recommendation. However, currently calculating cycling stress is slow and data-intensive, which hinders its broader application. In this paper, We propose a deep learning framework to support accurate, fast, and large-scale cycling stress assessments for urban road networks based on street-view images. Our framework features i) a contrastive learning approach that leverages the ordinal relationship among cycling stress labels, and ii) a post-processing technique that enforces spatial smoothness into our predictions. On a dataset of 39,153 road segments collected in Toronto, Canada, our results demonstrate the effectiveness of our deep learning framework and the value of using image data for cycling stress assessment in the absence of high-quality road geometry and motor traffic data.
## 1 Introduction
Safety and comfort concerns have been repeatedly identified as major factors that inhibit cycling uptake in cities around the world. A range of metrics, such as the level of traffic stress (LTS) [11, 12] and bicycle level of service index [13], have been proposed to quantify cyclists' perceived stress imposed by the built environment and motor traffic. These metrics are predictive of cycling behaviors [1, 10] and accidents [15], and thus have been applied to support cycling infrastructure planning [10, 11, 12] and route recommendation [15, 16]. However, calculating these metrics typically requires high-resolution road network data, such as motor traffic speed, the locations of on-street parking, and the presence/type of cycling infrastructure on each road segment. The practical challenge of collecting accurate and up-to-date data hinders the broader application of cycling stress assessment and tools built on it.
To tackle this challenge, we propose AutoLTS, a deep learning framework for assessing cycling stress of urban road networks based on street-view images. AutoLTS can facilitate timely, accurate, and large-scale assessments of cycling stress because up-to-date street-view images are easy to access via the Google StreetView API. Using a dataset of 39,153 road segments collected in Toronto, Canada, we focus on automating the calculation of the LTS metric. Specifically, as shown in Figure 1, road segments are classified into four classes, i.e., LTS 1, 2, 3 and 4 [14], corresponding to the cycling suitability of four types of cyclists, where LTS 1 is the least stressful road and LTS 4 is the most stressful. This metric has been applied to investigate the connectivity [10, 11] and equity [12] of urban cycling networks and to evaluate cycling interventions during the COVID-19 pandemic [13]. While we focus on LTS for demonstration, our approach applies to any cycling stress metric.
Formulating this task as a simple image classification
Figure 1: Example images with the four LTS labels: LTS1 roads are safe for all cyclists including children, LTS2 roads are for most adults, LTS3 and LTS4 are for “enthused and confident” and “strong and fearless” cyclists, respectively.
problem may not utilize the training dataset to its full potential because it ignores i) the causal relationship between road features and LTS, ii) the ordinal relationships among LTS labels, and iii) the spatial structure of urban road networks. It is critical to leverage i)-iii) to improve the prediction performance as our dataset, limited by the practical data collection challenge and the number of road segments in a city, is relatively small for a computer vision task. Item ii) is of particular importance as misclassifications between different pairs of LTS labels carry different empirical meanings. For example, predicting an LTS1 road as LTS3 is considered worse than predicting it as LTS2 because LTS2 corresponds to the cycling stress tolerance of most adults [14]. The former may lead to redundant cycling infrastructure on a low-stress road and or recommended cycling routes that exceed most adults' stress tolerance.
As illustrated in Figure 2, to capture i), we formulate the LTS assessment as a two-step learning task. We first predict LTS related road features based on the input image and learn high-quality representations of the image. We then combine the image embedding with the predicted and available road features to produce the final LTS prediction. This two-step framework allows us to capture ii) and iii) via _contrastive learning_ and a _spatial post-processing_ technique, respectively. Specifically, to address ii), we propose a contrastive learning approach to learn an image embedding space where images are clustered based on their LTS labels, and where these clusters are positioned according to the ordinal relationship among these labels. To tackle iii), we develop a post-processing technique to enforce spatial smoothness into road feature predictions. We opt not to directly enforce spatial smoothness into LTS predictions because it may smooth over important local patterns, which are critical for downstream applications such as cycling network design that aims to fix the disconnections between low-stress sub-networks. Our contributions are summarized below.
1. **A novel application.** We introduce the first dataset and the first computer vision framework for automating cycling stress assessment.
2. **New methodologies.** We propose a new contrastive loss for ordinal classification that generalizes the supervised contrastive loss [15]. We develop a post-processing technique that adjusts the road feature predictions considering the spatial structure of the road network. Both can be easily generalized to other tasks.
3. **Strong performance.** Through comprehensive experiments using a dataset collected in Toronto, Canada, we demonstrate i) the value of street-view images for cycling stress assessment, and ii) the effectiveness of our approach in a wide range of real-world settings.
## 2 Literature Review
**Computer vision for predicting urban perceptions.** Street view images have been used to assess the perceived safety, wealth, and uniqueness of neighborhoods [2, 1, 16, 17] and to predict neighborhood attributes such as crime rate, housing price, and voting preferences [1, 15]. We contribute to this stream of literature by i) proposing the first dataset and the deep-learning framework for assessing cycling stress, and ii) developing the first post-processing technique to enforce spatial smoothness in model predictions. Our proposal of automating cycling stress assessment via a computer vision approach is similar to the work of [16] who use pre-trained image segmentation and object detection models to extract road features and then construct a bike-ability index based on them. In contrast, we focus on automating the calculation of a cycling stress metric that is well-validated in the transportation literature. The approach proposed by [16]
Figure 2: An overview of AutoLTS. The input image is encoded to an image embedding and is used to predict missing road features. The image encoder is trained using a contrastive learning approach (Section 3.2). The predicted road features go through a post-processing module (Section 3.3) that enforces spatial smoothness into the predictions. Finally, a feedforward network predicts the the image’s LTS label based on the image embedding, and the predicted and available road features.
and Biljecki 2021) does not apply because many LTS-related road features are i) unlabeled in the dataset on which the segmentation and object detection models were trained (e.g. road and cycling infrastructure types) or ii) not observable in street-view images (e.g. motor traffic speed).
**Contrastive learning.** Contrastive learning, which learns data representations by contrasting similar and dissimilar data samples, has received growing attention in computer vision. Such techniques usually leverage a contrastive loss to guide the data encoder to pull together similar samples in an embedding space, which has been shown to facilitate downstream learning in many applications (Zhao et al., 2021; Bengar et al., 2021; Bjorck et al., 2021), especially when data labels are unavailable or scarce. To date, most contrastive learning approaches are designed in unsupervised settings (Gutmann and Hyvarinen, 2010; Sohn, 2016; Oord, Li, and Vinyals, 2018; Hjelm et al., 2018; Wu et al., 2018; Bachman, Hjelm, and Buchwalter, 2019; He et al., 2020; Chen et al., 2020). They typically generate "similar" data by applying random augmentations to unlabeled data samples. More recently, Khosla et al. (2020) apply contrastive learning in a supervised setting where they define "similar" data as data samples that share the same image label. Linear classifiers trained on the learned embeddings outperform image classifiers trained directly based on images. We extend the supervised contrastive loss (Khosla et al., 2020) by augmenting it with terms that measure the similarity of images with "neighboring" labels. Consequently, the relative positions of the learned embeddings reflect the similarity between their class labels, which helps to improve our model performance.
## 3 Method
### Data Collection and Pre-processing
Training and testing our model requires three datasets: i) road network topology, ii) ground-truth LTS labels for all road segments, and iii) street-view images that clearly present the road segments. We collect all the data in Toronto, Canada via a collaboration with the City of Toronto. Data sources and pre-processing steps are summarized below.
**Road network topology**: We retrieve the centerline road network from City of Toronto (2020). Geospatial coordinates of both ends of each road segment are presented. We exclude roads where cycling is legally prohibited, e.g., expressways. The final network has 59,554 road segments.
**LTS label.** The LTS calculation requires detailed road network data. For each road segment in Toronto, we collect road features as summarized in Table 1 and calculate its LTS label following Furth, Mekuria, and Nixon (2016) and Imani, Miller, and Saxe (2019) (detailed in Appendix B).
**Street-view image.** We collect street-view images using the Google StreetView API. We opt not to collect images for road segments that are shorter than 50 meters because a significant portion of those images typically present adjacent road segments that may have different LTS labels. For each of the remaining road segments, we collect one image using the geospatial coordinate of its mid-point. We manually examine the collected images to ensure that they clearly present the associated road segments. If an image fails the human screening, we manually recollect the image when possible. Images are missing for roads where driving is prohibited, such as trails and narrow local passageways.
Our final image dataset consists of 39,153 high-quality street-view images, with 49.0%, 34.5%, 6.9%, and 9.7% of them labeled as LTS 1, 2, 3 and 4, respectively.
### Supervised Contrastive Learning for Ordinal Classification
We propose a contrastive learning approach to train the image encoder. The novelty lies in the development of a new contrastive loss that considers the ordinal relationship among LTS labels. We adopt a contrastive learning framework (Figure 3) similar to MoCo (He et al., 2020) to train the image encoder \(f\) on a pretext task where the encoder learns to pull together "similar" images in the embedding space. Given a batch of \(n\) road segments indexed by \(\mathcal{N}\), let \(\mathbf{x}_{i}\) and \(y_{i}\) denote the street view image and the label of segment \(i\in\mathcal{N}\), respectively. We assume \(y_{i}\in[m]\) are discrete and ordered for all \(i\in\mathcal{N}\). We create \(l\) virtual labels \((y_{i}^{1},y_{i}^{2},\dots,y_{i}^{l})\) for each image \(\mathbf{x}_{i}\) where \(y_{i}^{u}=[y_{i}/u]\) for all \(u\in[l]\). In words, these virtual labels are created by grouping the "neighboring" real labels at different granularities. Consequently, images with "similar" real labels have more overlapping virtual labels. We create two views \(\tilde{\mathbf{x}}_{i}\) and \(\tilde{\mathbf{x}}_{i}\) of each image \(\mathbf{x}_{i}\) by applying a random augmentation module twice. We create a momentum encoder \(g\) that has the same structure as \(f\) and whose parameters are updated using the momentum update (He et al., 2020) as we train \(f\). The image views \(\{\tilde{\mathbf{x}}_{i}\}_{i\in[n]}\) and \(\{\tilde{\mathbf{x}}_{k}\}_{k\in\mathcal{K}}\) are encoded by \(f\) and \(g\), respectively, where \(\mathcal{K}\) is a fixed-length queue that stores previously generated image views. Let \(\tilde{\mathbf{z}}_{i}=f(\tilde{\mathbf{x}}_{i})\) and \(\tilde{\mathbf{z}}_{i}=g(\tilde{\mathbf{x}}_{i})\) denote the embedding generated by these two encoders. During training, these embeddings are further fed into a projection layer, which is discarded during inference following Khosla et al. (2020) and Chen et al. (2020). The encoder network \(f\) is trained to minimize the following loss that applies to the projected embeddings:
\[L^{\text{ord}}=-\frac{1}{N}\sum_{i\in\mathcal{N}}\sum_{u\in[l]}\frac{w^{u}}{| \mathcal{K}_{i}^{u}|}\sum_{j\in\mathcal{K}_{i}^{u}}\log\frac{\exp\left[\text{p }(\tilde{\mathbf{z}}_{i})^{\intercal}\text{p}(\tilde{\mathbf{z}}_{j})/\tau \right]}{\sum_{k\in\mathcal{K}}\exp\left[\text{p}(\tilde{\mathbf{z}}_{i})^{ \intercal}\text{p}(\tilde{\mathbf{z}}_{k})/\tau\right]}, \tag{1}\]
where \(\mathcal{K}_{i}^{u}=\{k\in\mathcal{K}:y_{i}^{u}=y_{k}^{u}\}\) for all \(u\in[l]\), \(w^{u}\) is a constant weight assigned to the \(u^{\text{th}}\) virtual label, \(\tau\) is a temperature hyper-parameter, and p is the projection function.
\begin{table}
\begin{tabular}{l l} \hline \hline Feature & Source \\ \hline Road type & (City of Toronto, 2020) \\ Road direction & (City of Toronto, 2020) \\ Number of lanes & (Government of Canada, 2020) \\ Motor traffic speed & (Travel Modelling Group, 2016) \\ Cycling infrastructure location & (City of Toronto, 2020) \\ On-street parking location & (Toronto Parking Authority, 2020) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of LTS-related road features.
**Comparison to other loss functions**. Compared to MoCo [14], our OrdCon takes advantage of label information. Consequently, as illustrated in Figure 3, our image embeddings form clusters that correspond to their image labels. Compared to the SupCon [11], our OrdCon considers the ordinal relationship among image labels by aggregating the real label at different granularities. As a result, the relative positions of our embedding clusters reflect the similarity between their corresponding labels. OrdCon recovers the SupCon when \(l=1\) and \(w^{1}=1\).
### Spatial Post-processing for Road Feature Predictions
Several LTS-related road features, e.g., motor traffic speed, have strong spatial correlations, meaning that the values associated with adjacent road segments are highly correlated. Such structure can be useful in regulating road feature predictions, which may lead to improved LTS predictions. However, it is often not obvious how spatial smoothness should be enforced. For example, consider a case where the motor traffic speeds of five consecutive road segments are predicted as 60, 40, 60, 40, and 60 km/h, respectively. It is likely that two of them are wrong, yet it is unclear if we should change the 40s to 60 or 60s to 40. In this section, we propose a principled way to address this problem.
A Causal ModelWe start by introducing a directed arc graph (DAG) (illustrated in Figure 4) that describes the relationships between the inputs \(\mathbf{x}_{i}\) (i.e., street view images) and targets \(a_{i}\in\mathcal{A}\) (i.e., the road feature of interest) of our road-feature prediction module (illustrated in Figure 2). We assume \(\mathcal{A}\) to be discrete. This is not restrictive because continuous road features can be categorized according to the LTS calculation scheme (detailed in Appendix C). Let \(\mathcal{I}\) denote the set of edges in the road network and \(\mathcal{J}(i)\subset\mathcal{I}\) denote the set of road segments that are adjacent to road segment \(i\in\mathcal{I}\). We make three assumptions as listed below.
1. For any \(i\in\mathcal{I}\) and \(k\in\mathcal{I}\backslash\mathcal{J}(i)\), \(a_{i}\) and \(a_{k}\) are conditionally independent given \(\{a_{j}\}_{j\in\mathcal{J}(i)}\).
2. For any \(i,j\in\mathcal{I}\) and \(j\neq i\), \(\mathbf{x}_{i}\) and \(a_{j}\) are conditionally independent given \(a_{i}\).
3. For any \(i\in\mathcal{I}\) and \(j,k\in\mathcal{J}(i)\) and \(j\neq k\), \(a_{j}\) and \(a_{k}\) are conditionally independent given \(a_{i}\).
The first and second assumptions state that the target \(a_{i}\) is directly influenced only by the input of the same road segment \(\mathbf{x}_{i}\) and the targets of its adjacent segments \(\{a_{j}\}_{j\in\mathcal{J}(i)}\). The third assumption states that when target \(a_{i}\) is known, its impacts on its adjacent targets \(\{a_{j}\}_{j\in\mathcal{J}(i)}\) are independent. This model naturally applies to several LTS-related road features. For example, the traffic speed on a road segment is affected by the built environment observable from its street-view image (\(\mathbf{x}_{i}\)) and the traffic speeds on its adjacent road segments (\(\{a_{j}\}_{j\in\mathcal{J}(i)}\)). The built environment and traffic speeds on other road segments may present indirect impacts on the road segment of interest, but such impacts must transmit through its adjacent road segments. Additionally, the impacts of the traffic speed on a road segment on the speeds of its adjacent road segments can be viewed as independent (or weakly dependent) because they usually correspond to motor traffics along different directions.
Enforcing Spatial SmoothnessGiven the DAG, target predictions can be jointly determined by maximizing the joint probability of all targets given all inputs, i.e., \(\text{maximize}_{\mathbf{a}}P\left(\{a_{i}\}_{i\in\mathcal{I}}|\{\mathbf{x}_{i }\}_{i\in\mathcal{I}}\right)\). However, evaluating the joint distribution of \(\{a_{i}\}_{i\in\mathcal{I}}\) is non-trivial because our DAG is cyclic. Instead, we look into determining the target of one road segment at a time assuming all other targets are fixed.
Figure 4: A causal model for road feature predictions. The blue lines indicate real-world road segments, black arrows represent causal impacts.
Figure 3: The contrastive learning framework and the learned image embeddings from different contrastive losses. MoCo indicates the self-supervised contrastive loss, SupCon indicates the supervised contrastive loss, and OrdCon indicates our contrastive loss. All the embeddings are projected to a two-dimensional space via T-SNE [13]. Each point corresponds to one street-view image and is color-coded according to the associated LTS label.
**Proposition 1**.: _Under assumptions 1-3, for any \(i\in\mathcal{I}\),_
\[P\left(a_{i}|\{\mathbf{x}_{i}\}_{i\in\mathcal{I}},\{a_{j}\}_{j\neq i \in\mathcal{I}}\right)\propto\prod_{j\in\mathcal{J}(i)}P\left(a_{j}|a_{i} \right)P(a_{i}|\mathbf{x}_{i}) \tag{2}\]
Proposition 1 decomposes the conditional probability of target \(a_{i}\) given all other targets and inputs. The transition probability \(P(a_{j}|a_{i})\) can be estimated from our training data, and \(P(a_{i}|\mathbf{x}_{i})\) can be produced by our deep learning model. The proof is presented in Appendix A. Inspired by Proposition 1, we next introduce an algorithm that iteratively updates the target predictions in the whole network until there are no further changes. The algorithm is summarized in Algorithm 1.
```
0: Initial predictions \(\{a_{i}\}_{i\in\mathcal{I}}\); Transition Probabilities \(\{P(a|a^{\prime})\}_{a,a^{\prime}\in\mathcal{A}}\); Model Predictions \(\{P(a_{i}|\mathbf{x}_{i})\}_{i\in\mathcal{I}}\); Adjacent sets \(\mathcal{J}(i)\) for any \(i\in\mathcal{I}\).
0: Updated predictions \(\{\hat{y}_{i}\}_{i\in\mathcal{I}}\).
1:repeat
2: set \(\hat{a_{i}}\gets a_{i}\) for all \(i\in\mathcal{I}\).
3:for\(i\in\mathcal{I}\)do
4: set \(a_{i}\leftarrow\arg\max_{a\in\mathcal{A}}\prod_{j\in\mathcal{J}(i)}P\left( \hat{a}_{j}|a\right)P(a|\mathbf{x}_{i})\)
5:until\(\hat{a}_{i}=a_{i}\) for all \(i\in\mathcal{I}\)
```
**Algorithm 1** An iterative target adaptation algorithm
## 4 Empirical Results
### Experiment Setup
**Evaluation scenarios.** We evaluate AutoLTS and baseline methods in three data-availability scenarios, each under four train-test-validation splits, totaling 12 sets of experiments.
For _data availability_, we consider LTS based on
1. Street view image
2. Street view image, road and cycling infrastructure types
3. Street view image, number of lanes, and speed limit.
The design of these scenarios is informed by the real data collection challenges we encountered in Toronto. The number of lanes and the speed limit of each road segment are accessed via Open Data Canada (Government of Canada, 2020). Road type and the location of cycling infrastructure are available via Open Data Toronto (City of Toronto, 2020). However, as the two data platforms use different base maps, combining data from these two sources requires considerable manual effort, echoing the data collection challenges in many other cities.
For the _train-test-validation split_, we consider
1. A random 70/15/15 train-test-validation split across all road segments in Toronto.
2. Three spatial splits, which use road segments in an area as the test set and performs a random 80/20 train-validation split for other road segments. As shown in Figure 5, we consider using road segments in three of Toronto's amalgamated cities, York, Etobicoke, and Scarborough as the test sets. These three areas have very different LTS distributions, which allows us to exam the generalization ability of AutoLTS in real-world settings.
The random split mimics the situation where we use AutoLTS to extrapolate manual LTS assessment or to update the LTS assessment in the city where the model is trained. The spatial split mimics the situation where we apply AutoLTS trained in one city to an unseen city.
**Evaluation Metrics.**
* LTS Prediction Accuracy \[\text{Acc}=\frac{1}{N_{\text{test}}}\sum_{i=1}^{N_{\text{test}}}\mathds{1}[y_{ i}=\hat{y}_{i}].\] (3)
* High/Low-Stress Prediction Accuracy \[\text{HLA}=\frac{1}{N_{\text{test}}}\sum_{i=1}^{N_{\text{test}}}\mathds{1}[h(y _{i})=h(\hat{y}_{i})]\] (4) where \(h\) is a function that takes a value of \(1\) if the input LTS label is low-stress (LTS1/2) and takes \(0\) if the LTS label is high-stress (LTS3/4).
* Average False High/Low-Stress Rate \[\text{AFR}=\frac{1}{2}\left\{\frac{\sum_{i=1}^{N_{\text{test}}}\mathds{1}[h( \hat{y}_{i})=1]}{n_{\text{test}}^{l}}+\frac{\sum_{i=1}^{N_{\text{test}}} \mathds{1}[h(\hat{y}_{i})=0]}{n_{\text{test}}^{h}}\right\}\] (5) where \(n_{\text{test}}^{l}\) and \(n_{\text{test}}^{h}\) denote, respectively, the numbers of test road segments that are low- and high-stress.
Acc and HLA measure the overall prediction performance, while AFR considers the fact that the dataset is imbalanced with a higher portion being low-stress. Ideally, we want a model that achieves high Acc and HLA and low AFR.
**Baselines.** To demonstrate the value of image data, in scenarios where road features are available, we use a classification and regression tree (CART) that predicts LTS based on available road features as a baseline. CART is selected because the LTS calculation scheme (Furth, Mekuria, and Nixon, 2016) can be summarized by a decision tree. We also compare AutoLTS with image-based supervised and contrastive learning methods. For supervised learning, we consider Res-50 (He et al., 2016) trained using the cross-entropy loss. For contrastive learning, we consider supervised contrastive learning (MoCo) (He et al., 2020) and self-supervised contrastive learning (SupCon) (Khosla et al., 2020), both implemented with the MoCo trick (He et al., 2020). Baselines are detailed in Appendix E.
**Model details.** We use ResNet-50 (He et al., 2016) as the image encoder. The normalized ReLU activations of the final pooling layer are used as the image embedding (\(\xi\)=2,048). We follow He et al. (2020) to set \(\tau=0.07\) and use the SimCLR augmentation (Chen et al., 2020) for training. We train one ResNet-50 to predict each missing road feature. All road features are discretized using the thresholds defined in the LTS calculation scheme (Furth, Mekuria, and Nixon, 2016) (Appendix C). In the LTS prediction module, we first train a CART model to predict a road segment's LTS
based on its predicted and available road features. We then use the LTS distribution in the leaf node that a road segment is assigned to as its road feature embedding, which is mapped to a \(\xi\)-dimensional space via a linear layer and averaged with the image embedding. Finally, a linear classifier predicts the road segment's LTS based on the averaged embedding. Training details are summarized in Appendix D.
### Main Results
The performance of AutoLTS and baselines are shown in Table 2. We summarize our findings below.
**The value of image data for cycling stress assessment**. AutoLTS achieves LTS prediction accuracy of 62.31%-73.41% and high/low-stress accuracy of 93.50%-94.69% only using street-view images. Such a model can be useful for cycling infrastructure planning and route recommendation tools that do not require the granularity of four LTS categories and focus solely on the difference between high-(LTS3/4) and low-stress (LTS1/2) road segments. In data-availability scenarios where partial road features are available (scenarios two and three), incorporating street-view images leads to increases of 0.43-32.39 percentage points in Acc with little to no increases in AFR. The improvements are particularly large in scenario two where the average increase in Acc due to the usage of street-view images is 23.10 percentage points across all train-test-validation splits considered. By combining street-view images with the speed limit and the number of lanes (scenario 3), the Acc is over 90% under all splits. These numbers demonstrate that street view images are valuable for cycling stress assessment with and without partial road features.
**The performance of AutoLTS and other image-based methods.** Overall, AutoLTS achieves the highest Acc, which is of primary interest, in all evaluation scenarios. Due to the limited sample size, unsupervised contrastive learning (MoCo) generally falls around 10% behind SupCon. Sup
\begin{table}
\begin{tabular}{l l c c c c c c c c c c c c} \hline \hline & & \multicolumn{3}{c}{Random} & \multicolumn{3}{c}{York} & \multicolumn{3}{c}{Etobicoke} & \multicolumn{3}{c}{Scarborough} \\ & & \multicolumn{3}{c}{(\(N_{\text{test}}=5\),873)} & \multicolumn{3}{c}{(\(N_{\text{test}}=2\),091)} & \multicolumn{3}{c}{(\(N_{\text{test}}=6\),667)} & \multicolumn{3}{c}{(\(N_{\text{test}}=8\),921)} \\ \cline{3-14} \multirow{-1}{*}{See.} & \multirow{2}{*}{Method} & Acc & HLA & AFR & Acc & HLA & AFR & Acc & HLA & AFR & Acc & HLA & AFR \\ \hline \multirow{4}{*}{1} & Cross-Entropy & 70.49 & 93.51 & 10.19 & 60.97 & 93.40 & 12.09 & 64.20 & 92.89 & 9.37 & 64.28 & 93.87 & 12.61 \\ & MoCo & 61.69 & 90.23 & 14.68 & 57.68 & 91.34 & 17.17 & 52.03 & 89.89 & 12.71 & 56.16 & 91.45 & 17.01 \\ & SupCon & 70.75 & 93.41 & 11.73 & 61.17 & 93.40 & 17.05 & 64.29 & 93.19 & 9.29 & 65.73 & 93.38 & 10.96 \\ & AutoLTS & **73.41** & **94.16** & 10.50 & **62.31** & **94.69** & 15.72 & **64.69** & **93.50** & 9.87 & **66.04** & **94.62** & **10.77** \\ \hline \multirow{4}{*}{2} & CART & 56.21 & 96.87 & 5.21 & 43.33 & 96.75 & 4.51 & 35.35 & 96.73 & 7.25 & 50.22 & 96.40 & 8.34 \\ & Cross-Entropy & 75.07 & 96.82 & 5.91 & 63.37 & 96.37 & 5.26 & 66.21 & 95.97 & 6.55 & 67.76 & 95.74 & 10.14 \\ & MoCo & 68.94 & 96.65 & 14.41 & 57.39 & 96.22 & 4.81 & 59.11 & 95.79 & 9.08 & 62.09 & 96.31 & 8.66 \\ & SupCon & 74.89 & 96.42 & 6.33 & 64.13 & 96.17 & 5.54 & 65.70 & 95.68 & 9.14 & 68.55 & 96.19 & 8.66 \\ & AutoLTS & **75.86** & 96.22 & 7.02 & **65.04** & 96.13 & 8.77 & **67.74** & 96.20 & 7.37 & **68.86** & **96.51** & **7.78** \\ \hline \multirow{4}{*}{3} & CART & 89.41 & 96.07 & 10.08 & 88.81 & 97.57 & 6.97 & 90.67 & 95.46 & 10.37 & 91.90 & 94.90 & 12.51 \\ & Cross-Entropy & 90.26 & 95.33 & 8.97 & 88.12 & 97.37 & 6.47 & 91.01 & 95.34 & 7.79 & 91.45 & 95.54 & 12.88 \\ \cline{1-1} & MoCo & 89.82 & 95.37 & 11.25 & 86.90 & 98.61 & 3.63 & 90.88 & 96.35 & 6.78 & 92.74 & 94.91 & 12.79 \\ \cline{1-1} & SupCon & 91.20 & 96.19 & 11.30 & 87.42 & 96.70 & 4.18 & 89.04 & 95.93 & 7.69 & 92.41 & 95.11 & 11.50 \\ \cline{1-1} & AutoLTS & **91.65** & **96.70** & **5.87** & **89.24** & 97.23 & 4.78 & **92.61** & **96.68** & **4.77** & **94.50** & **97.28** & **5.81** \\ \hline \hline \end{tabular}
\end{table}
Table 2: The out-of-sample performance of AutoLTS and baselines. The three blocks (top to bottom) correspond to data-availability scenarios 1, 2, and 3, respectively (Section 4.1). The four groups of columns (left to right) correspond to the train-test-validation splits defined in Section 4.1. Numbers in boldface are cases where our approach achieves the best performance.
Figure 5: Illustration of the three spatial splits. York has a similar LTS distribution as the overall city-wide distribution. Etobicoke has the majority of the road segments being LTS2 and more roads being LTS4 compared to the city’s average. Scarborough has an even higher LTS4 percentage.
Con outperforms the simple image classification formulation (Cross-Entropy) in 8 out of 12 scenarios yet is inferior to AutoLTS in all scenarios. However, we observe that when there is a significant domain shift from training to test data (spatial splits), all methods including AutoLTS are more prone to overfitting the training data, and thus have worse out-of-sample performance than in random splits.
### Ablation Studies
Next, we present ablation studies using data-availability scenario one to demonstrate the values of our two-step learning framework, ordinal contrastive learning loss, and the post-processing module, The results are summarized in Table 3.
**The value of the two-step learning framework.** We compare AutoLTS with an alternative approach that replaces the LTS prediction module with the exact LTS calculation scheme (2Step-Exact and 2Step-Spatial-Exact). This change leads to reductions of 6.16-40.83 percentage points in Acc due to the compounded errors from the first step, highlighting the importance of second-step learning. Moreover, AutoLTS outperforms all baselines that predict LTS based only on image (MoCo-, SupCon-, and OrdCon-NN), demonstrating the value of incorporating road feature predictions.
**The value of ordinal contrastive learning.** We compare the three contrastive learning methods using the AutoLTS framework and the linear classification protocol [1]. When used to predict LTS without road features (MoCo-, SupCon-, and OrdCon-NN), OrdCon and SupCon are competitive in Acc, yet OrdCon constantly achieves higher HLA and lower AFR because it considers the relationship among LTS labels. This is practically important because the ability to distinguish between low- and high-stress roads plays a vital role in most adults' cycling decision makings [13]. When combined with road features, all contrastive learning methods perform reasonably well. Nevertheless, Auto-OrdCon consistently outperforms others by a meaningful margin.
**The value of spatial post-processing.** Applying the spatial post-processing technique to road feature predictions generally leads to an increase of around 1% in road feature prediction accuracy (presented in Appendix C) which can be translated into improvements in LTS prediction Acc (2Step-Exact versus 2Step-Spatial-Exact). While the improvement seems to be limited, it corresponds to correctly assessing the LTS of 21-162 road segments in the studied area, which can have a significant impact on the routing and cycling infrastructure planning decisions derived based on the assessment.
## 5 Conclusion
In this paper, we present a deep learning framework, AutoLTS, that uses streetview images to automate cycling stress assessment. AutoLTS features i) a contrastive learning approach that learns image representations that preserve the ordinal relationship among image labels and ii) a post-processing technique that enforces spatial smoothness into the predictions. We show that AutoLTS can assist in accurate, timely, and large-scale cycling stress assessment in the absence of road network data.
Our paper has three limitations, underscoring potential future research directions. First, we observe performance degradation when the training and test data have very different label distributions (spatial splits). Future research may apply domain adaptation methods to boost the performance of AutoLTS in such scenarios. Second, AutoLTS does not consider the specific needs of downstream applications. For instance, in cycling route recommendations, under-estimation of cycling stress may be more harmful than over-estimation because the former may lead to cycling routes that exceed cyclists' stress tolerance and result in increased risks of cycling accidents. In cycling network design, cycling stress predictions might be more important on major roads than on side streets because cycling infrastructure is typically constructed on major roads. Such impacts may be captured by modifying the loss function to incorporate decision errors. Finally, all our experiments are based on a dataset collected in Toronto. Future research may collect a more comprehensive dataset to further assess the generalizability of our model. We hope this work will open the door to using deep learning to support the broader application of cycling stress assessment and to inform real-world decision makings that improve transportation safety and efficiency.
\begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline & \multicolumn{3}{c}{Random} & \multicolumn{3}{c}{York} & \multicolumn{3}{c}{Etobicoke} & \multicolumn{3}{c}{Scarborough} \\ & \multicolumn{3}{c}{(\(N_{\text{test}}=5\),873)} & \multicolumn{3}{c}{(\(N_{\text{test}}=2\),091)} & \multicolumn{3}{c}{(\(N_{\text{test}}=6\),667)} & \multicolumn{3}{c}{(\(N_{\text{test}}=8\),921)} \\ \cline{2-13} Model & Acc & HLA & AFR & Acc & HLA & AFR & Acc & HLA & AFR & Acc & HLA & AFR \\ \hline
2Step-Exact & 41.97 & 52.41 & 33.89 & 57.05 & 84.41 & 29.92 & 23.23 & 42.31 & 39.09 & 24.18 & 35.90 & 45.31 \\
2Step-Spatial-Exact & 43.10 & 54.20 & 32.28 & 58.06 & 85.41 & 21.60 & 23.46 & 44.10 & 38.81 & 25.39 & 37.79 & 43.12 \\ \hline MoCo-NN & 61.69 & 90.23 & 14.68 & 57.68 & 91.34 & 17.17 & 52.03 & 89.89 & 12.71 & 56.16 & 91.45 & 17.01 \\ SupCon-NN & 70.75 & 93.41 & 11.73 & 61.17 & 93.40 & 17.05 & 64.29 & 93.59 & 9.45 & 65.73 & 93.38 & 10.96 \\ OrdCon-NN & 71.11 & 93.96 & 9.95 & 60.74 & 93.93 & 11.62 & 64.02 & 93.98 & 9.29 & 65.95 & 94.54 & 10.55 \\ \hline AutoLTS-MoCo & 72.21 & 93.70 & 10.11 & 61.60 & 94.02 & 13.52 & 64.69 & 92.94 & 9.84 & 64.40 & 94.05 & 11.62 \\ AutoLTS-SupCon & 73.30 & 94.16 & 10.61 & 62.17 & 94.38 & 15.76 & 64.63 & 93.42 & 10.66 & 65.86 & 94.37 & 11.43 \\ AutoLTS-OrdCon & 73.41 & 94.16 & 10.50 & 62.31 & 94.69 & 15.72 & 64.69 & 93.50 & 9.87 & 66.04 & 94.62 & 10.77 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of ablation studies. |
2306.11896 | The Witten Diagram Bootstrap for Holographic Defects | We study the AdS/CFT correspondence with a brane extending in AdS, a setup
which is dual to CFT in the presence of a defect. We focus on the correlation
function of two local operators and the defect, which is the simplest
observable with non-trivial dependence on kinematical invariants. We propose a
method to bootstrap this observable which relies on supersymmetry, but does not
require detailed knowledge of the supergravity and brane effective actions.
After developing the method in full generality, we turn to the case of two
chiral-primary operators and a half-BPS Wilson loop in $\mathcal N=4$ SYM.
Working in the leading supergravity approximation, we determine the correlator
in closed form for chiral-primary operators of arbitrary length. The result has
elegant expressions in position and Mellin space, and it agrees with
localization and an explicit calculation up to contact terms. More generally,
we expect our method to be suitable in other holographic setups in the presence
of supersymmetric defects. | Aleix Gimenez-Grau | 2023-06-20T21:18:32Z | http://arxiv.org/abs/2306.11896v1 | # The Witten Diagram Bootstrap
###### Abstract
We study the AdS/CFT correspondence with a brane extending in AdS, a setup which is dual to CFT in the presence of a defect. We focus on the correlation function of two local operators and the defect, which is the simplest observable with non-trivial dependence on kinematical invariants. We propose a method to bootstrap this observable which relies on supersymmetry, but does not require detailed knowledge of the supergravity and brane effective actions. After developing the method in full generality, we turn to the case of two chiral-primary operators and a half-BPS Wilson loop in \({\cal N}=4\) SYM. Working in the leading supergravity approximation, we determine the correlator in closed form for chiral-primary operators of arbitrary length. The result has elegant expressions in position and Mellin space, and it agrees with localization and an explicit calculation up to contact terms. More generally, we expect our method to be suitable in other holographic setups in the presence of supersymmetric defects.
###### Contents
* 1 Introduction
* 2 Preliminaries
* 2.1 Setup
* 2.2 Effective action and Witten diagrams
* 2.3 Conformal symmetry and defect CFT
* 2.4 Bootstrap approach
* 3 Witten diagrams in position space
* 3.1 Simple diagrams
* 3.2 Contact diagram
* 3.3 Bulk-exchange diagram
* 3.4 Defect-exchange diagram
* 3.5 A detour: connection to analytic functionals
* 4 Witten diagrams in Mellin space
* 4.1 Mellin space
* 4.2 Contact diagram
* 4.3 Bulk-exchange diagram
* 4.4 Defect-exchange diagram
* 5 Half-BPS Wilson line in \(\mathcal{N}=4\) SYM
* 5.1 Setup
* 5.2 Effective action at strong coupling
* 5.3 Explicit calculation (up to contact terms)
* 5.4 Position space bootstrap
* 5.5 Mellin space
* 6 Conclusions
* A Spin-two bulk exchange
* B Details on the \(\mathrm{AdS}_{5}\) effective action
* C Spherical harmonics
* C.1 Basic definition
* C.2 Relation to index-free notation
* C.3 Index contractions
* D Comparison to topological sector
* E Ward identity in Mellin space
Introduction
One of the most exciting discoveries in high energy theory is that certain quantum gravity theories in Anti de-Sitter (AdS) are dual to conformal field theories [1, 2, 3]. This goes under the name AdS/CFT correspondence, and it has been a fruitful line of research for the last 25 years. A natural observable, which is studied since the early days of AdS/CFT, is the correlation function of local operators [4, 5, 6]. These correlators are the AdS analog of scattering amplitudes, an analogy that can be made precise in Mellin space [7]. More specifically, a simple formula relates the flat-space limit of Mellin amplitudes to flat-space string amplitudes [8, 9, 10]. In AdS/CFT, correlators of local operators can be calculated as sums of Witten diagrams, although this is notoriously hard due to the proliferation of diagrams and the complicated form of the supergravity effective actions [11, 12, 13, 14, 15]. Several years ago, the work of Rastelli and Zhou [16, 17] bypassed these complications by making an ansatz for four-point correlators, and fixing it with basic consistency conditions, such as crossing symmetry and supersymmetry. Since then, the study of correlators in the AdS/CFT correspondence has received renewed attention. To mention only some of the many developments, there are now results going beyond the supergravity approximation [18, 19, 20, 21, 22, 23, 24, 25], results for five-point functions [26, 27, 28], or results in setups without maximal supersymmetry [29, 30, 20].
In the present work, we generalize the standard setup in yet another way: we scatter local operators off an extended object. We refer to the extended object as a brane, which could be for example a long fundamental string, a probe D\(p\)-brane, etc. More precisely, we consider a \((p+1)\)-dimensional brane that extends in AdS\({}_{p+1}\subset\) AdS\({}_{d+1}\). From the point of view of the dual CFT, there is an extended operator where the brane intersects with the boundary of AdS\({}_{d+1}\). Because extended operators are often called defects, we refer to this setup as a holographic defect, see figure 1.
To probe the system, we prepare local operators at the boundary of AdS\({}_{d+1}\), and scatter them off the brane in the bulk. These correlation functions admit an expansion in Witten diagrams, with extra vertices due to interactions between bulk and
Figure 1: The bulk AdS\({}_{d+1}\) space is enclosed by the black circle. The brane is represented in blue and it extends in AdS\({}_{p+1}\subset\) AdS\({}_{d+1}\). The boundary theory is a CFT in the presence of a \(p\)-dimensional conformal defect.
focuses on the case of two local operators, for which some low-lying Witten diagrams are shown in figure 2. This setup is the AdS/CFT analog of a \(1\to 1\) scattering experiment off a fixed extended target. This analogy is particularly compelling in Mellin space, see [31] and section 4 below. We also expect a connection between Mellin amplitudes and flat-space string amplitudes, see figure 3, although this connection has not been worked out in the literature.
To make the discussion more concrete, let us consider a prototypical example of a holographic defect, namely a fundamental string in AdS\({}_{5}\times S^{5}\). This string intersects the boundary of AdS\({}_{5}\) at a contour \(\gamma\), and has Dirichlet boundary conditions on the \(S^{5}\). As shown in [32, 33], this configuration is dual to \(\mathcal{N}=4\) SYM with a half-BPS Wilson loop in the fundamental representation. The Wilson operator \(W\) is supported on the contour \(\gamma\), and when this contour is a straight line, the fundamental string extends in the bulk in an AdS\({}_{2}\subset\) AdS\({}_{5}\). This setup has been studied extensively in the literature, since it provides a rich interplay of techniques such as holography [34, 35, 36, 37, 38], localization [39, 40], integrability [41, 42, 43, 44], perturbation theory [45, 46, 47] or bootstrap [48, 49, 50, 44, 44].1 Many of these works focus on observables like the expectation value of the loop \(\langle W\rangle\), one-point functions of local operators \(\langle\mathcal{O}W\rangle\), or correlators of fields inserted on the defect \(\langle W[\widehat{\mathcal{O}}_{1}\widehat{\mathcal{O}}_{2}\widehat{\mathcal{ O}}_{3}\widehat{\mathcal{O}}_{4}]\rangle\). However, the observable that concerns us is the two-point function of local operators \(\langle\mathcal{O}_{1}\mathcal{O}_{2}W\rangle\), which has received comparatively little attention, but see e.g. [51, 52, 53, 54].
Footnote 1: This list covers only a small fraction of the literature, and we apologize for omissions.
Motivated by this lack of results, here we present an efficient bootstrap method to compute two-point functions of local operators in the presence of a holographic defect. The idea is that of the position-space method of Rastelli and Zhou [16, 17]: one makes an ansatz in terms of Witten diagrams as in figure 2, and fixes the relative coefficients demanding that superconformal Ward identities are satisfied. The use of an ansatz allows to bypass detailed knowledge of the bulk and brane effective actions, which can be quite complicated. Because the method relies on supersymmetry, we expect it to work for half-BPS branes in maximally supersymmetric holography, but it is unclear whether it will work with less supersymmetry.
Figure 2: Low-lying Witten diagrams for a scattering process of two local operators and a brane. Black dashed lines are fields propagating in the bulk, and blue dashed lines are fields propagating on the brane.
Two ingredients are necessary to implement the bootstrap. The first ingredient are closed form expressions for Witten diagrams like the ones in figure 2. Witten diagrams were studied for codimension-one branes, namely \(p=d-1\), in [55, 56, 57]. That case is simpler, because internal lines contain only scalar fields, and the result depends on only one cross-ratio. Some of these results were generalized to \(p<d-1\) in [31]. However, for arbitrary \(p\), one needs diagrams with spinning fields in internal lines, which have not been worked out yet. To fill this gap in the literature, sections 3 and 4 present a comprehensive computation of Witten diagrams in position and Mellin space respectively. The second ingredient of the bootstrap method are the superconformal Ward identities. Superconformal Ward identities are known in several interesting cases, such as half-BPS interfaces and lines in \(\mathcal{N}=4\) SYM [58], or surface defects in \(6d\)\((2,0)\) theory [59]. Although no general results are available, we expect that the need to derive Ward identities on a case-by-case basis will not be an obstacle to bootstrap other setups.
Section 5 combines the results of sections 3 and 4 to bootstrap \(\mathcal{N}=4\) SYM correlators in the presence of a half-BPS Wilson line. More precisely, we bootstrap \(\langle S_{p_{1}}S_{p_{2}}W\rangle\), the correlator of two local operators and a half-BPS Wilson line. The local operators \(S_{k}\) are superprimaries of half-BPS multiplets, and transform in rank-\(k\) symmetric-traceless representations of \(SO(6)_{R}\). The half-BPS Wilson line \(W\) was described above, and is dual to a fundamental string. Because we work in the supergravity approximation, the correlator is computed in the planar limit \(N\to\infty\) at fixed but large 't Hooft coupling \(\lambda=g^{2}N\to\infty\). The perturbative expansion is a double power series in \(\frac{1}{N^{2}}\) and \(\frac{1}{\sqrt{\lambda}}\)
\[\langle S_{p_{1}}S_{p_{2}}W\rangle_{c}=\frac{1}{N^{2}}\left(\sqrt{\lambda}\, \mathcal{F}_{-\frac{1}{2},2}+\mathcal{F}_{0,2}+\ldots\right)+\frac{1}{N^{4}} \left(\lambda\,\mathcal{F}_{-1,4}+\sqrt{\lambda}\,\mathcal{F}_{-\frac{1}{2}, 4}+\ldots\right)+\ldots\,. \tag{1}\]
Loosely speaking, powers of \(\frac{1}{N^{2}}\) count loops in the bulk, and powers of \(\frac{1}{\sqrt{\lambda}}\) count loops on the string worldsheet. In this work, we determine the leading term \(\mathcal{F}_{-\frac{1}{2},2}\), hoping this will
Figure 3: The connected correlator, to be defined below, should map in the flat-space limit to a scattering amplitude of closed strings off an extended brane.
be the first step towards determining higher-order corrections, see [40].
In equation (1) we are considering the connected part of the correlator \(\langle S_{p_{1}}S_{p_{2}}W\rangle_{c}\), which is the sum of connected Witten diagrams, compare figures 2 and 3. The disconnected correlator is somewhat trivial, and will be ignored in most of this work. Our main focus is \(\mathcal{F}_{-\frac{1}{2},2}\), which consists of a non-trivial function of three kinematical invariants. For chiral-primaries of length \(p_{1},p_{2}\leq 4\), the correlators \(\mathcal{F}_{-\frac{1}{2},2}\) were obtained recently in [53, 54] using analytic bootstrap.2 Here we improve their results in several ways. On one hand, thanks to the more powerful methods in the present work, we succeed in determining the correlator in closed form for arbitrary \(p_{1},p_{2}\). The result is particularly elegant in Mellin space, see equation (137). On the other hand, the analytic bootstrap method [53, 54] suffered from low-spin ambiguities that were fixed in a somewhat ad-hoc way. Here that calculation is put on solid ground by showing that low-spin ambiguities are due to Witten diagrams that exchange fields on the brane, as in the fourth diagram of figure 2. Finally, we also do a first-principles computation of the correlators, which requires us to determine the interaction vertices of the effective action. The result agrees with the bootstrap, up to contact terms that we do not know how to obtain in the first-principles calculation.
Footnote 2: The correlator \(\langle S_{2}S_{2}W\rangle\) was also studied in [52], but in the planar limit at weak ’t Hooft coupling \(\lambda\ll 1\). Other works [60, 36] have computed \(\langle S_{p_{1}}S_{p_{2}}W\rangle\) with localization, but their results apply to very special kinematics, where the cross-ratio dependence drops out, and the correlator reduces to a constant. Unlike their results, our correlator contains non-trivial information of long operators through OPE expansions.
The main sections of our work are 3-5, the contents of which we just outlined. Besides them, section 2 provides a more thorough description of the setup and its kinematics, reviews standard material and sets the notation. We also outline possible future work in section 6, including a list of setups where we expect our methods to be applicable. Finally, we relegate a number of technical details to appendix A-E.
## 2 Preliminaries
### Setup
As explained in the introduction, this work studies \((d+1)\)-dimensional anti-de Sitter space containing a \((p+1)\)-dimensional brane for \(p<d\), see figure 1. We restrict our attention to a brane that extends in an AdS\({}_{p+1}\) submanifold, so in particular the brane breaks the isometries of AdS\({}_{d+1}\) as follows
\[SO(d+1,1)\ \to\ SO(p+1,1)\oplus SO(d-p)\,. \tag{2}\]
In the standard AdS/CFT dictionary, the isometry group \(SO(d+1,1)\) maps to the conformal symmetry that acts on the boundary theory. In the present case, the symmetry of the
boundary theory is (2), which is precisely the group preserved by CFT in the presence of a flat conformal defect [61]. Borrowing common terminology of defect CFT, we use the words brane and defect interchangeably for the AdS\({}_{p+1}\). The connection to defect CFT is important for our purposes, and we discuss it in more detail in section 2.3.
To be more precise, we work in Euclidean signature and parametrize AdS\({}_{d+1}\) in Poincare coordinates \(z^{\mu}\in\text{AdS}_{d+1}\), with \(\mu=0,1,\ldots,d\). The conformal boundary of AdS\({}_{d+1}\) can be identified with \(\mathbb{R}^{d}\), and is approached in the limit \(z^{0}\to 0\). In particular, we denote boundary points as \(x\in\mathbb{R}^{d}\). It is convenient to split the coordinates of AdS\({}_{d+1}\) in directions parallel and orthogonal to the brane
\[z^{\mu}=(z^{0},z^{a},z^{i})\quad\text{ with }\quad a=1,\ldots,p\,,\quad i=p+1, \ldots,d\,. \tag{3}\]
For points in AdS\({}_{p+1}\) we use notation \(\widehat{z}^{\alpha}=(\widehat{z}^{0},\widehat{z}^{a})\), and points at the boundary are denoted \(x^{a}\in\mathbb{R}^{p}\). In Poincare coordinates, the \((d+1)\)- and \((p+1)\)-dimensional metrics read respectively
\[g_{\mu\nu}=\frac{\delta_{\mu\nu}}{(z^{0})^{2}}\,,\qquad\widehat{g}_{\alpha \beta}=\frac{\delta_{\alpha\beta}}{(\widehat{z}^{0})^{2}}\,. \tag{4}\]
These metrics define covariant derivatives and an integration measure in the standard way.
### Effective action and Witten diagrams
The focus of this paper are correlation functions of local operators in the setup just described, which are computed perturbatively with Witten diagrams. Here we present a simple example of a bulk scalar \(\phi\) and a brane scalar \(\widehat{\phi}\), which illustrates the main ingredients necessary in the rest of this work.
Let's start with the bulk scalar \(\phi\), which has mass \(m^{2}=\Delta(\Delta-d)\) and a cubic self-coupling
\[S_{\text{bulk}}=\int_{\text{AdS}_{d+1}}\!\frac{d^{d+1}z}{(z_{0})^{d+1}}\left( \frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi+\frac{1}{2}m^{2}\phi^{2}+\frac{1}{ 3!}g\phi^{3}\right)\,. \tag{5}\]
Besides the bulk field, we also consider a brane scalar \(\widehat{\phi}\) of mass \(\widehat{m}^{2}=\widehat{\Delta}(\widehat{\Delta}-p)\). The bulk field can couple directly to the brane or it can couple to \(\widehat{\phi}\). In this simplified example, we take the brane action to be
\[S_{\text{brane}}=\int_{\text{AdS}_{p+1}}\!\!\frac{d^{p+1}\widehat{z}}{( \widehat{z}_{0})^{p+1}}\left(\frac{1}{2}\nabla^{\alpha}\widehat{\phi}\,\nabla _{\alpha}\widehat{\phi}+\frac{1}{2}\widehat{m}^{2}\widehat{\phi}^{2}+\lambda \phi+\theta\phi^{2}+\mu\phi\widehat{\phi}\right)\,. \tag{6}\]
Although a toy model, this action resembles realistic examples such as the half-BPS Wilson line of section 5. The major difference is that the action for the half-BPS Wilson line contains many more terms, some of which are bulk spin-two fields or brane fields charged under \(SO(d-p)\), see (2). Here we ignore these complications.
In what follows, we treat the couplings \(g\), \(\lambda\), \(\theta\) and \(\mu\) as small perturbations. We can then compute correlation functions of the fields \(\phi\) and \(\widehat{\phi}\) using position space Feynman rules. In AdS/CFT, position-space diagrams are conventionally called Witten diagrams, and are built with free propagators in the standard way. For example, the propagator for \(\phi\) is
\[\left(-\nabla^{2}+m^{2}\right)G_{\Delta}(z_{1},z_{2})=(z_{0})^{d+1}\,\delta^{( d+1)}(z_{1}-z_{2})\,. \tag{7}\]
The solution \(G_{\Delta}\) is called bulk-to-bulk propagator, and it can be obtained in closed form [62, 63]
\[G_{\Delta}(z_{1},z_{2}) =\frac{{\cal C}_{\Delta}\left(z_{1}^{0}z_{2}^{0}\right)^{\Delta}} {(z_{1}-z_{2})^{2\Delta}}\,_{2}F_{1}\left(\Delta,\Delta-\frac{d}{2}+\frac{1}{2 },2\Delta-d+1,-\frac{4z_{1}^{0}z_{2}^{0}}{(z_{1}-z_{2})^{2}}\right)\,, \tag{8}\] \[{\cal C}_{\Delta} =\frac{\Gamma(\Delta)}{2\pi^{d/2}\Gamma(\Delta-d/2+1)}\,. \tag{9}\]
Similarly, the propagator for \(\widehat{\phi}\) is called brane-to-brane propagator. It is defined as in (8) with \(d\to p\), and we denote it \(\widehat{G}_{\widehat{\Delta}}(\widehat{z}_{1},\widehat{z}_{2})\).
In the computation of Witten diagrams, one often encounters fields that propagate from the boundary of AdS into the bulk. One then uses the bulk-to-boundary propagator, which follows from (8) sending one point to the boundary \(z\to\partial\text{AdS}_{d+1}\) and rescaling the result
\[K_{\Delta}(x,z)=\left(\frac{z_{0}}{(z-x)^{2}}\right)^{\Delta}\,. \tag{10}\]
In the literature, the bulk-to-boundary propagator sometimes includes a normalization factor, but we find more convenient to use it unnormalized. There is also a brane-to-boundary propagator, but it is given exactly by the same expression (10).
It is well known that processes that start with \(\phi\) at point \(x\) in the boundary of AdS\({}_{d+1}\) can be interpreted as correlation functions of a dual CFT operator \({\cal O}(x)\). When the mass of the scalar is \(m^{2}=\Delta(\Delta-d)\), then the CFT operator has scaling dimension \(\Delta\). In the present context, we compute processes with a brane in the bulk, which in CFT corresponds to inserting a non-local defect operator \({\cal D}\) in the expectation value. In the example of a fundamental string in AdS\({}_{5}\times S^{5}\), the defect operator is the half-BPS Wilson line operator \({\cal D}=W\sim\text{tr}\,\text{P}e^{\int A_{\tau}+i\phi_{6}}\). More generally, depending on the type of brane in the bulk, \({\cal D}\) is instead a 't Hooft line operator, surface operator, etc.
The simplest process that involves both local and non-local operators is the one-point function \(\langle{\cal OD}\rangle\). The first few Witten diagrams are
\[\langle{\cal OD}\rangle\ \ =\ \ \raisebox{-14.226378pt}{\includegraphics[width=14.226378pt]{figs.eps}}\ +\ \raisebox{-14.226378pt}{\includegraphics[width=14.226378pt]{figs.eps}}\ +\ \raisebox{-14.226378pt}{\includegraphics[width=14.226378pt]{figs.eps}}\ +\ \raisebox{-14.226378pt}{\includegraphics[width=14.226378pt]{figs.eps}}\ +\ \ldots\,, \tag{11}\]
where the vertices follow from the action (5)-(6). Each diagram corresponds to a position-space integral with propagators \(G_{\Delta}\), \(\widehat{G}_{\widehat{\Delta}}\) and \(K_{\Delta}\), which in general can be highly non-trivial to compute.
In this work, we focus mostly on two-point functions of local operators in the presence of the brane, and the first few diagrams read
\[\langle\mathcal{O}\mathcal{O}\mathcal{D}\rangle\ \ =\ \ \raisebox{-14.226378pt}{ \includegraphics[scale=0.5]{fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/ fig/ fig/ fig/ fig// fig// fig// fig// fig// fig// fig// fig// fig/// fig// fig// fig/// fig/// fig// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig/// fig//// fig/// fig//// fig/// fig//// fig//// fig//// fig//// fig//// fig//// fig//// fig//// fig///// fig///// fig///// fig///// fig//// fig///// fig///// fig///// fig///// fig///// fig///// fig///// fig///// fig////// fig////// fig////// fig////// fig/////// fig/////// fig//////// fig/////// fig///////// fig
Note that we normalize all correlators by the expectation value of the defect \(\langle\mathcal{D}\rangle\). This ensures that in the limit \(x_{1}\to x_{2}\), the two-point function (14) reduces to a unit-normalized two-point function without a defect. Equivalently, the correlator of equal operators is normalized as \(F(\xi,\eta)\sim\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}\) in the limit \(\xi\to 0\).
An important property of the correlator \(F(\xi,\eta)\) is that it admits two different OPE decompositions [61]. Although these OPEs only play a tangential role in the present work, it is still useful to discuss them, because sometimes they help interpret certain results. The first OPE is the standard one, which replaces the product of two bulk local operators by a sum of local operators, schematically \(\mathcal{O}_{1}\mathcal{O}_{2}\sim\sum_{\mathcal{O}}\lambda_{\mathcal{O}_{1} \mathcal{O}_{2}\mathcal{O}}\mathcal{O}\). The second OPE replaces the product of a bulk local operator with the defect \(\mathcal{D}\) by a sum of local operators on top of the defect, schematically \(\mathcal{O}\mathcal{D}\sim\sum_{\widehat{\mathcal{O}}}b_{\mathcal{O}\widehat{ \mathcal{O}}}\widehat{\mathcal{O}}\mathcal{D}\). These two OPEs can be resumed into two different conformal block expansions
\[F(\xi,\eta)=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{\mathcal{O}}2^{-\ell }\lambda_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}a_{\mathcal{O}}f_{\Delta,\ell}(\xi,\eta)=\sum_{\widehat{\mathcal{O}}}2^{-s}b_{\mathcal{O}_{1}\widehat {\mathcal{O}}}b_{\mathcal{O}_{2}\widehat{\mathcal{O}}}\widehat{f}_{\widehat{ \Delta},s}(\xi,\eta)\,, \tag{17}\]
where \(a_{\mathcal{O}}\) are the one-point coefficients (13). See appendix A of [66] for a pedagogical introduction, including our normalization conventions and explicit formulas for conformal blocks \(f_{\Delta,\ell}\), \(\widehat{f}_{\widehat{\Delta},s}\).
Another important concept is the connected correlator \(\langle\ldots\mathcal{D}\rangle_{c}\), which for the two-point function is defined by
\[\frac{\langle\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})\mathcal{D}\rangle}{ \langle\mathcal{D}\rangle}=\langle\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2} )\rangle+\frac{\langle\mathcal{O}_{1}(x_{1})\mathcal{D}\rangle}{\langle \mathcal{D}\rangle}\frac{\langle\mathcal{O}_{2}(x_{2})\mathcal{D}\rangle}{ \langle\mathcal{D}\rangle}+\frac{\langle\mathcal{O}_{1}(x_{1})\mathcal{O}_{2} (x_{2})\mathcal{D}\rangle_{c}}{\langle\mathcal{D}\rangle}\,. \tag{18}\]
Equivalently, if we use the one-point function (13) and factor out the prefactor in (14), the connected correlator \(F_{c}\) is
\[F(\xi,\eta)=\delta_{\mathcal{O}_{1},\mathcal{O}_{2}}\xi^{-\frac{\Delta_{1}+ \Delta_{2}}{2}}+a_{\mathcal{O}_{1}}a_{\mathcal{O}_{2}}+F_{c}(\xi,\eta)\,. \tag{19}\]
The main reason to decompose the correlator in this way is that the first two terms appear in mean-field theory (MFT), while \(F_{c}\) contains the truly non-trivial part of the correlator. A second reason is that Mellin amplitudes, to be defined in section 4, make sense only for connected correlators. For example, the connected part of equation (12) is
\[\langle\mathcal{O}\mathcal{O}\mathcal{O}\rangle_{c}\ \ =\ \raisebox{-14.226378pt}{ \includegraphics[scale=0.5]{fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/ fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig/fig
### Bootstrap approach
To summarize the discussion so far, we have considered a theory in AdS\({}_{d+1}\) described by an effective action, that in our example is given in (5). Furthermore, we added a brane that extends in an AdS\({}_{p+1}\) submanifold. The brane has degrees of freedom of its own, that interact with bulk fields through another effective action, in our example (6). With these two actions we can compute Witten diagrams, which define correlation functions in a defect CFT that lives at the boundary of AdS\({}_{d+1}\).
In examples arising from string theory, the bulk and brane effective actions are significantly more involved, see for example [11, 12, 13, 14, 15]. For this reason, it is extremely valuable to have a method that bypasses detailed knowledge of the effective action. The idea of [16, 17] is that one only needs to know what fields appear in the effective action, but not their precise interactions. In this case, one can write all Witten diagrams that contribute to a certain correlator, with unspecified coefficients. The combination of Witten diagrams needs to satisfy certain consistency conditions, such as superconformal Ward identities, or agreement with the flat-space limit or localization. In many examples, the free coefficients in the ansatz are fully fixed by these requirements. In this work, we apply the same idea in the presence of a brane in the bulk. We show in section 5 that the method works perfectly for a half-BPS Wilson line in \(\mathcal{N}=4\) SYM. More generally, it is plausible that this method applies to many (if not all) half-BPS defects in maximally supersymmetric theories. We discuss some of these possible setups in the conclusions.
## 3 Witten diagrams in position space
In this section, we compute Witten diagrams for a two-point functions of local operators in the presence of a brane. The interested reader can find related work in [55, 31]. We restrict our attention to diagrams that appear at leading order in the supergravity approximation. In the future, it would of course be very interesting to extend the analysis to loop diagrams.
### Simple diagrams
As a first example, we compute tree-level diagrams for the bulk one-point function \(\langle\mathcal{OD}\rangle\) and the bulk-defect two-point function \(\langle\mathcal{O}\widehat{\mathcal{OD}}\rangle\). Besides the pedagogical value, these results are needed to calculate Mellin amplitudes in section 4.
#### 3.1.1 Bulk one-point diagram
We consider a scalar \(\phi\) dual to an operator \(\mathcal{O}\) of dimension \(\Delta\). If the scalar couples to a brane as \(\int_{\text{AdS}_{p+1}}\phi\), then the leading diagram to the one-point function \(\langle\mathcal{O}\mathcal{D}\rangle\) is
\[I_{\Delta}(x)\ \ =\ \ \ \raisebox{-28.452756pt}{\includegraphics[]{fig/d1.eps}}\ =\ \ \int_{\widehat{z}}K_{\Delta}(x,\widehat{z})\,. \tag{21}\]
The integral is over \(\widehat{z}\in\text{AdS}_{p+1}\) with appropriate measure, and the bulk-boundary propagator is defined in (10), so we have to compute
\[I_{\Delta}(x)=\int_{0}^{\infty}\frac{d\,\widehat{z}_{0}}{\widehat{z}_{0}^{p+1} }\int_{\mathbb{R}^{p}}d^{p}\widehat{z}^{\,a}\left(\frac{\widehat{z}_{0}}{ \widehat{z}_{0}^{\,2}+(x^{i})^{2}+(x^{a}-\widehat{z}^{\,a})^{2}}\right)^{ \Delta}\,. \tag{22}\]
The dependence on \(x^{a}\) drops out by translation invariance, so we can use spherical coordinates in the \(\widehat{z}^{\,a}\) directions. Introducing a Schwinger parameter \(s\), we find
\[I_{\Delta}(x)=\frac{1}{\Gamma(\Delta)}\int_{0}^{\infty}\frac{ds}{s}\,s^{ \Delta}e^{-s(x^{i})^{2}}\int_{0}^{\infty}\frac{d\,\widehat{z}_{0}}{\widehat{z }_{0}}\,\widehat{z}_{0}^{\Delta-p}e^{-s\widehat{z}_{0}^{2}}\,\frac{2\pi^{ \frac{p}{2}}}{\Gamma(p/2)}\int_{0}^{\infty}\frac{dr}{r}\,r^{p}e^{-sr^{2}}\,. \tag{23}\]
The integrals over \(\widehat{z}_{0}\) and \(r=|\widehat{z}^{\,a}|\) are straightforward, and converge provided \(0<p<\operatorname{Re}\Delta\). The final integral over \(s\) is elementary, and we find [31, 55]3
Footnote 3: There is a minor typo in equation (4.2) of [55], where \(d-2\) should read \(d-1\).
\[I_{\Delta}(x)=\frac{\pi^{\frac{p}{2}}\Gamma\big{(}\frac{\Delta-p}{2}\big{)}} {2\Gamma(\Delta)}\int_{0}^{\infty}\frac{ds}{s}\,s^{\frac{\Delta}{2}}e^{-s(x^{ i})^{2}}=\frac{a_{\Delta}}{|x^{i}|^{\Delta}}\,,\quad a_{\Delta}\equiv\pi^{p/2}\, \frac{\Gamma\big{(}\frac{\Delta}{2}\big{)}\,\Gamma\big{(}\frac{\Delta-p}{2} \big{)}}{2\Gamma(\Delta)}\,. \tag{24}\]
This has the correct form dictated by conformal symmetry [61].
#### 3.1.2 Bulk-defect diagram
Now let's imagine a brane scalar \(\widehat{\phi}\), dual to a defect operator \(\widehat{\mathcal{O}}\) of dimension \(\widehat{\Delta}\). The scalar \(\phi\) can couple to this brane field as \(\int_{\text{AdS}_{p+1}}\phi\,\widehat{\phi}\). In this case, the tree-level correlator \(\langle\mathcal{O}\widehat{\mathcal{O}}\mathcal{D}\rangle\) consists of the diagram
\[I_{\Delta,\widehat{\Delta}}(x_{1},x_{2})\ =\ x_{1}\raisebox{-28.452756pt}{\includegraphics[]{fig/d2.eps}}\ =\ \int_{ \widehat{z}}K_{\Delta}(x_{1},\widehat{z})K_{\widehat{\Delta}}(x_{2},\widehat{ z})\,. \tag{25}\]
Note that point \(x_{2}\) lives in the boundary of AdS\({}_{p+1}\), meaning that \(x_{2}^{i}=0\). As before, we introduce Schwinger parameters and integrate over \(\widehat{z}\). The integrals are elementary and
converge provided \(0<p<\operatorname{Re}\Delta+\operatorname{Re}\widehat{\Delta}\):
\[I_{\Delta,\widehat{\Delta}}(x_{1},x_{2})=\frac{\pi^{p/2}\Gamma\!\left(\frac{ \Delta+\widehat{\Delta}-p}{2}\right)}{2\Gamma(\Delta)\Gamma(\widehat{\Delta})} \,\int_{0}^{\infty}\frac{ds\,dt}{s\,t}\frac{s^{\Delta}t^{\widehat{\Delta}}}{(s +t)^{\frac{\Delta+\widehat{\Delta}}{2}}}\exp\left(-\frac{st}{s+t}(x_{12}^{a}) ^{2}-s(x_{1}^{i})^{2}\right)\,. \tag{26}\]
To integrate over \(s,t\) we employ a method that we also use repeatedly below. First multiply by the identity \(1=\int_{0}^{\infty}d\lambda\,\delta(\lambda-t)\) and change variables to \(s,t\to\lambda s,\lambda t\). Now use \(\delta(\lambda(1-t))=\lambda^{-1}\delta(1-t)\), so both the \(t\) and \(\lambda\) integrals become elementary. Integrating \(t\) and \(\lambda\) gives
\[I_{\Delta,\widehat{\Delta}}(x_{1},x_{2})=\frac{\pi^{p/2}\Gamma\!\left(\frac{ \Delta+\widehat{\Delta}}{2}\right)\Gamma\!\left(\frac{\Delta+\widehat{\Delta} -p}{2}\right)}{2\Gamma(\Delta)\Gamma(\widehat{\Delta})}\int_{0}^{\infty} \frac{ds}{s}\,\frac{s^{\frac{\Delta-\widehat{\Delta}}{2}}}{\left((x_{12}^{a} )^{2}+(s+1)(x_{1}^{i})^{2}\right)^{(\Delta+\widehat{\Delta})/2}}\,. \tag{27}\]
The integral over \(s\) converges for \(\operatorname{Re}\Delta>\operatorname{Re}\widehat{\Delta}\), giving [31, 55]
\[I_{\Delta,\widehat{\Delta}}(x_{1},x_{2})=\frac{b_{\Delta,\widehat{\Delta}}}{ |x_{1}^{i}|^{\Delta-\widehat{\Delta}}\big{(}(x_{1}^{i})^{2}+(x_{12}^{a})^{2} \big{)}^{\widehat{\Delta}}}\,,\qquad b_{\Delta,\widehat{\Delta}}=\pi^{p/2}\, \frac{\Gamma\!\left(\frac{\Delta-\widehat{\Delta}}{2}\right)\Gamma\!\left( \frac{\Delta+\widehat{\Delta}-p}{2}\right)}{2\Gamma(\Delta)}\,. \tag{28}\]
Once again, the result has the form dictated by conformal symmetry [61].
We can also have brane fields charged under \(SO(d-p)\) rotations. For example, a spin-\(s\) field \(\widehat{\phi}^{i_{1}\ldots i_{s}}\) couples to a bulk scalar as
\[S_{\phi,\widehat{\phi}}\,\sim\,\int\frac{d^{p+1}\widehat{z}}{\widehat{z}_{0}^ {p+1}}\,\widehat{z}_{0}^{s}\,\widehat{\phi}^{i_{1}\ldots i_{s}}\partial_{i_{ 1}}\ldots\partial_{i_{s}}\phi\,, \tag{29}\]
where as before \(i=p+1,\ldots d\) and the factor \(\widehat{z}_{0}^{s}\) is required by dimensional analysis. As a result, the correlation function \(\langle\mathcal{O}\widehat{\mathcal{O}}^{i_{1}\ldots i_{s}}\mathcal{D}\rangle\) is
\[I_{\Delta,\widehat{\Delta}}^{i_{1}\ldots i_{s}}(x_{1},x_{2})=\ \int_{\widehat{z}}\widehat{z}_{0}^{s}\,\partial^{\{i_{1} \ldots\partial^{i_{s}}\}}K_{\Delta}(x_{1},\widehat{z})K_{\widehat{\Delta}}(x_ {2},\widehat{z})\propto x_{1}^{\{i_{1}\ldots x_{1}^{i_{s}}\}}I_{\Delta+s, \widehat{\Delta}}(x_{1},x_{2})\,. \tag{30}\]
The bracket \(\{i\ldots j\}\) denotes the symmetric traceless combination. In the second step we take derivatives of the bulk-to-boundary propagator (10), and observe that the resulting integral is proportional to (25) with \(\Delta\to\Delta+s\). The upshot is that correlators with defect operators charged under \(SO(d-p)\) are simple shifts of the scalar results. When we consider two-point functions below we will observe the same phenomena.
### Contact diagram
Next we consider a coupling between two bulk scalars on the brane \(\int_{\text{AdS}_{p+1}}\phi_{1}\phi_{2}\), which contributes to the two-point function \(\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{D}\rangle\) of operators dual to \(\phi_{1}\) and \(\phi_{2}\). We call the
resulting Witten diagram a contact diagram:
\[\raisebox{-14.226378pt}{\includegraphics[]{fig/contact_diagram.pdf}} \tag{31}\] \[\raisebox{-14.226378pt}{\includegraphics[]{fig/contact_diagram.pdf}} = \int_{\widetilde{z}}K_{\Delta_{1}}(x_{1},\widetilde{z})K_{\Delta_{ 2}}(x_{2},\widehat{z})\;=\;\frac{C_{\Delta_{1}\Delta_{2}}(\xi,\eta)}{|x_{1}^{ i}|^{\Delta_{1}}|x_{2}^{i}|^{\Delta_{2}}}\,. \tag{32}\]
As explained in equation (14), the result depends on two conformal cross-ratios \(\xi\) and \(\eta\). The calculation proceeds as in section 3.1.2, but now the result also depends on the parallel distance \(x_{12}^{a}\), and the \(s\) integral is not elementary:
\[C_{\Delta_{1}\Delta_{2}}(\xi,\eta) =\frac{\pi^{\frac{p}{2}}\Gamma\big{(}\frac{\Delta_{1}+\Delta_{2} }{2}\big{)}\,\Gamma\big{(}\frac{\Delta_{1}+\Delta_{2}-p}{2}\big{)}}{2\Gamma( \Delta_{1})\Gamma(\Delta_{2})}\] \[\qquad\times\int_{0}^{\infty}\frac{ds}{s}\,\frac{s^{\Delta_{1}}| x_{1}^{i}|^{\Delta_{1}}|x_{2}^{i}|^{\Delta_{2}}}{\Big{(}(\xi+2\eta-2)s|x_{1}^{i}| |x_{2}^{i}|+(s|x_{1}^{i}|+|x_{2}^{i}|)^{2}\Big{)}^{\frac{\Delta_{1}+\Delta_{2 }}{2}}}\,. \tag{33}\]
To simplify the denominator, we used the definition of the cross-ratios \(\xi\) and \(\eta\) in (15). We shall encounter many similar integrals below. A convenient way to proceed is to split the denominator with
\[\frac{1}{(A+B)^{\Delta}}=\frac{1}{\Gamma(\Delta)}\int_{-i\infty}^{i\infty} \frac{d\tau}{2\pi i}\frac{\Gamma(\tau)\Gamma(\Delta-\tau)}{A^{\tau}B^{\Delta -\tau}}\,,\qquad 0<\operatorname{Re}\tau<\Delta\,. \tag{34}\]
This identity makes the \(s\) integral elementary, giving
\[C_{\Delta_{1}\Delta_{2}}(\xi,\eta)=\frac{\pi^{\frac{p+1}{2}}\Gamma\big{(} \frac{\Delta_{1}+\Delta_{2}-p}{2}\big{)}}{2^{\Delta_{1}+\Delta_{2}}\Gamma( \Delta_{1})\Gamma(\Delta_{2})}\int\frac{d\tau}{2\pi i}\frac{\Gamma(\tau) \Gamma(\Delta_{1}-\tau)\Gamma(\Delta_{2}-\tau)}{\Gamma\big{(}\frac{\Delta_{1 }+\Delta_{2}+1}{2}-\tau\big{)}}\left(\frac{\xi+2\eta-2}{4}\right)^{-\tau}\,. \tag{35}\]
The remaining \(\tau\) integral is the Mellin-Barnes representation of the hypergeometric function, and we find
\[C_{\Delta_{1}\Delta_{2}}(\xi,\eta)=\frac{\pi^{\frac{p+1}{2}}}{2^{\Delta_{1}+ \Delta_{2}}}\frac{\Gamma\big{(}\frac{\Delta_{1}+\Delta_{2}-p}{2}\big{)}}{ \Gamma\big{(}\frac{\Delta_{1}+\Delta_{2}+1}{2}\big{)}}\,_{2}F_{1}\left(\Delta_ {1},\Delta_{2},\frac{\Delta_{1}+\Delta_{2}+1}{2};-\frac{\xi+2\eta-2}{4}\right)\,. \tag{36}\]
An analogous formula was found for boundary CFT in [55], and we just showed that the result continues to hold with minor modifications for arbitrary codimension. It is important to note that the contact diagram only depends on \(\xi+2\eta\), or equivalently, it depends on the cross-ratio \(r\) defined in (16). In fact, for integer \(\Delta_{1},\Delta_{2}\) the hypergeometric function reduces to a rational function of \(r\) and \(\log r\), for example
\[C_{11}(r) \propto \frac{r\log r}{r^{2}-1}\,,\qquad C_{22}(r)\,\propto\,\frac{r^{2} \left(r^{2}+1\right)}{\left(r^{2}-1\right)^{3}}\log r-\frac{r^{2}}{\left(r^{2} -1\right)^{2}}\,, \tag{37a}\] \[C_{21}(r) \propto \frac{r}{\left(1+r\right)^{2}}\,,\qquad C_{31}(r)\,\propto\,\frac{ r^{3}+r}{2\left(r^{2}-1\right)^{2}}-\frac{2r^{3}}{\left(r^{2}-1\right)^{3}}\log r\,. \tag{37b}\]
One could compute contact diagrams for interactions with derivatives, but we do not need the results below.
As a final comment, contact diagrams \(C_{\Delta_{1}\Delta_{2}}\) are analogs of \(D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}\) functions for four-point correlators, see their definition in [5, 67]. However, the functions \(C_{\Delta_{1}\Delta_{2}}\) are simpler than \(D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}\), because they depend on a single cross ratio. Moreover, the \(D\)-functions for integer \(\Delta_{i}\) contain dilogarithms, as opposed to just logarithms in (36).
### Bulk-exchange diagram
The simplest interaction in the bulk is a three-point vertex \(\int_{\text{AdS}_{d+1}}\phi_{1}\phi_{2}\phi\), which combined with the one-point coupling to the brane \(\int_{\text{AdS}_{p+1}}\phi\), leads to the scalar-exchange Witten diagram
(37)
The calculation of this integral is significantly more involved than above, because the bulk-to-bulk propagator \(G_{\Delta}\) is the hypergeometric function (8). Fortunately, reference [62] developed an efficient method to reduce exchange Witten diagram to sums of contact diagrams, leading to a relation of the form
(38)
Although in general the sum contains infinitely many terms, for many cases of interest it truncates. More specifically, whenever \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}\) the exchange diagram reads
\[E_{\Delta_{1}\Delta_{2}}^{\Delta,0}(\xi,\eta)=\sum_{n=1}^{\frac{ \Delta_{1}+\Delta_{2}-\Delta}{2}}\frac{\left(-\frac{\Delta_{1}+\Delta_{2}- \Delta-2}{2}\right)_{n-1}\left(-\frac{\Delta_{1}+\Delta_{2}+\Delta-d-2}{2} \right)_{n-1}}{4(1-\Delta_{1})_{n}(1-\Delta_{2})_{n}}\frac{C_{\Delta_{1}-n, \Delta_{2}-n}(\xi,\eta)}{\xi^{n}}\,. \tag{39}\]
We do not repeat the derivation of this formula here, because it follows easily from [62], and it was reviewed for Witten diagrams with branes in [55]. For general scaling dimensions, we calculate the bulk-exchange diagram in Mellin space in section 4.3.
Besides scalars, there can also be higher-spin fields in the AdS\({}_{d+1}\) effective action. For example, a spin-two field \(\phi_{\mu\nu}\) couples to the brane as \(\int_{\text{AdS}_{p+1}}g^{ij}\phi_{ij}\), where we contract orthogonal indices \(i,j=p+1,\ldots,d\). As a result, the CFT operator dual to \(\phi_{\mu\nu}\) acquires a one-point function. Furthermore, if the spin-two field couples to the scalars \(\sim\int_{\text{AdS}_{d+1}}\phi_{\mu\nu}\nabla^{\mu}\phi_{1}\nabla^{\nu}\phi_ {2}\)
then it can be exchanged in a two-point function
(40)
In many situations of interest, the spin-two diagram reduces to a sum of contact diagrams. This has been discussed for equal external fields in [17, 68], and in appendix A we generalize the discussion to unequal fields.4 Furthermore, we also explain how to apply the truncation in the presence of branes, which is the focus of this work. Because the appendix is somewhat technical, we attach to this publication a Mathematica notebook that computes spin-two diagrams \(E^{\Delta,2}_{\Delta_{1}\Delta_{2}}\).
Footnote 4: Spin-two diagrams for unequal fields were also discussed in [69], but because our formulation is more streamlined than the reference, we believe it is valuable to include a detailed discussion in the appendix.
As a final comment, note that the diagram \(E^{d,2}_{\Delta_{1}\Delta_{1}}\), which corresponds to the exchange of a bulk graviton, contains spurious divergences. However, we found that the limit \(E^{d,2}_{\Delta_{1}\Delta_{1}}\equiv\lim_{\varepsilon\to 0}E^{d+2 \varepsilon,2}_{\Delta_{1}+\varepsilon,\Delta_{1}+\varepsilon}\) leads to sensible results. To motivate this choice of limit, note that it keeps \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}\), which is required for the formulas in appendix A to make sense. As illustration of graviton- and massive-exchange diagrams, for \(d=4\) and \(p=1\) we have
\[E^{42}_{22}=\frac{4}{3\xi}\big{(}\eta C_{22}-C_{13}\big{)}\,,\qquad E^{52}_{32 }=\frac{3}{5\xi}\big{(}2\eta C_{23}-C_{23}-C_{14}\big{)}\,. \tag{41}\]
### Defect-exchange diagram
#### 3.4.1 Scalar diagram
Lastly, we consider a diagram where a brane field is excited by couplings \(\int_{\text{AdS}_{p+1}}\phi_{i}\,\widehat{\phi}\). More specifically, we introduce the defect exchange diagram
(42)
where for now we assume the defect field is not charged under transverse \(SO(d-p)\) rotations. The result depends only on \(\xi+2\eta\), or equivalently, it depends only on the cross-ratio \(r\) defined in (16). This follows by observing that the integrand (42) contains no terms of the form \(x_{1}^{i}x_{2}^{i}\).
Since the exchange diagram depends on a single cross-ratio \(r\), it is very efficient to compute it by solving a differential equation. To obtain the differential equation in question, we
employ the method of [56, 70]. Start giving a name to the \(\widehat{z}_{1}\) integral in (42)
\[I(x_{1},\widehat{z}_{2})=\int_{\widehat{z}_{1}}\widehat{G}_{\widehat{\Delta}}( \widehat{z}_{1},\widehat{z}_{2})K_{\Delta_{1}}(x_{1},\widehat{z}_{1})\,. \tag{43}\]
Because this integral preserves the \(p\)-dimensional conformal group, we have the relation
\[\left(\widehat{\mathbf{L}}_{1}+\widehat{\mathcal{L}}_{\widehat{z}_{2}}\right) _{AB}I(x_{1},\widehat{z}_{2})=0\,, \tag{44}\]
where \(\widehat{\mathbf{L}}_{1}\) is a generator of \(SO(p+1,1)\) acting on operator \(1\), and \(\widehat{\mathcal{L}}_{\widehat{z}}\) is the AdS\({}_{p+1}\) symmetry generator. It then follows that
\[\left(\frac{1}{2}\,\widehat{\mathbf{L}}_{1}^{2}+\widehat{\Delta}(\widehat{ \Delta}-p)\right)I(x_{1},\widehat{z}_{2})=\left(-\nabla_{\widehat{z}_{2}}^{2} +\widehat{\Delta}(\widehat{\Delta}-p)\right)I(x_{1},\widehat{z}_{2})=K_{ \Delta_{1}}(x_{1},\widehat{z}_{2})\,, \tag{45}\]
where the last equality is a consequence of the equations of motion for the brane-to-brane propagator, see (7) with \(d\to p\). The final step is to multiply (45) by \(K_{\Delta_{2}}(x_{2},\widehat{z}_{2})\) and integrate over \(\widehat{z}_{2}\). The left-hand side gives a differential operator acting on \(\widehat{E}_{\Delta_{1}\Delta_{2}}^{\Delta,0}\), and the right-hand side gives a contact diagram
\[\left[-r\partial_{r}r\partial_{r}+\frac{pr(r^{2}+1)}{(1-r)(1+r)}\,\partial_{r} +\widehat{\Delta}(\widehat{\Delta}-p)\right]\widehat{E}_{\Delta_{1},\Delta_{ 2}}^{\widehat{\Delta},0}(r)=C_{\Delta_{1},\Delta_{2}}(r)\,. \tag{46}\]
In deriving the differential operator, we used that the diagram depends only on \(r\) to simplify expressions. The most general solution of (46) depends on two free parameters, but they can always be fixed demanding the following behavior
\[\widehat{E}_{\Delta_{1},\Delta_{2}}^{\widehat{\Delta},0}(r)=\frac{\pi^{p/2} \Gamma(\widehat{\Delta})\Gamma\!\left(\frac{\Delta_{1}-\widehat{\Delta}}{2} \right)\Gamma\!\left(\frac{\Delta_{2}-\widehat{\Delta}}{2}\right)\Gamma\! \left(\frac{\Delta_{1}+\widehat{\Delta}-p}{2}\right)\Gamma\!\left(\frac{ \Delta_{2}+\widehat{\Delta}-p}{2}\right)}{8\Gamma(\Delta_{1})\Gamma(\Delta_{ 2})\Gamma(\widehat{\Delta}+1-\frac{p}{2})}\,r^{\widehat{\Delta}}+O(r^{ \widehat{\Delta}+1})\,. \tag{47}\]
To derive this equation we used the split representation, as in section 4.4. First one computes (88) exactly in \(r\), and then closes the \(\nu\) contour to either left or right. Picking the leading residue gives the contribution (47).
In practice, we have found that solving the ODE provides a very efficient computational method. Recall that for integer \(\Delta_{1}\) and \(\Delta_{2}\) the contact diagram is a rational function of \(r\) and \(\log r\), see (36). In the examples relevant to section 5, the solution to the ODE also turns out to be a rational function, but now including \(\log r\) and \(\log(r+1)\). For example, in \(d=4\) and \(p=1\) we obtain
\[\widehat{E}_{2,2}^{1,0} \propto \log(r+1)-\frac{r^{2}}{r^{2}-1}\log r\,, \tag{48}\] \[\widehat{E}_{3,3}^{2,0} \propto \frac{-2r^{4}+r^{3}+4r^{2}+r-2}{2\left(r^{2}-1\right)^{2}}-\frac {\left(r^{4}-2r^{2}+3\right)r^{3}}{\left(r^{2}-1\right)^{3}}\,\log r+\frac{1+ r^{2}}{r}\log(r+1)\,. \tag{49}\]
Although the ODE works perfectly for the calculations in section 5, let us mention that when \(\Delta_{1}-\widehat{\Delta}\in 2\mathbb{Z}_{>0}\), we can instead use a truncation method:
\[\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}}(\xi,\eta)=\sum_{n=1}^{ \frac{\Delta_{1}-\widehat{\Delta}}{2}}\frac{\left(\!-\frac{\Delta_{1}-\widehat {\Delta}-2}{2}\right)_{n-1}\!\left(\!-\frac{\Delta_{1}+\widehat{\Delta}-p-2}{ 2}\right)_{n-1}}{4(1-\Delta_{1})_{2n}}C_{\Delta_{1}-2n,\Delta_{2}}(\xi,\eta)\,. \tag{50}\]
(If \(\Delta_{2}-\widehat{\Delta}\in 2\mathbb{Z}_{>0}\), one uses the same formula with \(\Delta_{1}\leftrightarrow\Delta_{2}\)). Let us stress that the truncation condition on \(\Delta_{1}\), \(\Delta_{2}\) and \(\widehat{\Delta}\) does not cover all cases of interest. As a result, formula (50) is of little use, and we do not derive it here (in any case, the derivation is identical to [55] with \(d-1\to p\)).
#### 3.4.2 Transverse-spin diagrams
The previous discussion has a simple generalization to operators that are charged under transverse spin, i.e. charged under \(SO(d-p)\) rotations. As a simple example, taking the bulk-brane coupling in (29) we find that the spin-one exchange diagram is given by
\[\int_{\widehat{z}_{1},\widehat{z}_{2}}\widehat{z}_{1}^{0}\widehat{z}_{2}^{0} \widehat{G}_{\widehat{\Delta}}(\widehat{z}_{1},\widehat{z}_{2})\partial_{i}K_ {\Delta_{1}}(x_{1},\widehat{z}_{1})\partial_{i}K_{\Delta_{2}}(x_{2},\widehat{ z}_{2})\ \equiv\ \frac{\widehat{E}^{\widehat{\Delta},1}_{\Delta_{1}\Delta_{2}}(\xi,\eta)}{|x_{1}^{i }|^{\Delta_{1}}|x_{2}^{i}|^{\Delta_{2}}}\,. \tag{51}\]
A simple calculation shows that the result factorizes as
\[\widehat{E}^{\widehat{\Delta},1}_{\Delta_{1}\Delta_{2}}(\xi,\eta)\equiv 4 \Delta_{1}\Delta_{2}\eta\,\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}+1, \Delta_{2}+1}(\xi,\eta)\,. \tag{52}\]
It is important to notice the shift in the arguments \(\Delta_{1},\Delta_{2}\to\Delta_{1}+1,\Delta_{2}+1\). More generally, it is not hard to check that a defect-exchange diagram with transverse spin reduces to a product of a scalar diagram with shifted arguments, and a factor with simple \(\eta\) dependence:
\[\widehat{E}^{\widehat{\Delta},s}_{\Delta_{1}\Delta_{2}}(\xi,\eta)\propto\eta ^{s}\,\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}+s,\Delta_{2}+s}(\xi,\eta)\,. \tag{53}\]
However, we only need the \(s=0,1\) cases below.
### A detour: connection to analytic functionals5
Footnote 5: This section lies somewhat outside the main scope of this work, and can be safely skipped by readers not interested in analytic functionals.
It is well known that, in CFT without defects, different approaches to analytic bootstrap are closely related to each other. Indeed, the Lorentzian inversion formula [71, 72] was resummed into a dispersion relation [73], which in turn generates the analytic functionals of [74]. Furthermore, the dispersion relations in position and Mellin space [75] are equivalent [76]. For boundary CFT, some of these connections have been explored in [55, 56, 57].
Motivated by these developments, we show here that in defect CFT there is also a connection between the dispersion relation, a basis of analytic functionals, and exchange Witten diagrams. Unlike the rest of this work, this section relies heavily on analytic bootstrap methods for conformal defects, and the required background can be found in [54, 65, 77, 78]. Recall that in defect CFT there are two dispersion relations [54, 78] (or equivalently two inversion formulas [65, 77]). Here we focus on the dispersion relation that reconstructs the correlator starting from a single discontinuity \(\operatorname{Disc}F(r,w^{\prime})=F(r,w^{\prime}+i0)-F(r,w^{\prime}-i0)\). We define the Polyakov-Regge block \(P^{\Delta,J}_{\Delta_{1}\Delta_{2}}\) as the dispersion relation applied to a bulk-channel conformal block \(f_{\Delta,J}\).6 The expression looks more elegant in the \(r\), \(w\) cross ratios of equation (16):
Footnote 6: In this section, \(J\) is the spin of an operator of dimension \(\Delta\), and \(\ell\) is the spin of a double-twist operator. Similarly, we also distinguish transverse-spin \(S\) and \(s\) below.
\[P^{\Delta,J}_{\Delta_{1}\Delta_{2}}(r,w)\equiv\int_{0}^{r}\frac{dw^{\prime}}{2 \pi i}\frac{w(1-w^{\prime})(1+w^{\prime})}{w^{\prime}(w^{\prime}-w)(1-ww^{ \prime})}\operatorname{Disc}\Bigl{[}\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{ \Delta,J}(r,w^{\prime})\Bigr{]}\,. \tag{54}\]
The discontinuity vanishes on double-twist operators \(\operatorname{Disc}\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{\Delta_{1}+ \Delta_{2}+\ell+2n,\ell}=0\), see e.g. [54]. As a result, the bulk-channel block expansion of the Polyakov-Regge block is of the form
\[P^{\Delta,J}_{\Delta_{1}\Delta_{2}}(r,w)=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2 }}\biggl{[}f_{\Delta,J}(r,w)+\sum_{n,\ell=0}^{\infty}a_{\ell,n}f_{\Delta_{1}+ \Delta_{2}+\ell+2n,\ell}(r,w)\biggr{]}\,. \tag{55}\]
To obtain the defect-channel block expansion, one can apply the inversion formula of [65] to the block \(f_{\Delta,J}\). It is not difficult to see that the expansion has the following structure
\[P^{\Delta,J}_{\Delta_{1}\Delta_{2}}(r,w)=\sum_{n,\ell=0}^{\infty}b_{s,n} \widehat{f}_{\Delta_{1}+s+2n,s}(r,w)+\sum_{n,\ell=0}^{\infty}c_{s,n}\widehat{ f}_{\Delta_{2}+s+2n,s}(r,w)\,, \tag{56}\]
although extracting \(b_{n,\ell}\) and \(c_{n,\ell}\) in closed form is more challenging. Finally, note that the Polyakov-Regge block decays in the Regge limit
\[\bigl{|}P^{\Delta,J}_{\Delta_{1}\Delta_{2}}(r,w)\bigr{|}<|w|^{-\varepsilon} \quad\text{for}\quad w\to\infty\,,\;\varepsilon>0\,, \tag{57}\]
a property that follows from definition (54). In the terminology of [56, 74], we say that the Polyakov-Regge block is superbounded.
At this point, we claim that the functions \(\{f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell},\widehat{f}_{\Delta_{1}+s+2n,s}, \widehat{f}_{\Delta_{2}+s+2n,s}\}\) form a basis in a suitable space of functions, and restrict our attention to CFT correlators in such a space.7 We do not attempt to prove this claim, although there is evidence in the literature for similar statements being true [56, 74, 79]. We can then introduce a dual basis \(\{\alpha_{\ell,n},\beta_{s,n},\gamma_{s,n}\}\)
in the natural way, called the basis of analytic functionals. The analytic functionals are useful in the bootstrap program, because when acting on the crossing equation, they generate sum rules obeyed by the CFT data. For example, we have
\[\sum_{\mathcal{O}}\lambda_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}a_{\mathcal{ O}}\,\alpha_{\ell,n}[f_{\Delta,J}]-\sum_{\widehat{\mathcal{O}}}b_{\mathcal{O}_{1} \widehat{\mathcal{O}}}b_{\mathcal{O}_{2}\widehat{\mathcal{O}}}\,\alpha_{\ell,n} [\widehat{f}_{\widehat{\Delta},s}]=0\,, \tag{58}\]
and two more expressions with \(\beta_{s,n}\) and \(\gamma_{s,n}\). These sum rules are valid in any CFT, although they are particularly useful to bootstrap theories perturbatively around mean-field theory. To compute the action of the functionals on a generic bulk block, note that acting with the functionals on (55) and (56) gives
\[\alpha_{\ell,n}[f_{\Delta,\ell}]=-a_{\ell,n}\,,\qquad\beta_{s,n}[f_{\Delta, \ell}]=b_{s,n}\,,\qquad\gamma_{s,n}[f_{\Delta,\ell}]=c_{s,n}\,. \tag{59}\]
Therefore, the expansion coefficients of the Polyakov-Regge block give the action of analytic functionals on bulk-channel conformal blocks. A similar story holds for defect-channel Polyakov-Regge blocks \(\widehat{P}^{\widehat{\Delta},S}_{\Delta_{1}\Delta_{2}}\). These are defined using the "defect-to-bulk" dispersion relation, namely \(\widehat{P}^{\widehat{\Delta},S}_{\Delta_{1}\Delta_{2}}=\int K\,\mathrm{dDisc }\,\widehat{f}_{\widehat{\Delta},S}\) with a kernel \(K\) that can be found in [78]. Their expansion is of the form
\[\widehat{P}^{\widehat{\Delta},S}_{\Delta_{1}\Delta_{2}}(r,w) =\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{n,\ell=0}^{\infty} \widehat{a}_{\ell,n}f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell}(r,w) \tag{60}\] \[=\widehat{f}_{\widehat{\Delta},S}+\sum_{n,\ell=0}^{\infty} \widehat{b}_{s,n}\widehat{f}_{\Delta_{1}+s+2n,s}(r,w)+\sum_{n,\ell=0}^{\infty }\widehat{c}_{s,n}\widehat{f}_{\Delta_{2}+s+2n,s}(r,w)\,, \tag{61}\]
which implies the action of analytic functionals
\[\alpha_{\ell,n}[\widehat{f}_{\widehat{\Delta},S}]=\widehat{a}_{\ell,n}\,, \qquad\beta_{s,n}[\widehat{f}_{\widehat{\Delta},S}]=-\widehat{b}_{s,n}\,, \qquad\gamma_{s,n}[\widehat{f}_{\widehat{\Delta},S}]=-\widehat{c}_{s,n}\,. \tag{62}\]
Besides the definition in terms of Polyakov-Regge blocks, the functionals \(\alpha_{\ell,n}[f]\), \(\beta_{s,n}[f]\) and \(\gamma_{s,n}[f]\) might also admit a representation as integrals of \(\mathrm{Disc}\,f\) or \(\mathrm{dDisc}\,f\) against certain kernels. As in [74, 56, 79], one should obtain the integral form of the functionals by expanding in conformal blocks the dispersion relation kernel. We do not perform this calculation here.
Having shown the utility of Polyakov-Regge blocks, we are faced with the question of how to compute them. One method uses that Polyakov-Regge blocks are exchange Witten diagrams, with appropriate addition of contact terms to make them Regge superbounded [56, 74]. For spin \(\ell=0,2\), the relations should be of the form
\[P^{\Delta,0}_{\Delta_{1}\Delta_{2}}=k_{1}E^{\Delta,0}_{\Delta_{1}\Delta_{2}}\,, \qquad P^{\Delta,2}_{\Delta_{1}\Delta_{2}}=k_{1}k_{2}\left(E^{\Delta,2}_{\Delta _{1}\Delta_{2}}+k_{3}C_{\Delta_{1}\Delta_{2}}\right)\,. \tag{63}\]
To extract the precise relation, we use the decomposition of exchange diagrams in terms of contact diagrams, see (39) and appendix A. We fix the overall normalization comparing to
the block expansion (55),8 and we fix the contact terms taking the Regge limit (57). In the end, we find
Footnote 8: This equation depends on the choice of normalization for the conformal blocks. Here we normalize blocks in the lightcone limit \(|1-z|\ll|1-\bar{z}|\ll 1\) as \(f_{\Delta,\ell}(z,\bar{z})=(1-z)^{\frac{\Delta-\ell}{2}}(1-\bar{z})^{\frac{ \Delta+\ell}{2}}+\ldots\).
\[k_{1} =\frac{2^{\Delta+2}(-1)^{\frac{\Delta_{1}+\Delta_{2}-\Delta+2}{2} }}{\pi^{\frac{p+1}{2}}\Gamma\big{(}\frac{\Delta-p}{2}\big{)}\,\big{(}\frac{ \Delta_{1}+\Delta_{2}-\Delta-2}{2}\big{)}!\,\big{(}\frac{d-\Delta-\Delta_{1}- \Delta_{2}+2}{2}\big{)}_{\frac{\Delta_{1}+\Delta_{2}-\Delta-2}{2}}}\,, \tag{64}\] \[k_{2} =\frac{8\,(\Delta^{2}-1)}{(\Delta_{1}+\Delta_{2}-\Delta)(\Delta_ {1}+\Delta_{2}+\Delta-d)(\Delta^{2}-\Delta_{12}^{2})}\,,\] (65) \[k_{3} =\frac{d+1}{2d}\!-\!\frac{\Delta_{12}^{2}(\Delta_{1}+\Delta_{2}- d)^{2}}{2d\Delta(\Delta-d)}+\frac{(\Delta_{12}^{2}-1)\,((\Delta_{1}+\Delta_{2}-d)^{2 }-1)}{2d(\Delta-1)(\Delta-d+1)}\,. \tag{66}\]
These formulas hold for \(\ell=0\), \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}\), and for \(\ell=2\), \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{\geq 0}\). For general dimensions, we expect the formulas to be more complicated, and perhaps they can be obtained in Mellin space in analogy to [56, 74].
Although the presentation in section 3.5 has been somewhat schematic, we hope it will encourage further exploration of these interesting issues.
## 4 Witten diagrams in Mellin space
This section studies the Mellin representation of the two-point function of local operators in the presence of a defect, initially introduced in [31]. We show how the Witten diagrams of section 3 translate to Mellin space, and in particular, we find closed formulas for exchange Mellin amplitudes with arbitrary scaling dimensions.
### Mellin space
The Mellin amplitude is only valid for the connected correlator \(F_{c}\), which we introduced in equation (19). Keeping this in mind, the Mellin amplitude \(\mathcal{M}(\delta,\rho)\) is
\[F_{c}(\xi,\eta)=\int\frac{d\delta\,d\rho}{(2\pi i)^{2}}\,\frac{\Gamma(\delta) \Gamma(\rho)\Gamma\big{(}\frac{\Delta_{1}-\delta-\rho}{2}\big{)}\,\Gamma \big{(}\frac{\Delta_{2}-\delta-\rho}{2}\big{)}}{\xi^{\delta}\,(2\eta)^{\rho}} \,\mathcal{M}(\delta,\rho)\,. \tag{67}\]
This construction can be generalized to any correlator in defect CFT [31], but we do not need it here. Although we are somewhat loose on what the integration contours are, suffice it to mention that they run parallel to the imaginary axis. Actually, it is not obvious whether non-perturbative correlators admit a Mellin representation, and if they do, what is the correct integration contour. This is not a problem to us, because we work in perturbation theory and apply the Mellin transform diagram by diagram, so it is possible to find a contour such
that all manipulations are well defined. However, it would be very interesting to analyze non-perturbative Mellin amplitudes along the lines of [75, 80].
Leaving these considerations aside, one motivation to work in Mellin space is that Mellin amplitudes are meromorphic in \(\delta\) and \(\rho\), at least in perturbation theory. Furthermore, the location of poles corresponds to dimensions of operators exchanged in the OPEs. More precisely, assume the OPE is of the form \(\mathcal{O}_{1}\times\mathcal{O}_{2}\sim\mathcal{O}+\dots\), where operator \(\mathcal{O}\) has scaling dimension \(\Delta\) and spin \(\ell\). Then the integrand of (67) has poles at
\[\delta=\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell-2n}{2}\,,\qquad n=0,1,2,\dots\,. \tag{68}\]
Similarly, if the defect expansion is of the form \(\mathcal{O}_{1}\times\mathcal{D}\sim\widehat{\mathcal{O}}\mathcal{D}+\dots\), with \(\widehat{\mathcal{O}}\) a local operator on the defect of dimension \(\widehat{\Delta}\) and transverse-spin \(s\), then the integrand in (67) has poles at
\[\delta+\rho=\widehat{\Delta}-s+2n\,,\qquad n=0,1,2,\dots\,. \tag{69}\]
Let us stress that these are poles of the integrand in (67). In holographic theories, the spectrum is approximately that of MFT
\[\text{Bulk:}\quad\mathcal{O}_{1}\square^{n}\partial_{\mu_{1}}\dots\partial_{ \mu_{\ell}}\mathcal{O}_{2}\,,\qquad\text{Defect:}\quad\square^{n}\partial_{ i_{1}}\dots\partial_{i_{s}}\mathcal{O}_{1}\,,\quad\square^{n}\partial_{i_{1}} \dots\partial_{i_{s}}\mathcal{O}_{2}\,. \tag{70}\]
The integrand in (67) is such that poles by the MFT operators are taken care of by the gamma functions. As a result, in holographic theories the Mellin amplitude \(\mathcal{M}(\delta,\rho)\) has a simpler analytic structure than the full integrand, as we show in examples below.
### Contact diagram
As a first application of Mellin space, let us obtain the Mellin transform of the contact diagram \(C_{\Delta_{1}\Delta_{2}}\). We start from (32) and use identity (33) twice, so the contact diagram has the form
\[C_{\Delta_{1}\Delta_{2}}(\xi,\eta)=\frac{\pi^{p/2}\Gamma\left( \frac{\Delta_{1}+\Delta_{2}-p}{2}\right)}{2\Gamma(\Delta_{1})\Gamma(\Delta_{2})} \int\frac{d\delta\,d\rho}{(2\pi i)^{2}}\frac{\Gamma(\delta)\Gamma(\rho) \Gamma\big{(}\frac{\Delta_{1}+\Delta_{2}}{2}-\delta-\rho\big{)}}{\xi^{\delta} (2\eta)^{\rho}}\] \[\times\int_{0}^{\infty}\frac{ds}{s}\,s^{\Delta_{1}-\delta-\rho} \frac{|x_{1}^{i}|^{\Delta_{1}-\delta-\rho}|x_{2}^{i}|^{\Delta_{2}-\delta-\rho} }{(s^{2}|x_{1}^{i}|^{2}+|x_{2}^{i}|^{2})^{\frac{\Delta_{1}+\Delta_{2}}{2}- \delta-\rho}}\,. \tag{71}\]
The \(s\) integral is elementary, and after factoring out the prefactor (67) we get a Mellin amplitude that is constant:
\[\mathcal{M}[C_{\Delta_{1},\Delta_{2}}]=\frac{\pi^{\frac{p}{2}}\Gamma\big{(} \frac{\Delta_{1}+\Delta_{2}-p}{2}\big{)}}{4\Gamma(\Delta_{1})\Gamma(\Delta_{2 })}\,. \tag{72}\]
This is completely analogous to four-point contact Mellin amplitudes, which are also constant. Regarding the integration contour, observe that the manipulations that led to (72)
are valid provided \(\operatorname{Re}\delta,\operatorname{Re}\rho>0\) and \(0<\operatorname{Re}\delta+\operatorname{Re}\rho<\min(\Delta_{1},\Delta_{2})\). As a result, we have to obey these constraints when choosing the integration contour in (67), or equivalently, the contour has to separate the "left" and "right" families of poles in the integrand.
The fact that contact amplitudes are constant is very desirable, and actually it motivates the choice of prefactors in the Mellin amplitude (67). One reason is that large classes of exchange diagrams are sums of contact diagrams, and in this case, the Mellin amplitudes become rational functions of \(\delta\) and \(\rho\). To combine contact amplitudes correctly, there are two observations to keep in mind. The first observation is that contact diagrams are often multiplied in position space by powers of the cross-ratios \(\xi^{n}\) and \(\eta^{m}\), which in Mellin space are shifts of the variables \(\delta\to\delta+n\) and \(\rho\to\rho+m\). However, because of our definition of the Mellin amplitude (67), the shift is accompanied by extra Pochhammer symbols:
\[\xi^{n}\eta^{m}F(\xi,\eta)\ \leftrightarrow\ \frac{(\rho)_{m}(\delta)_{n}}{2^{m}} \left(\frac{\Delta_{1}-\delta-\rho}{2}\right)_{\frac{m-n}{2}}\left(\frac{ \Delta_{2}-\delta-\rho}{2}\right)_{\frac{m-n}{2}}\mathcal{M}(\delta+n,\rho+m)\,. \tag{73}\]
The second observation is that the integrand in (67) depends on \(\Delta_{1}\) and \(\Delta_{2}\), which gives an extra contribution when combining contact diagrams of different dimensions. These two observations are best illustrated in practice. For example, the Mellin transform of (41) reads
\[\mathcal{M}\big{[}E_{2,2}^{4,2}\big{]}=\frac{\pi}{24}\left(1+\frac{3\rho-1}{ \delta-1}\right)\,,\qquad\mathcal{M}\big{[}E_{3,2}^{5,2}\big{]}=\frac{\sqrt{ \pi}}{16}\left(\frac{4}{5}+\frac{2\rho-1}{\delta-1}\right)\,. \tag{74}\]
As anticipated, the exchange diagrams are remarkably simple rational functions of the Mellin variables.
### Bulk-exchange diagram
We already mentioned in section 3.3 that exchange Witten diagrams are hard to calculate due to the complicated formula for the bulk-to-bulk propagator (8). Fortunately, there exists a "split representation" of the propagator as a double integral of bulk-to-boundary propagators
\[G_{\Delta}(z_{1},z_{2})=\int_{-i\infty}^{i\infty}\frac{d\nu}{2\pi i}\,\frac{2 \nu^{2}\,\mathcal{C}_{d/2+\nu}\,\mathcal{C}_{d/2-\nu}}{\nu^{2}-(\Delta-\frac{ d}{2})^{2}}\int_{\partial\operatorname{AdS}_{d+1}}\!d^{d}x\,K_{\frac{d}{2}+\nu}(z_{1}, x)K_{\frac{d}{2}-\nu}(z_{2},x)\,. \tag{75}\]
Here \(\mathcal{C}_{\Delta}\) is the normalization of the bulk-bulk propagator (9), the \(\nu\) integration goes along the contour \(\operatorname{Re}\nu=0\), and the \(x\) integral is over the boundary of \(\operatorname{AdS}_{d+1}\), namely \(\mathbb{R}^{d}\). This formula is motivated and proven in the appendix of [7]. For our purposes, it suffices to note that thanks to this formula, exchange diagram factorize as products of lower-point diagrams, at the expense of adding extra integrations.
For example, the bulk-exchange diagram defined in (37) is schematically
\[E^{\Delta,0}_{\Delta_{1}\Delta_{2}}\ \ =\
We finally have all the ingredients for the bulk-exchange diagram (76). The combination of the split representation (75), the one-point function (21), the three-point function (78) and the position integral (81) has to be compared to the Mellin integral (67). For the comparison to work, we shift the Mellin variable as \(\bar{\delta}=\delta-\frac{\Delta_{1}+\Delta_{2}-\nu-d/2}{2}\). All in all, the bulk-exchange amplitude \(\mathcal{M}^{\Delta,0}_{\Delta_{1}\Delta_{2}}\equiv\mathcal{M}\big{[}E^{ \Delta,0}_{\Delta_{1}\Delta_{2}}\big{]}\) is given by
\[\mathcal{M}^{\Delta,0}_{\Delta_{1}\Delta_{2}}(\delta,\rho)=\frac{ \pi^{p/2}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\delta)\Gamma\big{(} \delta-\frac{\Delta_{1}+\Delta_{2}-d+p}{2}\big{)}}\int_{-i\infty}^{i\infty} \frac{d\nu}{2\pi i}\frac{l_{b}(\nu)l_{b}(-\nu)}{(\Delta-\frac{d}{2})^{2}-\nu^ {2}}\,, \tag{82}\]
where
\[l_{b}(\nu)=\frac{\Gamma\big{(}\frac{2\nu+d-2p}{4}\big{)}\,\Gamma \big{(}\frac{2\nu+2\Delta_{1}+2\Delta_{2}-d}{4}\big{)}\,\Gamma\big{(}\delta+ \frac{2\nu-2\Delta_{1}-2\Delta_{2}+d}{4}\big{)}}{4\Gamma(\nu)}\,. \tag{83}\]
Note that the scalar Mellin amplitude only depends on \(\delta\). This is a consequence of our choice of prefactor in (67), because the prefactor captures the poles in \(\delta+\rho\) corresponding to defect operators with MFT dimensions. On the other hand, (82) captures new poles in \(\delta\) due to the exchange of a bulk operator of dimension \(\Delta\), recall (68).
Note also the similarity between our formula and the Mellin amplitude of a four-point scalar Witten diagram, see equation (38) in [7]. In fact, we can follow verbatim the analysis of reference [7], which observed that poles in \(\delta\) arise when two poles in the integrand pinch the \(\nu\) contour. A simple application of the residue theorem then gives
\[\mathcal{M}^{\Delta,0}_{\Delta_{1}\Delta_{2}}(\delta,\rho)=\sum_{n=0}^{\infty }\frac{R_{n}}{\delta-\frac{\Delta_{1}+\Delta_{2}-\Delta-2n}{2}}\,, \tag{84}\]
where the residues are9
Footnote 9: To obtain this formula, we use that for a meromorphic function \(g(z)=\sum_{i}\frac{f_{i}(z)}{z-a_{i}}=\sum_{i}\frac{f_{i}(a_{i})}{z-a_{i}}\), provided \(g(z)\) decays at infinity. We have checked numerically in examples that (82) and (84) agree, which justifies using this identity.
\[R_{n}=\frac{\pi^{p/2}}{16n!}\frac{\Gamma\big{(}\frac{\Delta-p}{2}\big{)}\, \Gamma\big{(}\frac{\Delta+\Delta_{1}+\Delta_{2}-d}{2}\big{)}\,\big{(}1+\frac {\Delta-d+p}{2}\big{)}_{n}\,\big{(}1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2} \big{)}_{n}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma\big{(}\Delta+n+1- \frac{d}{2}\big{)}}\,. \tag{85}\]
Here we see that when \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}\), the infinite sum truncates to a finite sum. This finite sum agrees with the sum over contact diagrams (39) that we found in position space. Although we shall not need it, the sum can be computed in terms of a generalized hypergeometric function
\[\mathcal{M}^{\Delta,0}_{\Delta_{1}\Delta_{2}}(\delta,\rho)=\frac{ \pi^{p/2}\Gamma\big{(}\frac{\Delta-p}{2}\big{)}\,\Gamma\big{(}\frac{\Delta_{1 }+\Delta_{2}+\Delta-d}{2}\big{)}}{16\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma \big{(}\frac{2\Delta-d+2}{2}\big{)}\big{(}\delta-\frac{\Delta_{1}+\Delta_{2} -\Delta}{2}\big{)}}\] \[\times\,\,_{3}F_{2}\!\left(\begin{array}{c}1+\frac{\Delta-d+p}{ 2},1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2},\delta+\frac{\Delta-\Delta_{1}- \Delta_{2}}{2}\\ \Delta+1-\frac{d}{2},\delta+1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2}\end{array} ;1\right)\,. \tag{86}\]
The \(\ell>0\) blocks for arbitrary dimensions can also be computed using the split representation, which was developed in [81]. However, that technology is somewhat involved and we choose not to explore it in the present work. Actually, for applications to section 5, all \(\ell=2\) exchanges truncate to a sum of contact diagrams, which can be easily Mellin transformed.
### Defect-exchange diagram
A similar calculation also applies to the defect-exchange diagram (42). The idea is to use the split representation for the brane-to-brane propagator, equation (75) with \(d\to p\). The exchange diagram factorizes as a product of bulk-brane diagrams
\[\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}} = \tag{87}\]
at the expense of adding integrations over \(\widehat{x}\) and \(\nu\). The bulk-to-brane Witten diagrams are kinematically fixed, and their normalization is computed in section 3.1.2. As a result, we need to compute the integral
\[H=\int\frac{d^{p}\widehat{x}_{3}}{\big{(}(x_{1}^{i})^{2}+(x_{1 \dot{3}}^{a})^{2}\big{)}^{\frac{p}{2}+\nu}\big{(}(x_{2}^{i})^{2}+(x_{2\dot{3}} ^{a})^{2}\big{)}^{\frac{p}{2}-\nu}}\,. \tag{88}\]
The technique is by now standard. Introduce two Schwinger parameters \(s\) and \(t\), integrate over \(\widehat{x}_{3}\), and integrate over \(t\) with the help of an auxiliary parameter \(\lambda\). The result takes the form
\[H =\frac{\pi^{\frac{p}{2}}\Gamma(\frac{p}{2})}{\Gamma(\frac{p}{2}- \nu)\Gamma(\frac{p}{2}+\nu)}\int\frac{ds}{s}\frac{s^{\frac{p}{2}+\nu}}{\big{(} s^{2}|x_{1}^{i}|^{2}+|x_{2}^{i}|^{2}+2\eta s|x_{1}^{i}||x_{2}^{i}|+\xi s|x_{1}^{i}| |x_{2}^{i}|\big{)}^{p/2}}\] \[=\frac{\pi^{\frac{p}{2}}}{|x_{1}^{i}|^{\frac{p}{2}+\nu}|x_{2}^{i} |^{\frac{p}{2}-\nu}}\int\frac{d\delta\,d\rho}{(2\pi i)^{2}}\frac{\Gamma(\delta )\Gamma(\rho)}{\xi^{\delta}(2\eta)^{\rho}}\frac{\Gamma\Big{(}\frac{p/2-\nu- \delta-\rho}{2}\Big{)}\,\Gamma\Big{(}\frac{p/2+\nu-\delta-\rho}{2}\Big{)}}{2 \Gamma(\frac{p}{2}-\nu)\Gamma(\frac{p}{2}+\nu)}\,, \tag{89}\]
where we used (33) twice to factorize the denominator, and then computed an elementary integral over \(s\). Again, the combination of the split representation (75), the bulk-brane diagram (28) and the boundary integral (89) is compared with the Mellin amplitude (67). Using notation \(\widehat{\mathcal{M}}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}}\equiv \mathcal{M}\big{[}\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}}\big{]}\), the defect-exchange amplitude reads
\[\widehat{\mathcal{M}}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}}(\delta, \rho)=\frac{\pi^{\frac{p}{2}}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma \big{(}\frac{\Delta_{1}-\delta-\rho}{2}\big{)}\,\Gamma\big{(}\frac{\Delta_{2} -\delta-\rho}{2}\big{)}}\int_{-i\infty}^{i\infty}\frac{d\nu}{2\pi i}\frac{l_{ d}(\nu)l_{d}(-\nu)}{(\widehat{\Delta}-\frac{p}{2})^{2}-\nu^{2}}\,, \tag{90}\]
where
\[l_{d}(\nu)=\frac{\Gamma\big{(}\frac{2\nu+2\Delta_{1}-p}{4}\big{)}\,\Gamma \big{(}\frac{2\nu+2\Delta_{2}-p}{4}\big{)}\,\Gamma\big{(}\frac{2\nu+p-2\delta -2\rho}{4}\big{)}}{4\Gamma(\nu)}\,. \tag{91}\]
In this case the amplitude depends only on \(\delta+\rho\), and the \(\delta\) dependence is accounted for by the prefactor in (67). The interpretation is similar as before, namely the bulk-channel block expansion contains operators with MFT dimension \(\Delta=\Delta_{1}+\Delta_{2}+\ell+2n\) which are completely captured by the prefactor. Instead, in the defect-channel expansion there is a new operator with \(\widehat{\Delta}\), whose poles are captured by (90).
Once again, we can obtain the poles in \(\delta+\rho\) by picking the residues when two poles pinch the \(\nu\) integration contour. We find the following sum
\[\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},0} (\delta,\rho)=\sum_{n=0}^{\infty}\frac{\widehat{R}_{n}}{\delta+\rho-\widehat{ \Delta}-2n}\,, \tag{92}\]
with residues
\[\widehat{R}_{n}=-\frac{\pi^{\frac{p}{2}}}{8n!}\frac{\Gamma\! \left(\frac{\widehat{\Delta}+\Delta_{1}-p}{2}\right)\Gamma\!\left(\frac{ \widehat{\Delta}+\Delta_{2}-p}{2}\right)\left(1+\frac{\widehat{\Delta}-\Delta _{1}}{2}\right)_{n}\left(1+\frac{\widehat{\Delta}-\Delta_{2}}{2}\right)_{n}}{ \Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\widehat{\Delta}+n+1-\frac{p}{2}) }\,. \tag{93}\]
As expected, the sum truncates when \(\Delta_{1}-\widehat{\Delta},\Delta_{2}-\widehat{\Delta}\in\mathbb{Z}_{>0}\), analogously to equation (50). More importantly, even when the sum does not truncate, our calculation goes through and we obtain
\[\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},0} =-\frac{\pi^{p/2}\Gamma\!\left(\frac{\Delta_{1}+\widehat{\Delta}-p}{2} \right)\Gamma\!\left(\frac{\Delta_{2}+\widehat{\Delta}-p}{2}\right)}{8\Gamma (\Delta_{1})\Gamma(\Delta_{2})\Gamma(\widehat{\Delta}+1-\frac{p}{2})(\delta+ \rho-\widehat{\Delta})}{}^{3}F_{2}\!\left(\!\!\!\begin{array}{c}1-\frac{ \Delta_{1}-\widehat{\Delta}}{2},1-\frac{\Delta_{2}-\widehat{\Delta}}{2}, \frac{\widehat{\Delta}-\delta-\rho}{2}\\ \widehat{\Delta}+1-\frac{p}{2},1+\frac{\widehat{\Delta}-\delta-\rho}{2}\end{array} \!\!\!;1\right)\,. \tag{94}\]
Although the discussion so far is restricted to scalar exchanges, we showed in section 3.4.2 that generalization to fields with transverse spin \(s>0\) is straightforward. Indeed, a generic exchange diagrams factorizes as \(\widehat{E}_{\Delta_{1},\Delta_{2}}^{\widehat{\Delta},0}\sim\eta^{s}\widehat{ E}_{\Delta_{1}+s,\Delta_{2}+s}^{\widehat{\Delta},0}\), where the scalar exchange has shifted arguments. For example, the Mellin transform of the spin-one exchange (52) reads
\[\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},1} (\delta,\rho)=2\Delta_{1}\Delta_{2}\rho\mathcal{M}_{\Delta_{1}+1,\Delta_{2}+1 }^{\widehat{\Delta},0}(\delta,\rho+1)\,, \tag{95}\]
where the factor \(\rho\) is explained by (73). More generally, the spin-\(s\) exchange diagram is of the form
\[\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},s} (\delta,\rho)\propto(\rho)_{s}\,\mathcal{M}_{\Delta_{1}+s,\Delta_{2}+s}^{ \widehat{\Delta},0}(\delta,\rho+s)\,. \tag{96}\]
## 5 Half-BPS Wilson line in \(\mathcal{N}=4\) Sym
In this section, we apply the position-space bootstrap to the half-BPS Wilson line in \(\mathcal{N}=4\) SYM. More precisely, we bootstrap the correlator of two chiral-primary operators and a half-BPS Wilson line at leading order in the supergravity approximation. We start reviewing the
setup from the CFT perspective, and then move on to the holographic description. We derive the AdS effective actions for bulk and brane, and input this information to the position-space bootstrap method. This allows us to determine the correlator in closed form. The result agrees with a first-principles Witten diagram calculation, up to contact diagrams that we do not know how to fix in the first-principles method. We conclude expressing our result in Mellin space, where it takes a remarkably simple form.
### Setup
Before presenting the holographic calculation, let us review the system using field-theoretic language. The same system was studied in [53, 54], where the interested reader can find further details.
We work with planar \(\mathcal{N}=4\) SYM theory in the strong 't Hooft coupling limit \(\lambda=g^{2}N\to\infty\). The local operators are chiral primaries in the \([0,k,0]\) representation of \(SU(4)_{R}\), that in field theory language read10
Footnote 10: Here \(\ldots\) contains multi-trace operators that ensure that \(S_{k}\) are operators dual to single-particle supergravity modes. The most important property of this choice of operators is that extremal three-point functions vanish, namely the OPE coefficient \(\lambda_{S_{k_{1}}S_{k_{2}}S_{k_{1}+k_{2}}}=0\). The difference between single-trace and \(S_{k}\) operators plays only a tangential role in our story, and we refer to section 2.1 of [18] for a more detailed exposition.
\[S_{k}(x,u)\propto\text{tr}\left(u\cdot\phi(x)\right)^{k}+\ldots\,. \tag{97}\]
As usual, we work with index-free notation, where \(u\) is a null six-component vector \(u^{2}=0\) that implements the tracelessness of \(S_{k}\). The chiral-primary operators are scalars with protected scaling dimension \(\Delta=k\).
Besides local operators, we also consider a straight half-BPS Wilson line
\[W(\theta)=\frac{1}{N}\text{ tr }\text{Pexp}\int_{-\infty}^{\infty}d\tau\left(i \dot{x}^{\mu}A_{\mu}+|\dot{x}|\,\theta\cdot\phi\right). \tag{98}\]
This is a one-dimensional defect that lives in four-dimensional space, or in other words, the dual setup is AdS\({}_{5}\) with a brane extending on AdS\({}_{2}\). Therefore, from now on we set \(d=4\) and \(p=1\). In (98) there is a choice of \(R\)-symmetry direction which we parametrize with the unit six-component vector \(\theta^{2}=1\). For a straight contour, the expectation value of the Wilson operator is \(\langle W(\theta)\rangle=1\), so unlike section 2.3, we do not need to normalize correlators by \(\langle W(\theta)\rangle\).
Thanks to the index-free notation, we can write the two-point function of chiral-primary operators as
\[\langle S_{p_{1}}(x_{1},u_{1})S_{p_{2}}(x_{2},u_{2})W(\theta) \rangle=\frac{(u_{1}\cdot\theta)^{p_{1}}(u_{2}\cdot\theta)^{p_{2}}}{|x_{1}^{i }|^{p_{1}}|x_{2}^{i}|^{p_{2}}}\,\mathcal{F}(\xi,\eta,\sigma)\,. \tag{99}\]
The function \(\mathcal{F}(\xi,\eta,\sigma)\) captures the non-trivial information of this correlator. The first two variables \(\xi\), \(\eta\) are the spacetime cross-ratios (15), while \(\sigma\) is an \(R\)-symmetry cross-ratio. The dependence on \(\sigma\) is polynomial, namely
\[\mathcal{F}(\xi,\eta,\sigma)=\sum_{n=0}^{p_{\rm min}}\sigma^{n}\,\mathcal{F}_{n }(\xi,\eta)\quad\text{ with }\quad\sigma=\frac{u_{1}\cdot u_{2}}{(u_{1}\cdot\theta)(u_{2} \cdot\theta)}\,. \tag{100}\]
Here and below we define \(p_{\rm min}=\min(p_{1},p_{2})\). We will be mostly concerned with the connected correlator \(\mathcal{F}_{c}\), which is defined analogously to (18). After extracting the overall prefactor in (99), the connected correlator is
\[\mathcal{F}(\xi,\eta,\sigma)=\Big{(}\frac{\sigma}{\xi}\Big{)}^{\frac{p_{1}+p_{ 2}}{2}}+a_{p_{1}}a_{p_{2}}+\mathcal{F}_{c}(\xi,\eta,\sigma)\,, \tag{101}\]
where \(a_{p}\) is the one-point function \(\langle S_{p}\,W\rangle=a_{p}\big{(}\frac{u_{1}\cdot\theta}{|x^{i}|}\big{)}^{p}\). For reference, observe that the one-point function is known explicitly
\[a_{p}=\frac{\sqrt{p\lambda}I_{p}(\sqrt{\lambda})}{2^{\frac{p}{2}+1}NI_{1}( \sqrt{\lambda})}+O\big{(}\tfrac{1}{N^{2}}\big{)}\,. \tag{102}\]
Also note that our definition of connected correlator is different than the one used e.g. in [82].
The Wilson line is a half-BPS object that breaks supersymmetry according to
\[PSU(2,2|4)\ \supset\ OSp(4^{*}|4)\ \supset\ SL(2,\mathbb{R})\oplus SO(3)_{\rm trans.}\oplus SO(5)_{R}\,. \tag{103}\]
The \(SL(2,\mathbb{R})\) is the group of global conformal transformation along the line, \(SO(3)_{\rm trans.}\) are rotation in the transverse spacetime directions, while \(SO(5)_{R}\) is the leftover \(R\)-symmetry. The preserved \(OSp(4^{*}|4)\) symmetry has important consequences for the correlation function under study, namely the superconformal Ward identity
\[\left(\partial_{z}+\frac{1}{2}\partial_{\alpha}\right)\mathcal{F}(z,\bar{z}, \alpha)\bigg{|}_{z=\alpha}=0 \tag{104}\]
must be obeyed [58]. This has to be supplemented by another equation with \(z\leftrightarrow\bar{z}\). The Ward identity is expressed in terms of variables \(z,\bar{z},\alpha\) defined as
\[\xi=\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\,,\qquad\eta=\frac{z+\bar{z}}{2 \sqrt{z\bar{z}}}\,,\qquad\sigma=-\frac{(1-\alpha)^{2}}{2\alpha}\,. \tag{105}\]
The superconformal Ward identity is key for the position-space bootstrap of section 5.4.
### Effective action at strong coupling
With this preliminary information in mind, we can now start the holographic calculation. The chiral-primary operator \(S_{k}\) is dual to a supergravity field \(s_{k}\), which arises from the KK
reduction of IIB supergravity on AdS\({}_{5}\times S^{5}\). As in the simple example of section 2, the first step is to obtain the effective action for \(s_{k}\). We start in section 5.2.1 reviewing the AdS\({}_{5}\) effective action, which is already known in the literature [11, 12, 13]. We then derive the AdS\({}_{2}\) effective action. Besides rederiving vertices that were previously known [34, 37], we also obtain new bulk-brane vertices that, to the best of our knowledge, have not been previously computed.
#### 5.2.1 Effective action in \(\mathrm{AdS}_{5}\)
We use the ten-dimensional metric \(G_{MN}=g_{MN}+h_{MN}\), where \(g_{MN}\) is the background \(AdS_{5}\times S_{5}\) metric, and \(h_{MN}\) are fluctuations around it. We use Poincare coordinates \(z^{\mu}\) on AdS\({}_{5}\), and stereographic coordinates \(y^{A}\) on the \(S^{5}\), so the background metric reads
\[ds^{2}_{\mathrm{AdS}_{5}\times S^{5}}=\frac{dz^{\mu}dz^{\mu}}{(z^{0})^{2}}+ \frac{dy^{A}dy^{A}}{(1+\frac{1}{4}y^{B}y^{B})^{2}}\,,\qquad\mu,\nu=0,1,\ldots, 4\,,\quad A,B=5,\ldots,9\,. \tag{106}\]
Let us consider the fluctuations \(h_{\mu\nu}\) of the AdS\({}_{5}\) metric, which are expanded in \(S^{5}\) spherical harmonics to perform the KK reduction
\[h_{\mu\nu}(z,y)=\sum_{k,I}(h_{k}^{I})_{\mu\nu}(z)Y_{k}^{I}(y)\,. \tag{107}\]
The spherical harmonic \(Y_{k}^{I}\) for \(k=0,1,2,\ldots\) transforms in the rank-\(k\) symmetric traceless representation of \(SO(6)\), with the index \(I=1,\ldots,d_{k}\) labeling the elements in the representation. Appendix C gives an explicit definition of spherical harmonics, and provides simple rules to map them to the index-free notation in equations (97)-(100).
Besides fluctuations of the AdS\({}_{5}\) metric in (107), there are fluctuations for the metric on the sphere \(h_{AB}\), and for the Ramond-Ramond four form \(C_{4}=\bar{C}_{4}+\delta C_{4}\), which are also expanded in spherical harmonics. Upon inserting the spherical harmonic expansions in the IIB supergravity action, one obtains mixing in the kinetic terms. The mixing problem was first solved in [11], where the reader can find a table of all fields, and their properties after unmixing. Of all the supergravity fields, only the ones with even spin and in symmetric-traceless irreps of \(SO(6)_{R}\) couple to the fundamental string. This leaves us with the \(s\), \(\varphi_{\mu\nu}\) and \(t\) fields, whose properties are summarized in table 1. The standard AdS/CFT dictionary shows that \(s_{k}\) is the supergravity field dual to the CFT operator \(S_{k}\), while \(\varphi_{\mu\nu}\) and \(t\) are superconformal descendants.
For future reference, let us mention that the fields \(s\), \(\varphi_{\mu\nu}\) and \(t\) are related to AdS\({}_{5}\) metric
fluctuations as [13]
\[(h_{k}^{I})_{\mu\nu} =(\varphi_{k}^{I})_{\mu\nu}+\frac{4}{k+1}\left(\nabla_{\mu}\nabla_{ \nu}-\frac{1}{2}k(k-1)g_{\mu\nu}\right)s_{k}^{I}\] \[\quad+\frac{4}{k+3}\left(\nabla_{\mu}\nabla_{\nu}-\frac{1}{2}(k+4 )(k+5)g_{\mu\nu}\right)t_{k}^{I}\,. \tag{108}\]
There exist similar formulas for the fluctuations \(h_{AB}\), \(\delta C\), but they will not play a role in our calculation. Inserting formula (108) and its analogs for \(h_{AB}\), \(\delta C\) in the IIB supergravity action, one extracts kinetic terms, cubic couplings, and so on:
\[S_{\rm IIB}=\frac{4N^{2}}{(2\pi)^{5}}\int\!\frac{d^{5}z}{(z_{0})^{5}}\left(L_{ \rm IIB}^{(2)}+L_{\rm IIB}^{(3)}+\dots\right). \tag{109}\]
Recall that we are interested in the scattering of \(s_{k}\) with a fundamental string, which is dual to the CFT correlator \(\langle S_{p_{1}}S_{p_{2}}W\rangle\). A simple power-counting argument shows that only cubic couplings contribute at leading order in the supergravity approximation. Since we shall not attempt to compute corrections beyond leading order, the quadratic and cubic terms in (109) are all we need.11 The kinetic terms for \(s\), \(\varphi\) and \(t\) are [11, 12, 13]
Footnote 11: This is a fortunate state of affairs, since quartic and higher couplings are remarkably complicated [15].
\[L_{\rm IIB}^{(2)}=\sum_{k,I}\left(\zeta_{k}^{s}\,L_{2}^{(0)}(s_{k}^{I})+\zeta_{ k}^{t}\,L_{2}^{(0)}(t_{k}^{I})+\zeta_{k}^{\varphi}\,L_{2}^{(2)}(\varphi_{k}^{I})+ \dots\right), \tag{110}\]
where \(\zeta_{k}^{X}\) are normalizations presented in appendix B. Besides the overall normalization, the kinetic terms are standard
\[L_{2}^{(0)}(s) =\frac{1}{2}\nabla_{\mu}s\nabla^{\mu}s+\frac{1}{2}m^{2}s^{2}\,, \tag{111}\] \[L_{2}^{(2)}(\varphi) =\frac{1}{4}\nabla_{\mu}\varphi_{\nu\rho}\nabla^{\mu}\varphi^{ \nu\rho}-\frac{1}{2}\nabla_{\mu}\varphi^{\mu\nu}\nabla^{\rho}\varphi_{\nu\rho }+\frac{1}{2}\nabla_{\mu}\varphi\nabla_{\rho}\varphi^{\mu\rho}\] \[\quad-\frac{1}{4}\nabla_{\mu}\varphi\nabla^{\mu}\varphi-\frac{1} {4}(2-m^{2})\varphi_{\mu\nu}\varphi^{\mu\nu}-\frac{1}{4}(2+m^{2})\varphi^{2}\,, \tag{112}\]
where the trace of the spin-two field is \(\varphi=g^{\mu\nu}\varphi_{\mu\nu}\), and the masses appear in table 1. In a similar way, one can write down the cubic couplings. For our purposes, we only need vertices
\begin{table}
\begin{tabular}{c||c|c|c|c} Field & \(m^{2}\) & \(\Delta\) & \(\ell\) & \(SU(4)\)-irrep \\ \hline \hline \(s\) & \(k(k-4)\) & \(k\) & \(0\) & \([0,k,0]\) \\ \hline \(\varphi_{\mu\nu}\) & \((k+2)(k-2)\) & \(k+2\) & \(2\) & \([0,k-2,0]\) \\ \hline \(t\) & \(k(k+4)\) & \(k+4\) & \(0\) & \([0,k-4,0]\) \\ \hline \end{tabular}
\end{table}
Table 1: Kaluza-Klein modes that couple to the string worldsheet
that contain two \(s\) fields, and one \(X=s,\varphi,t\) field:
\[L^{(3)}_{\rm IIB}=-\sum_{k,p,q}\left(S^{IJK}_{kpq}s^{I}_{k}s^{J}_{p}s^{K}_{q}+T^{ IJK}_{kpq}s^{I}_{k}s^{J}_{p}t^{K}_{q}+G^{IJK}_{kpq}T_{\mu\nu}(s^{I}_{k},s^{J}_{p})( \varphi^{K}_{q})^{\mu\nu}+\dots\right). \tag{113}\]
The precise form of the vertices was obtained in [14] and is reviewed in appendix B.
#### 5.2.2 Effective action in \(\mathrm{AdS}_{2}\)
Let us now derive the effective action of fluctuations on top of the fundamental string F1, which lives in the background \(\mathrm{AdS}_{5}\times S^{5}\) just described. Because in this background the NS-NS forms are turned-off, the string action reads
\[S_{\rm F1}=\frac{\sqrt{\lambda}}{2\pi}\int d^{2}\widehat{z}\,\sqrt{\det\left(G _{MN}(X)\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\right)}\,, \tag{114}\]
where we also ignore fermions. The worldsheet coordinates are \(\widehat{z}=(\widehat{z}^{\,0},\widehat{z}^{\,1})\), and \(X^{M}(\widehat{z})\) describes the embedding of the string in ten dimensions. The string intersects the boundary of \(\mathrm{AdS}_{5}\) in a contour \(\gamma\), and sits at a particular point in the \(S^{5}\). From the CFT perspective, \(\gamma\) is the location of the Wilson loop, while the point in \(S^{5}\) is related to the polarization \(\theta\) in (98). The expectation value of the Wilson loop is computed by the string path integral, which in the limit \(\lambda\gg 1\) is dominated by the saddle point where the string has minimal area.12 Here we consider small fluctuations around the minimal-area configuration.
Footnote 12: See section 2 of [38] for a recent pedagogical review of a similar setup.
From now on, we restrict our attention to the case when \(\gamma\) is a straight line, a configuration that preserves the \(OSp(4^{*}|4)\) group, recall equation (103). More precisely, the string intersects the boundary of \(\mathrm{AdS}_{5}\) along the \(x^{1}\) direction, and is located at \(x^{i=2,3,4}=0\). Furthermore, we choose the string to be located at point \(y^{A}=0\) on \(S^{5}\). It is convenient to work in static gauge, in which case the embedding reads
\[X^{0}=\widehat{z}^{\,0}\,,\quad X^{1}=\widehat{z}^{\,1}\,,\quad X^{i=2,3,4}=X^ {A=5,\dots,9}=0\,. \tag{115}\]
This embedding represent the static configuration of the string. However, we are interested in allowing small fluctuations around this configuration. In particular, we allow the orthogonal directions \(X^{i}\) and \(X^{A}\) to fluctuate, which can be implemented in a change of the boundary
conditions (115). Motivated by [37],13
Footnote 13: To be precise, the authors of [37] split the coordinates as \(z^{\mu}=(z^{\alpha},z^{i})\) for \(\alpha=0,1\) and \(i=2,3,4\). They used the metric
\[ds^{2}_{\text{AdS}_{5}\times S^{5}}=\left(\frac{1+\frac{1}{4}(z^{i})^{2}}{1- \frac{1}{4}(z^{i})^{2}}\right)^{2}\,\frac{dz^{\alpha}dz^{\alpha}}{(z^{0})^{2}} +\frac{dz^{i}dz^{i}}{(1-\frac{1}{4}(z^{i})^{2})^{2}}+\frac{dy^{4}dy^{A}}{(1+ \frac{1}{4}y^{B}y^{B})^{2}}\,, \tag{116}\]
and chose static gauge with \(X^{\alpha}=\widehat{z}^{\alpha}\), \(X^{i}=x^{i}\) and \(X^{A}=y^{A}\). This is equivalent to our choice (117) because we work in Poincare coordinates, see (106). we require the string to be embedded as
\[X^{0}=\widehat{z}^{0}\,\frac{1-\frac{1}{4}(x^{i})^{2}}{1+\frac{1}{4}(x^{i})^ {2}}\,,\quad X^{1}=\widehat{z}^{1}\,,\quad X^{i}=\frac{x^{i}\widehat{z}^{0}}{ 1+\frac{1}{4}(x^{i})^{2}}\,,\quad X^{A}=y^{A}\,. \tag{117}\]
For \(x^{i}=y^{A}=0\) this naturally reduces to the static configuration (115). However, we instead treat \(x^{i}(\widehat{z})\) and \(y^{A}(\widehat{z})\) as fields that live on the worldsheet, and expand (114) around the background configuration to find their action. The fields \(x^{i}\) and \(y^{A}\) have standard kinetic terms, as well as self-interactions and interactions with bulk fields. The general structure of the action is
\[S_{\text{F}1}=\frac{\sqrt{\lambda}}{2\pi}\int\frac{d^{2}\widehat{z}}{( \widehat{z}^{0})^{2}}\left(1+L^{(2,0)}_{\text{F}1}+L^{(0,1)}_{\text{F}1}+L^{( 1,1)}_{\text{F}1}+L^{(0,2)}_{\text{F}1}+\dots\right), \tag{118}\]
where \(L^{(m,n)}_{\text{F}1}\) corresponds to terms with \(m\) worldsheet fields and \(n\) bulk fields. For the kinetic part, we find
\[L^{(2,0)}_{\text{F}1}=\frac{1}{2}g_{2}^{\alpha\beta}\partial_{\alpha}x^{i}\, \partial_{\beta}x^{i}+x^{i}x^{i}+\frac{1}{2}g_{2}^{\alpha\beta}\partial_{ \alpha}y^{A}\,\partial_{\beta}y^{A}\,,\qquad g_{2}^{\alpha\beta}=\delta^{ \alpha\beta}(\widehat{z}^{0})^{2}\,. \tag{119}\]
This result is in perfect agreement with [37]. Similarly to the bulk case, these fields are part of a short multiplet of \(OSp(4^{*}|4)\), with quantum numbers reported in table 2. In defect CFT terminology, the field \(y^{A}\) is dual to the tilt operator, which appears in the Ward identity for the broken \(R\)-symmetry \(SO(6)_{R}\to SO(5)_{R}\)[83, 84]. Similarly, the field \(x^{i}\) is dual to the displacement operator, which appears in the Ward identity for the the broken translation symmetry [61].
One can proceed to extract the rest of couplings in a similar manner. For example, reference [37] computed all quartic vertices of worldsheet fields, in our notation \(L^{(4,0)}_{\text{F}1}\). These terms do not contribute to the two-point function of chiral-primary operators at leading
\begin{table}
\begin{tabular}{c||c|c|c|c} Field & \(m^{2}\) & \(\Delta\) & \(SO(3)_{\text{trans.}}\) & \(SO(5)_{R}\) \\ \hline \hline \(y^{A}\) & \(0\) & \(1\) & singlet & vector \\ \hline \(x^{i}\) & \(2\) & \(2\) & vector & singlet \\ \end{tabular}
\end{table}
Table 2: Bosonic open-string modes on the fundamental string worldsheet.
order, so we do not present the formulas here. The first vertex of interest couples a bulk field directly to the string worldsheet
\[L^{(0,1)}_{\rm F1}=\sum_{k,I}\left(\frac{1}{2}(\varphi_{k}^{I})_{\alpha}^{\alpha}- \frac{2k(k-1)}{k+1}s_{k}^{I}-\frac{2(k+4)(k+5)}{k+3}t_{k}^{I}+\ldots\right)Y_{k }^{I}\,. \tag{120}\]
These vertices allow bulk fields to be "absorbed" by the string, recall (11). The vertex for \(s_{k}^{I}\) has appeared multiple times in the literature, see e.g. [34, 35, 36], but to the best of our knowledge, the vertices for \(\varphi\) and \(t\) are new. In (120) we neglect total derivatives \(\partial_{\alpha}V^{\alpha}\), since they drop from Witten diagrams. It is also implicit in the notation that the spherical harmonic is evaluated at zero \(Y_{k}^{I}=Y_{k}^{I}(y=0)\), and the bulk fields live on the worldsheet, for example \(s_{k}^{I}=s_{k}^{I}(\widehat{z}^{\,0},\widehat{z}^{\,1},0,0,0)\).
Finally, we need vertices where the bulk field \(s\) excites a worldsheet fluctuation. These vertices have not appeared before in the literature, and a calculation exactly as above gives
\[L^{(1,1)}_{\rm F1}=-\sum_{k,I}\frac{2k(k-1)}{k+1}\Big{(}\,y^{A}\partial_{A}+ \widehat{z}_{0}\,x^{i}\partial_{i}\,+\ldots\Big{)}s_{k}^{I}Y_{k}^{I}\,. \tag{121}\]
To obtain this formula we used multiple integration by parts, and dropped total derivatives and terms proportional to the equations of motion (EOM). When computing the correlator \(\langle S_{p_{1}}S_{p_{2}}W\rangle\), the EOM terms will contribute contact diagrams like the one in (31). However, in section 5.3 we neglect all contact diagrams, and only fix them later with the bootstrap approach of section 5.4. A more careful first-principle calculation, that aims at computing the correlator including contact terms, should keep track of the EOM terms in (121).
### Explicit calculation (up to contact terms)
Having derived the AdS\({}_{5}\) and AdS\({}_{2}\) effective actions, we are ready for a first attempt to calculate \(\langle S_{p_{1}}S_{p_{2}}W\rangle_{c}\). Recall that we are interested in the connected correlator, so we neglect disconnected diagrams as in (20). Figure 4 shows the connected diagrams in the leading supergravity approximation, which consist of exchange diagrams with \(s\), \(\varphi_{\mu\nu}\) and \(t\) in AdS\({}_{5}\)
Figure 4: Diagrams that contribute to the connected correlator \(\langle S_{p_{1}}S_{p_{2}}W\rangle_{\rm c}\) at leading order in the supergravity limit. The quantum numbers of exchanged fields appear in tables 1 and 2.
exchange diagrams with \(x\), \(y\) in AdS\({}_{2}\), and contact diagrams. The quantum numbers of the exchanged fields are summarized in tables 1 and 2, while contact interactions are ignored in this section for reasons to be explained later.
To calculate \(\langle S_{p_{1}}S_{p_{2}}W\rangle\), one should view the on-shell supergravity action with certain boundary conditions as the generating functional of CFT correlators [2, 3], and then take functional derivatives. In practice, however, this amounts to computing the correlator with standard position-space Feynman rules, using propagators (8)-(10) and the effective actions of section 5.2. For the bulk-to-boundary propagator, we choose the normalization [34]
\[\Pi_{p}(x,z)=\mathcal{N}_{p}K_{p}(x,z)\,,\qquad\mathcal{N}_{p}=2^{p/2-2}\, \frac{p+1}{N\sqrt{p}}\,, \tag{122}\]
which ensures that in the absence of the defect, the two-point function \(\langle S_{p}S_{p}\rangle\) is unit normalized. Besides this, there are several numerical factors that need to be accounted for. For example, because of (109), bulk-to-bulk propagators come with a factor \(\frac{(2\pi)^{5}}{4N^{2}\zeta}\), where the normalizations \(\zeta\) are shown in (168). Similarly, brane-to-brane propagators come with a factor \(\frac{2\pi}{\sqrt{\lambda}}\), see (118). Regarding bulk three-point vertices, they contain an overall factor \(\frac{4N^{2}}{(2\pi)^{5}}\). There are three bulk vertices \(X=S,\,T,\,G\), which come with extra factors \(6,\,2\) and \(2\) respectively, that count the number of different Wick contractions. Each three-point vertex is of the form
\[X^{IJK}_{p_{1}p_{2}k}Y^{I}_{k}(0)\;\to\;X_{p_{1}p_{2}k}h^{k}_{p_{1}p_{2}}( \sigma)\,, \tag{123}\]
where \(X_{p_{1}p_{2}k}\) are found in (171). In equation (123) we have mapped the spherical harmonic indices \(I,J,K\) to the index-free notation of section 5.1. The derivation of this map is presented in appendix C, but here it suffices to note that the result is exactly the \(R\)-symmetry block
\[h^{k}_{p_{1}p_{2}}(\sigma)=\sigma^{\frac{p_{1}+p_{2}-k}{2}}\,_{2 }F_{1}\!\left(-\frac{p_{12}+k}{2},\frac{p_{12}-k}{2};-k-1;\frac{\sigma}{2} \right)\,. \tag{124}\]
For integer values of \(p_{1}\), \(p_{2}\) and \(k\) the \(R\)-symmetry block is a polynomial in \(\sigma\). Finally, the bulk-boundary vertices are proportional to \(\frac{\sqrt{\lambda}}{2\pi}\), with the precise values in (120). As before, appendix C translates the \(R\)-symmetry contractions to index-free notation
\[Y^{I}_{p_{1}}(0)Y^{J}_{p_{2}}(0)\;\to\;1\,,\quad\partial_{A}Y^{I }_{p_{1}}(0)\partial_{A}Y^{J}_{p_{2}}(0)\;\to\;p_{1}p_{2}(\sigma-1)\,. \tag{125}\]
All in all, we have the following contributions to the connected correlator
\[\mathcal{F}_{c}(\xi,\eta,\sigma) \,\sim\,\mathcal{N}_{p_{1}}\mathcal{N}_{p_{2}}\,\frac{\sqrt{\lambda} }{2\pi}\Bigg{[}\sum_{k}\frac{6S_{p_{1}p_{2}k}}{\zeta_{k}^{s}}\frac{2k(k-1)}{k+1 }E_{p_{1}p_{2}}^{k,0}n_{p_{1}p_{2}}^{k}\] \[\,+\sum_{k}\frac{2T_{p_{1}p_{2}k}}{\zeta_{k}^{\ell}}\frac{2(k+4)(k +5)}{k+3}E_{p_{1}p_{2}}^{k+8,0}h_{p_{1}p_{2}}^{k}+\sum_{k}\frac{2G_{p_{1}p_{2} k}}{\zeta_{k}^{\varphi}}\frac{1}{2}E_{p_{1}p_{2}}^{k+4,2}h_{p_{1}p_{2}}^{k}\] \[\,+\frac{4p_{1}p_{2}(p_{1}-1)(p_{2}-1)}{(p_{1}+1)(p_{2}+1)}\left( p_{1}p_{2}(\sigma-1)\widehat{E}_{p_{1}p_{2}}^{1,0}+\widehat{E}_{p_{1}p_{2}}^{ 2,1}\right)\Bigg{]}\,, \tag{126}\]
where the sum runs over \(k\)'s allowed by \(SU(4)_{R}\) symmetry. The couplings \(S\), \(T\), \(G\) and \(\zeta\) are found in appendix B, while \(E_{\Delta_{1}\Delta_{2}}^{\Delta,\ell}\) and \(\widehat{E}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},s}\) are bulk-exchange and brane-exchange Witten diagrams respectively, which we computed in section 3.
Combining all these ingredients, one can check that (126) cannot be the correct result, because it does not obey the superconformal Ward identity (104). The reason (126) is incorrect is that we have completely ignored contact diagrams. Instead, the correct result takes the form
\[\mathcal{F}_{c}(\xi,\eta,\sigma) \,=\,\eqref{eq:216}\,+\,C_{p_{1}p_{2}}(\xi,\eta)\sum_{n=0}^{p_{ \rm min}}c_{n}\,\sigma^{n}\,. \tag{127}\]
Unfortunately, starting from the IIB and fundamental string actions, we have been unable to determine coefficients \(c_{n}\) that generate correlators consistent with the Ward identities. The naive way to obtain \(c_{n}\) is to expand the string action up to terms like \(\sim\int_{{\rm AdS}_{2}}s_{p_{1}}s_{p_{2}}\), and keep track of the EOM terms neglected in (121). Because of their \(R\)-symmetry structure, these terms can contribute to \(c_{0}\) and \(c_{1}\), but as we show below, all \(c_{n}\) are non-vanishing in the supersymmetric correlator. This suggests there are other sources of contact terms, and raises the question of how to do a first-principles calculation that correctly accounts for them. One possibility is that in the derivation of the IIB effective action, one generates the missing contact terms when integrating by parts or in doing certain field redefinitions. We hope future work will clarify this question, while in the rest of the paper we determine \(c_{n}\) with a bootstrap method.
### Position space bootstrap
Although we got far using traditional methods, in the end we failed to find a correlator consistent with the supersymmetry of the system. To solve this problem, in this section we determine the correlator with a bootstrap method. The main idea was put forth in [17]: write an ansatz with all Witten diagrams allowed by the effective action, and fix the relative coefficients with the superconformal Ward identity.
Let us spell out the method in detail. First, recall there are bulk-exchange diagrams with the fields \(s\), \(\varphi\) and \(t\) in table 1. These fields belong to a superconformal multiplet, where the superprimary \(s\) transforms in the representation \([0,k,0]\) of \(SU(4)_{R}\). The value of \(k\) is restricted by \(SU(4)_{R}\) symmetry to be \(|p_{12}|\leq k\leq p_{1}+p_{2}\) and to run in steps of two. Furthermore, when the superprimary satisfies \(k=p_{1}+p_{2}\) or \(k=|p_{12}|\), the three-point vertices \(S\), \(T\), \(G\) in (171) vanish, so we take \(|p_{12}|<k<p_{1}+p_{2}\). As a result, each multiplet \((s,\varphi,t)\) in table 1 contributes for \(k\in\mathcal{S}\), where
\[\mathcal{S}=\left\{|p_{12}|+2,|p_{12}|+4,\ldots,p_{1}+p_{2}-4,p_{1}+p_{2}-2 \right\}. \tag{128}\]
This concludes the discussion of bulk-exchange diagrams.
Regarding defect-exchange diagrams, the fields \(x\), \(y\) in table 2 always contribute, because they are always allowed by \(R\)-symmetry. Finally, there can be contact diagrams, so the full ansatz for the connected correlator is14
Footnote 14: One issue we glossed over is what contact diagrams to include in the ansatz. In fact, (129) ignores contact diagrams with derivatives, which would come from \(\int_{\mathrm{AdS}_{2}}\nabla^{n}s_{1}\nabla^{m}s_{2}\). One justification is that spin-two diagrams \(E^{\Delta,2}_{\Delta_{1}\Delta_{2}}\) and contact diagrams \(C_{\Delta_{1}\Delta_{2}}\) have the same Regge behavior. One the other hand, derivative contact diagrams grow faster in the Regge limit, and because we do not expect the Regge limit to be controlled only by contact diagrams, they should be absent. We thank Xinan Zhou for this argument.
\[\mathcal{F}_{c}(\xi,\eta,\sigma) =\sum_{k\in\mathcal{S}}\left(A_{k}^{s}\,E_{p_{1}p_{2}}^{k,0}(\xi, \eta)\,h_{p_{1}p_{2}}^{k}(\sigma)+A_{k}^{\varphi}\,E_{p_{1}p_{2}}^{k+2,2}(\xi,\eta)\,h_{p_{1}p_{2}}^{k-2}(\sigma)\right.\] \[\qquad\qquad+A_{k}^{t}\,E_{p_{1}p_{2}}^{k+4,0}(\xi,\eta)\,h_{p_{ 1}p_{2}}^{k-4}(\sigma)\right)+C_{p_{1}p_{2}}(\xi,\eta)\sum_{n=0}^{p_{\min}}A_{ n}^{ss}\,\sigma^{n}\] \[\quad+A^{y}\,\widehat{E}_{p_{1}p_{2}}^{1,0}(\xi,\eta)(\sigma-1)+ A^{x}\,\widehat{E}_{p_{1}p_{2}}^{2,1}(\xi,\eta)\,, \tag{129}\]
with the sum ranges in (128) and \(A\)'s numerical coefficients to be determined. The bulk-exchange diagrams \(E^{\Delta,\ell}_{\Delta_{1}\Delta_{2}}\) can be evaluated as sums of contact diagrams, see (39) and appendix A. Furthermore, contact diagrams are known in closed form (35). Finally, the defect-exchange diagrams \(\widehat{E}^{\widehat{\Delta},s}_{\Delta_{1}\Delta_{2}}\) can be computed from the differential equation (46).
For a given choice of \(p_{1}\) and \(p_{2}\), we build the ansatz as just explained and impose the superconformal Ward identity (104) order by order in an expansion around \(r=0\). The resulting equations can be solved, and they fully determine the correlator (129) up to an overall normalization, which of course cannot be fixed by the Ward identity only. In order to fix the overall normalization, note that in the strong 't Hooft coupling limit we have [53]
\[\lim_{\lambda\to\infty}\lim_{N\to\infty}\lambda_{S_{p_{1}}S_{p_{2}}S_{k}}a_{ S_{k}}=\frac{\sqrt{\lambda}}{N^{2}}\frac{\sqrt{p_{1}p_{2}}\,k}{2^{k/2+1}}+\ldots\,, \tag{130}\]
where \(a_{\mathcal{O}}\propto\langle\mathcal{O}W\rangle\) and \(\lambda_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}\propto\langle\mathcal{ O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\). The Euclidean OPE limit \(\xi\to 0\), \(\eta\to 1\) is dominated by the lowest-dimension chiral primary operator \(S_{k_{\min}}\), and we find
\[\lim_{\xi\to 0}\lim_{\eta\to 1}\mathcal{F}_{c}(\xi,\eta,\sigma)=\frac{\sqrt{ \lambda}}{N^{2}}\frac{\sqrt{p_{1}p_{2}}\,k_{\min}}{2^{k_{\min}/2+1}}\,\xi^{ \frac{k_{\min}-p_{1}-p_{2}}{2}}\,h_{p_{1}p_{2}}^{k_{\min}}(\sigma)+\ldots\,. \tag{131}\]
Comparing this to ansatz (129) fixes the overall normalization of \(\mathcal{F}_{c}\).
We bootstrapped many correlators with \(p_{1},p_{2}\lesssim 30\), and in every case all coefficients were fixed. All cases with \(p_{1},p_{2}\leq 4\) were computed previously in [53, 54], and we found perfect agreement.15 At this point, it is interesting to compare our method to the analytic bootstrap in [53, 54]. The analytic bootstrap calculation reconstructed the correlator \(\mathcal{F}_{c}\) using a dispersion relation or inversion formula. These formulas suffer from low-spin ambiguities, which in practice means that only part of the correlator is reconstructed. In the present language, the dispersion relation reconstructs only the sum over bulk-exchange diagrams, because both defect-exchange and contact diagrams have vanishing discontinuity. In [53, 54], the missing part of the correlator was determined making a very general ansatz and fixing it from superconformal Ward identities. Remarkably, this ad-hoc procedure managed to reconstruct precisely the defect-exchange diagrams for the \(x^{i}\) and \(y^{A}\) fluctuations. Needless to say, the present method is superior to the one employed in [53, 54]. On one hand, here we have a more clear physical picture of all the terms that form the correlator. On the other, the present method is easier and faster, thanks to the efficient methods to compute Witten diagrams.
Footnote 15: The careful reader will note that our correlator for \((p_{1},p_{2})=(2,4)\) differs with the one in [54]. The difference is only the contribution \(\lambda_{242}a_{2}f_{2,0}^{-2}\) in the bulk OPE. The reason is that [54] considered the two-point function of single-trace operators \(\mathcal{O}_{p}\), for which \(\mathcal{O}_{2}\times\mathcal{O}_{4}\sim\mathcal{O}_{2}\). However, we work in the basis natural for supergravity, such that \(\lambda_{S_{2}S_{4}S_{2}}=0\), recall footnote 10.
Because of the efficiency of our method, it is possible to generate many correlation functions, and by looking at all the examples, we could guess a closed formula
\[\mathcal{F}_{c}(\xi,\eta,\sigma) =\frac{\sqrt{\lambda}}{N^{2}}\sum_{k\in\mathcal{S}}\frac{\sqrt{p _{1}p_{2}}\,k}{2^{\frac{k}{2}+1}}\Bigg{(}P_{p_{1}p_{2}}^{k,0}h_{p_{1}p_{2}}^{ k}+\frac{(p_{12}^{2}-k^{2})\big{(}p_{12}^{2}-(k+2)^{2}\big{)}}{128k(k+1)^{2}(k+3)}P_{p_ {1}p_{2}}^{k+2,2}h_{p_{1}p_{2}}^{k-2}\] \[\qquad\qquad+\frac{(p_{12}^{2}-(k+2)^{2})(p_{12}^{2}-(k-2)^{2})(p _{12}^{2}-k^{2})^{2}}{16384(k-2)(k-1)^{2}k^{2}(k+1)(k+2)(k+3)}P_{p_{1}p_{2}}^{ k+4,0}h_{p_{1}p_{2}}^{k-4}\Bigg{)}\] \[\quad+\frac{\sqrt{\lambda}}{N^{2}}\frac{(p_{1}p_{2})^{\frac{1}{2 }}(p_{1}-1)(p_{2}-1)}{2^{3-\frac{p_{1}+p_{2}}{2}}\pi}\bigg{(}p_{1}p_{2}( \sigma-1)\widehat{E}_{p_{1}p_{2}}^{1,0}+\widehat{E}_{p_{1},p_{2}}^{2,1}+(1-2 \sigma)C_{p_{1}p_{2}}\bigg{)}\,. \tag{132}\]
The sum over \(k\) is described in (128), and \(P_{\Delta_{1}\Delta_{2}}^{\Delta\ell}\) are Polyakov-Regge blocks. Recall that, up to contact terms, Polyakov-Regge blocks are bulk-exchange Witten diagrams with a different normalization, see (63). The bootstrap result (132) agrees with the explicit calculation (126), up to the contact terms that we could not determine in the first-principles calculation. This provides a strong consistency check for our results. Let us stress that to obtain (132), the only input from section 5.2 is the field content in the effective actions, but not the precise value of vertices.
The position-space correlator (132) has a simple interpretation in terms of conformal blocks. Because Polyakov-Regge blocks are normalized as \(P^{\Delta\ell}_{\Delta_{1}\Delta_{2}}=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{ \Delta,\ell}+\ldots\), the coefficient of \(P^{k,0}_{p_{1}p_{2}}h^{k}_{p_{1}p_{2}}\) in (132) is the OPE coefficient \(\lambda_{S_{p_{1}}S_{p_{2}}S_{k}}a_{S_{k}}\), in perfect agreement with the known value (130). Also note that \(\varphi_{\mu\nu}\) and \(t\) are superdescendants of \(s\), so their OPE coefficients are fixed in terms of the superprimary. This is also explicit in the correlator, because the relative coefficients of the first two lines of (132) precisely form a superblock (compare with equation (5.12) of [54]). Finally, the last line of (132) contains bulk-defect OPE coefficients of chiral-primaries \(S_{k}\) with tilt \(\widehat{t}\) and displacement \(\widehat{D}^{i}\) operators. To present the OPE coefficients, we use index-free notation \(t(0,\widehat{v})=t^{A}(0)\widehat{v}_{A}\), with the polarization vector satisfying \(\widehat{v}^{\,2}=\widehat{v}\cdot\theta=0\), and we find16
Footnote 16: We use the tilt and displacement operators with unit-normalized two-point functions. As a result, the Ward identities that define these operators have extra normalization factors
\[\partial_{\mu}T^{\mu i}=C_{D}^{1/2}\,\widehat{D}^{i}\,\delta_{W} \,,\qquad\partial_{\mu}J^{\mu}_{A}=C_{t}^{1/2}\,\widehat{t}_{A}\,\delta_{W}\,. \tag{133}\]
\[\langle S_{p}(x,u)\,\widehat{t}(0,\widehat{v})\,W(\theta)\rangle =\frac{(u\cdot\theta)^{p-1}(u\cdot\widehat{v})}{x^{2}|x^{i}|^{p- 1}}\left(\frac{\lambda^{1/4}}{N}\frac{p^{3/2}}{2^{\frac{p+1}{2}}}+\ldots \right)\,, \tag{134}\] \[\langle S_{p}(x,u)\,\widehat{D}^{i}(0)\,W(\theta)\rangle =\frac{x^{i}(u\cdot\theta)^{p}}{x^{4}|x^{j}|^{p}}\left(\frac{ \lambda^{1/4}}{N}\frac{p^{3/2}}{2^{\frac{p}{2}}\sqrt{3}}+\ldots\right)\,. \tag{135}\]
To obtain this formula, we used the normalization of defect-exchange Witten diagrams in (47). The correlator \(\langle S_{p}\,\widehat{t}\,W\rangle\) is captured by the topological sector of [41, 42], and we find perfect agreement. Summarizing, the correlator (132) is a sum over protected operators, each of them contributing an exchange Witten diagram, supplemented by extra contact diagrams fixed by superconformal Ward identities. Besides these highly non-trivial sanity checks, in appendix D we successfully compare to the localization calculation of [82], which captures only the topological part of the correlator.
### Mellin space
There are at least three good reasons to map the above results to Mellin space. First, it would be good to have a bootstrap method directly in Mellin space, since this could be necessary in setups where the position-space bootstrap does not work. For example, bulk-exchange diagrams do not truncate in theories with an AdS\({}_{4}\) bulk, making the position space bootstrap unfeasible. Second, the Mellin amplitude \(\mathcal{M}\) should be closely related to a flat-space scattering amplitudes of supergravitons off a string or brane, see [40] for a recent discussion. By making this link more precise, one could input information of flat-space amplitudes to bootstrap subleading corrections in \(\frac{1}{\sqrt{\lambda}}\), see similar examples [85, 22, 25].
Third, thanks to the properties of Mellin amplitudes, it is plausible the result might be simpler in Mellin space, a hypothesis that we confirm below.
Keeping this motivation in mind, we Mellin transform the position-space correlator (132). The first two lines in (132) contain Polyakov-Regge blocks, which are combinations of bulk-exchange and contact diagrams, see (63). The Mellin transforms of bulk-exchange and contact diagrams are presented in (86) and (72) respectively, and they always reduce to rational functions of the Mellin variables \(\delta\) and \(\rho\). The last line in (132) contains defect-exchange diagrams, that in Mellin space do not always reduce to rational functions, recall (94). For example, for \(p_{1}=p_{2}=2\), the Mellin amplitude reads
\[\mathcal{M}(\delta,\rho,\sigma)=\frac{\sqrt{\lambda}}{8N^{2}} \Bigg{(}\frac{(\rho\sigma-\sigma+4)\sigma}{\delta-1}-\frac{2\rho+4 \sigma-4}{\delta+\rho}+\frac{\rho}{\delta+\rho+2}+(\sigma-1)^{2}\] \[\qquad\qquad+\frac{2^{\delta+\rho+2}\,\Gamma(-\delta-\rho)}{ \Gamma\big{(}\frac{2-\delta-\rho}{2}\big{)}^{2}}\left(\frac{\delta+2}{\delta+ \rho+2}-\sigma\right)\Bigg{)}\,. \tag{136}\]
In this case, the defect-exchange amplitudes \(\widehat{\mathcal{M}}^{1,0}_{2,2}\) and \(\widehat{\mathcal{M}}^{2,1}_{2,2}\) simplify in terms of gamma functions, but we have not checked if this holds for arbitrary \(p_{1},p_{2}>2\).
In a similar way, we can Mellin transform the correlator for many values \(p_{1}\), \(p_{2}\), and guess a closed formula for \(\mathcal{M}\). We observe that the Mellin transform of the first two lines in (132) simplifies drastically, but there are no important simplifications in the last line:
\[\mathcal{M}(\delta,\rho,\sigma)=\sum_{n=1}^{p_{\min}}\sum_{i=1}^{n}\frac{b_{n,i}\,\rho+c_{n,i}}{\delta-i}\,\sigma^{n}+e\,p_{1}p_{2}(\sigma-1)\widehat{ \mathcal{M}}^{1,0}_{p_{1}p_{2}}+e\widehat{\mathcal{M}}^{2,1}_{p_{1}p_{2}}+a \,(\sigma-1)^{2}\,. \tag{137}\]
The numerical constants \(a\), \(b_{n,i}\), \(c_{n,i}\) and \(e\) were guessed in closed form, and take a relatively simple form:
\[a =-\frac{\sqrt{\lambda}}{N^{2}}\,\frac{2^{\frac{p_{1}+p_{2}}{2}} \sqrt{p_{1}p_{2}}\,\big{(}-\frac{1}{2}\big{)}_{\frac{p_{1}+p_{2}}{2}}}{16(p_{ 1}-2)!(p_{2}-2)!}\,,\] \[b_{n,i} =a\,\frac{(2-n)_{i-1}(2-p_{1})_{n-2}(2-p_{2})_{n-2}}{2^{n-2}(i-1)! (n-2)!\,\big{(}\frac{3-p_{1}-p_{2}}{2}\big{)}_{n-2}}\,,\] \[c_{n,i} =\frac{b_{n,i}}{4}\left(\frac{p_{12}^{\frac{1}{2}-1-8(n-p_{1})(n- p_{2})(2n-p_{1}-p_{2})/(i-n)}}{p_{1}+p_{2}-2n+1}+4i+6n-5p_{1}-5p_{2}+1\right)\,,\] \[e =\frac{\sqrt{\lambda}}{N^{2}}\,\frac{2^{\frac{p_{1}+p_{2}}{2}}}{8 \pi}\,(p_{1}p_{2})^{1/2}(p_{1}-1)(p_{2}-1)\,. \tag{138}\]
The formula for \(c_{n,i}\) is singular for \(n=i\), but the value of \(c_{n,n}\) in (137) is finite and given by
\[c_{n,n}=a\,\frac{\big{(}\frac{p_{1}+p_{2}}{2}-n\big{)}\,(2-p_{1})_{n-1}(2-p_{2 })_{n-1}}{(-2)^{n-3}(n-1)!\,\big{(}\frac{3-p_{1}-p_{2}}{2}\big{)}_{n-1}}\,. \tag{139}\]
The Mellin amplitude (137) has a simple interpretation in terms of the OPE. The finite sum captures contributions of single-trace operators in the OPE \(S_{p_{1}}\times S_{p_{2}}\sim S_{k}+\ldots\), compare with the general formula (68). Similarly, the last two terms correspond to the exchange of tilt and displacement in the defect expansion \(S_{p}\times W\sim\widehat{t}\,W+\widehat{D}^{i}W+\ldots\). On the other hand, we do not have a clear interpretation for the term \(a(\sigma-1)^{2}\), which should correspond to a contact interaction, because it is constant.
The result (137) deserves several comments. First, it would be interesting to find the prescription for taking the flat-space limit and comparing to flat-space string amplitudes. Second, the sum in (137) is remarkably simple, and is reminiscent of the elegant formula found in [16, 17] for the four-point function \(\langle S_{p_{1}}S_{p_{2}}S_{p_{3}}S_{p_{4}}\rangle\). In that case, the simplicity is related to the hidden ten-dimensional conformal symmetry [86], and it would be fascinating if a similar structure can be found here. Finally, it is unclear to us whether (137) could have been bootstrapped directly in Mellin space. In any case, if such bootstrap approach is possible, it certainly requires superconformal Ward identities in Mellin space, a subject that we now discuss.
#### 5.5.1 Ward identity
Superconformal Ward identities were first implemented in Mellin space in [87], see also [29, 88]. We adapt these methods to the present setup in appendix E. The result is that given the decomposition (100) of the correlation function, the \(R\)-symmetry channels satisfy
\[\sum_{j=0}^{p_{\text{min}}}\left[\zeta_{\pm}^{(j)}\big{(}4\xi \partial_{\xi}-\xi\partial_{\eta}+4j\big{)}+\zeta_{\pm}^{(j+1)}(\xi\partial_{ \xi}+\eta\partial_{\eta}+j)\right]\frac{\mathcal{F}_{j}(\xi,\eta)}{(-2)^{j}}= 0\,, \tag{140}\]
where \(\zeta_{+}^{(i)}\) are polynomials of the cross-ratios
\[\zeta_{+}^{(i)} =\sum_{j=0}^{\lfloor i/2\rfloor}\binom{i}{2j}\left(\eta^{2}-1 \right)^{j}\left((2\eta+\xi)^{2}-4\right)^{j}(2\eta^{2}+\xi\eta-2)^{i-2j}\,, \tag{141a}\] \[\zeta_{-}^{(i)} =\sum_{j=0}^{\lfloor i/2\rfloor}\binom{i}{2j+1}\left(\eta^{2}-1 \right)^{j}\left((2\eta+\xi)^{2}-4\right)^{j}(2\eta^{2}+\xi\eta-2)^{i-2j-1}\,. \tag{141b}\]
This form of the superconformal Ward identity has a direct translation in Mellin space, because derivatives map to
\[\xi\partial_{\xi}\ \leftrightarrow\ -\delta\,,\qquad\eta\partial_{\eta}\ \leftrightarrow\ -\rho\,, \tag{142}\]
while products by \(\xi^{m}\eta^{n}\) act as shifts, see (73). As a result, the superconformal Ward identity maps into equations that involve shifts of the Mellin amplitudes and products by rational
functions of the Mellin variables. For example, when \(p_{1}=p_{2}=2\) the Ward identity for \(\zeta_{-}^{(j)}\) reads
\[0= \,(\delta+\rho)\mathcal{M}_{0}(\delta,\rho)+2\rho\mathcal{M}_{1}( \delta,\rho)+4\rho\mathcal{M}_{2}(\delta,\rho)-\frac{2\rho(\rho+1)(\delta+\rho+ 1)}{(\delta+\rho)^{2}}\mathcal{M}_{1}(\delta,\rho+2)\] \[+\delta\mathcal{M}_{1}(\delta+1,\rho-1)+2\delta\mathcal{M}_{2}( \delta+1,\rho-1)-\frac{2\delta\rho(\delta+\rho+1)}{(\delta+\rho)^{2}}\mathcal{ M}_{1}(\delta+1,\rho+1)\] \[-\frac{4\rho(\rho+1)(\delta+2\rho+2)}{(\delta+\rho)^{2}}\mathcal{ M}_{2}(\delta,\rho+2)-\frac{2\delta\rho(2\delta+5\rho+3)}{(\delta+\rho)^{2}} \mathcal{M}_{2}(\delta+1,\rho+1)\] \[-\frac{\delta(\delta+1)(\delta+3\rho)}{(\delta+\rho)^{2}} \mathcal{M}_{2}(\delta+2,\rho)+\frac{4\rho(\rho+1)(\rho+2)(\rho+3)}{(\delta+ \rho)^{2}(\delta+\rho+2)}\mathcal{M}_{2}(\delta,\rho+4)\] \[+\frac{8\delta\rho(\rho+1)(\rho+2)}{(\delta+\rho)^{2}(\delta+\rho+ 2)}\mathcal{M}_{2}(\delta+1,\rho+3)+\frac{4\delta(\delta+1)\rho(\rho+1)}{( \delta+\rho)^{2}(\delta+\rho+2)}\mathcal{M}_{2}(\delta+2,\rho+2)\,. \tag{143}\]
It is not hard to check that the Mellin amplitude (136) indeed satisfies the Ward identity. Note that \(\mathcal{M}_{0}\) can be solved directly as a function of \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\). This seems to be a general feature, although we leave a more detailed analysis of the Ward identities for future work.
## 6 Conclusions
In this work we proposed a bootstrap method for holographic defect correlators, inspired by the position-space bootstrap of [16, 17]. Our focus was on two-point functions of local operators in the presence of the holographic defect, a setup that is relatively unexplored in the literature. The most important ingredient for the bootstrap is knowledge of certain Witten diagrams, that were partially studied in [55, 56, 57, 31]. The present paper extends these results, by analyzing systematically all Witten diagrams that should contribute in the leading supergravity approximation to setups of arbitrary dimension and codimension. In this respect, the spin-two bulk-exchange diagram is the most challenging diagram, and its computation in appendix A constitutes an important new technical result.
To illustrate the position-space bootstrap, we revisited the half-BPS Wilson line in \(\mathcal{N}=4\) SYM. References [53, 54] initiated the study of bulk two-point functions with the Wilson line using analytic bootstrap. Our calculation clarifies the results in [53, 54], because we show that their "low-spin ambiguities" correspond to defect-exchange and contact Witten diagrams. Furthermore, because the present method is more streamlined, we have succeeded in obtaining closed formulas for correlator of chiral-primary operators of arbitrary length. These formulas involve finite sums with known coefficients, see the position space result (132) and its Mellin counterpart (137). The position-space formula agrees with the explicit calculation (126), up to contact terms that we were unable to fix without the bootstrap.
Although this provides a strong sanity check for our method, it still begs the question of how to obtain the correlators from first principles. We hope this will be clarified in future work.
Our results open a number of new research directions. One possibility is to extend the calculation in \(\mathcal{N}=4\) SYM with the half-BPS Wilson line to subleading orders in the large-\(N\) or large-\(\lambda\) expansion. Presumably this requires the use of integrated correlators and a better understanding of the flat-space limit, as [40] proposes inspired by the success of similar work, e.g. [22, 25]. Another option is to study other holographic defects at leading order in the supergravity limit. Some examples in \(\mathcal{N}=4\) SYM are Wilson lines in symmetric or antisymmetric representations [89, 90, 35, 91], 't Hooft lines [92, 93], surface operators [94, 95, 96], or interfaces [97, 98, 99]. The present methods are not restricted to four dimensions, and they should also apply to the \(3d\) ABJM or \(6d\)\((2,0)\) theories. In this respect, recently [59] studied the M2-brane surface defect of \(6d\)\((2,0)\) with the analytic bootstrap of [53, 54], and it seems this setup is also amenable to the Witten diagram bootstrap [100].
Finally, we have outlined in section 3.5 how some of our results are interconnected with other analytic bootstrap methods, such as analytic functionals. However, these links could be extended and made more precise. For example, it would be good to understand under what circumstances the Mellin representation is valid for non-perturbative defect correlators, in the spirit of [75, 80], and also to derive dispersion relations directly in Mellin space. Alternatively, one could explore the crossing-symmetric Mellin bootstrap of [101, 102, 103], or the factorization properties of Mellin amplitudes in analogy with [104, 105, 106]. Finally, one might attempt to bootstrap higher-point correlators, see [107] for recent progress in BCFT.
## Acknowledgements
I am particularly grateful to Julien Barrat and Pedro Liendo for collaborations that inspired this work. I am also really thankful to Yifan Wang and Xinan Zhou for many useful discussions, Simone Giombi for correspondence and comments on the draft, and Junding Chen and Xinan Zhou for collaboration on related work. I thank Pietro Ferrero, Vasco Goncalves, Apratim Kaviraj, Leonardo Rastelli, Victor Rodriguez and Junchen Rong for stimulating discussions. Preliminary versions of this work were presented in Porto, SCGP, NYU and Torino, and I would like to thank these groups for their interesting questions and comments. This work was supported by a grant from the Simons Foundation (915279, IHES) and (733758, Simons Bootstrap Collaboration).
Spin-two bulk exchange
In this appendix we compute the spin-two bulk-exchange diagram with the method of [62]. The original paper considered only graviton exchange, and later this was generalized to massive spin-two exchanges [68, 17] and unequal external scalars [69]. By a slightly modification of their method, we compute the Witten diagram much more explicitly than reference [69]. This result is relevant for setups with and without defects, and we hope it will find use in future calculations.
Our initial goal is to compute
\[A_{\mu\nu}(w,x_{1},x_{2})=\int\frac{d^{d+1}z}{z_{0}^{d+1}}G_{\mu \nu\mu^{\prime}\nu^{\prime}}(w,z)T^{\mu^{\prime}\nu^{\prime}}(z,x_{1},x_{2})\,, \tag{144}\]
where \(G_{\mu\nu\mu^{\prime}\nu^{\prime}}(w,z)\) is the bulk-to-bulk propagator for a massive spin-two particle. The form of the tensor \(T_{\mu\nu}\) is dictated by the coupling of the tensor and the scalars (172). More specifically
\[T_{\mu\nu}(z,x_{1},x_{2})=\frac{1}{2}\nabla_{(\mu}K_{\Delta_{1} }\nabla_{\nu)}K_{\Delta_{2}}-\frac{g_{\mu\nu}}{2}\Big{[}\nabla_{\rho}K_{\Delta _{1}}\nabla^{\rho}K_{\Delta_{2}}+\frac{1}{2}(m_{1}^{2}+m_{2}^{2}-f)K_{\Delta_{ 1}}K_{\Delta_{2}}\Big{]}\,, \tag{145}\]
where the mass of the external scalars is \(m_{i}^{2}=\Delta_{i}(\Delta_{i}-d)\), the mass of the exchanged tensor is \(f=\Delta(\Delta-d)\), and we normalize symmetrization as \(a_{(\mu}b_{\nu)}=a_{\mu}b_{\nu}+a_{\nu}b_{\mu}\). For the sake of clarity, we suppressed arguments of bulk-to-boundary propagators.
The result for \(A_{\mu\nu}\) allows to compute exchange diagrams for four-point functions, see for example [17]. In this paper, we are interested instead in a process where the spin-two particle is absorbed by a brane in the bulk. Because the absorption vertex is \(\int_{\text{AdS}_{p+1}}g^{ij}\varphi_{ij}\), the diagram is
\[\int\frac{d^{p+1}w}{w_{0}^{p+1}}\,g^{ij}A_{ij}(w,x_{1},x_{2})\equiv\frac{E_{ \Delta_{1},\Delta_{2}}^{\Delta,2}(\xi,\eta)}{|x_{1}^{i}|^{\Delta_{1}}|x_{2}^{ i}|^{\Delta_{2}}}\,. \tag{146}\]
Recall that indices \(i,j=p+1,\ldots,d\) are orthogonal to the brane.
The computation of \(A_{\mu\nu}\) proceeds in several steps. First act with the equations of motion on the integral (144), so the bulk-to-bulk propagator reduces to a delta function and the integral trivializes. Next make an ansatz for the result of integral (144), and compute the action of the equations of motion on this ansatz. Comparing the two sides gives differential equations for the ansatz that can be solved in closed form. Finally, compute the defect integral (146) using the value of \(A_{\mu\nu}\) previously determined. We carry out the key steps of the calculation, but more detail can be found in the original references.
EOM on the RHS.Instead of working with \(A_{\mu\nu}\), it is convenient to exploit conformal invariance to simplify its form. Using a translation \(w\to w-x_{1}\) and \(x_{2}\to x_{21}\), and a conformal inversion \(x_{\mu}^{\prime}=x_{\mu}/x^{2}\) and \(z_{\mu}^{\prime}=z_{\mu}/z^{2}\), one finds
\[A_{\mu\nu}(w,x_{1},x_{2})=\frac{1}{(x_{12}^{2})^{\Delta_{2}}}\frac{J_{\mu \lambda}(w)J_{\nu\rho}(w)I^{\lambda\rho}(w^{\prime}-x_{12}^{\prime})}{w^{4}}\,, \quad J_{\mu\nu}(w)=\delta_{\mu\nu}-\frac{2w_{\mu}w_{\nu}}{w^{2}}\,. \tag{147}\]
The function \(I_{\mu\nu}(w)\) is essentially the integral \(A_{\mu\nu}\)
\[I_{\mu\nu}(w)=\int\frac{d^{d+1}z}{z_{0}^{d+1}}G_{\mu\nu\mu^{\prime}\nu^{\prime} }(w,z)\bar{T}^{\mu^{\prime}\nu^{\prime}}(z)\,, \tag{148}\]
but the "stress-tensor" no longer depends on the position of external operators:
\[\bar{T}_{\mu\nu}=\frac{1}{2}\nabla_{(\mu}z_{0}^{\Delta_{1}}\nabla_{\nu)} \left(\frac{z_{0}}{z^{2}}\right)^{\Delta_{2}}\!\!\!-\frac{g_{\mu\nu}}{2}\! \left[\nabla_{\rho}z_{0}^{\Delta_{1}}\nabla^{\rho}\left(\frac{z_{0}}{z^{2}} \right)^{\Delta_{2}}\!\!\!+\frac{1}{2}(m_{1}^{2}+m_{2}^{2}-f)z_{0}^{\Delta_{1} }\left(\frac{z_{0}}{z^{2}}\right)^{\Delta_{2}}\right]. \tag{149}\]
Now act with the equations of motion on \(I_{\mu\nu}(w)\). The operator \(W_{\mu\nu}^{\phantom{\mu\nu}\lambda\rho}\) implements the equation of motion for the spin-two propagator, and in particular
\[W_{\mu\nu}^{\phantom{\mu\nu}\lambda\rho}[G_{\lambda\rho\mu^{\prime}\nu^{\prime} }]=\left(g_{\mu\mu^{\prime}}g_{\nu\nu^{\prime}}+g_{\mu\nu^{\prime}}g_{\nu\mu^{ \prime}}-\frac{2}{d-1}g_{\mu\nu}g_{\mu^{\prime}\nu^{\prime}}\right)\delta(w,z)\,. \tag{150}\]
Comparing with (149) and evaluating the derivatives one finds
\[W_{\mu\nu}^{\phantom{\mu\nu}\lambda\rho}[I_{\lambda\rho}(w)]=w_{0}^{\Delta_{ 12}}t^{\Delta_{2}}\left(\frac{m_{1}^{2}+m_{2}^{2}-f}{d-1}\,g_{\mu\nu}-2\Delta_ {1}\Delta_{2}\frac{P_{(\mu}w_{\nu)}}{w^{2}}+2\Delta_{1}\Delta_{2}P_{\mu}P_{ \nu}\right)\,. \tag{151}\]
Here and below we use the notation \(P_{\mu}=\delta_{0\mu}/w_{0}\).
EOM on the LHS.In order to proceed, we make an ansatz for the integral \(I_{\mu\nu}(w)\)
\[I_{\mu\nu}(w)=w_{0}^{\Delta_{12}}\big{(}g_{\mu\nu}h(t)+P_{\mu}P_{\nu}\phi(t) \big{)}+\nabla_{\mu}\nabla_{\nu}\big{[}w_{0}^{\Delta_{12}}X(t)\big{]}+\nabla_{ (\mu}\big{[}P_{\nu)}w_{0}^{\Delta_{12}}Y(t)\big{]}\,, \tag{152}\]
where \(t=w_{0}^{2}/w^{2}\). This ansatz is somewhat different than in [69], and it leads to cleaner equations that can be solved explicitly. Now act with the differential operator \(W_{\mu\nu}^{\phantom{\mu\nu}\lambda\rho}\) on the ansatz using
\[W_{\mu\nu}^{\phantom{\mu\nu}\lambda\rho}[\phi_{\lambda\rho}]=-\nabla_{\rho} \nabla^{\rho}\phi_{\mu\nu}+\nabla_{(\nu}|\nabla^{\rho}\phi_{\rho|\mu)}-\nabla_ {\mu}\nabla^{\nu}\phi_{\rho}^{\rho}-(2-f)\phi_{\mu\nu}+\frac{2d-2+f}{d-1}g_{ \mu\nu}\phi_{\rho}^{\rho}\,, \tag{153}\]
and then compare the result with (151). This calculation is tedious, but fortunately it can be implemented in Mathematica. It is not hard to solve the resulting differential equations,
and we find
\[h(t)=\frac{fX(t)-\phi(t)}{d-1}\,, \tag{154}\]
\[fY(t)=2(t-1)t\phi^{\prime}(t)+(d-\Delta_{12}-2)\phi(t)+\Delta_{1}t^{\Delta_{2}}+a\,, \tag{155}\]
\[\frac{df(d+f-1)}{d-1} X(t)=2(t-1)(2t-1)t^{2}\phi^{\prime\prime}(t)+t\big{(}(4t-3)(d+2t-2 )-2\Delta_{12}(t-1)\big{)}\phi^{\prime}(t)\] \[+\left(\frac{(d+1)f}{2(d-1)}-\frac{\Delta_{12}(3d-\Delta_{12}-4)} {2}+(d-2)(d-1)\right)\phi(t)+a(2d-\Delta_{12}-1)\] \[+\frac{t^{\Delta_{2}}}{2}\big{(}\Delta_{2}(\Delta_{2}-d)-\Delta_{ 1}(\Delta_{1}+2\Delta_{2}-3d+2)+4\Delta_{1}\Delta_{2}t-f\big{)}\,. \tag{156}\]
Here \(a\) is an integration constant that will drop from the final result. The function \(\phi(t)\) is still the solution to the differential equation
\[4(t-1)t^{2}\phi^{\prime\prime}(t)+t\big{(}2d+12t-12+4\Delta_{12} (t-1)\big{)}\phi^{\prime}(t)\] \[\quad+\big{(}\Delta_{12}(d-4-\Delta_{12})+2d+f-4\big{)}\phi(t)+2a (\Delta_{12}+1)+2\Delta_{1}(\Delta_{1}+1)t^{\Delta_{2}}=0\,. \tag{157}\]
Provided \(\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{\geq 0}\), the solution is a finite power series in \(t\):
\[\phi(t)=\frac{2a(\Delta_{12}+1)}{(\Delta-\Delta_{12}-2)(d-\Delta-\Delta_{12}- 2)}-\sum_{n=1}^{\frac{\Delta_{1}+\Delta_{2}-\Delta+2}{2}}\frac{\big{(}\frac{ \Delta-\Delta_{1}-\Delta_{2}}{2}\big{)}_{n-1}\,\big{(}\frac{d-\Delta_{1}- \Delta_{2}}{2}\big{)}_{n-1}}{2(1-\Delta_{1})_{n-2}(1-\Delta_{2})_{n}}\,t^{ \Delta_{2}-n}\,. \tag{158}\]
Restoring \(A_{\mu\nu}\).Now we have an explicit formula for \(\phi(t)\), \(h(t)\), \(X(t)\) and \(Y(t)\). To obtain a formula for \(I_{\mu\nu}\), we should simply unwrap the ansatz (152) using
\[\nabla_{\mu}\nabla_{\nu}\big{[}w_{0}^{\Delta_{12}}X(t)\big{]}=w_ {0}^{\Delta_{12}}\Big{[}-g_{\mu\nu}\left(2t\partial_{t}+\Delta_{12}\right)- \frac{P_{(\mu}w_{\nu)}}{w^{2}}\left(4t^{2}\partial_{t}^{2}+(2\Delta_{12}+6)t \partial_{t}\right)\] \[\qquad+\frac{w_{\mu}w_{\nu}}{w^{4}}\left(4t^{2}\partial_{t}^{2}+ 8t\partial_{t}\right)+P_{\mu}P_{\nu}\left(4t^{2}\partial_{t}^{2}+(4\Delta_{12} +6)t\partial_{t}+\Delta_{12}(\Delta_{12}+1)\right)\Big{]}X(t)\,, \tag{159}\]
\[\nabla_{(\mu}\big{[}P_{\nu)}w_{0}^{\Delta_{12}}Y(t)\big{]}=w_{0}^{\Delta_{12}} \Big{[}2(\Delta_{12}+1)P_{\mu}P_{\nu}-2g_{\mu\nu}+4P_{\mu}P_{\nu}\,t\partial_ {t}-\frac{2P_{(\mu}w_{\nu)}}{w^{2}}t\partial_{t}\Big{]}Y(t)\,. \tag{160}\]
Finally, to obtain \(A_{\mu\nu}\) from \(I_{\mu\nu}\) we need to undo the coordinate transformation. The transformation is simple for the cross-ratio \(t\) and \(w_{0}\)
\[t^{\prime}\;\to\;x_{12}^{2}K_{1}(x_{1},w)K_{1}(x_{2},w)\,,\qquad w_{0}^{\prime} \;\to\;K_{1}(x_{1},w)\,, \tag{161}\]
where \(K_{\Delta}(x,w)\) is the bulk-to-boundary propagator (10). Furthermore, the contractions with \(J_{\mu\nu}(w)\) were worked out in [17]:
\[P_{\nu}^{\prime}\frac{J_{\mu\nu}}{w^{2}}\;\to\;R_{\mu}\equiv P_{ \mu}-2\frac{(w-x_{1})_{\mu}}{(w-x_{1})^{2}}\,, \tag{162}\] \[\frac{J_{\mu\nu}}{w^{2}}\frac{(w^{\prime}-x^{\prime})_{\mu}}{(w^{ \prime}-x^{\prime})^{2}}\;\to\;Q_{\mu}\equiv-\frac{(w-x_{1})_{\mu}}{(w-x_{1})^{ 2}}+\frac{(w-x_{2})_{\mu}}{(w-x_{2})^{2}}\,,\] (163) \[\frac{J_{\mu\rho}}{w^{2}}g_{\rho\lambda}^{\prime}\frac{J_{\lambda \nu}}{w^{2}}\;\to\;g_{\mu\nu}\,. \tag{164}\]
For applications to four-point functions, it is now straightforward to express spin-two exchange Witten diagram as sums of \(D\)-functions, as explained in more detail in [17].
Computing the defect integral.Finally, to obtain the exchange diagram \(E^{\Delta,2}_{\Delta_{1}\Delta_{2}}\) we need to contract \(g^{ij}A_{ij}\), where \(i,j=p+1,\ldots,d\) are directions orthogonal to the brane. Since we integrate \(w\) along the brane, the orthogonal components of \(w\) vanish \(w^{i}=0\), and we find
\[g_{i}^{\phantom{i}i} = d-p\,,\qquad R_{i}R^{i}\,=\,4|x_{1}^{i}|^{2}K_{2}(x_{1},w)\,, \tag{165}\] \[Q_{i}Q^{i} = |x_{1}^{i}|^{2}K_{2}(x_{1},w)+|x_{2}^{i}|^{2}K_{2}(x_{2},w)-2x_{1 }^{i}x_{2}^{i}K_{1}(x_{1},w)K_{1}(x_{2},w)\,,\] (166) \[Q_{i}R^{i} = 2|x_{1}^{i}|^{2}\,K_{2}(x_{1},w)-2x_{1}^{i}x_{2}^{i}\,K_{1}(x_{1 },w)K_{1}(x_{2},w)\,. \tag{167}\]
Finally, the \(w\)-integral is given by the contact diagram formula (31), and the terms with \(x_{1,2}^{i}\) in (165)-(167) lead to powers of the cross-ratios \(\xi\) and \(\eta\), see (15). This process gives a closed-form expression for \(E^{\Delta,2}_{\Delta_{1}\Delta_{2}}\) as a finite sum of contact diagrams, but because the expression is not particularly illuminating, we give it in an ancillary notebook.
## Appendix B Details on the \(\mathrm{AdS}_{5}\) effective action
This appendix gives formulas omitted in section 5.2.1. The normalizations of kinetic terms read
\[\zeta_{k}^{s}=\frac{32\pi^{3}k(k-1)}{2^{k-1}(k+1)^{2}}\,,\quad\zeta_{k}^{t}= \frac{32\pi^{3}(k+4)(k+5)}{2^{k-1}(k+1)(k+3)}\,,\quad\zeta_{k}^{\varphi}=\frac {\pi^{3}}{2^{k-1}(k+1)(k+2)}\,. \tag{168}\]
For the three-point couplings, it is convenient to introduce
\[\Sigma=k_{1}+k_{2}+k_{3}\,,\quad\alpha_{1}=\frac{k_{2}+k_{3}-k_{1}}{2}\,, \quad\alpha_{2}=\frac{k_{1}+k_{3}-k_{2}}{2}\,,\quad\alpha_{3}=\frac{k_{1}+k_{ 2}-k_{3}}{2}\,. \tag{169}\]
All three-point couplings have the same \(R\)-symmetry structure, which we factor out
\[X^{IJK}_{k_{1}k_{2}k_{3}}=X_{k_{1}k_{2}k_{3}}\langle\mathcal{C}^{J}_{k_{1}} \mathcal{C}^{J}_{k_{2}}\mathcal{C}^{K}_{k_{3}}\rangle\qquad\text{for}\quad X= S,T,G\,. \tag{170}\]
The factored term is defined in equation (180) below, while the couplings read
\[S_{k_{1}k_{2}k_{3}} =\frac{2^{7}\Sigma((\Sigma/2)^{2}-1)((\Sigma/2)^{2}-4)\alpha_{1} \alpha_{2}\alpha_{3}}{3(k_{1}+1)(k_{2}+1)(k_{3}+1)}\,a_{k_{1}k_{2}k_{3}}\,, \tag{171a}\] \[T_{k_{1}k_{2}k_{3}} =\frac{2^{7}(\Sigma+4)(\alpha_{1}+2)(\alpha_{2}+2)\alpha_{3}( \alpha_{3}-1)(\alpha_{3}-2)(\alpha_{3}-3)(\alpha_{3}-4)}{(k_{1}+1)(k_{2}+1)(k_ {3}+3)}\,a_{k_{1}k_{2}k_{3}}\,,\] (171b) \[G_{k_{1}k_{2}k_{3}} =\frac{4(\Sigma+2)(\Sigma+4)\alpha_{3}(\alpha_{3}-1)}{(k_{1}+1)(k _{2}+1)}\,a_{k_{1}k_{2}k_{3}}\,,\] (171c) \[a_{k_{1}k_{2}k_{3}} =\frac{\pi^{3}}{(\Sigma/2+2)!2^{(\Sigma-2)/2}}\frac{k_{1}!k_{2}! k_{3}!}{\alpha_{1}!\alpha_{2}!\alpha_{3}!}\,. \tag{171d}\]
The massive spin-two field \(\varphi_{\mu\nu}\) couples to the tensor
\[T_{\mu\nu}(s_{1},s_{2})=\frac{1}{2}\nabla_{(\mu}s_{1}\nabla_{\nu)}s_{2}-\frac{1} {2}g_{\mu\nu}\Big{[}\nabla_{\rho}s_{1}\nabla^{\rho}s_{2}+\frac{1}{2}(m_{1}^{2}+ m_{2}^{2}-f)s_{1}s_{2}\Big{]}\,. \tag{172}\]
## Appendix C Spherical harmonics
This appendix discusses \(S^{5}\) spherical harmonics. The ultimate goal is to map the spherical harmonics \(Y^{I}\) often found in the supergravity literature, to the index-free notation of section 5.1, that is more convenient in field-theory calculations.
### Basic definition
Spherical harmonics provide a basis of scalar functions in \(S^{5}\), that can be organized in irreducible representations of \(SO(6)\). For each \(k=0,1,2,\ldots\), we introduce a basis of tensors \(\mathcal{C}^{I}_{m_{1}\ldots m_{k}}\) that are symmetric and traceless in the indices \(m_{i}\), where each index runs over \(m_{i}=1,\ldots,6\). The index \(I=1,\ldots,d_{k}\) labels the \(d_{k}=\frac{1}{12}(k+1)(k+2)^{2}(k+3)\) elements in the irrep. We choose the basis \(\mathcal{C}\) such that
\[\mathcal{C}^{I}_{m_{1}\ldots m_{k}}\mathcal{C}^{J}_{m_{1}\ldots m_{k}}=\delta ^{IJ}\,,\qquad\mathcal{C}^{I}_{m_{1}\ldots m_{k}}\mathcal{C}^{I}_{n_{1}\ldots n _{k}}=\Pi^{m_{1}\ldots m_{k}}_{n_{1}\ldots n_{k}}\,, \tag{173}\]
where \(\Pi\) is normalized as \(\Pi^{2}=\Pi\), and it projects tensors to their symmetric-traceless part. Then, we define the spherical harmonics as
\[Y^{I}_{k}(y)=\mathcal{C}^{I}_{m_{1}\ldots m_{k}}\theta^{m_{1}}\ldots\theta^{m _{k}}\qquad\text{for}\qquad\theta^{m}=\left(\frac{y^{A}}{1+\frac{1}{4}y^{2}}, \frac{1-\frac{1}{4}y^{2}}{1+\frac{1}{4}y^{2}}\right)\,. \tag{174}\]
Here \(\theta^{m}\) is a six-dimensional unit vector \(\theta^{2}=1\) that labels a point in \(S^{5}\), while \(y^{A}\) for \(A=1,\ldots,5\) are the stereographic coordinates of the sphere used in section 5.2. In particular, we took the string solution in (115) to be located at \(y^{A}=0\), which corresponds to \(\theta=(0,\ldots,0,1)\).
### Relation to index-free notation
In section 5 we study correlators of chiral-primary operators \(S_{k}\) in rank-\(k\) symmetric-traceless representations of \(SO(6)\). In \(\mathcal{N}=4\) SYM these operators are constructed from \(k\) fundamental scalars. One possible description of \(S_{k}\) consists on contracting with the tensors \(\mathcal{C}^{I}\)
\[S^{I}_{k}(x)\,\propto\,\mathcal{C}^{I}_{m_{1}\ldots m_{k}}\operatorname{tr} \left[\phi^{m_{1}}(x)\ldots\phi^{m_{k}}(x)\right]. \tag{175}\]
The constant of proportionality is fixed requiring unit-normalized two-point function. On the other hand, chiral-primary operators can also be described using index-free notation
\[S_{k}(x,u)\,\propto\,u_{m_{1}}\ldots u_{m_{k}}\operatorname{tr}\left[\phi^{m_{1}} (x)\ldots\phi^{m_{k}}(x)\right], \tag{176}\]
where \(u_{m}\) is a null six-component vector \(u^{2}=0\). It follows from the completeness relation (173) that the two notations are related as
\[S_{k}(x,u)=u^{m_{1}}\ldots u^{m_{k}}\mathcal{C}^{I}_{m_{1}\ldots m_{k}}S^{I}_{k }(x)\,. \tag{177}\]
Finally, given an operator in index-free notation, we can free the vector indices [108]
\[S_{m_{1}\ldots m_{k}}(x)=\frac{1}{k!(k+1)!}D_{m_{1}}\ldots D_{m_{k}}S_{k}(x,u)\,, \tag{178}\]
where we use the Todorov operator
\[D_{m}=\left(2+u\cdot\frac{\partial}{\partial u}\right)\frac{\partial}{ \partial u^{m}}-\frac{u_{m}}{2}\frac{\partial^{2}}{\partial u\cdot\partial u}\,. \tag{179}\]
### Index contractions
Using this preliminary information, we are ready to map the \(R\)-symmetry structures that use spherical harmonics to index free notation.
Bulk-exchange diagramsIn the holographic calculation, one encounters bulk-bulk-bulk vertices (170) that are proportional to
\[\langle\mathcal{C}^{I}_{k_{1}}\mathcal{C}^{J}_{k_{2}}\mathcal{C}^{K}_{k_{3}} \rangle=\mathcal{C}^{I}_{m_{1}\ldots m_{\alpha_{3}}n_{1}\ldots n_{\alpha_{2}}} \mathcal{C}^{J}_{m_{1}\ldots m_{\alpha_{3}}l_{1}\ldots l_{\alpha_{1}}} \mathcal{C}^{K}_{n_{1}\ldots n_{\alpha_{2}}l_{1}\ldots l_{\alpha_{1}}}\,, \tag{180}\]
where \(\alpha_{i}\) were defined in (169). As a result, the correlator of interest contains terms
\[\langle S^{I}_{k_{1}}S^{J}_{k_{2}}W\rangle\supset\langle\mathcal{C}^{I}_{p_{1 }}\mathcal{C}^{J}_{p_{2}}\mathcal{C}^{K}_{k}\rangle Y^{K}_{k}(0)\,. \tag{181}\]
When we map this formula to index-free notation using (177), we encounter contractions of the form \(C^{I}C^{I}=\Pi\), with the projector \(\Pi\) introduced in (173). This projector can be written in terms of the Todorov operator [108] in terms of an auxiliary null polarization vector \(u\):
\[\Pi^{n_{1}\ldots n_{k}}_{m_{1}\ldots m_{k}}=\frac{1}{k!(k+1)!}D_{m_{1}}\ldots D _{m_{k}}u^{n_{1}}\ldots u^{n_{1}}\,. \tag{182}\]
Using this information, one sees that the tensor structure in (181) is equivalent to the following expression
\[\langle\mathcal{C}^{I}_{p_{1}}\mathcal{C}^{J}_{p_{2}}\mathcal{C}^ {K}_{k}\rangle Y^{K}_{k}(0) \to \frac{(u_{1}\cdot u_{2})^{\frac{p_{1}+p_{2}-k}{2}}}{k!(k+1)!}\,(u_ {1}\cdot D)^{\frac{k+p_{12}}{2}}(u_{2}\cdot D)^{\frac{k-p_{12}}{2}}(u\cdot \theta)^{k} \tag{183}\] \[= (u_{1}\cdot\theta)^{p_{1}}(u_{2}\cdot\theta)^{p_{2}}\times h^{k} _{p_{1}p_{2}}(\sigma)\,.\]
Here \(\sigma\) is the \(R\)-symmetry cross-ratio in (100), the \(R\)-symmetry block is defined in (124), and we use \(\theta=\theta(y=0)\). The right-hand side of (183) was previously encountered in equation (A.6) of [53], where it was observed that it equals an \(R\)-symmetry block.
Defect-exchange diagramsThe calculation for the defect exchanges, proceeds in exactly the same way. The exchange of a \(x^{i}\) mode comes with tensor structure
\[Y^{I}_{p_{1}}(y)Y^{J}_{p_{2}}(y) \to (u_{1}\cdot\theta)^{p_{1}}(u_{2}\cdot\theta)^{p_{2}}\times 1\,. \tag{184}\]
For the exchange of \(y^{A}\), the tensor structure is instead
\[\left.\frac{\partial Y^{I}_{p_{1}}}{\partial y^{A}}\frac{\partial Y^{J}_{p_{2} }}{\partial y^{A}}\right|_{y=0} \to \left.\frac{\partial(u_{1}\cdot\theta)^{p_{1}}}{\partial y^{A}} \frac{\partial(u_{2}\cdot\theta)^{p_{2}}}{\partial y^{A}}\right|_{y=0}=(u_{1} \cdot\theta)^{p_{1}}(u_{2}\cdot\theta)^{p_{2}}\times p_{1}p_{2}(\sigma-1)\,. \tag{185}\]
As expected, these results are proportional to \(R\)-symmetry block for defect operator that are scalars and vectors of \(SO(5)_{R}\). This follows simply from the formula for the blocks \(\widehat{h}_{\widehat{K}}(\sigma)\) given in [53, 54], evaluated at \(\widehat{K}=0,1\).
## Appendix D Comparison to topological sector
In this appendix, we compare our prediction for the two-point function in the supegravity limit (132), with results from supersymmetric localization [36, 82].
The main observation is that correlators of chiral-primary operators and Wilson loops restricted to \(S^{2}\subset\mathbb{R}^{4}\) become topological when applying an appropriate twist. This topological subsector is the reduction of the full theory to the cohomology of a certain nilpotent supercharge [109]. The topological sector admits an OPE, which is essentially the truncation of the usual OPE to chiral-primary operators. The truncated OPE leads to so-called micro-bootstrap equations, that impose non-trivial constraints on the CFT data of chiral-primary operators in the presence of a defect [110]. The story is completely analogous to other protected sectors in the literature, see for example [111, 112, 113]. Here, one has the added feature that the topological sector is described by a two-dimensional topological field theory, and correlation functions can be evaluated in terms of Gaussian matrix models [36, 41, 42, 82].
Figure 5: Comparison of the closed-chain (left) and open-chain (right) topology.
To describe the topological sector, we put the Wilson line at \(x_{2}=x_{3}=x_{4}=0\), so that it extends in the \(\mathrm{t}=x_{1}\) direction, and also choose \(\theta=(0,\ldots,0,1)\). Furthermore, chiral-primary operators are restricted to the plane \((\mathrm{t},\mathrm{x})\), where \(\mathrm{x}=x_{2}\). We call the topological operators \(\mathcal{S}_{p}\), which are chiral-primary operators with \(R\)-symmetry polarization correlated with the spacetime position
\[\mathcal{S}_{p}(\mathrm{t},\mathrm{x})\equiv S_{p}(x(\mathrm{t}, \mathrm{x});u(\mathrm{t},\mathrm{x}))\,. \tag{186}\]
The form of \(u(\mathrm{t},\mathrm{x})\) can be determined by studying the cohomology of the nilpotent supercharge, and it reads [109]
\[x(\mathrm{t},\mathrm{x}) =(\mathrm{t},\mathrm{x},0,0)\,, \tag{187}\] \[u(\mathrm{t},\mathrm{x}) =\left(1-\mathrm{t}^{2}-\mathrm{x}^{2},i(1+\mathrm{t}^{2}+ \mathrm{x}^{2}),0,0,-2\mathrm{t},-2\mathrm{x}\right). \tag{188}\]
For the correlator we are interested in \(\langle\mathcal{S}_{p_{1}}(\mathrm{t}_{1},\mathrm{x}_{1})\mathcal{S}_{p_{2}} (\mathrm{t}_{2},\mathrm{x}_{2})W(\theta)\rangle\), we can use conformal transformations to set \(\mathrm{t}_{1}=\mathrm{t}_{2}=0\) and \(\mathrm{x}_{2}=1\). Then the only free parameter is \(\mathrm{x}_{1}\equiv z\), and the correlator reads
\[\mathcal{F}_{\mathrm{top}}(\vartheta)=\mathcal{F}_{c}\left(\xi= \frac{(1-z)^{2}}{|z|},\,\eta=\frac{z}{|z|},\,\sigma=-\frac{(1-z)^{2}}{2z}\right)\,. \tag{189}\]
The statement that the correlator becomes topological is that it only depends on
\[\vartheta=\mathrm{sgn}\,z=\begin{cases}-1&\text{for open chain}\\ +1&\text{for closed chain}\end{cases}\,. \tag{190}\]
In other words, the topological correlator only depends on whether the two local operators are on the same side (closed chain) or opposite sides (open chain) of the Wilson loop. The open/closed chain terminology, which we borrow from [82], is explained in figure 5. At this point, it is possible to check that the correlator (132) indeed reduces to a constant in the kinematics (189), giving different results depending on \(\vartheta=\mathrm{sgn}\,z\).
We can perform an even stronger check, because the topological correlator was computed in the planar limit at arbitrary 't Hooft coupling in [82]. To compare to their result, we need to note two important differences. First, their definition of connected correlator differs from ours by the factors of \(a_{p_{1}}a_{p_{2}}\) in (101). Second, they consider the two-point function of single-trace operators \(\mathcal{O}_{p}=\mathrm{tr}(u\cdot\phi)^{p_{1}}\) in \(U(N)\) gauge theory, while we consider fields \(S_{p}\) dual to supergravity modes. The difference is that for supergravity fields extremal OPE coefficients vanish, in particular \(\lambda_{S_{p_{1}}S_{p_{2}}S_{|p_{12}|}}=0\), whereas for single-trace operators \(\lambda_{\mathcal{O}_{p_{1}}\mathcal{O}_{p_{2}}\mathcal{O}_{|p_{12}|}}\neq 0\). These two differences with the analysis of Giombi and Pestun (GP), can be easily compensated as \(\mathcal{F}_{\mathrm{top}}\sim\mathcal{F}_{\mathrm{GP}}-a_{p_{1}}a_{p_{2}}- \lambda_{p_{1}p_{2}|p_{12}|}a_{|p_{12}|}\). More precisely, the prediction of [82] becomes
in our conventions
\[\mathcal{F}_{\rm top}(\vartheta)=\frac{\sqrt{p_{1}p_{2}}}{2^{\frac{p_ {1}+p_{2}+2}{2}}I_{1}(\sqrt{\lambda})}\frac{\sqrt{\lambda}}{N^{2}}\Bigg{[}- \frac{\sqrt{\lambda}I_{p_{1}}(\sqrt{\lambda})I_{p_{2}}(\sqrt{\lambda})}{2I_{1} (\sqrt{\lambda})}-(-\vartheta)^{\frac{p_{1}+p_{2}-|p_{12}|}{2}}|p_{12}|I_{p_{1 2}|}(\sqrt{\lambda})\\ +\frac{\sqrt{\lambda}}{2}I_{1+\varepsilon}(\sqrt{\lambda})+\sum_{ k=1}^{p_{\rm min}}(-\vartheta)^{k}(p_{1}+p_{2}-2k)I_{p_{1}+p_{2}-2k}(\sqrt{ \lambda})-\sum_{k=1}^{\frac{p_{1}+p_{2}-\varepsilon-2}{2}}(2k+\varepsilon)I_{2 k+\varepsilon}(\sqrt{\lambda})\Bigg{]}, \tag{191}\]
where we introduced \(\varepsilon=p_{1}+p_{2}\ (\mathrm{mod}\,2)\). The first term in the first line corrects for the extra \(a_{p_{1}}a_{p_{2}}\), the second term in the first line corrects for the extra \(\lambda_{p_{1}p_{2}|p_{12}|}a_{|p_{12}|}\). The last line corresponds to equation (4.42) in [82], where we rewrote the infinite sums as finite sums, see also their equation (4.46). The result (191) should be valid at any \(\lambda\), and at leading order as \(\lambda\to\infty\), one checks that it is in perfect agreement with our prediction (132).
## Appendix E Ward identity in Mellin space
In this appendix, we rewrite the superconformal Ward identity in Mellin space, adapting the method of [87]. Recall that the superconformal Ward identity has a simple expression in terms of \(z,\bar{z},\alpha\) defined in (105):
\[W(z,\bar{z})=\left.\left(\partial_{z}+\frac{1}{2}\partial_{\alpha}\right) \mathcal{F}(z,\bar{z},\alpha)\right|_{z=\alpha}=0\,. \tag{192}\]
Furthermore, the correlator \(\mathcal{F}\) decomposes as a sum of \(R\)-symmetry channels
\[\mathcal{F}(z,\bar{z},\alpha)=\sum_{j=0}^{p_{\rm min}}\frac{(1-\alpha)^{2j}}{ \alpha^{j}}\frac{\mathcal{F}_{j}(z,\bar{z})}{(-2)^{j}}\,. \tag{193}\]
It is clear that (192) relates the different channels \(\mathcal{F}_{j}\) through linear differential equations. In particular, we want the result to be polynomial in \(\xi\) and \(\eta\), because polynomials have simple action on the Mellin representation (67). A straightforward application of the chain rule gives
\[W(z,\bar{z})=\sum_{j=0}^{p_{\rm min}}\frac{(1-z)^{2j}}{z^{j}}\left(\frac{ \partial_{\eta}}{2\sqrt{z\bar{z}}}+\frac{\bar{z}-1}{\sqrt{z\bar{z}}}\partial_ {\xi}-\frac{\xi\partial_{\xi}+\eta\partial_{\eta}}{2z}+\frac{j(z+1)}{2(z-1)z} \right)\frac{\mathcal{F}_{j}(\xi,\eta)}{(-2)^{j}}=0\,. \tag{194}\]
The crucial observation is that \(\xi\) and \(\eta\) are invariant under \(z\leftrightarrow\bar{z}\) and \(z,\bar{z}\leftrightarrow\frac{1}{z},\frac{1}{\bar{z}}\), but the Ward identity (194) is not. Thus, to rewrite the Ward identity as a polynomial in \(\xi\) and \(\eta\), we have to take linear combinations of (194) that respect these symmetries. First we
implement the \(z\to\frac{1}{z}\) invariance by considering
\[W(z,\!\bar{z})+W\left(\tfrac{1}{z},\tfrac{1}{\bar{z}}\right)=\sum_{j=0}^{p_{\rm min }}\frac{(1-z)^{2j}}{2z^{j}}\left(\frac{(1-z)^{2}}{z}(\eta\partial_{\eta}+\xi \partial_{\xi}+j)-\xi\partial_{\eta}+4\xi\partial_{\xi}+4j\right)\frac{{\cal F }_{j}(\xi,\eta)}{(-2)^{j}}\,, \tag{195}\]
and then we implement \(z\leftrightarrow\bar{z}\) invariance taking the two linear combinations
\[W(z,\bar{z})+W(\tfrac{1}{z},\tfrac{1}{\bar{z}})\pm\left(W(\bar{z},z)+W(\tfrac {1}{\bar{z}},\tfrac{1}{z})\right)=0\,. \tag{196}\]
The two equations in (196) are equivalent to formula (140) in the main text. This identification follows from the definition of \(\zeta^{(j)}_{\pm}\) in terms of \(z,\bar{z}\):
\[\zeta^{(j)}_{+}=\frac{(1-z)^{2j}}{2z^{j}}+\frac{(1-\bar{z})^{2j}}{2\bar{z}^{j }}\,,\quad\zeta^{(j)}_{-}=\frac{z\bar{z}}{(\bar{z}-z)(1-z\bar{z})}\left(\frac{ (1-z)^{2j}}{z^{j}}-\frac{(1-\bar{z})^{2j}}{\bar{z}^{j}}\right)\,. \tag{197}\]
Although not manifestly so, these objects are polynomial in \(\xi\) and \(\eta\). In fact, the prefactor of \(\zeta^{(j)}_{-}\) ensures it is indeed a polynomial. Finally, a combination of (105), (197) and the binomial theorem gives formula (141) for \(\zeta^{(j)}_{\pm}\).
|
2302.03965 | Dual-interest Factorization-heads Attention for Sequential
Recommendation | Accurate user interest modeling is vital for recommendation scenarios. One of
the effective solutions is the sequential recommendation that relies on click
behaviors, but this is not elegant in the video feed recommendation where users
are passive in receiving the streaming contents and return skip or no-skip
behaviors. Here skip and no-skip behaviors can be treated as negative and
positive feedback, respectively. With the mixture of positive and negative
feedback, it is challenging to capture the transition pattern of behavioral
sequence. To do so, FeedRec has exploited a shared vanilla Transformer, which
may be inelegant because head interaction of multi-heads attention does not
consider different types of feedback. In this paper, we propose Dual-interest
Factorization-heads Attention for Sequential Recommendation (short for DFAR)
consisting of feedback-aware encoding layer, dual-interest disentangling layer
and prediction layer. In the feedback-aware encoding layer, we first suppose
each head of multi-heads attention can capture specific feedback relations.
Then we further propose factorization-heads attention which can mask specific
head interaction and inject feedback information so as to factorize the
relation between different types of feedback. Additionally, we propose a
dual-interest disentangling layer to decouple positive and negative interests
before performing disentanglement on their representations. Finally, we evolve
the positive and negative interests by corresponding towers whose outputs are
contrastive by BPR loss. Experiments on two real-world datasets show the
superiority of our proposed method against state-of-the-art baselines. Further
ablation study and visualization also sustain its effectiveness. We release the
source code here: https://github.com/tsinghua-fib-lab/WWW2023-DFAR. | Guanyu Lin, Chen Gao, Yu Zheng, Jianxin Chang, Yanan Niu, Yang Song, Zhiheng Li, Depeng Jin, Yong Li | 2023-02-08T09:42:45Z | http://arxiv.org/abs/2302.03965v3 | # Dual-interest Factorization-heads Attention for Sequential Recommendation
###### Abstract.
Accurate user interest modeling is vital for recommendation scenarios. One of the effective solutions is the sequential recommendation that relies on click behaviors, but this is not elegant in the video feed recommendation where users are passive in receiving the streaming contents and return skip or no-skip behaviors. Here skip and no-skip behaviors can be treated as negative and positive feedback, respectively. With the mixture of positive and negative feedback, it is challenging to capture the transition pattern of behavioral sequence. To do so, FeedRec has exploited a shared vanilla Transformer, which may be inelegan because head interaction of multi-heads attention does not consider different types of feedback. In this paper, we propose **D**ual-interest **F**actorization-heads **A**ttention for Sequential **R**ecommendation (short for DFAR) consisting of feedback-aware encoding layer, dual-interest disentangling layer and prediction layer. In the feedback-aware encoding layer, we first suppose each head of multi-heads attention can capture specific feedback relations. Then we further propose factorization-heads attention which can mask specific head interaction and inject feedback information so as to factorize the relation between different types of feedback. Additionally, we propose a dual-interest disentangling layer to decouple positive and negative interests before performing disentanglement on their representations. Finally, we evolve the positive and negative interests by corresponding towers whose outputs are contrastive by BPR loss. Experiments on two real-world datasets show the superiority of our proposed method against state-of-the-art baselines. Further ablation study and visualization also sustain its effectiveness. We release the source code here: [https://github.com/tsinghua-fib-lab/WWW2023-DFAR](https://github.com/tsinghua-fib-lab/WWW2023-DFAR).
Sequential recommendation, User feedback, Contrastive Learning +
Footnote †: [
far more complex due to negative feedback. A user may provide negative feedback only because she has consumed a very similar item before, which makes accurate modeling of transition essential and challenging.
* **Mixed interest in one behavioral sequence.** The negative feedback in the behavioral sequence brings significant challenges to interest learning. The traditional methods of sequential recommendation always conduct a pooling operation on user sequence to obtain the users' current interest, which will fail when the sequence is hybrid with positive and negative signals.
To address the above challenges, in this work, we propose a model named **D**ual-interest **F**actorization-heads **A**ttention for Sequential **R**ecommendation (short for DFAR), further extracting the transition pattern and pair-wise relation between positive and negative interests. To address the first challenge, in the feedback-aware encoding layer, we assume each head of multi-head attention (Wang et al., 2017) tends to capture specific relations of certain feedback (Wang et al., 2017). As different heads of multi-head attention (Wang et al., 2017) are independent, it may fail to capture the transition pattern between different feedback when positive feedback and negative feedback are indeed not independent of each other. Thus we exploit talking-heads attention (Wang et al., 2017) to implicitly extract the transition pattern between positive and negative historical items. However, talking-heads attention may mix different heads too much without sufficient prior knowledge. To explicitly extract the transition pattern between positive and negative historical items, we further propose feedback-aware factorization-heads attention which can even incorporate the feedback information into the head interaction. To address the second challenge, we propose a dual-interest disentangling layer and prediction layer, respectively, to disentangle and extract the pair-wise relation between positive and negative interests. Specifically, we first mask and encode the sequence hybrid with positive feedback and negative feedback into two single interest representations before performing disentanglement on them to repel the dissimilar interests. Then we perform a prediction of each interest with the corresponding positive or negative tower and apply contrastive loss on them to extract their pair-wise relation.
In general, we make the following contributions in this work.
* We have taken the pioneering step of fully considering the modeling of negative feedback, along with its impact on transition patterns, to enhance sequential recommendation.
* We propose a feedback-aware encoding layer to capture the transition pattern, dual-interest disentangling layer and prediction layer to perform disentanglement and capture the pair-wise relation between positive and negative historical items.
* We conduct experiments on one benchmark dataset and one collected industrial dataset, where the results show the superiority of our proposed method. A further ablation study also sustains the effectiveness of our three components.
## 2. Problem Formulation
**Click-based Sequential Recommendation.** Given item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\) with only positive feedback, the goal of traditional click-based sequential recommendation is accurately predicting the probability that **given user \(u\)** will click the target item _i.e._, \(i_{t+1}\). The traditional click-based sequential recommendation can be formulated as follows.
_Input_: Item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\) with only positive feedback for a **given user \(u\)**.
_Output_: The predicted score that the **given user \(u\)** will click the target item \(i_{t+1}\).
_Input_: Item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\) with only positive feedback for a **given user \(u\)**.
_Output_: The predicted score that the **given user \(u\)** will click the target item \(i_{t+1}\).
_Input_: Item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\) with positive and negative feedback, the dual-interest sequential recommendation aims to better predict the probability that **given user \(u\)** will skip or not skip the target item _i.e._, \(i_{t+1}\). The dual-interest sequential recommendation with both positive and negative feedback can be formulated as follows.
_Input_: Item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\) with positive and negative feedbacks for a **given user \(u\)**.
_Output_: The predicted score that the **given user \(u\)** will skip or do not skip the target item \(i_{t+1}\).
## 3. Methodology
Our model captures the relation between positive feedback and negative feedback at the transition level and interest level of sequential recommendation, respectively, by the proposed Feedback-aware Encoding Layer, Dual-interest Disentangling Layer and Prediction Layer, as shown in Figure 2.
* **Feedback-aware Encoding Layer**. We build item embeddings by item IDs and label embeddings by item feedback and further propose feedback-aware factorization-heads attention to capture the transition pattern between different feedback.
* **Dual-interest Disentangling Layer**. We mask the sequence hybrid with both positive and negative feedback into two sequences with solely positive or negative feedback. After encoding two split sequences with independent factorization-heads attention to extract the positive and negative interests, we then disentangle them to repel the dissimilar interests.
* **Dual-interest Prediction Layer**. We further extract the positive and negative interests with independent towers and then perform contrastive loss on them to extract the pair-wise relation.
### Feedback-aware Encoding Layer
In the feedback-aware encoding layer, we first inject each historical item embedding with corresponding feedback embeddings to incorporate the feedback information into each historical item embedding. Then we further propose talking-heads attention and feedback-aware factorization-heads attention to capture the transition pattern between positive and negative historical items.
#### 3.1.1. **Feedback-aware Embedding Layer**
To fully distinguish positive and negative feedback, we build a label embedding matrix \(\mathbf{L}\in\mathbb{R}^{2\times D}\), besides the item embedding matrix \(\mathbf{E}\in\mathbb{R}^{m\times D}\). Here \(m\) denotes the number of items, and \(D\) is the dimensionality for the hidden state. Then we inject the feedback information into the item embedding and obtain the feedback-aware input embeddings as the model input. Therefore, given item sequence \(\mathcal{I}_{u}=(i_{1},i_{2},\ldots,i_{t})\), we can obtain the feedback-aware item embeddings \(\mathbf{E}^{f}\in\mathbb{R}^{T\times D}\) as:
\[\mathbf{E}^{f}=[\mathbf{E}_{i_{1}},\mathbf{E}_{i_{2}},\ldots,\mathbf{E}_{i_{t }}]+[\mathbf{L}_{y_{u_{1}}},\mathbf{L}_{y_{u_{2}}},\ldots,\mathbf{L}_{y_{u_{t }}}], \tag{1}\]
where \(\{y_{u_{i},i_{1}},y_{u_{i},i_{2}},\cdots,y_{u_{t},i_{t}}\}\) are feedback of items \(\{i_{1},i_{2},\cdots,i_{t}\}\). Here \(y_{u_{i},i_{1}}=1\) if \(i_{1}\) is the no-skip item, and \(y_{u_{i},i_{1}}=0\) if \(i_{1}\) is the skip item. Note that if the sequence length is less than \(t\), we can pad \(\mathbf{E}^{f}\) with zero embedding (Kang et al., 2017).
#### 3.1.2. **Talking-Heads Attention**
After obtaining the input embeddings for positive and negative historical items, we then capture the transition pattern between them. The existing work, FeedbackRec (Wang et al., 2017), exploits vanilla Transformer to roughly capture this transition pattern, of which multi-head attention (Kang et al., 2018) is the essential part, having the following equation:
\[\mathbf{S}=\mathrm{MHA}(\mathbf{Q},\mathbf{K},\mathbf{V})=\left[\mathbf{A}_{1} ^{\text{MHA}}\mathbf{V}_{1},\dots,\mathbf{A}_{H}^{\text{MHA}}\mathbf{V}_{H} \right]\mathbf{W}_{0}, \tag{2}\]
\[\mathbf{A}_{h}^{\text{MHA}}=\mathrm{softmax}\left(\frac{\mathbf{Q}_{h}\mathbf{K }_{h}{}^{T}}{\sqrt{d}}\right), \tag{3}\]
Figure 2. Illustration of DEAR. (_i_) Feedback-aware Encoding Layer is linked after the Feedback-aware Embedding Layer where each historical item is injected with a label embedding according to the corresponding feedback; It consists of linear transformation and feedback-aware factorization-heads attention. In the linear transformation, input embeddings are transformed into query, key and value matrices. In feedback-aware factorization-heads attention, the transition relation between different items is factorized into different heads which are masked according to the positive or negative feedback. (_ii_) Dual-interest Disentangling Layer decouples positive and negative interests and performs disentanglement to repel the dissimilar representations of different feedback; (_iii_) Dual-interest Prediction Layer evolves positive and negative interests with corresponding towers and perform BPR loss to capture the pair-wise relation.
\[\mathbf{Q}_{h}=\mathbf{Q}\mathbf{W}_{h}^{Q},\mathbf{K}_{h}=\mathbf{K}\mathbf{W}_{h}^ {K},\mathbf{V}_{h}=\mathbf{V}\mathbf{W}_{h}^{V}, \tag{4}\]
where \(h\in\{1,2,\cdots,H\}\) is the number of heads. \(\mathbf{W}_{0}\in\mathbb{R}^{HD\times D}\) and \(\mathbf{W}_{h}^{Q}\), \(\mathbf{W}_{h}^{K}\), \(\mathbf{W}_{h}^{V}\in\mathbb{R}^{D\times D}\) are parameters to be learned. MHA means multi-heads attention (Kang et al., 2017). However, different heads of multi-head attention are independent of each other, sharing no information across heads. If assuming different heads capture specific relations between different feedback, then this means there is no information sharing across different feedback. Thus we first propose talking-heads attention (Kang et al., 2017) to address this issue as below.
(5) \[\mathbf{S}=\mathrm{THA}(\mathbf{Q},\mathbf{K},\mathbf{V})=\left[\mathbf{A}_{ 1}^{THA}\mathbf{V}_{1},\ldots,\mathbf{A}_{H}^{THA}\mathbf{V}_{H}\right]\mathbf{ W}_{0},\] (6)
\[\left[\begin{array}{c}\mathbf{A}_{1}^{THA}\\ \mathbf{A}_{2}^{THA}\\ \vdots\\ \mathbf{A}_{H}^{THA}\end{array}\right]=\mathbf{W}_{THA}^{S}\left[\begin{array} []{c}\mathrm{softmax}\left(\mathbf{A}_{1}\right)\\ \mathrm{softmax}\left(\mathbf{A}_{2}\right)\\ \vdots\\ \mathrm{softmax}\left(\mathbf{A}_{H^{\prime}}\right)\end{array}\right], \tag{7}\]
where \(\mathbf{W}_{THA}\in\mathbb{R}^{H\times H}\), \(\mathbf{W}_{THA}^{S}\in\mathbb{R}^{H\times H^{\prime}}\) and \(\mathbf{W}_{0}\in\mathbb{R}^{HD\times D}\) are parameters to be learned. Here THA refers to talking-heads attention. However, the interaction between different heads in talking-heads attention is implicit, which may confuse the task for each head and result in overfitting. Not to mention, the two additional linear transformations (i.e. Eq.(6) and Eq.(7)) of talking-heads attention will increase the computation cost.
#### 3.1.3. **Feedback-aware Factorization-heads Attention**
In this part, we factorize the interaction between positive and negative feedback. Traditional multi-heads attention assigns similar items with higher attention weights. However, in our problem with both positive and negative feedback, two similar items may have different attention weights due to the feedback they have. For example, an NBA fan skips the recommended video about basketball when he/she has watched a lot of basketball videos. But he/she engages in the video about basketball when he/she only has watched a few videos about basketball. In the first case we should repel the representations between historical basketball videos and target basketball videos, while in the second case we should attract them. That is to say, it is necessary to inject the user's feedback into the transition pattern between different feedback. Here we suppose different heads can represent different transition patterns for different feedback (Shen et al., 2017). To explicitly factorize interaction across different heads, we further propose factorization-heads attention as:
(8) \[\mathbf{S}=\mathrm{FHA}(\mathbf{Q},\mathbf{K},\mathbf{V})=\left[\mathbf{A}_{ 1,1}^{FHA}\mathbf{V}_{1},\ldots,\mathbf{A}_{H,H}^{FHA}\mathbf{V}_{H}\right] \mathbf{W}_{0},\] (9)
\[\mathbf{A}_{h_{1},h_{2}}^{FHA}=\mathrm{softmax}\left(\frac{\mathbf{Q}_{h_{1}} \mathbf{K}_{h_{2}}^{T}}{\sqrt{d}}\right),\]
where \(h_{1},h_{2}\in\{1,2,\cdots,H\}\). \(\mathbf{W}_{0}\in\mathbb{R}^{HD\times D}\) are parameters to be learned. Here FHA is our proposed factorization-heads attention. The factorization-heads attention can represent \(H\times H\) relations by \(H\) heads. That is to say, our factorization-heads attention can reduce \(\sqrt{H}\) times parameters if we want to represent \(H\) head interaction relations like talking-heads attention or multi-heads attention. Besides, to further inject the prior feedback knowledge into the factorization-heads attention, we propose feedback-aware factorization-heads attention with a label mask \(\mathbf{M}_{h_{1},h_{2}}\in\{0,1\}^{D\times L}\) as:
\[\mathbf{S}=\mathrm{FFHA}(\mathbf{Q},\mathbf{K},\mathbf{V})=\left[\mathbf{A}_{ 1,1}^{FFHA}\mathbf{V}_{1},\ldots,\mathbf{A}_{H,H}^{FFHA}\mathbf{V}_{H}\right] \mathbf{W}_{0}, \tag{10}\]
\[\mathbf{A}_{h_{1},h_{2}}^{FFHA}=\mathrm{softmax}\left(\mathbf{M}_{h_{1},h_{2}} \frac{\mathbf{Q}_{h_{1}}\mathbf{K}_{h_{2}}^{T}}{\sqrt{d}}\right), \tag{11}\]
where \(\mathbf{M}_{h_{1},h_{2},i,j}=1,if\)\(h_{1}\in\{\frac{y_{u},j}{2}H+1,\frac{y_{u},j}{2}H+2,\cdots,\frac{(y_{u}+1)H}{2}\},h_{2} \in\{\frac{y_{u},j}{2}H+1,\frac{y_{u},j}{2}H+2,\cdots,\frac{(y_{u}+1)H}{2}\}, i\in\{1,2,\cdots,t\},j\in\{1,2,\cdots,t\}\) and \(\mathbf{M}_{h_{1},h_{2},i,j}=0\), otherwise. Here the first half of heads w.r.t. \(\{1,2,\cdots,\frac{H}{2}\}\) represent negative heads and second half of heads w.r.t. \(\{\frac{H}{2}+1,\frac{H}{2}+2,\cdots,H\}\) represent positive heads. For example, as shown in Figure 3, if item \(i\) is positive and item \(j\) is negative (i.e., \(y_{u,i}=1\) and \(y_{u,j}=0\)), \(h_{1}\) in positive half and \(h_{2}\) in negative half will be preserved, i.e., \(\mathbf{M}_{2,1,i,j}=1\), and \(\mathbf{M}_{1,1,i,j}\), \(\mathbf{M}_{1,2,i,j}\), \(\mathbf{M}_{2,2,i,j}=0\).
Besides, FFHA is our proposed feedback-aware factorization-heads attention. Apart from the advantage of explicit interaction between different heads, unlike talking-heads attention, our factorization-heads attention also improves the multi-heads attention without high computation cost. We feed the input embedding into the feedback-aware factorization attention module as:
\[\mathbf{S}=\mathrm{FFHA}(\mathbf{E}^{f},\mathbf{E}^{f},\mathbf{E}^{f}), \tag{12}\]
where \(\mathbf{S}\) are the obtained feedback-aware sequential representations. We put the pseudocode of FHA at Appendix A.1 and compare its complexity with MHA and THA at Appendix A.1.5.
### Dual-interest Disentangling Layer
Though feedback-aware factorization-heads attention has factorized the transition relation between positive feedback and negative feedback, their interest-level relations require further extracting. In this part, we decouple the positive and negative interests and then perform disentanglement on them to repel the dissimilar interests.
#### 3.2.1. **Dual-interest Decoupling Attention**
After capturing the transition pattern between positive feedback and negative feedback, we then filter out each feedback by a corresponding feedback mask
Figure 3. Illustration of label mask \(\mathbf{M}_{h_{1},h_{2}}\) on head interaction. Here we show the comprehensible case with two heads, where the first half of heads, i.e. head 1, represents negative head and second half of heads, i.e. head 2, represents positive head.
as follows,
\[\begin{split}\mathbf{S}^{P}&=[\mathbf{S}_{i_{1}}, \mathbf{S}_{i_{2}},\ldots,\mathbf{S}_{i_{t}}]*[y_{u_{i},i_{t}},y_{u_{i},b_{i}}, \ldots,y_{u_{i},t}],\\ \mathbf{S}^{N}&=[\mathbf{S}_{i_{1}},\mathbf{S}_{i_{2 }},\ldots,\mathbf{S}_{i_{t}}]*(1-[y_{u_{i},i_{t}},y_{u_{i},i_{2}},\ldots,y_{u_{ i},t}]),\end{split} \tag{13}\]
which are then fed into the corresponding factorization-heads attention modules to enhance the transition pattern learning for each feedback as:
\[\mathbf{S}^{P}=\mathrm{FHA}(\mathbf{S}^{P},\mathbf{S}^{P},\mathbf{S}^{P}), \mathbf{S}^{N}=\mathrm{FHA}(\mathbf{S}^{N},\mathbf{S}^{N},\mathbf{S}^{N}), \tag{14}\]
where \(\mathbf{S}^{P}\) (or \(\mathbf{S}^{N}\)) are the single-feedback sequential representations for positive feedback (or negative feedback). In the subsequent section, we will exploit these filtered representations to further extract the interest-level relations.
#### Dual-interest Aggregation and Disentanglement
The positive and negative interests of a given user should be distinguished from each other. Hence we aim to repel the positive and negative representations of corresponding interests. Specifically, we assume the target item is possibly either positive or negative. Then we assign the target item with positive and negative label embeddings, respectively, in positive and negative assumed cases. To calculate the attention scores of positive and negative historical items, we fuse them with the target item in assumed positive and negative cases as below.
\[\mathbf{A}^{P}=\mathrm{MLP}\left((\mathbf{E}_{i_{t+1}}+\mathbf{L}_{1})|| \mathbf{S}^{P}\right),\mathbf{A}^{N}=\mathrm{MLP}\left((\mathbf{E}_{i_{t+1}} +\mathbf{L}_{0})||\mathbf{S}^{N}\right), \tag{15}\]
where \(\mathbf{A}^{P}\) and \(\mathbf{A}^{N}\in\mathbb{R}^{t\times D}\) are the positive and negative attention scores. \(\mathrm{MLP}\) is the multi-layer perceptron. Here \(\mathbf{L}_{1}\) and \(\mathbf{L}_{0}\) are the label embeddings for positive and negative feedback, respectively. With the calculated attention scores by (15), we can then obtain the single-feedback aggregated representations for positive and negative items, respectively, as,
\[\mathbf{F}^{P}=\sum_{j=1}^{t}\mathbf{F}_{j}^{P},\mathbf{f}^{N}=\sum_{j=1}^{t} \mathbf{F}_{j}^{N}, \tag{16}\]
which are then further disentangled with cosine distance as:
\[\mathcal{L}^{D}=\frac{\mathbf{f}^{P}\cdot\mathbf{f}^{N}}{\left\|\mathbf{f}^{P }\right\|\times\left\|\mathbf{f}^{N}\right\|}. \tag{18}\]
where \(\|\cdot\|\) is the L2-norm. By this disentangling loss, we can repel the aggregated positive and negative representations so as to capture the dissimilar characteristics between them.
### Dual-interest Prediction Layer
In this section, we predict the next item of different interests by positive and negative towers. Finally, we further perform contrastive loss on the outputs of positive and negative towers so as to extract the pair-wise relation between them.
#### 3.3.1. **Dual-interest Prediction Towers**
To extract the positive and negative interests, we fuse the feedback-aware sequential representations, single-feedback sequential representations, and single-feedback aggregated representations into the corresponding positive or negative prediction tower. Before feeding different representations into the final prediction towers, we first aggregate part of them by the sum pooling as:
\[\mathbf{s}=\sum_{j=1}^{t}\mathbf{S}_{j},\mathbf{s}^{P}=\sum_{j=1}^{t}\mathbf{ S}_{j}^{P},\mathbf{s}^{N}=\sum_{j=1}^{t}\mathbf{S}_{j}^{N},\]
which are then finally fed into the positive and negative prediction towers as:
\[logit_{u,t}^{P}=\mathbf{MLP}\left(\mathbf{s}||\mathbf{s}^{P}||\mathbf{f}^{P} ||(\mathbf{E}_{i_{t+1}}+\mathbf{L}_{1})\right), \tag{19}\]
\[logit_{u,t}^{N}=\mathbf{MLP}\left(\mathbf{s}||\mathbf{s}^{N}||\mathbf{f}^{N} ||(\mathbf{E}_{i_{t+1}}+\mathbf{L}_{0})\right). \tag{20}\]
where \(logit_{u,t}^{P}\) and \(logit_{u,t}^{N}\) are positive and negative predicted logits for user \(u\) on time step \(t\), aiming to capture the positive and negative interests, respectively. Here \(\mathbf{f}^{P}\) and \(\mathbf{f}^{N}\) are pooled at Eq.(17).
#### 3.3.2. **Pair-wise Contrastive Loss**
When the target item is positive, the prediction logit of the positive tower will be greater than that of the negative tower, and vice versa. After obtaining the positive and negative prediction logits, we then perform BPR loss (Kang et al., 2017) on them as:
\[\mathcal{L}^{BPR}=\begin{cases}-\log(\sigma(logit_{u,t}^{P}-logit_{u,t}^{N})), &y_{u,t}=1,\\ -\log(\sigma(logit_{u,t}^{N}-logit_{u,t}^{P})),&y_{u,t}=0.\end{cases} \tag{21}\]
where \(\sigma\) denotes the sigmoid function. With this BPR loss, we can extract the pair-wise relations between positive and negative logits.
### Joint Optimization
Though we have positive and negative towers, in the optimization step, we only need to optimize the next item prediction loss with the positive tower as:
\[\mathcal{L}=-\frac{1}{|\mathcal{R}|}\sum_{(u,t)\in\mathcal{R}}\left(y_{u,t}\log \hat{y}_{u,t}^{P}+(1-y_{u,t})\log\left(1-\hat{y}_{u,t}^{P}\right)\right), \tag{22}\]
where \(\hat{y}_{u,t}^{P}=\sigma(logit_{u,t}^{P})\) and \(\mathcal{R}\) is the training set. The negative prediction tower \(\hat{y}_{u,t}^{N}\) indeed will be self-supervised and optimized by the contrastive loss of Eq.(21). After obtaining the main loss for the next item prediction, disentangling loss for repelling representations and BPR loss for pair-wise learning, we can then jointly optimize them as:
\[\mathcal{L}^{J}=\mathcal{L}+\lambda^{BPR}\mathcal{L}^{BPR}+\lambda^{D}\mathcal{L} ^{D}+\lambda\|\Theta\|, \tag{23}\]
where \(\lambda^{BPR}\) and \(\lambda^{D}\) are hyper-parameters for weighting each loss. Here \(\lambda\) is the regularization parameter, and \(\Theta\) denotes the model parameters to be learned.
## 4. Experiments
In this section, we experiment on a public dataset and an industrial dataset, aiming to answer the following research questions (RQ):
* **RQ1**: Is the proposed DFAR effective when compared with the state-of-the-art sequential recommenders?
* **RQ2** : What is the effect of our proposed feedback-aware encoding layer, dual-interest disentangling layer and prediction layer?
* **RQ3** : How do the heads of proposed feedback-aware factorization-heads attention capture the transition pattern between different feedback?
* **RQ4**: How does the proposed method perform compared with the sequential recommenders under different sequence lengths? We also look into the question: 'how do the auxiliary loss for disentanglement and pair-wise contrastive learning perform under different weights?' in Appendix A.4.
### Experimental Settings
#### 4.1.1. **Datasets**
The data statistics of our experiments are illustrated in Table 1 where Micro-video is a collected industrial dataset and Amazon is the public benchmark dataset which is widely used in existing work for sequential recommendation (Kumar et al., 2017). The detailed descriptions of them are as below.
**Micro-video** This is a popular micro-video application dataset, which is recorded from September 11 to September 22, 2021. In this platform, users passively receive the recommended videos, and their feedbacks are mostly either skip or no-skip. Skip can be treated as a form of negative feedback, and no-skip can be treated as a form of positive feedback. That is to say, we have hybrid positive and negative feedback in this sequential data which is very rare in modern applications.
**Amazon1** This is Toys domain from a widely used public e-commerce dataset in recommendation. The rating score in Amazon ranges from 1 to 5, and we treat the rating score over three and under two as positive and negative feedback, respectively, following existing work DenoisingRec (Song et al., 2019) which is not for the sequential recommendation.
Footnote 1: [https://www.amazon.com](https://www.amazon.com)
For the Micro-video dataset, interactions before and after 12 pm of the last day are split as the validation and test sets, respectively, while interactions before the last day are used as the training set. For the Amazon dataset, we split the last day as the test set and the second last day as the validation set, while other days are split as the training set.
#### 4.1.2. **Baselines and Evaluation Metrics**
We compare our DFAR with the following state-of-the-art methods for sequential recommender systems.
* **DIN**(Wang et al., 2019): It aggregates the historical items via attention score with the target item.
* **Caser**(Wang et al., 2019): It captures the transition between historical items via convolution.
* **GRU4REC**(Kumar et al., 2017): It captures the transition between historical items via GRU (Chen et al., 2019).
* **DIEN**(Wang et al., 2019): It captures the transition between historical items via interest extraction and evolution GRUs (Chen et al., 2019).
* **SARec**(Wang et al., 2019): It captures the transition between historical items via multi-heads attention (Wang et al., 2019).
* **THA4Rec**: It means talking-heads attention (Wang et al., 2019) for the sequential recommendation, which is firstly applied in the recommendation by us.
* **DFN**(Wang et al., 2019): It purifies unclick (weak feedback) by click (strong positive feedback) and dislike (strong positive feedback).
* **FeedRec**(Wang et al., 2019): It further performs disentanglement on the weak positive and negative feedback.
Besides, Widely-used AUC and GAUC (Chen et al., 2019) are adopted as accuracy metrics here while MRR@10 and NDCG@10 (Kumar et al., 2017) are used as ranking metrics for performance evaluation. The detailed illustration of them is in Appendix A.2.
#### 4.1.3. **Hyper-parameter Settings**
Hyper-parameters are generally set following the default settings of baselines. We strictly follow existing work for sequential recommendation (Kumar et al., 2017) and leverage Adam (Kingmae and Ba, 2014) with the learning rate of 0.0001 to weigh the gradients. The embedding sizes of all models are set as 32. We use batch sizes 20 and 200, respectively, on the Micro-video and Amazon datasets. We search the loss weights for pair-wise contrastive loss in \([10^{-4},10^{-3},10^{-2},10^{-1}]\).
### Overall Performance Comparison(RQ1)
We compare our proposed method with eight competitive baselines, and the results are shown as Table 2, where we can observe that:
* **Our method achieves the best performance.** The results on two datasets show that our DFAR model achieves the best performance compared with these seven baselines on all metrics. Specifically, GAUC is improved by about 2.0% on the Micro-video dataset and 0.5% on the Amazon dataset and when comparing DFAR with other baselines. Please note that 0.5% improvement on GAUC could be claimed as significant, widely acknowledged by existing works (Wang et al., 2019). Besides, the improvement is more significant in the Micro-video with more negative feedback, which means incorporating the negative feedback into the historical item sequence can boost the recommendation performance.
* **Existing work roughly captures the relation between positive feedback and negative feedback.** FeedRec and DFN even underperform some traditional sequential recommendation models like GRU4REC and Caser in Amazon dataset. Besides, though they outperform other baselines in Micro-video dataset, the improvement is still slight. In other words, their designs fail to capture the relation between positive feedback and negative feedback, which motivates us to further improve them from transition and interest perspectives.
### Ablation Study (RQ2)
We further study the impact of four proposed components as Table 3, where FHA represents the factorization-heads attention, the MO represents the mask operation on factorized heads for factorization-heads attention, IDL means the interest disentanglement loss on the positive and negative interest representations, and IBL means the interest BPR loss on the positive and negative prediction logits. From this table, we can have the following observations.
* **Factorization of heads for transition attention weights is important**. Removing FHA and MO both show significant performance drops, which means these two components are both
\begin{table}
\begin{tabular}{c|c|c} \hline
**Dataset** & **Micro-video** & **Amazon** \\ \hline \#Users** & 37,497 & 6,919 \\ \hline \#Items & 129,092 & 28,695 \\ \hline
**Positive** & 6,413,396 & 99,753 \\ \hline
**Negative** & 5,448,693 & 20,581 \\ \hline
**Avg. records per user** & 316.35 & 17.39 \\ \hline \end{tabular}
\end{table}
Table 1. Micro-video and Amazon data statistics.
necessary to each other. Specifically, removing FHA means it is impossible to apply the mask on the implicit head interaction of either multi-heads attention or talking-heads attention. At the same time, removing MO on FHA will cause it to fail to exploit the prior knowledge of labels for historical items and degenerate to even as poor as multi-heads attention or talking-heads attention in the Amazon dataset.
* **Pair-wise interest is more important than disentangling interest**. Removing IDL and IBL will both drop the performance, while removing IBL is more significant. This is because contrastive learning by BPR loss can indeed inject more self-supervised signals, while disentanglement solely tends to repel the dissimilar representations of positive feedback and negative feedback.
### Visualization for Attention Weights of Heads (RQ3)
As illustrated in Eq.(8), our proposed factorization-heads attention can factorize the relation between different feedback, which makes it possible for us to study the attention weights between them. Therefore, we perform visualization on the attention weights between positive and negative heads in Figure 4, where \(h_{1}\) and \(h_{2}\) (defined at (11)) represent heads for source and target behaviors, respectively, with corresponding feedback. From this figure, we can observe that: (1) For the collected Micro-video dataset, users are still willing to watch videos even after they receive the disliked videos. This may be because the negative recommended videos are of low cost for users as they can easily skip the disliked videos, making no significant impact on their later preferred videos; (2) For the e-commerce dataset about Amazon, we can discover that when the source feedback is negative, the probability of target feedback being negative will increase sharply. This may be because the negative purchased items are of high cost in e-commerce for users as it will waste their money, increasing their unsatisfied emotion sharply.
\begin{table}
\begin{tabular}{c|c|c c c c c c c|c|c} \hline \hline \multicolumn{2}{c|}{**Models**} & \multicolumn{1}{c}{**DIN**} & \multicolumn{1}{c}{**Caser**} & \multicolumn{1}{c}{**GRUREC**} & \multicolumn{1}{c}{**DIEN**} & \multicolumn{1}{c}{**SASRec**} & \multicolumn{1}{c}{**THA4Rec**} & \multicolumn{1}{c}{**DFN**} & \multicolumn{1}{c|}{**FeedRec**} & \multicolumn{1}{c|}{**Ours**} & \multicolumn{1}{c}{**Improv.**} \\ \hline \multirow{4}{*}{**Micro-video**} & **AUC** & 0.7345 & 0.8113 & 0.7983 & 0.7446 & 0.8053 & 0.8104 & 0.8342 & 0.8119 & **0.8578** & 2.83\% \\ & **MRR** & 0.5876 & 0.6138 & 0.5927 & 0.5861 & 0.6046 & 0.6080 & 0.6321 & 0.6095 & **0.6568** & 3.91\% \\ & **NDCG** & 0.6876 & 0.7079 & 0.6916 & 0.6861 & 0.7009 & 0.7035 & 0.7222 & 0.7047 & **0.7410** & 2.60\% \\ & **GAUC** & 0.7703 & 0.8211 & 0.8041 & 0.7753 & 0.8120 & 0.8138 & 0.8362 & 0.8180 & **0.8545** & 2.19\% \\ \hline \multirow{4}{*}{**Amazon**} & **AUC** & 0.6595 & 0.7192 & 0.7278 & 0.6688 & 0.6903 & 0.7069 & 0.6998 & 0.7037 & **0.7333** & 0.76\% \\ & **MRR** & 0.4344 & 0.4846 & 0.4901 & 0.4547 & 0.4604 & 0.4599 & 0.4743 & 0.4675 & **0.4980** & 1.61\% \\ & **NDCG** & 0.5669 & 0.6073 & 0.6114 & 0.5832 & 0.5883 & 0.5879 & 0.5990 & 0.5938 & **0.6175** & 1.00\% \\ & **GAUC** & 0.6618 & 0.7245 & 0.7266 & 0.6859 & 0.7029 & 0.7021 & 0.7120 & 0.7079 & **0.7305** & 0.54\% \\ \hline \hline \end{tabular}
\end{table}
Table 2. Overall evaluations for DFAR against baselines under Micro-video and Amazon datasets on four metrics. Here Improv. is the improvement. Bold is the highest result and underline is the second highest result.
Figure 4. Visualization of accumulated attention weights between different heads. Here \(h_{1}\) and \(h_{2}\) represent the heads for the source and target behaviors, respectively (i.e., if the source behavior is negative and target behavior is positive, we have \(h_{1}=0\) and \(h_{2}=1\)). This illustrates our method can factorize and extract the relation between different feedback based on the proposed factorization-heads attention.
Figure 5. AUC performance comparisons under different sequence lengths on the Micro-video and Amazon datasets.
\begin{table}
\begin{tabular}{c|c c c c|c} \hline \hline \multicolumn{2}{c|}{**Dataset**} & \multicolumn{4}{c}{**Micro-video**} \\ \hline
**Methods** & **w/o FHA** & **w/o MO** & **w/o IDL** & **w/o IBL** & **Ours** \\ \hline
**AUC** & 0.8360 & 0.8473 & 0.8475 & 0.8364 & **0.8578** \\ \hline
**MRR** & 0.6198 & 0.6378 & 0.6377 & 0.6324 & **0.6568** \\ \hline
**NDCG** & 0.7127 & 0.7264 & 0.7264 & 0.7212 & **0.7410** \\ \hline
**GAUC** & 0.8319 & 0.8428 & 0.8436 & 0.8283 & **0.8545** \\ \hline
**Dataset** & \multicolumn{4}{c}{**Amazon**} \\ \hline
**AUC** & 0.7133 & 0.7141 & 0.7284 & 0.7137 & **0.7333** \\ \hline
**MRR** & 0.4782 & 0.4883 & 0.4855 & 0.4839 & **0.4980** \\ \hline
**NDCG** & 0.6016 & 0.6095 & 0.6073 & 0.6057 & **0.6175** \\ \hline
**GAUC** & 0.7054 & 0.7137 & 0.7128 & 0.7047 & **0.7305** \\ \hline \hline \end{tabular}
\end{table}
Table 3. Effectiveness study of our proposed components. FHA means factorization-heads attention; MO means label mask operation on heads; IDL means interest disentangling loss on positive and negative representations; IBL means interest BPR loss on positive and negative logits.
### The Impact of Sequence Length (RQ4)
On large-scale online platforms, active users often observe a lot of items and generate very long historical item sequences, while cold-start users are recorded with very short sequences. Long historical item sequences can bring them more information but the problem of gradient vanishing will increase, while short historical item sequence brings limited information and tends to overfit the model. Thus, we divide historical item sequences into five groups based on their lengths and further study how DFAR outperforms the attention-based models under different lengths, under Micro-video and Amazon datasets, as illustrated in Figure 5. From the visualization, we can observe that:
* **DFAR is superior under different sequence lengths**. It is obvious that there is always a significant performance gap between DFAR and other methods. In the Amazon dataset, where the sequence length is relatively short, the AUC performances increase with the growth of sequence length for all methods. This means a longer sequence can bring more information. However, in the Micro-video dataset where the sequence length is relatively long, the performances of all methods improve with the increase of sequence length and reach their peak at around 50-100. But then they all decline with the further increase in length. Most importantly, our DFAR outperforms other methods significantly throughout various sequence lengths.
* **DFAR is stable under different sequence lengths**. DFAR is more stable with the sequence length increasing or decreasing, even into very long or short. In the Amazon dataset, other methods first increase with the sequence length but fluctuate at 15-20 while DFAR increases steadily with the sequence length. In the Micro-video dataset, All methods drop sharply when the sequence length is too short or long, but our DFAR is more stable and still keeps a decent AUC performance at 0.8382.
In summary, our DFAR is superior and robust under both long and short historical item sequences.
## 5. Related Work
**Sequential Recommendation** Sequential Recommendation (Shen et al., 2017) predicts the next interacted item of the given user based on his/her historical items. As the early work, FPMC (Zhou et al., 2018) exploits the Markov chain to capture the transition pattern of historical item sequence in the recommendation. Then some advanced deep learning methods such as RNN (Han et al., 2017; He et al., 2017) and attentive network (Zhou et al., 2018) are applied in recommendation (He et al., 2017; Li et al., 2018; Li et al., 2018; Li et al., 2018) to capture the chronological transition patterns between historical items. While the evolution of RNN-based methods should forward each hidden state one by one and are difficult to parallel, attention-based methods can directly capture the transition patterns among all historical items at any time step. Furthermore, researchers also attempt to leverage convolution neural network (He et al., 2018) to capture the union and point levels sequential pattern in recommendation (Zhou et al., 2018). Compared with CNN-based methods, attention-based methods are more effective for their non-local view of self-attention (Wang et al., 2019). However, the most existing sequential recommendation is based on click behavior. Recently, there have been some methods of achieving sequential recommendations beyond click behaviors (Liu et al., 2019). For example, DFN (Zhou et al., 2019) captures the sequential patterns among click, unclick and dislike behaviors by an internal module for each behavior and an external module to purify noisy feedback under the guidance of precise but sparse feedback. CPRS (Zhou et al., 2019) derives reading satisfaction from the completion of users on certain news to facilitate click-based modeling. Based on them, FeedRec (Zhou et al., 2019) further enhances sequential modeling by a heterogeneous transformer framework to capture the transition patterns between user feedback such as click, dislike, follow, etc. However, these works mainly focus on exploiting the auxiliary feedback to enhance the modeling in the sequential recommendation, which does not consider the most important characteristic - the transition patterns between historical positive and negative feedback. Differently from them, our approach can factorize the transition patterns between different feedback, achieving more accurate modeling for sequential recommendation with both positive and negative feedback. Additionally, our approach extracts the relation between positive and negative feedback at interest level.
**Explainable Attention** Attention methods are popular in many machine learning fields such as recommender systems (Li et al., 2018; Li et al., 2018; Li et al., 2018), computer vision (Han et al., 2017; He et al., 2017; Li et al., 2018; Li et al., 2018) and natural language processing (He et al., 2017; Li et al., 2018), etc. Attention mechanisms are often explainable and have been widely used in deep models to illustrate the learned representation by visualizing the distribution of attention scores or weights under specific inputs (Li et al., 2018; Li et al., 2018; Li et al., 2018). Some explainable attention methods are also generalizable and can be equipped with many backbones. For example, L2X (Li et al., 2018) exploits Gumbel-softmax (Li et al., 2018) for feature selection by instance, with its hard attention design (Zhou et al., 2019). Moreover, VIBI (Li et al., 2018) further propose a feature score constraint in a global prior so as to simplify and purify the explainable representation learning. As self-attention is popular (Li et al., 2018; Li et al., 2018), there is also a work that explains what heads learn and concludes that some redundant heads can be pruned (Zhou et al., 2019). In this work, we propose feedback-aware factorization-heads attention to explicitly capture the transition pattern between positive and negative feedback. The feedback mask matrix in our attention module can be treated as hard attention based on feedback.
## 6. Conclusions and Future Work
In this work, we considered the positive and negative feedback in the historical item sequence for the sequential recommendation, while existing works were mostly click-based and considered solely positive feedback. Such exploration addressed the challenge of current multi-head attention for different feedback interactions in one sequence. More specifically, we first applied talking-heads attention in the sequential recommendation and further proposed feedback-aware factorization-heads attention to explicitly achieve interaction across different heads for self-attention. Secondly, we proposed disentanglement and pair-wise contrastive learning to repel the dissimilar interests and capture the pair-wise relation between positive and negative feedback. In the future, we plan deploy the model in industrial applications to validate online performance.
## Acknowledgment
This work is supported in part by the National Key Research and Development Program of China under 2022YFB3104702, the National Natural Science Foundation of China under 62272262, 61971267, U1936217 and 61972223, BNRist, the Fellowship of China Postdoctoral Science Foundation under 2021TQ0027 and 2022M710006, and the Tsinghua University Guoqiang Institute under 2021GQG1005.
## Appendix A Appendix for Reproducibility
### Pseudocode
```
1defMultiHeadAttention(
2X[n, d_X], # nvectors with dimensionality d_X
3M[m, d_M], # mvectors with dimensionality d_M
4P_[d_X, d_k, h], # learned linear projection to produce queries
5P_[d_M, d_k, h], # learned linear projection to produce values
6P_[d_Y, d_v, h]): # learned linear projection of output
7Q[n, d_k, h] = einsum (X[n, d_X], P_d[d_X, d_k, h])
8K[m, d_k, h] = einsum (M[m, d_M], P_k[d_M, d_k, h])
10V[n, d_v, h] = einsum (M[m, d_M], P_v[d_M, d_v, h])
11L[m, m, h] = einsum (Q[n, d_k, h], K[m, d_k, h]) # logits h==m d_k
13
14W[n, m, h] = softmax (L[n, m, h], reduced_dim =m) # weights
15
16O[n, d_v, h] = einsum (W[n, m, h], V[m, d_v, h]) # h=einsum (Q[n, d_v, h], P_of[d_v, d_v, h]) # output h=m& d_v * d_v
17returnY[n, d_Y]
```
**Listing 1**Pseudocode for Multi-heads Attention
We follow talking-heads attention [25] and present the following notation and pseudocode.
#### a.1.1. **Notation**
In our pseudocode, we follow talking-heads attention [25] and have a notation as below.
* The capital letters represent the variable names, and lower-case letters represent the number of dimensions. Each variable of a tensor is presented with its dimensions. For example, a tensor for an item sequence with batch size \(b\), sequence length \(n\), hidden state \(d\) is written as: X[b, n, d] [25].
* The einsum represents the generalized contractions between tensors without any constraint on their dimension. Its computation process is: (1) Broadcasting each input to have the union of all dimensions, (2) multiplying component-wise, and (3) summing across all dimensions not in the output. The dimensions are identified by the dimension-list annotations on the arguments and on the result instead of being identified by an equation, as in TensorFlow and NumPy. For example, multiplying two matrices is written as: Z[a, c] = einsum (X[a, b], W[b, c]) [25].
#### a.1.2. **Multi-heads Attention**
The pseudocode for multi-heads attention [28] is as shown in Pseudocode 1, where different heads for Q and K do not interact with each other on line 12.
#### a.1.3. **Talking-heads Attention**
The pseudocode for talking-heads attention [25] is as shown in Pseudocode 2, where different heads for Q and K achieve implicit interaction by lines 15 and 18.
#### a.1.4. **Factorization-heads Attention**
The pseudocode for our proposed factorization-heads attention is as shown in Pseudocode 3, where different heads for Q and K achieve explicit interaction by line 16.
```
1defTalkingHeadAttention(
2X[n, d_X], # nvectors with dimensionality d_X
3M[m, d_M], # mvectors with dimensionality d_M
4P_[d_X, d_k, h_k], # learned linear projection to produce queries
5P_[d_M, d_k, h_k], # learned linear projection to produce keys
6P_v[d_M, d_v, h_v], # learned linear projection to produce values
7P_of_d_v, h_v]
8P_[h_k, h], # talking - heads projection for logits
9P_w[h, h_v]): # talking - heads projection for weights
10Q[n, d_k, h_k] = einsum (X[n, d_X], P_of[d_X, d_k, h_k])
11K[m, d_k, h_k] = einsum (M[m, d_M], P_k[d_M, d_k, h_k])
12V[m, d_v, h_v] = einsum (M[m, d_M], P_v[d_M, d_v, h_v])
13J[n, m, h_k] = einsum (Q[n, d_k, h_k], K[m, d_k, h_k]) # dot prod. num* d_k * h_k
14L[n, m, h] = einsum (J[n, m, h,k], P_[h_k, h]) # Talking - heads proj. num* h_k
15W[n, m, h] = softmax (L[n, m, h], reduced_dim=m) # Attention weights
16U[n, h_v] = einsum (V[n, m, h], P_w[h, h_v]) # Talking - heads proj. num* h_v
17O[n, d_v, h_v] = einsum (U[n, m, h_v], V[m, d_v, h_v]) # num* d_v * h_v
18Y[n, d_v] = einsum (Q[n, d_v, h_v], P_o[d_Y, d_v, h_v]) # n_d_v * d_v * h_v
19returnY[n, d_v]
```
**Listing 2**Pseudocode for Talking-heads Attention
#### a.1.5. **Comparison**
From these three Python pseudocodes, we can discover that our factorization-heads attention achieves head interaction at a low cost. The comparison of it with multi-heads attention and talking-heads attention are as below.
* **Comparing with Multi-heads Attention**: our factorization-heads attention incorporates the interaction between different heads with additional four lines at lines 12-14 and 17, which are transpose and reshape operations and with only \(O(1)\) temporal complexity. 2. Footnote 2: [https://stackoverflow.com/questions/58279082/time-complexity-of-numpy-transpose](https://stackoverflow.com/questions/58279082/time-complexity-of-numpy-transpose)
* **Comparing with Talking-heads Attention**: our factorization-heads attention achieves explicit interaction with additional transpose and reshape operations at \(O(1)\) temporal complexity while talking-heads attention achieves implicit interaction with two matrix multiplication operations at \(O(m\times h_{k}\times h)\) and \(O(m\times h\times h_{b})\) temporal complexities 3, respectively.
### Evaluation Metrics
The detailed illustration of adopted evaluation metrics is as follows.
* **AUC**: Randomly selecting one positive item and one negative item, it represents the probability that the predicted score of the positive item is higher than that of the negative item. It tests the model's ability to classify the positive and negative items.
* **GAUC**: It weighs each user's AUC based on his/her test set size. It tests the model's personalized classification ability on each user as recommender systems indeed tend to rank preferred items for users individually.
* **MRR@K**: It is the average of the reciprocal of the first hit item ranking.
* **NDCG@K**: It assigns hit items that rank higher with more weights and thus tests the model's ability to rank the hit items in higher and more confident positions.
### Implementation Details
We implement all the models by a Microsoft 4 TensorFlow 5 framework in Python, which is accessible here 6. We will publish the Micro-video dataset to benefit the community in the future, and the public Amazon dataset is accessible at this website 7.
Footnote 4: [https://github.com/microsoft/recommenders](https://github.com/microsoft/recommenders)
Footnote 5: [https://www.tensorflow.org](https://www.tensorflow.org)
Footnote 6: [https://amonymous.dopen.science/r/DFAE-887B](https://amonymous.dopen.science/r/DFAE-887B)
Footnote 7: [http://momalley.ucsd.edu/data/amazon/index](http://momalley.ucsd.edu/data/amazon/index), 2014.html
The environment is as below.
* Anaconda 3
* Python 3.7.7
* TensorFlow 1.15.0
Besides, for other parameters, we stop the model training with early stop step 2 and leverage the MLP layer sandwiched between two normalization layers as the prediction tower for each model.
### Hyper-parameter Study (RQ5)
We perform hyper-parameter study on the weights for loss of disentanglement and pair-wise contrastive Learning (w.r.t. \(\lambda^{\textit{BPR}}\) and \(\lambda^{D}\) at Eq.(23)) as Figure 6, varying the loss weights from \(10^{-4}\) to \(10^{-1}\). From the figure, we can observe that the AUC performance reaches the peak at \(10^{-3}\) under the Amazon dataset while that reaches the peak at \(10^{-2}\) under the Micro-video dataset. This is because the rating for Amazon is a discrete value, but the playing time for Micro-video is a continuous value. The partition of positive and negative feedback based on continuous value is unclear and thus requires more contrastive learning. Based on the above observation, we finally choose \(10^{-3}\) and \(10^{-2}\) as the best values for the loss weights under Amazon and Micro-video datasets, respectively.
```
defFactorizationHeadAttention( X[n, d_X], # nvectors with dimensionality d_X M[m, d_M], # mvectors with dimensionality d_M P_q[d_X, d_k, h], # learned linear projection to produce queries P_k[d_M, d_k, h], # learned linear projection to produce keys P_v[d_M, d_v, h], # learned linear projection to produce values P_q[d_V, d_v, h]): # learned linear projection of output Q[n, d_k, h] = einsum (X[n, d_X], P_q[d_X, d_k, h]) K[m, d_k, h] = einsum (M[m, d_M], P_k[d_M, d_k, h]) V[m, d_v, h] = einsum (M[m, d_M], P_v[d_M, d_v, h])
12 Q[n, h, d_k] = reshape(transpose(Q, [@, 2, 1]), [n* h, d_k]) # queries h== d_X * d_k K[d_k, h, m] = reshape(transpose(K, [1, 2, 0]), [d_k, h += m]) # keys h== d_M * d_k V[m, d_v, h] = tile(V[n, d_v, h], [1, 1, h]) # values h== d_M * d_v L[n* h, h * m] = einsum (Q[n* h, d_k], K[d_k, h * m]) L[n, h * h, m] = transpose(reshape(L, [n, h * h, m]), [0, 2, 1]) # logits h==m* d_k W[m, m, h * h] = softmax (L[n, m, h * h], reduced_dim=m) # weights O[n, d_v, h * h] = einsum (V[n, m, h * h], V[m, d_v, h * h]) # h==einsum (Q[n, d_v, h * h], P_of[d_Y, d_v, h * h]) # output h==e d_Y * d_v return Y[n, d_Y]
```
Figure 6: AUC performance of different auxiliary loss weights w.r.t \(\lambda^{\textit{BPR}}\) and \(\lambda^{D}\) under Micro-video and Amazon datasets. |
2304.01224 | Optimizing Data Shapley Interaction Calculation from O(2^n) to O(t n^2)
for KNN models | With the rapid growth of data availability and usage, quantifying the added
value of each training data point has become a crucial process in the field of
artificial intelligence. The Shapley values have been recognized as an
effective method for data valuation, enabling efficient training set
summarization, acquisition, and outlier removal. In this paper, we introduce
"STI-KNN", an innovative algorithm that calculates the exact pair-interaction
Shapley values for KNN models in O(t n^2) time, which is a significant
improvement over the O(2^n)$ time complexity of baseline methods. By using
STI-KNN, we can efficiently and accurately evaluate the value of individual
data points, leading to improved training outcomes and ultimately enhancing the
effectiveness of artificial intelligence applications. | Mohamed Karim Belaid, Dorra El Mekki, Maximilian Rabus, Eyke Hüllermeier | 2023-04-02T06:15:19Z | http://arxiv.org/abs/2304.01224v1 | Optimizing Data Shapley Interaction Calculation from \(\mathcal{O}(2^{n})\) to \(\mathcal{O}(tn^{2})\) for KNN models
###### Abstract
With the rapid growth of data availability and usage, quantifying the added value of each training data point has become a crucial process in the field of artificial intelligence. The Shapley values have been recognized as an effective method for data valuation, enabling efficient training set summarization, acquisition, and outlier removal. In this paper, we introduce "STI-KNN", an innovative algorithm that calculates the exact pair-interaction Shapley values for KNN models in \(\mathcal{O}(tn^{2})\) time, which is a significant improvement over the \(\mathcal{O}(2^{n})\) time complexity of baseline methods. By using STI-KNN, we can efficiently and accurately evaluate the value of individual data points, leading to improved training outcomes and ultimately enhancing the effectiveness of artificial intelligence applications.
Keywords:Data Valuation Shapley Value Exact values KNN model.
## 1 Introduction
Data valuation, in data science, is the process that aims to quantify the added value of a train data point given a specific test dataset. Considering that data points are sometimes expensive to acquire or difficult to label, data valuation could help in making the right decision while investing time and effort in expanding the training set, removing mislabeled points, or summarizing the training set.
#### 1.0.1 Shapley vs. LOO method
Leave-one-out (LOO) is a known method for estimating the contribution of an element [6]. Let \(N\) be the training set. LOO estimates the contribution of a data point \(i\) as the difference between the test score after training on \(N\) and the test score after training on \(N\setminus\{i\}\). On the other hand, the Shapley method considers all subsets \(S\subseteq N\). The Shapley method estimates the contribution of train point \(i\) as the average difference between the test score when training on \(S\) and the test score after training on \(S\setminus\{i\}\). It was proven in related work that the Shapley method is more accurate in estimating the contribution of a point than LOO [6, 2, 3].
In previous works, researchers tackle data valuation by introducing several methods to estimate the value of train data points [3, 5, 7]. Cited methods are based on the Shapley values [9]. Consequently, they disregard the interaction term.
#### 1.0.1 KNN-Shapley
[5] computes the exact Shapley values if the model is a K-Nearest-Neighbor (KNN). Despite that KNN is a simple ML model, it is still possible to work on complex tasks like image classification thanks to the usage of pre-trained models: given a pre-trained feature extractor for images that is independent of the training set to valuate, the KNN model is trained each time on the extracted feature rather than the initial image. Moreover, Jia et al. proved that the Shapley values proposed by the KNN model are transferable to other models like Gradient Boosting. With an execution time of \(\mathcal{O}(n\log n)\)[5], KNN-Shapley is the fastest data valuation algorithm while it scales to complex tasks. Jia et al. reduced the complexity of the Shapley values by defining the valuation function as the likelihood of the right label.
Our contribution is the following: We propose, **STI-KNN**, the first algorithm that calculates the exact pair-interaction Shapley values in \(\mathcal{O}(tn^{2})\) rather than \(\mathcal{O}(2^{n})\). STI-KNN is the first algorithm that allows studying the exact interaction on large real-world datasets. This research paper is the first to consider two disjoint fields: Data valuation and Interaction in Explainable AI. Finally, we study various cases of positive and negative data interactions using STI-KNN.
## 2 Setting and Notation
### Valuation function of KNN model
\(N\) is the training set of size \(n\).
\(v:N\rightarrow\mathbb{R}\) is a valuation function that trains on \(S\subseteq N\) and returns the test score. It varies with the model and the metric.
Equations (1) and (2) are used in literature as the valuation function of KNN models [5, 6]. The test score is defined as the likelihood of the right label.
\[v(S)=\frac{1}{t}\sum_{y_{test}\in T}u_{y_{test}}(S) \tag{1}\]
\[u_{y_{test}}(S)=\frac{1}{k}\sum_{i=1}^{min(k,s)}\mathds{1}_{[y_{i}=y_{test}]} \tag{2}\]
\(T\) is the test set of size \(t\).
\(k\) is the parameter of KNN.
\(y_{test}\in T\) is the label of a test data point.
\(y_{i}\in S\) is the train point label, sorted starting from the nearest to \(y_{test}\).
\(\mathds{1}_{[y_{i}=y_{test}]}\) returns 1 if both train and test labels are equal, otherwise returns 0.
**Example** Consider a KNN model with parameter \(k=3\), \(t=1\) test point denoted \(y_{test}\), \(n=4\) train points sorted starting from the closest to the test point, \(N=\{1,2,3,4\}\), See Figure 1.
\(v(N)=2/3\)
Proof: \(v(N)=u_{y_{test}}(N)\) because the test score \(v(N)\) is the average over the test scores on each \(y_{test}\) but in this example, there is only one test point.
\(u_{y_{test}}(N)=u_{y_{test}}(\{1,2,3\})\) because the KNN model considers only the \(k=3\) closest points. Finally, there are only two data points of the same class
\(u_{y_{test}}(\{1,2,3\})=2/k=2/3\)
The calculation of the Shapley values requires training and testing the model with various training sets. In the following examples, we illustrate the calculation of the valuation function for alternative training sets: \(u_{y_{test}}(\{1\})=1/3\)
\(u_{y_{test}}(\{2\})=0/3\)
\(u_{y_{test}}(\{1,3,4\})=3/3\)
### Shapley Taylor Interaction \(\mathcal{O}(2^{n})\)
\(\phi_{ij}\) is the Shapley interaction value between data points \(i\) and \(j\)[10]. It is the average interaction over different training sets \(S\subseteq N\).
\[\phi_{ij}(v)=\frac{2}{n}\sum_{S\subseteq N\setminus\{i,j\}}\frac{1}{\binom{n -1}{s}}(v(S\cup\{i,j\})-v(S\cup\{i\})-v(S\cup\{j\})+v(S)) \tag{3}\]
#### 2.2.1 Example of a simple interaction
Consider the simple dataset in Figure 2. Let \(k=2\) and \(N=\{1,2,3,4\}\). To calculate \(\phi_{1,2}\), the interaction between \(i=1\) and \(j=2\), we consider all subsets \(S\subseteq N\setminus\{i,j\}\) and calculate the term \(\phi_{i,j}\)
\(v(S\cup\{i,j\})-v(S\cup\{i\})-v(S\cup\{j\})\)
For \(S=\{3,4\}\):
\(v(S\cup\{i,j\})-v(S\cup\{i\})-v(S\cup\{j\})+v(S)=1/2-1/2-0+1/2=1/2\)
Figure 1: Example of a simple dataset to explain the test score calculation for KNN model.
#### 3.1.1 NP Complexity
The Calculation of the matrix of interactions \(\phi_{ij}\), \(i\neq j\) is challenging because Equation (3) requires \(\mathcal{O}(2^{n})\) trainings of the model, each time over a subset \(S\subseteq N\). Therefore, there are no real-world applications at this level [6]. We propose to simplify \(\phi_{ij}\) for a specific \(v(S)\).
#### 3.1.2 Main term
The calculation of the main terms \(\phi_{ii}\) can be done in polynomial time and does not represent an issue. For this reason, the focus of this paper is the interaction term \(\phi_{ij}\).
\[\phi_{ii}=v(i)-v(\emptyset) \tag{4}\]
#### 3.1.3 Acquired linearity relative to the test set
Equation (1) involves a linearity relative to the test set: \(\phi_{ij}(v)=\frac{1}{t}\sum_{y_{test}\in T}\phi_{ij}(u_{y_{test}})\). Moreover, Equation (2) is linear for train sets of size \(s\leq k\), which would allow simplifying the Shapley Interaction equation.
#### 3.1.4 Simple notation
In the following sections, we will use \(u_{y_{test}}(S)\) with only a singleton. Therefore, we simplify the notation by discarding the curved parenthesis. Note that in this specific case, the sum is discarded and the function returns only 0 or \(1/k\) to express weather train point \(i\) has the same label as test point \(y_{test}\).
\[u_{y_{test}}(i)=\frac{\mathds{1}_{[y_{i}=y_{test}]}}{k} \tag{5}\]
Figure 2: Example of a simple dataset to explain a pair interaction between train data points.
## 3 Method: STI-KNN \(\mathcal{O}(tn^{2})\)
We study the interaction between data points by calculating the pair-interaction terms using STI [10]. The complexity of the STI algorithm (or any Shapley-based algorithm) is \(\mathcal{O}(2^{n})\). This hinders researchers from applying this xAI algorithm to real-world datasets. We propose STI-KNN, a faster exact calculation method adapted to the KNN model. STI-KNN is a recursive method that calculates the STI in an \(\mathcal{O}(tn^{2})\) execution time. The proof is in Appendix 0.A
### Algorithm
```
0: Training data \(N=\{(x_{i},y_{i})\}_{i=1}^{n}\), test data \(T=\{(x_{test,p},y_{test,p})\}_{p=1}^{t}\),
0: Valuation function \(u:N\rightarrow\mathbb{R}\) as defined in Equation (5), \(k\) is the parameter of KNN. output A matrix representing the pair interaction Shapley values \(\{\phi_{i,j}\}_{1\leq i,j\leq n}\)
1:function STI-KNN-one-test\((x_{test},y_{test})\)
2:\((\alpha_{1},...,\alpha_{n})\leftarrow\) Index sorting of training data starting from the closest to \(x_{test}\)
3:\(\phi_{\alpha_{n-1},\alpha_{n}}(u_{y_{test}})\leftarrow-\frac{2(n-k)}{n(n-1)} u_{y_{test}}(\alpha_{n})\)
4:for\(j\gets n\) to \(1\)do
5:if\(j>k+1\)then
6:\(\phi_{\alpha_{j-2},\alpha_{j-1}}(u)\leftarrow\phi_{\alpha_{j-1},\alpha_{j}}(u _{y_{test}})+\frac{2(j-k-1)}{(j-2)(j-1)}(u_{y_{test}}(\alpha_{j})-u_{y_{test}}( \alpha_{j-1}))\)
7:else
8:\(\phi_{\alpha_{j-2},\alpha_{j-1}}(u)\leftarrow\phi_{\alpha_{j-1},\alpha_{j}}(u _{y_{test}})\)
9:endif
10:endfor
11:for\(j\gets 3\) to \(n\)do
12:for\(i\gets j-2\) to \(1\)do
13:\(\phi_{\alpha_{i},\alpha_{j}}(u_{y_{test}})\leftarrow\phi_{\alpha_{i+1},\alpha_ {j}}(u_{y_{test}})\)
14:endfor
15:endfor
16:return\(\{\phi_{i,j}(u_{y_{test}})\}_{1\leq i,j\leq n}\) The pair-interaction matrix relative to one test point
17:endfunction
18:for\(p\gets 1\) to \(t\)do
19:\(\{\phi_{i,j}(u_{y_{test,p}})\}_{1\leq i,j\leq n}\leftarrow\) STI-KNN-one-test\((x_{test,p},y_{test,p})\)
20:endfor
21:\(\{\phi_{i,j}\}_{1\leq i,j\leq n}\leftarrow\)mean over \(p:\{\phi_{i,j}(u_{y_{test,p}})\}_{1\leq i,j\leq n}\) endmain
```
**Algorithm 1** STI-KNN for calculating the matrix of pair interaction Shapley values for a KNN classifier.
#### 3.1.1 Explaining the STI-KNN algorithm
Consider the valuation function of KNN models defined in Equations (1) and (2). The Shapley-Taylor Interaction index (\(\phi_{ij}\)) can be calculated recursively using Algorithm 1.
We explain first the function **STI-KNN-one-test** that considers one test point and a train set \(N=\{1,...,n\}\). This function will return \(\phi_{ij}(u_{y_{test}})\) the pair interaction matrix considering only one test point.
Line 2: the train points are sorted from \(\alpha_{1}\) to \(\alpha_{n}\), starting from the closest to the test point.
Line 3: calculates the pair interaction between the two points that are the most far away from \(x_{test}\).
\[\phi_{n-1,n}(u_{y_{test}})=-\frac{2(n-k)}{n(n-1)}u_{y_{test}}(\alpha_{n}) \tag{6}\]
Line 4: The loop allows the calculation of the superdiagonal of the matrix recursively.
\[\phi_{j-2,j-1}(u_{y_{test}})=\begin{cases}\phi_{j-1,j}(u_{y_{test}})+\frac{2(j- k-1)}{(j-2)(j-1)}(u_{y_{test}}(\alpha_{j})-u_{y_{test}}(\alpha_{j-1})),&\text{if $j>k+1$},\\ \phi_{j-1,j}(u_{y_{test}}),&\text{otherwise}.\end{cases} \tag{7}\]
Line 6: The value of pair interaction changes during the recursive calculation if \(u_{y_{test}}(j)\neq u_{y_{test}}(j-1)\), i.e., when the label of data point \(\alpha_{j}\) differ from \(\alpha_{j-1}\).
Line 8: The pair-interactions between the train points that are too close to the test point do not differ. This can be explained by the fact that the KNN model does not differentiate them.
Line 11: The elements of the superdiagonal were calculated in the steps before. In the nested loop, we calculate the remaining elements by repeating the superdiagonal. Indeed, the elements of each column of the upper triangle (without the diagonal) are equal, i.e.,
\(\forall a,b\) with \(a<j,b<j\)
\[\phi_{aj}(u_{y_{test}})=\phi_{bj}(u_{y_{test}}) \tag{8}\]
Line 16: The function returns a matrix of pair interaction considering only one test point.
Line 18: All matrices are calculated and saved.
Line 21: Equations (6) to (8) calculate \(\phi_{ij}(u)\) index for one test point. To obtain the pair interaction matrix \(\phi_{ij}(v)\) for all test points \(T\), we average over all calculated \(\phi_{ij}(u_{y_{test}})\)
\[\phi_{ij}(v)=\frac{1}{t}\sum_{y_{test}\in T}\phi_{ij}(u_{y_{test}}) \tag{9}\]
Proof: See Appendix A
### Analysis of the Data Interaction matrix
The STI indices are symmetric. Therefore, we study only the upper triangle, i.e., \(i<j\).
#### 4.2.2 Unexpected Independence property
The interaction \(\phi_{ij}(u_{y_{test}})\) is independent of point \(i\) in the case of one test point (demonstrated in Equation (8)). This is explained by the fact that KNN does not consider each point individually, but it considers only the order in which the train points come. Nevertheless, once we average over multiple test points, \(\phi_{ij}\) becomes dependent on both \(i\) and \(j\).
#### 4.2.3 The parameter \(k\) has a negligible effect on the pair-interaction matrix
Choosing to work with KNN as a surrogate model did speed up the computation. But, on the other hand, it did introduce another parameter: \(k\). We prove that, in practice, \(k\) does not change the overall shape of the interaction matrix. We conduct an empirical experiment based on 16 simple datasets described in Appendix C. We consider the following range for the \(k\) parameters: \(3\leq k\leq 20\). For each \(k_{1},k_{2}\) in the selected range, we find that the Pearson's correlation between the two STI-KNN matrices is each time higher than 0.99. This involves an insignificant change between pair-interaction matrices. Moreover, visual comparison of the matrices does not reveal any difference, see Appendix B.
#### 4.2.4 Complexity
\(n\) is the training set size. \(t\) is the testing set size. The complexity of the function **STI-KNN-one-test** (Line 1), is \(\mathcal{O}(n^{2})\) because, first, it sorts the training points in \(\mathcal{O}(n\log n)\) (Line 2), second, it calculates recursively the super-diagonal in \(\mathcal{O}(n)\)(Line 4), and third it repeats the values in \(\mathcal{O}(n^{2})\). The main script loop through the testing set and apply the function in \(\mathcal{O}(tn^{2})\) (Line 18), then it averages over the list of matrices by reducing the dimensions from (n,n,t) to (n,n) which costs \(\mathcal{O}(tn^{2})\) (Line 21). Thus, the complexity is \(\mathcal{O}(tn^{2})\).
#### 4.2.5 The baseline algorithm's complexity considering \(t\).
The baseline algorithm calculates the simple Shapley values, i.e., a one-dimensional array of Shapley values representing one value per train point. It was declared with a complexity of \(\mathcal{O}(n\log n)\) without considering the size of the testing set. While the baseline algorithm does sort the train points each time with respect to a specific test point resulting in a complexity of \(\mathcal{O}(tn\log n)\).
#### 4.2.6 Effect of \(t\) on the complexity.
Depending on the use case, we can consider \(t\ll n\) then the complexity of the baseline is again \(\mathcal{O}(n\log n)\) and the complexity of STI-KNN is \(\mathcal{O}(n^{2})\). On the other hand, \(t\) could have a significant effect on the time complexity, especially when using an 80/20 train test split as recommended in literature [1]. Then \(t\sim n\) and the baseline complexity becomes \(\mathcal{O}(n^{2}logn)\) while the complexity of STI-KNN becomes \(\mathcal{O}(n^{3})\).
#### 4.2.7 The STI-KNN values are approximately centered
i.e., \(mean(\{\phi_{ij}\})\approx 0\)
Proof: Consider the efficiency axiom fulfilled by STI which states that the sum of the values is equal to \(a_{test}\), the test accuracy when trained on the entire train set, i.e., \(\sum\phi_{ij}=a_{test}\). Thus,
\(mean(\{\phi_{ij}\})=\frac{1}{n^{2}}\sum\phi_{ij}=\frac{a_{test}}{n^{2}}\)
Given \(0<a_{test}\leq 1\) and \(n\gg 1\), \(\frac{a_{test}}{n^{2}}\approx 0\)
#### 3.3.2 The main terms are always positive
Proof.: Consider Equation (4): \(\phi_{ii}=v(i)-v(\emptyset)\). \(v(i)\) is defined as the likelihood [5], then it is always positive, and \(v(\emptyset)=0\).
#### 3.3.3 Similar pair interaction algorithms
The obtained result for STI could be applied to SII [4], a similar pair interaction algorithm. The only difference would be in the coefficient. For example for SII: \(\phi_{n-1,n}(u_{y_{test}})=-\frac{1}{n-1}u_{y_{test}}(\alpha_{n})\).
We further discuss the visual interpretation of the STI-KNN matrix in Section 4.
## 4 Discussion: Example of Data Interaction matrices
In this section, we analyze different types of interaction. We select the _Circle dataset_[8]. Its input features are 2D points and it is a binary classification task. The distribution of the two classes represents two concentric circles, See Figure 3. The two classes are balanced. Each class contains 300 generated points.
In the matrix, the points are first sorted by their class (0 or 1). Within each class, the points are sorted based on their input feature \(x_{1}\), and then by \(x_{2}\). For example, the point at index 0 is a blue point (class 0) with the smallest \(x_{1}\) value.
#### 4.0.1 In-class vs. Out-of-class interaction.
We observe that points in the same group heavily interact (negatively), while pairs of points formed by both groups almost do not interact. Clusters are clearly visible in the interaction matrix. A pair interaction Shapley value reflects the contribution of a training point to the test score as defined initially in the valuation function. This value is the result of many factors like the correctness of the input feature (is the point an outlier?), the correctness of the target class (is the point mislabeled?), the redundancy of the training points, etc. We can emphasize one of these effects with the following variation of the circle dataset.
Figure 3: Example of interaction with a balanced dataset.
Redundancy decreases in-class interaction.Consider one training point \(P_{1}\) with a positive Shapley value. With a redundant point \(P_{2}\), both points will have the same Shapley value (symmetry axiom of the Shapley values).
The sum of the interaction matrix is the test accuracy (efficiency axiom of the STI). Without loss of generality, suppose the test accuracy is equal to 1. By adding redundant training points and having the same accuracy, the individual contribution of each point or each pair of points will decrease. Figure 4 illustrates this effect with fewer blue points and an accuracy equal to 1. The removed blue points are not perfectly overlapping (perfectly symmetric) but considering the KNN algorithm, the distance is not significant.
Mislabeled points behave like the opposite class.Noisy points or mislabeled points like in Figure 5 tend to behave differently compared to the majority of the points in the same class. The interaction matrix in Figure 5 (right) helps to identify mislabeled points as their pattern corresponds more to the opposite class.
Figure 4: Example of interaction with an unbalanced dataset.
Figure 5: Example of interaction with mislabeled training points.
## 5 Conclusions and Future Work
The considerable value of data creation in industries such as medicine and automotive has led to an increased focus on data generation, crowdsourcing, and simulation techniques. These latter provide a variety of benefits such as improving decision-making and enhancing efficiency. On the other hand, these techniques raise the urgent need for data valuation to confirm the value of data and its role in business and society. The value of data can be difficult to quantify and is often influenced by factors such as quality, relevance, and accessibility. Thus, it is crucial to develop methods for accurately valuating data and maximizing its contribution to AI models.
This paper bridges the gap between Shapley interaction and data valuation by introducing STI-KNN, the first algorithm that calculates the exact pair-interaction Shapley values with a complexity of \(\mathcal{O}(tn^{2})\) rather than \(\mathcal{O}(2^{n})\).
|
2303.12599 | Stability approach to torsion pairs on abelian categories | In this paper we introduce a local-refinement procedure to investigate
stability data on an abelian category, and provide a sufficient and necessary
condition for a stability data to be finest. We classify all the finest
stability data for the categories of coherent sheaves over certain weighted
projective curves, including the classical projective line, smooth elliptic
curves and certain weighted projective lines. As applications, we obtain a
classification of torsion pairs for these categories via stability data
approach. As a by-product, a new proof for the classification of torsion pairs
in any tube category is also provided. | Mingfa Chen, Yanan Lin, Shiquan Ruan | 2023-03-22T14:37:38Z | http://arxiv.org/abs/2303.12599v1 | # Stability approach to torsion pairs on abelian categories
###### Abstract.
In this paper we introduce a local-refinement procedure to investigate stability data on an abelian category, and provide a sufficient and necessary condition for a stability data to be finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and certain weighted projective lines. As applications, we obtain a classification of torsion pairs for these categories via stability data approach. As a by-product, a new proof for the classification of torsion pairs in any tube category is also provided.
Key words and phrases:Stability data; torsion pair; weighted projective line; elliptic curve; tube category 2020 Mathematics Subject Classification: 18E40, 18E10, 16G20, 16G70, 14F06 \({}^{*}\) the corresponding author
## 1. Introduction
Torsion pairs in abelian categories were introduced by Dickson [14], which become increasingly important in a wide range of research areas. Mizuno and Thomas [24] found a close relationship between \(c\)-sortable elements of a Coxeter group and torsion pairs, in terms of the representation theory of preprojective algebras, and described the cofinite torsion classes in the context of the Coxeter group. Tattar [34] defined torsion pairs for quasi-abelian categories and gave several characterizations, he showed that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories.
The wall and chamber structure of a module category was introduced by Bridgeland in [9] to give an algebraic interpretation of scattering diagrams studied in mirror symmetry by Gross, Hacking, Keel and Kontsevich, see [19]. It is shown in [11] that all functorially finite torsion classes of an algebra can be obtained from its wall and chamber structure. Asai [2] proved that chambers coincide with the so-called TF equivalence classes (defined via numerical torsion pairs) by using Koenig-Yang correspondences in silting theory. Research highlights about torsion pairs are also studied in [4, 6, 7, 15, 20, 26].
The notion of stability data arising from Geometric Invariant Theory was introduced by Mumford [25] in order to construct the moduli spaces of vector bundles on an algebraic curve. This new approach was adapted by a great deal of mathematicians to different branches of mathematics. Such is the case of Schofield, who did it for quiver representations in [31]; King, for representation of finite dimensional algebras in [21]; Rudakov, for abelian categories in [30]; and Bridgeland, for triangulated categories in [8]. The stability data is generalized to t-stability in [17], which provides an effective approach to classify the bounded t-structures on the derived categories of coherent sheaves on the classical projective line and on an smooth elliptic curves.
The stability data on an abelian category has a close relation with torsion theory. Namely, for an abelian category \(\mathcal{A}\), any torsion pair gives a stability data; and for any stability data on \(\mathcal{A}\), there exists a family of torsion pairs, c.f. [12]. More precisely, any stability data induces a chain of torsion classes in \(\mathcal{A}\). A finite non-refinable increasing chain of torsion classes starting with the zero class and ending in \(\mathcal{A}\) is called a maximal green sequence in \(\mathcal{A}\). Brustle-Smith-Treffinger [12] characterized which stability data induce maximal green sequences in \(\mathcal{A}\).
The stability data have played a key role in the calculations of Donaldson-Thomas invariants, which have deep implications in algebraic geometry and theoretical physics. As is described in [10], generating functions for Donaldson-Thomas invariants can be deduced from certain factorisation of a distinguished element \(e_{\mathcal{A}}\) in the completed Hall algebra associated to \(\mathcal{A}\) via an integration map. The equalities between different factorisations of \(e_{\mathcal{A}}\) induced by distinct stability data are known under the name of wall-crossing formulas.
In the present paper we study the finest stability data on abelian categories and apply them to the classification of torsion pairs. We define a partial ordering for the set of stability data on an abelian category, minimal elements with respect to this partial ordering will be called the finest stability data. We give a sufficient and necessary condition to determine when a stability data is finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and the weighted projective line of weight type (2). As an application, we obtain the classification of torsion pairs for these categories. Moreover, for a nilpotent representation category of a cyclic quiver (i.e., a tube category), Baur-Buan-Marsh [5] classified the torsion pairs by using maximal rigid objects in the extension of the tube category. We provide a new proof via stability data approach.
This paper is organized as follows. In Section 2, we recall the definition and some basic results of stability data on an abelian category \(\mathcal{A}\) in the sense of [17], and establish the connections between stability data and torsion pairs. In Section 3, we introduce a partial order relation on the set of stability data. We describe the finest stability data, and introduce a procedure to refine the stability data locally. In Section 4, we investigate the finest stability data on a tube category, and obtain the classification of torsion pairs via the stability data approach. In Section 5, we show that each stability data can be refined to a finest one for tame hereditary algebras. The classification of finest stability data for the projective line and elliptic curves are given in Section 6 and Section 7 respectively. In the final Section 8, we investigate the stability data for weighted projective lines. In particular, we obtain classification results for finest stability data and torsion pairs for weight type (2).
_Notation._ Throughout this paper, let \(\mathbf{k}\) be an algebraically closed field, let \(\mathcal{A}\) be an abelian category. All subcategories are assumed to be full and closed under isomorphisms. For \(X,Y\in\mathcal{A}\), we simply denote \(\operatorname{Hom}\left(X,Y\right):=\operatorname{Hom}_{\mathcal{A}}(X,Y)\) and \(\operatorname{Ext}^{i}(X,Y):=\operatorname{Ext}^{i}_{\mathcal{A}}(X,Y)\) for \(i\geq 1\). Given a set \(\mathcal{S}\) of objects in \(\mathcal{A}\), we write \(\langle\mathcal{S}\rangle\) for the subcategory of \(\mathcal{A}\) generated by the objects in \(\mathcal{S}\) closed _under extensions and direct summands_, and denote by add \(\mathcal{S}\) the subcategory of \(\mathcal{A}\) consisting of direct summands of finite direct sums of objects in \(\mathcal{S}\). For two subcategories \(\mathcal{T},\mathcal{F}\) in \(\mathcal{A}\), we denote \(\mathcal{T}^{\perp}:=\{Y\in\mathcal{A}\mid\operatorname{Hom}\left(X,Y\right)= 0,\forall\ X\in\mathcal{T}\}\) and \({}^{\perp}\mathcal{F}:=\{X\in\mathcal{A}\mid\operatorname{Hom}\left(X,Y\right) =0,\forall\ Y\in\mathcal{F}\}\). For any \(n\in\mathbb{N}\), the set \(\Phi=\{i\,|\,1\leq i\leq n\}\) is always viewed as a linearly ordered set in the sense that \(1<2<\cdots<n\).
## 2. Preliminary
In this section, we recall the definitions of stability data and torsion pair for an abelian category, and explain the close relations between them.
### Stability data
The notion of stability data on an abelian category \(\mathcal{A}\) is introduced by Gorodentsev-Kuleshov-Rudakov in [17]. The most important feature of a stability data is the fact that they create a distinguished subclass of objects in \(\mathcal{A}\) called semistable factors.
**Definition 2.1**.: ([17, Def. 2.4]) _Suppose that \(\mathcal{A}\) is an abelian category, \(\Phi\) is a linearly ordered set, and an extension-closed subcategory \(\Pi_{\varphi}\subset\mathcal{A}\) is given for every \(\varphi\in\Phi\). A pair \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is called a stability data if_
1. \(\operatorname{Hom}(\Pi_{\varphi^{\prime}},\Pi_{\varphi^{\prime\prime}})=0\) _for all_ \(\varphi^{\prime}>\varphi^{\prime\prime}\) _in_ \(\Phi\)_;_
2. _every non-zero object_ \(X\in\mathcal{A}\) _has a Harder-Narasimhan filtration_ (2.1) _with non-zero factors_ \(A_{i}=X_{i}/X_{i-1}\in\Pi_{\varphi_{i}}\) _and strictly decreasing_ \(\varphi_{i}>\varphi_{i+1}\)_._
The categories \(\Pi_{\varphi}\) are called the semistable subcategories of the stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\). The nonzero objects in \(\Pi_{\varphi}\) are said to be semistable of _phase_\(\varphi\), while the minimal objects are said to be stable. The filtration (2.1) is called the Harder-Narasimhan filtration (_HN-filtration_ for short) of \(X\), which is unique up to isomorphism. The factors \(A_{i}\)'s are called the _HN-factors_ of \(X\), where \(A_{1}\) is called the _maximal semistable subobject_ of \(X\), and \(A_{n}\) is called the _minimal semistable quotient object_ of \(X\). Define \(\boldsymbol{\phi}^{+}(X):=\varphi_{1}\) and \(\boldsymbol{\phi}^{-}(X):=\varphi_{n}\). Then \(X\in\Pi_{\varphi}\) if and only if \(\boldsymbol{\phi}^{+}(X)=\boldsymbol{\phi}^{-}(X)=\varphi=:\boldsymbol{\phi}(X)\). For simplification, we often omit the trivial exact sequence \(0\to X_{0}\stackrel{{ 0}}{{\longrightarrow}}X_{1}\stackrel{{ q_{1}}}{{ \longrightarrow}}A_{1}\to 0\) in (2.1) in the following.
For an extension-closed subcategory \(\mathcal{B}\) of \(\mathcal{A}\), if there exists \(\Pi_{\varphi}\subset\mathcal{B}\) for any \(\varphi\in\Phi\) satisfying Definition 2.1 (1) and (2.1) for any non-zero object \(X\in\mathcal{B}\), then we call the pair \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) a _local stability data_ on \(\mathcal{B}\).
The following observation is important for our later use.
**Lemma 2.2**.: _Keep the notations in (2.1), then for each \(i\),_
1. \(X_{i+1}/X_{i}\) _is the maximal semistable subobject of_ \(X/X_{i}\)_;_
2. \(A_{i}\) _is the minimal semistable quotient object of_ \(X_{i}\)_._
Proof.: We only prove the second statement, since the proof for the first one is dual. Assume there is a non-zero semistable quotient object \(Y\) of \(X_{i}\) with \(\boldsymbol{\phi}(Y)<\boldsymbol{\phi}(A_{i})\). Then \(\operatorname{Hom}\left(A_{j},Y\right)=0\) for any \(j\leq i\) since \(\boldsymbol{\phi}(Y)<\boldsymbol{\phi}(A_{i})\leq\boldsymbol{\phi}(A_{j})\). Now applying \(\operatorname{Hom}\left(-,Y\right)\) to the exact sequence \(0\to X_{j-1}\stackrel{{ p_{j-1}}}{{\longrightarrow}}X_{j} \stackrel{{ q_{j}}}{{\longrightarrow}}A_{j}\to 0\) for \(1<j\leq i\), we obtain that \(\operatorname{Hom}\left(X_{j},Y\right)=0\) for any \(j\leq i\). But \(Y\) is a quotient object of \(X_{i}\), we have \(\operatorname{Hom}\left(X_{i},Y\right)\neq 0\), a contradiction.
**Remark 2.3**.: We can obtain the HN-filtration of \(X\) by considering the maximal semistable subobject of \(X/X_{i}\) step by step for \(i=0,1,\cdots,n\), or by considering the minimal semistable quotient object of \(X_{i}\) step by step for \(i=n,n-1,\cdots,0\).
The following lemma shows that each semistable subcategory is closed direct summands.
**Lemma 2.4**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}\). Then_
\[\Pi_{\varphi}=\big{(}\bigcap_{\psi>\varphi}\Pi_{\psi}^{\perp}\big{)}\cap\big{(} \bigcap_{\psi<\varphi}{}^{\perp}\Pi_{\psi}\big{)},\]
_where_
\[\Pi_{\psi}^{\perp}=\{X\in\mathcal{A}\,|\operatorname{Hom}\left(A,X\right)=0, \;\forall\,A\in\Pi_{\psi}\}\quad\text{and}\quad{}^{\perp}\Pi_{\psi}=\{X\in \mathcal{A}\,|\operatorname{Hom}\left(X,A\right)=0,\;\forall\,A\in\Pi_{\psi}\}.\]
_Consequently, \(\Pi_{\varphi}\) is closed under direct summands._
Proof.: By Definition 2.1 (1), we have \(\operatorname{Hom}(\Pi_{\varphi^{\prime}},\Pi_{\varphi^{\prime\prime}})=0\) for all \(\varphi^{\prime}>\varphi^{\prime\prime}\) in \(\Phi\). Hence \(\Pi_{\varphi}\subseteq\Pi_{\psi}^{\perp}\) for any \(\psi>\varphi\), and \(\Pi_{\varphi}\subseteq{}^{\perp}\Pi_{\psi}\) for any \(\psi<\varphi\). Therefore, \(\Pi_{\varphi}\subseteq\big{(}\bigcap_{\psi>\varphi}\Pi_{\psi}^{\perp}\big{)} \cap\big{(}\bigcap_{\psi<\varphi}{}^{\perp}\Pi_{\psi}\big{)}\).
On the other hand, for any \(X\in\big{(}\bigcap_{\psi>\varphi}\Pi_{\psi}^{\perp}\big{)}\cap\big{(}\bigcap_{ \psi<\varphi}{}^{\perp}\Pi_{\psi}\big{)}\). Since \(\operatorname{Hom}\left(\Pi_{\psi},X\right)=0\) for any \(\psi>\varphi\), we have \(\boldsymbol{\phi}^{+}(X)\leq\varphi\). Similarly, since \(\operatorname{Hom}\left(X,\Pi_{\psi}\right)=0\) for any \(\psi<\varphi\), we have \(\boldsymbol{\phi}^{-}(X)\geq\varphi\). Hence \(\boldsymbol{\phi}^{-}(X)=\boldsymbol{\phi}^{+}(X)=\varphi\), and then \(X\in\Pi_{\varphi}\). Therefore, \(\Pi_{\varphi}\supseteq\big{(}\bigcap_{\psi>\varphi}\Pi_{\psi}^{\perp}\big{)} \cap\big{(}\bigcap_{\psi<\varphi}{}^{\perp}\Pi_{\psi}\big{)}\). We are done.
Given a set \(\mathcal{S}\) of objects in \(\mathcal{A}\), recall that \(\langle\mathcal{S}\rangle\) denotes the subcategory of \(\mathcal{A}\) generated by the objects in \(\mathcal{S}\) closed under extensions and direct summands. For any interval \(I\subseteq\Phi\), define \(\Pi_{I}:=\langle\Pi_{\varphi}\,|\,\varphi\in I\rangle\) to be the subcategory of \(\mathcal{A}\) generated by \(\Pi_{\varphi}\) for all \(\varphi\in I\). Note that non-zero objects in \(\Pi_{I}\) consists precisely of those objects \(X\in\mathcal{A}\) which satisfy \(\boldsymbol{\phi}^{\pm}(X)\in I\).
### Torsion pair
The notion of torsion pair in an abelian category was first introduced by Dickson [14], generalizing properties of abelian groups of finite rank. We recall the definition as follows.
**Definition 2.5**.: _A pair \((\mathcal{T},\mathcal{F})\) of subcategories in an abelian category \(\mathcal{A}\) is a torsion pair if the following conditions are satisfied:_
1. \(\operatorname{Hom}\left(\mathcal{T},\mathcal{F}\right)=0\)_;_
2. _for any_ \(Z\in\mathcal{A}\)_, there is an exact sequence_ \(0\to X\to Z\to Y\to 0\) _with_ \(X\in\mathcal{T},Y\in\mathcal{F}\)_._
By definition, there are two trivial torsion pairs in \(\mathcal{A}\), namely, \((\mathcal{A},0)\) and \((0,\mathcal{A})\). All the other torsion pairs are called non-trivial. A pair \((\mathcal{T},\mathcal{F})\) in \(\mathcal{A}\) is called a _tilting torsion pair_ if there is a tilting object \(T\) (c.f. [13]) such that
\[\mathcal{T}=\{E\in\mathcal{A}\ |\ \text{Ext}\,^{1}(T,E)=0\},\quad\mathcal{F}=\{E \in\mathcal{A}\ |\ \text{Hom}\left(T,E\right)=0\}.\]
The following result is well-known, see for example [14, Prop. 3.3], [27, Lem. 1.1.3] and [34, Prop. 5.14].
**Proposition 2.6**.: _Let \(\mathcal{A}\) be an abelian category. Then a pair \((\mathcal{T},\mathcal{F})\) of subcategories in \(\mathcal{A}\) is a torsion pair if and only if \(\mathcal{T}=\,^{\perp}\mathcal{F}\) and \(\mathcal{F}=\mathcal{T}^{\perp}\)._
A torsion pair gives a natural stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\), where \(\Phi=\{-,+\}\) with the order \(-<+\), \(\Pi_{-}=\mathcal{F}\) and \(\Pi_{+}=\mathcal{T}\). In fact, a stability data can be thought as a kind of refinement of a torsion pair.
**Proposition 2.7**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}\). Let \(\Phi=\Phi_{-}\cup\Phi_{+}\) be an arbitrary decomposition with \(\varphi_{-}<\varphi_{+}\) for any \(\varphi_{\pm}\in\Phi_{\pm}\). Then the subcategories_
\[\mathcal{T}=\langle\Pi_{\varphi}\ |\ \varphi\in\Phi_{+}\rangle\quad\text{and} \quad\mathcal{F}=\langle\Pi_{\varphi}\ |\ \varphi\in\Phi_{-}\rangle\]
_give a torsion pair \((\mathcal{T},\mathcal{F})\) in \(\mathcal{A}\)._
Proof.: By construction we have \(\operatorname{Hom}\left(\mathcal{T},\mathcal{F}\right)=0\). In order to prove the second axiom of a torsion pair, we consider the HN-filtration (2.1) for any non-zero object \(X\in\mathcal{A}\). If \(\phi^{-}(X)\in\Phi_{+}\) or \(\phi^{+}(X)\in\Phi_{-}\), then \(X\) belongs to the subcategory \(\mathcal{T}\) or \(\mathcal{F}\) respectively. Otherwise, we find some \(k\), such that \(\varphi_{k}\in\Phi_{+}\) but \(\varphi_{k+1}\in\Phi_{-}\). Then we get the following exact sequence \(0\to X_{k}\to X\to X/X_{k}\to 0\), where \(X_{k}\in\mathcal{T}\) and \(X/X_{k}\in\mathcal{F}\). This finishes the proof.
As a consequence of the previous proposition, we have the following result which provides a method to construct torsion pairs in \(\mathcal{A}\) using stability data.
**Corollary 2.8**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}\). Then each \(\psi\in\Phi\) determines two torsion pairs \((\Pi_{\geq\psi},\Pi_{<\psi})\) and \((\Pi_{>\psi},\Pi_{\leq\psi})\), where_
\[\Pi_{\geq\psi} =\langle\Pi_{\varphi}\ |\ \varphi\geq\psi\rangle,\quad\Pi_{<\psi}= \langle\Pi_{\varphi}\ |\ \varphi<\psi\rangle;\] \[\Pi_{>\psi} =\langle\Pi_{\varphi}\ |\ \varphi>\psi\rangle,\quad\Pi_{\leq\psi}= \langle\Pi_{\varphi}\ |\ \varphi\leq\psi\rangle.\]
## 3. Finset stability data
In this section, we define a partial ordering for the set of all stability data on an abelian category \(\mathcal{A}\). We introduce a procedure to refine a stability data, and give a sufficient and necessary condition to justify a stability data to be finest. Finally, we show that any stability data can be refined to a finest one for representation-directed algebras.
### Local refinement
In order to classify the torsion pairs via the stability data approach, we only need to consider the equivalent classes of stability data in the following sense.
**Definition 3.1**.: _Two stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi}),(\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) on \(\mathcal{A}\) are called equivalent if there exists an order-preserved bijective map \(r:\Phi\to\Psi\) such that \(P_{r(\varphi)}=\Pi_{\varphi}\) for any \(\varphi\in\Phi\)._
Now let us define a partial ordering for the set of all stability data on an abelian category \(\mathcal{A}\) as follows.
**Definition 3.2**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi}),(\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) be stability data on \(\mathcal{A}\). We say that the stability data \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\), or \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is coarser than \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) and write \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\preceq(\Psi,\{P_{\psi}\}_{\psi\in\Psi})\), if there exists a surjective map \(r:\Psi\to\Phi\) such that_
1. \(\psi^{\prime}>\psi^{\prime\prime}\) _implies_ \(r(\psi^{\prime})\geq r(\psi^{\prime\prime})\)_;_
2. _for any_ \(\varphi\in\Phi\)_,_ \(\Pi_{\varphi}=\langle P_{\psi}\ |\ \psi\in r^{-1}(\varphi)\rangle\)_._
In other words, a coarser stability data is obtained from a finer one by fusing certain blocks of consecutive semistable subcategories. The relation "finer-coarser" defines a partial ordering on the set of all stability data on a given abelian category. Minimal elements with respect to this partial ordering will be called the _finest_ stability data. It is clear that the finest stability data contain the most complete information about the torsion pairs.
For a given stability data on an abelian category, we introduce a local-refinement method as follows.
**Proposition 3.3**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}\). For any \(\varphi\in\Phi\), assume \((I_{\varphi},\{P_{\psi}\}_{\psi\in I_{\varphi}})\) is a local stability data on \(\Pi_{\varphi}\). Let \(\Psi=\cup_{\varphi\in\Phi}I_{\varphi}\), which is a linearly ordered set containing each \(I_{\varphi}\) as a linearly ordered subset, and \(\psi_{1}>\psi_{2}\) whenever \(\psi_{i}\in I_{\varphi_{i}}\) with \(\varphi_{1}>\varphi_{2}\). Then \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) is a stability data on \(\mathcal{A}\), which is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\)._
Proof.: By definition we have \(\operatorname{Hom}\left(P_{\psi^{\prime}},P_{\psi^{\prime\prime}}\right)=0\) for any \(\psi^{\prime}>\psi^{\prime\prime}\in\Psi\). Moreover, for any non-zero object \(X\in\mathcal{A}\), there is a HN-filtration (2.1) with respect to the stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\). Since each \((I_{\varphi},\{P_{\psi}\}_{\psi\in I_{\varphi}})\) is a local stability data on \(\Pi_{\varphi}\), each factor \(A_{i}\in\Pi_{\varphi_{i}}\) has a filtration of the following form
with non-zero factors \(B_{i,k}=A_{i,k}/A_{i,k-1}\in P_{\psi_{k}}\) and strictly decreasing \(\psi_{k}>\psi_{k+1}\in I_{\varphi_{i}}\).
By (2.1) we have a short exact sequence \(\xi_{i}:0\to X_{i-1}\xrightarrow{p_{i-1}}X_{i}\xrightarrow{q_{i}}A_{i}\to 0\) for any \(i=1,2,\cdots,n\). Taking pullback along \(f_{i,m_{i}-1}\), we obtain the following commutative diagram
Then we obtain the following two short exact sequences:
which fit together as follows:
By replacing \(\xi_{i}\) by \(\xi_{i,m_{i}-1}\), and taking pullback along \(f_{i,m_{i}-2}\), and keeping the procedure going on step by step, we finally obtain that each \(X_{i}\) admits a finite filtration as follows:
Therefore, \(X\) admits a subobject filtration with factors \((B_{1,1},\cdots,B_{1,m_{1}},B_{2,1},\cdots,B_{2,m_{2}},\cdots,B_{n,1},\cdots,\)\(B_{n,m_{n}})\), having strictly decreasing order by the definition of ordering in \(\Psi\). Hence the subobject filtration of \(X\) is in fact a HN-filtration, which ensures that \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) is a stability data on \(\mathcal{A}\).
Recall that \(\Psi=\cup_{\varphi\in\Phi}I_{\varphi}\). This induces a well-defined surjective map \(r:\Psi\to\Phi\) such that \(r(\psi)=\varphi\) for any \(\psi\in I_{\varphi}\). In order to show \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\), it remains to show the statements (1) and (2) in Definition 3.2 hold. In fact, if \(\psi_{1}>\psi_{2}\in\Psi\), we claim \(r(\psi_{1})\geq r(\psi_{2})\). Otherwise, we write \(r(\psi_{i})=\varphi_{i}\), that is \(\psi_{i}\in I_{\varphi_{i}}\) for \(i=1,2\). Then \(\varphi_{1}<\varphi_{2}\), which follows that \(\psi_{1}<\psi_{2}\) by definition of ordering in \(\Psi\), a contradiction. This proves the claim. On the other hand, for any \(\varphi\in\Phi\),
\[\Pi_{\varphi}=\langle P_{\psi}\ |\ \psi\in I_{\varphi}\rangle=\langle P_{\psi} \ |\ \psi\in r^{-1}(\varphi)\rangle.\]
We are done.
The stability data \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) obtained in the above way will be called a _local refinement_ of \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\).
### Finest stability data
In this subsection, we provide a criterion for a stability data to be finest on arbitrary abelian category.
**Theorem 3.4**.: _Let \(\mathcal{A}\) be an abelian category. A stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\mathcal{A}\) is finest if and only if for any \(\varphi\in\Phi\) and non-zero objects \(X,Y\in\Pi_{\varphi}\), \(\operatorname{Hom}\left(X,Y\right)\neq 0\neq\operatorname{Hom}\left(Y,X\right)\)._
Proof.: To prove the " if " part, we assume \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is not finest. Then there exists a stability data \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) which is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\). Hence there is a surjective map \(r:\Psi\to\Phi\), which is not a bijection. So there exists \(\varphi\in\Phi\) and \(\psi_{1}>\psi_{2}\in\Psi\) such that \(r(\psi_{1})=r(\psi_{2})=\varphi\). Then \(P_{\psi_{1}},P_{\psi_{2}}\subseteq\Pi_{\varphi}\), but \(\operatorname{Hom}\left(P_{\psi_{1}},P_{\psi_{2}}\right)=0\), a contradiction.
For the " only if " part, we assume there exists some \(\varphi\in\Phi\) and non-zero \(X,Y\in\Pi_{\varphi}\) such that \(\operatorname{Hom}\left(X,Y\right)=0\). We will show that \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is not finest.
For this we first need to construct a new torsion pair in \(\mathcal{A}\). Define
\[\Pi_{\varphi_{-}}=\{Z\in\Pi_{\varphi}\ |\ \operatorname{Hom}\left(X,Z\right)=0\} \quad\text{and}\quad\Pi_{\varphi_{+}}=\{W\in\Pi_{\varphi}\ |\ \operatorname{Hom}\left(W,Z\right)=0,\forall Z\in\Pi_{\varphi_{-}}\}.\]
Then obviously \(Y\in\Pi_{\varphi_{-}}\), \(X\in\Pi_{\varphi_{+}}\) and \(\operatorname{Hom}\left(\Pi_{\varphi_{+}},\Pi_{\varphi_{-}}\right)=0\). Let
\[\mathcal{T}=\langle\Pi_{\psi},\psi>\varphi;\ \Pi_{\varphi_{+}}\rangle\quad \text{and}\quad\mathcal{F}=\langle\Pi_{\psi},\psi<\varphi;\ \Pi_{\varphi_{-}}\rangle.\]
We claim that \((\mathcal{T},\mathcal{F})\) forms a torsion pair in \(\mathcal{A}\). According to Proposition 2.6, it suffices to show that \(\mathcal{T}={}^{\perp}\mathcal{F}\) and \(\mathcal{F}=\mathcal{T}^{\perp}\). We only show the first statement, the second one can be obtained similarly.
By definition, we have \(\operatorname{Hom}\left(\mathcal{T},\mathcal{F}\right)=0\). Hence \(\mathcal{T}\subseteq{}^{\perp}\mathcal{F}\). On the other hand, for any \(Z\in{}^{\perp}\mathcal{F}\), we need to show that \(Z\in\mathcal{T}\). Assume the HN-filtration of \(Z\) is given by
with factors \(A_{i}\in\Pi_{\varphi_{i}}\). Now \(\operatorname{Hom}\left(Z,\mathcal{F}\right)=0\) implies \(\varphi_{n}\geq\varphi\). If \(\varphi_{n}>\varphi\), then \(Z\in\langle\Pi_{\psi}\ |\ \psi>\varphi\rangle\subseteq\mathcal{T}\). If \(\varphi_{n}=\varphi\), then \(\operatorname{Hom}\left(Z,\Pi_{\varphi_{-}}\right)=0\) implies \(\operatorname{Hom}\left(A_{n},\Pi_{\varphi_{-}}\right)=0\) since there is a surjection \(q_{n}:Z\twoheadrightarrow A_{n}\). Hence \(A_{n}\in({}^{\perp}\Pi_{\varphi_{-}})\cap\Pi_{\varphi}=\Pi_{\varphi_{+}} \subseteq\mathcal{T}\) and then \(Z\in\mathcal{T}\). Hence, \(\mathcal{T}={}^{\perp}\mathcal{F}\).
Since \((\mathcal{T},\mathcal{F})\) is a torsion pair, for any object \(W\in\Pi_{\varphi}\subset\mathcal{A}\), there exists a decomposition \(0\to W_{t}\to W\to W_{f}\to 0\) with \(W_{t}\in\mathcal{T}\) and \(W_{f}\in\mathcal{F}\). For any \(\psi>\varphi\), \(\operatorname{Hom}\left(\Pi_{\psi},W\right)=0\) implies \(\operatorname{Hom}\left(\Pi_{\psi},W_{t}\right)=0\), which ensures that \(W_{t}\in\langle\Pi_{\varphi}\mid\psi\leq\varphi\rangle\cap\mathcal{T}=\Pi_{ \varphi_{+}}\). Similarly, one can show that \(W_{f}\in\Pi_{\varphi_{-}}\). Therefore, \(\Pi_{\varphi}=\langle\Pi_{\varphi_{-}},\Pi_{\varphi_{+}}\rangle\).
Let \(\Psi=(\Phi\setminus\{\varphi\})\cup\{\varphi_{-},\varphi_{+}\}\), which is a linearly ordered set with the relations \(\varphi^{\prime}>\varphi_{+}>\varphi_{-}>\varphi^{\prime\prime}\) for any \(\varphi^{\prime}>\varphi>\varphi^{\prime\prime}\in\Phi\). By Proposition 3.3, we know that \((\Psi,\{\Pi_{\psi}\}_{\psi\in\Psi})\) is a stability data, which is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\), yielding a contradiction. We are done.
We know that stability data is a refinement of torsion pairs. In fact, for certain cases, all the torsion pairs can be obtained from finest stability data.
**Proposition 3.5**.: _Assume that each stability data on \(\mathcal{A}\) is coarser than a finest one. Then for any torsion pair \((\mathcal{T},\mathcal{F})\), there exists a stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) and a decomposition \(\Phi=\Phi_{-}\cup\Phi_{+}\), such that_
\[\mathcal{T}=\langle\Pi_{\varphi}\mid\varphi\in\Phi_{+}\rangle,\quad\mathcal{F} =\langle\Pi_{\varphi}\mid\varphi\in\Phi_{-}\rangle.\]
_Moreover, \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) can be taken from the set of finest stability data._
Proof.: Recall that each torsion pair gives a stability data. Then the result follows from Proposition 2.7.
Let \(A\) be a representation-directed algebra. Recall from [28] that the indecomposable modules in \(\operatorname{mod}\) - \(A\) can be ordered by a linearly ordered set \(\Phi_{A}\), namely, \(\operatorname{ind}\left(\operatorname{mod}\text{-}A\right)=\{M_{\varphi} \mid\varphi\in\Phi_{A}\}\), such that \(\operatorname{Hom}\left(M_{\varphi^{\prime}},M_{\varphi^{\prime\prime}}\right)=0\) for any \(\varphi^{\prime}>\varphi^{\prime\prime}\in\Phi_{A}\). Let \(\Pi_{\varphi}=\langle M_{\varphi}\rangle\) for any \(\varphi\in\Phi_{A}\), then by Theorem 3.4, \((\Phi_{A},\{\Pi_{\varphi}\}_{\varphi\in\Phi_{A}})\) is a finest stability data on \(\operatorname{mod}\text{-}A\). Clearly, any finest stability data on \(\operatorname{mod}\text{-}A\) has the above form (depends on \(\Phi_{A}\)), and any stability data can be refined to a finest one.
Note that the path algebra \(\mathbf{k}Q\) of a Dynkin quiver \(Q\) is a representation-directed algebra, hence each stability data on \(\operatorname{mod}\text{-}\mathbf{k}Q\) is coarser than a finest one. By Proposition 3.5, we can obtain the classification of torsion pairs in the module category \(\operatorname{mod}\text{-}\mathbf{k}Q\) via the stability data approach. In the following we provide two concrete examples for path algebras of Dynkin type \(A_{2}\) and \(A_{3}\).
**Example 3.6**.: Let \(Q:1\to 2\) and \(A_{2}\):=\(\mathbf{k}Q\). Then there are only three indecomposable modules in \(\operatorname{mod}\text{-}A_{2}\), the simple modules \(S_{1},S_{2}\) and the projective modules \(P_{1},P_{2}(=S_{2})\). There are only two equivalent classes of finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{mod}\text{-}A_{2}\) as follows:
1. \(\Phi=\{1,2,3\}\), and \(\Pi_{1}=\langle S_{2}\rangle,\Pi_{2}=\langle P_{1}\rangle,\Pi_{3}=\langle S_{ 1}\rangle\); in this case, each indecomposable module is semistable;
2. \(\Phi=\{1,2\}\), and \(\Pi_{1}=\langle S_{1}\rangle,\Pi_{2}=\langle S_{2}\rangle\); in this case, only \(S_{1},S_{2}\) are semistable indecomposable modules. The indecomposable module \(P_{1}\) is not semistable, whose HN-filtration is induced from the exact sequence \[0\to S_{2}\to P_{1}\to S_{1}\to 0.\]
As a consequence of Proposition 3.5, we obtain a classification of torsion pairs in \(\operatorname{mod}\text{-}A_{2}\) as below:
\begin{table}
\begin{tabular}{|c|c|} \hline \(\mathcal{T}\) & \(\mathcal{F}\) \\ \hline \(\langle S_{1}\rangle\) & \(\langle P_{1},S_{2}\rangle\) \\ \hline \(\langle S_{2}\rangle\) & \(\langle S_{1}\rangle\) \\ \hline \(\langle P_{1},S_{1}\rangle\) & \(\langle S_{2}\rangle\) \\ \hline \end{tabular}
\end{table}
Table 1. Non-trivial torsion pairs in \(\operatorname{mod}\text{-}A_{2}\)
**Example 3.7**.: Let \(Q:1\to 2\to 3\) and \(A_{3}\):=\(\mathbf{k}Q\). Then the Auslander-Reiten quiver \(\Gamma(\operatorname{mod}\)-\(A_{3})\) of the module category \(\operatorname{mod}\)-\(A_{3}\) has the form
Let \(\Phi=\{1,2,3\}\) be a linear ordered set in the natural sense \(1<2<3\), and let \(\Pi_{1}=\langle S_{1}\rangle,\Pi_{2}=\langle S_{2}\rangle,\Pi_{3}=\langle S_{ 3}\rangle\), then \((\Phi,\{\Pi_{\varphi}\}_{\varphi\,\in\,\Phi})\) is a finest stability data on \(\operatorname{mod}\)-\(A_{3}\). In this case, \(\boldsymbol{\phi}(S_{3})>\boldsymbol{\phi}(S_{2})>\boldsymbol{\phi}(S_{1})\) and only \(S_{3},S_{2},S_{1}\) are semistable indecomposable modules. The indecomposable module \(P_{2},I_{3},I_{2}\) are not semistable, whose HN-filtration are given as below respectively:
Similarly, we can obtain all the other finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\,\in\,\Phi})\) on \(\operatorname{mod}\)-\(A_{3}\) as follows:
1. \(\Phi=\{1,2,3,4\}\);
\begin{tabular}{|c|c|c|c|} \hline \(\Pi_{1}\) & \(\Pi_{2}\) & \(\Pi_{3}\) & \(\Pi_{4}\) \\ \hline \(\langle S_{1}\rangle\) & \(\langle S_{3}\rangle\) & \(\langle P_{2}\rangle\) & \(\langle S_{2}\rangle\) \\ \hline \(\langle S_{2}\rangle\) & \(\langle I_{2}\rangle\) & \(\langle S_{1}\rangle\) & \(\langle S_{3}\rangle\) \\ \hline \(\langle S_{2}\rangle\) & \(\langle I_{2}\rangle\) & \(\langle S_{3}\rangle\) & \(\langle S_{1}\rangle\) \\ \hline \(\langle S_{3}\rangle\) & \(\langle S_{1}\rangle\) & \(\langle P_{2}\rangle\) & \(\langle S_{2}\rangle\) \\ \hline \end{tabular}
2. \(\Phi=\{1,2,3,4,5\}\);
3. \(\Phi=\{1,2,3,4,5,6\}\);
As a consequence of Proposition 3.5, we obtain a classification of torsion pairs in \(\operatorname{mod}\)-\(A_{3}\) as below:
## 4. **Finest stability data on tube categories**
In this section we focus on the nilpotent representation category of cyclic quiver \(C_{n}\) with \(n\) vertices, that is, on the _tube_ category \(\mathbf{T}_{n}\) of rank \(n\geq 1\) in the sense of [28, Sect. 4.6] and [32, Chap. X], whose associated Auslander-Reiten quiver (AR-quiver for short) \(\Gamma(\mathbf{T}_{n})\) is a _stable tube_ of rank \(n\). We investigate the stability data for tube category \(\mathbf{T}_{n}\), and provide a new method to classify torsion pairs on \(\mathbf{T}_{n}\) via stability data approach.
### Homological properties for tubes
Recall that \(\mathbf{T}_{n}\) is a hereditary finite length abelian category with \(n\) simple objects \(S_{0},\cdots,S_{n-1}\), equipped with an Auslander-Reiten translation \(\tau\) satisfying \(\tau(S_{i})=S_{i-1}\), where the index is (always) considered module \(n\). A stable tube is called _homogeneous_ if it has rank one and it is called _non-homogeneous_ otherwise. For any simple object \(S_{j}\) in the tube category \(\mathbf{T}_{n}\) and any \(t\in\mathbb{Z}_{\geq 1}\), there is a unique object \(S_{j}^{(t)}\) of length \(t\) and top \((S_{j}^{(t)})=S_{j}\), and any indecomposable object in \(\mathbf{T}_{n}\) has this form. The tube category \(\mathbf{T}_{n}\) is a uniserial category in the sense that all subobjects of \(S_{j}^{(t)}\) form a chain with respect to the inclusion:
\[0:=S_{j-t}^{(0)}\subseteq S_{j-t+1}^{(1)}\subseteq S_{j-t+2}^{(2)}\subseteq \cdots\subseteq S_{j-1}^{(t-1)}\subseteq S_{j}^{(t)}.\]
Consequently, the socle of \(S_{j}^{(t)}\) is given by soc \((S_{j}^{(t)})=S_{j-t+1}\). Moreover, \(S_{j-t+r}^{(r)}/S_{j-t+r-1}^{(r-1)}=S_{j-t+r}\) for \(1\leq r\leq t\), and the _composition factor sequence_ of \(S_{j}^{(t)}\) is given by \((S_{j-t+1},\cdots,S_{j-1},S_{j})\). The set of pairwise distinct simple objects appearing in the sequence is called the _composition factor set_ of \(S_{j}^{(t)}\). It is easy to see that any \(S_{j}^{(t)}\) with \(t\geq n\) has the same composition factor set \(\{S_{0},S_{1},\cdots,S_{n-1}\}\).
We denote by \(u_{j,t}:S_{j-1}^{(t-1)}\to S_{j}^{(t)}\) and \(p_{j,t}:S_{j}^{(t)}\to S_{j}^{(t-1)}\) the irreducible injection map and surjection map respectively. For convenience, in the following we will simply denote them by \(u\) and \(p\) respectively if no confusions appear. For example, the composition
\[\begin{CD}S_{j}^{(t)}@>{p_{j,t}}>{}>S_{j}^{(t-1)}@>{p_{j,t-1}}>{}>S_{j}^{(t-2) }@>{p_{j,t-2}}>{}>\cdots @>{p_{j,3}}>{}>S_{j}^{(2)}@>{p_{j,2}}>{}>S_{j}\end{CD}\]
will be just denoted by \(p^{t-1}:S_{j}^{(t)}\to S_{j}\). With this simplified notations, we have \(u\circ p=p\circ u\) whenever it makes sense. We have the following fundamental exact sequences in the tube category \(\mathbf{T}_{n}\), see for example [32, Thm. 2.2] and [29, Lem. A.1].
**Lemma 4.1**.: _The following are exact sequences in \(\mathbf{T}_{n}\) for any \(j\in\mathbb{Z}/n\mathbb{Z}\) and \(t\geq 1\):_
1. \(0\)\(S_{j-t}^{(t)}\)\(S_{j-1}^{(t)}\)\(S_{j}^{(t+1)}\)\(S_{j}^{(t+1)}\)\(S_{j}^{(t)}\)\(S_{j}\)\(0\);
2. \(0\)\(S_{j-t}\)\(S_{j}^{(t)}\)\(S_{j-t}^{(t+1)}\)\(S_{j}^{(t+1)}\)\(S_{j}^{(t-1)}\)\(S_{j}^{(t)}\)\(S_{j+1}^{(t)}\)\(0\);
3. \(0\)\(S_{j}^{(t)}\)\(S_{j}^{(t)}\)\(S_{j+1}^{(t+1)}\)\(S_{j}^{(t-1)}\)\(S_{j+1}^{(t)}\)\(S_{j+1}^{(t)}\)\(0\);
4. \(0\)\(S_{j-t+1}\)\(S_{j}^{(t)}\)\(S_{j+1}^{(t)}\)\(S_{j+1}^{(t)}\)\(S_{j+1}^{(t)}\)\(S_{j+1}\)\(0\).
In the following we consider the non-zero morphisms in the tube category \(\mathbf{T}_{n}\).
**Proposition 4.2**.: _For any objects \(S_{j_{i}}^{(t_{1})}\)\((i=1,2)\) in \(\mathbf{T}_{n}\),_
1. _if_ \(t_{1}\geq t_{2}\)_, then_ \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\) _if and only if_ \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)\) _belongs to the composition factors set of_ \(S_{j_{2}}^{(t_{2})}\)
_._
2. _if_ \(t_{1}\leq t_{2}\)_, then_ \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\) _if and only if_ \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\) _belongs to the composition factors set of_ \(S_{j_{1}}^{(t_{1})}\)_._
Proof.: We only prove the second statement, since the proof for the first one is dual. First we assume there exists \(0\neq f\in\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\). Then there is a decomposition of \(f:S_{j_{1}}^{(t_{1})}\twoheadrightarrow\operatorname{im}f\hookrightarrow S_{j _{2}}^{(t_{2})}\). It follows that \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)=\operatorname{soc}\left( \operatorname{im}f\right)\), which is a composition factor of \(\operatorname{im}f\), and hence a composition factor of \(S_{j_{1}}^{(t_{1})}\).
On the other hand, assume \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\) belongs to the composition factors set of \(S_{j_{1}}^{(t_{1})}\). Let \(S_{j_{1}}^{(t)}\) with \(t\leq n\) be the unique object such that \(\operatorname{soc}\left(S_{j_{1}}^{(t)}\right)=\operatorname{soc}\left(S_{j _{2}}^{(t_{2})}\right)\). By assumption, \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\) belongs to the composition factors set of \(S_{j_{1}}^{(t_{1})}\), we get \(t\leq t_{1}\), hence there is an epimorphism \(S_{j_{1}}^{(t_{1})}\twoheadrightarrow S_{j_{1}}^{(t)}\). Since \(t\leq t_{1}\leq t_{2}\) and \(\operatorname{soc}\left(S_{j_{1}}^{(t_{1})}\right)=\operatorname{soc}\left(S_ {j_{2}}^{(t_{2})}\right)\), there is a monomorphism \(S_{j_{1}}^{(t)}\hookrightarrow S_{j_{2}}^{(t_{2})}\). Therefore, there is a non-zero morphism \(S_{j_{1}}^{(t_{1})}\twoheadrightarrow S_{j_{1}}^{(t)}\hookrightarrow S_{j _{2}}^{(t_{2})}\). Hence \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\).
**Proposition 4.3**.: _For any objects \(S_{j_{i}}^{(t_{i})}\)\((i=1,2)\) in \(\mathbf{T}_{n}\), \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\) if and only if the following hold:_
1. \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)\) _belongs to the composition factors set of_ \(S_{j_{2}}^{(t_{2})}\)_;_
2. \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\) _belongs to the composition factors set of_ \(S_{j_{1}}^{(t_{1})}\)_._
Proof.: First we assume there exists \(0\neq f\in\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\). Then there is a decomposition of \(f:S_{j_{1}}^{(t_{1})}\twoheadrightarrow\operatorname{im}f\hookrightarrow S_{j _{2}}^{(t_{2})}\). It follows that \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)=\operatorname{top}\left( \operatorname{im}f\right)\), which is a composition factor of \(\operatorname{im}f\), and hence a composition factor of \(S_{j_{2}}^{(t_{2})}\). Similarly, \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)=\operatorname{soc}\left( \operatorname{im}f\right)\), which is a composition factor of \(\operatorname{im}f\), and hence a composition factor of \(S_{j_{1}}^{(t_{1})}\). Hence (1) and (2) hold.
On the other hand, if \(t_{1}\geq t_{2}\), since \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)\) belongs to the composition factors set of \(S_{j_{2}}^{(t_{2})}\), by Proposition 4.2 (1), we get \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\); otherwise, \(t_{1}<t_{2}\), since \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\) belongs to the composition factors set of \(S_{j_{1}}^{(t_{1})}\), by Proposition 4.2 (2), we get \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\). Then we are done.
**Corollary 4.4**.: _For any objects \(S_{j_{i}}^{(t_{i})}\)\((i=1,2)\) in \(\mathbf{T}_{n}\) with \(t_{1}\geq n\),_
1. \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\) _if and only if_ \(S_{j_{1}}\) _belongs to the composition factors set of_ \(S_{j_{2}}^{(t_{2})}\)_;_
2. \(\operatorname{Hom}\left(S_{j_{2}}^{(t_{2})},S_{j_{1}}^{(t_{1})}\right)\neq 0\) _if and only if_ \(S_{j_{1}-t_{1}+1}\) _belongs to the composition factors set of_ \(S_{j_{2}}^{(t_{2})}\)_._
_In particular, if \(t_{2}\geq n\), then \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0\neq \operatorname{Hom}\left(S_{j_{2}}^{(t_{2})},S_{j_{1}}^{(t_{1})}\right)\)._
Proof.: The assumption \(t_{1}\geq n\) implies that the composition factor set of \(S_{j_{1}}^{(t_{1})}\) is \(\{S_{0},S_{1},\cdots,S_{n-1}\}\), which already contains \(\operatorname{top}\left(S_{j_{2}}^{(t_{2})}\right)\) and \(\operatorname{soc}\left(S_{j_{2}}^{(t_{2})}\right)\). Observe that \(\operatorname{soc}\left(S_{j_{1}}^{(t_{1})}\right)=S_{j_{1}-t_{1}+1}\) and \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)=S_{j_{1}}\), then the result follows from Proposition 4.3.
The following result plays a key role in this section.
**Lemma 4.5**.: _There are only finitely many subcategories of \(\mathbf{T}_{n}\) which are closed under extensions and direct summands._
Proof.: For any \(0\leq j\leq n-1\) and \(1\leq l\leq n\), using Lemma 4.1, one obtains the following exact sequence
\[0\to S_{j}^{((r-1)n+l)}\to S_{j}^{(rn+l)}\to S_{j}^{(n)}\to 0\quad(r\geq 1).\]
Then the following pushout-pullback commutative diagram
yields a short exact sequence
\[0\to S_{j}^{(rn+l)}\to S_{j}^{((r-1)n+l)}\oplus S_{j}^{((r+1)n+l)}\to S_{j}^{(rn+l)}\to 0.\]
Hence \(S_{j}^{((r\pm 1)n+l)}\in\langle S_{j}^{(rn+l)}\rangle\). Then for \(r\geq 1\), we obtain that \(\langle S_{j}^{(rn)}\rangle=\langle S_{j}^{(n)}\rangle\), and \(\langle S_{j}^{(rn+l)}\rangle=\langle S_{j}^{(n+l)}\rangle\) for \(1\leq l\leq n-1\). Note that there are only finitely many indecomposable objects in \(\mathbf{T}_{n}\) with length smaller than \(2n\). Then the result follows.
### Finest stability data on the tube categories
In this subsection, we will describe the semistable subcategories for finest stability data on the tube category \(\mathbf{T}_{n}\).
**Proposition 4.6**.: _Any stability data on \(\mathbf{T}_{n}\) can be refined to a finest one._
Proof.: Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathbf{T}_{n}\). If it is not finest, then we can make local refinement to obtain a finer stability data. But there are only finitely many ways to make refinement since there are only finitely many candidates for semistable subcategories of \(\mathbf{T}_{n}\) by Lemma 4.5. Hence the local refinement procedures will stop after finite steps, which yields a finest stability data.
From now onward, we always fix a finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\mathbf{T}_{n}\). Then we have the following results.
**Lemma 4.7**.: _For any two distinct semistable objects \(S_{j_{i}}^{(t_{i})}\) in \(\mathbf{T}_{n}\) with \(t_{i}\leq n,\,i=1,2\), we have \(\boldsymbol{\phi}(S_{j_{1}}^{(t_{1})})\neq\boldsymbol{\phi}(S_{j_{2}}^{(t_{2})})\)._
Proof.: For contradiction we assume there exist \(S_{j_{i}}^{(t_{i})}\) with \(t_{i}\leq n\)\((i=1,2)\) belong to the same semistable subcategory \(\Pi_{\varphi}\) for some \(\varphi\in\Phi\). By Theorem 3.4, we have
\[\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t_{2})}\right)\neq 0 \neq\operatorname{Hom}\left(S_{j_{2}}^{(t_{2})},S_{j_{1}}^{(t_{1})}\right). \tag{4.1}\]
Let \(0\neq f\in\operatorname{Hom}\left(S_{j_{2}}^{(t_{2})},S_{j_{1}}^{(t_{1})}\right)\). Then \(\operatorname{im}f=S_{j_{2}}^{(t)}\) for some \(t\leq\min\{t_{1},t_{2}\}\). If \(t=t_{1}\), then \(\operatorname{soc}\left(S_{j_{2}}^{(t)}\right)=\operatorname{soc}\left(S_{j_{ 1}}^{(t_{1})}\right)\) implies \(j_{1}=j_{2}\), yielding a contradiction to (4.1). Hence \(t<t_{1}\). Now we have the following commutative diagram:
which yields a short exact sequence
\[0\to S_{j_{2}}^{(t_{2})}\to S_{j_{2}}^{(t)}\oplus S_{j_{1}}^{(t_{2}+t_{1}-t)} \to S_{j_{1}}^{(t_{1})}\to 0.\]
Since \(\Pi_{\varphi}\) is closed under extensions and direct summands, we see that \(S_{j_{2}}^{(t)}\in\Pi_{\varphi}\). Since \(\operatorname{soc}\left(S_{j_{2}}^{(t)}\right)=\operatorname{soc}\left(S_{j_{ 1}}^{(t_{1})}\right)\) and \(t<t_{1}\), \(\operatorname{top}\left(S_{j_{1}}^{(t_{1})}\right)\) does not belong to the composition factor set of \(S_{j_{2}}^{(t)}\). Then by Proposition 4.2 (1), we see that \(\operatorname{Hom}\left(S_{j_{1}}^{(t_{1})},S_{j_{2}}^{(t)}\right)=0\), a contradiction. Then we are done.
**Lemma 4.8**.: _For any \(1\leq t\leq n\), there are at most \((n-t+1)\)-many indecomposable semistable objects of length \(t\) in \(\mathbf{T}_{n}\)._
Proof.: Note that any simple object \(S_{j}\)\((0\leq j\leq n-1)\) is stable since it has no non-trivial subobjects. Then the result holds for \(t=1\). For contradiction we assume there exists \(2\leq t\leq n\), such that the number of indecomposable semistable objects of length \(t\) is greater than \(n-t+1\). Assume they are given by \(S_{j_{i}}^{(t)}\)\((1\leq j\leq n-t+k)\) for some \(k\geq 2\), where \(0\leq j_{1}<j_{2}<\cdots<j_{n-t+k}\leq n-1\). Clearly, \(1\leq j_{i+1}-j_{i}\leq t-1\) for each \(i\) since \((t-1)+(n-t+k)>n\). It follows that \(\operatorname{Hom}\left(S_{j_{i}}^{(t)},S_{j_{i+1}}^{(t)}\right)\neq 0\) since \(\operatorname{top}\left(S_{j_{i}}^{(t)}\right)\) belongs to the composition factors set of \(S_{j_{i+1}}^{(t)}\). Since \(j_{n-t+k}<j_{1}+n\), we have \(\operatorname{Hom}\left(S_{j_{n-t+k}}^{(t)},S_{j_{1}}^{(t)}\right)\neq 0\) by similar
arguments as above. Then there is a chain of non-zero morphisms \(S_{j_{1}}^{(t)}\to S_{j_{2}}^{(t)}\to\cdots\to S_{j_{n-t+k}}^{(t)}\to S_{j_{1}}^{(t)}\), which implies \(\boldsymbol{\phi}(S_{j_{1}}^{(t)})=\boldsymbol{\phi}(S_{j_{2}}^{(t)})=\cdots= \boldsymbol{\phi}(S_{j_{n-t+k}}^{(t)})\), a contradiction to Lemma 4.7. We are done.
**Lemma 4.9**.: _There exists a unique integer \(0\leq j\leq n-1\), such that \(S_{j}^{(n)}\) is semistable._
Proof.: First we assume that each \(S_{j}^{(n)}\) is not semistable for \(0\leq j\leq n-1\). Note that any simple object is semistable. Then there exist \(0\leq j\leq n-1\) and \(1\leq t<n\), such that \(S_{j}^{(t)}\) is semistable, but \(S_{i}^{(k)}\) is not semistable for any \(t<k\leq n\) and \(0\leq i\leq n-1\). Consider the following exact sequences
\[0\to S_{j-t}^{(n-t)}\to S_{j}^{(n)}\to S_{j}^{(t)}\to 0,\quad\text{and}\quad 0\to S_{j}^{(t)} \to S_{j-t}^{(n)}\to S_{j-t}^{(n-t)}\to 0.\]
Since \(S_{j}^{(t)}\) is the minimal semistable quotient object of \(S_{j}^{(n)}\), we get \(\boldsymbol{\phi}(S_{j}^{(t)})<\boldsymbol{\phi}^{-}(S_{j-t}^{(n-t)})\). Similarly, since \(S_{j}^{(t)}\) is the maximal semistable subobject of \(S_{j-t}^{(n)}\), we get \(\boldsymbol{\phi}^{+}(S_{j-t}^{(n-t)})<\boldsymbol{\phi}(S_{j}^{(t)})\). Hence \(\boldsymbol{\phi}(S_{j}^{(t)})<\boldsymbol{\phi}^{-}(S_{j-t}^{(n-t)})\leq \boldsymbol{\phi}^{+}(S_{j-t}^{(n-t)})<\boldsymbol{\phi}(S_{j}^{(t)})\), a contradiction. Therefore, there exists at least one \(S_{j}^{(n)}\) which is semistable. The uniqueness of \(S_{j}^{(n)}\) follows from Lemma 4.8. We are done.
**Lemma 4.10**.: _For any \(0\leq i\leq n-1\), \(1\leq s<n\) and \(r\geq 1\), \(S_{i}^{(rn+s)}\) is not semistable._
Proof.: Assume \(S_{i}^{(rn+s)}\) is semistable with phase \(\varphi\) for some \(0\leq i\leq n-1\), \(1\leq s<n\) and \(r\geq 1\). Using Lemma 4.1, one can obtain the following short exact sequence
\[0\to S_{i}^{(rn+s)}\to S_{i}^{(s)}\oplus S_{i}^{(2rn+s)}\to S_{i}^{(rn+s)}\to 0.\]
Since \(\Pi_{\varphi}\) is closed under extensions and direct summands, we see that \(S_{i}^{(s)}\in\Pi_{\varphi}\).
On the other hand, by Lemma 4.9, there exists a unique \(0\leq j\leq n-1\), such that \(S_{j}^{(n)}\) is semistable. By Corollary 4.4, we know that \(\operatorname{Hom}\left(S_{j}^{(n)},S_{i}^{(rn+s)}\right)\neq 0\neq \operatorname{Hom}\left(S_{i}^{(rn+s)},S_{j}^{(n)}\right)\). Then we obtain that \(\boldsymbol{\phi}(S_{j}^{(n)})=\boldsymbol{\phi}(S_{i}^{(rn+s)})=\varphi= \boldsymbol{\phi}(S_{i}^{(s)})\), a contradiction to Lemma 4.7. We are done.
Denote by \(f(t)\) the number of indecomposable semistable objects of length \(t\) in \(\mathbf{T}_{n}\). Combining with Lemmas 4.8, 4.9 and 4.10, we have:
\[f(t)=\begin{cases}n,&\text{if $t=1$;}\\ 1,&\text{if $n|t$;}\\ \leq n-t+1,&\text{if $2\leq t<n$;}\\ 0,&\text{if else.}\end{cases}\]
Moreover, if \(\boldsymbol{\phi}(S_{0})<\boldsymbol{\phi}(S_{1})<\cdots<\boldsymbol{\phi}(S_ {n-1})\), then \(f(t)=n-t+1\) holds for any \(1\leq t\leq n\).
**Proposition 4.11**.: _Each semistable subcategory \(\Pi_{\varphi}\) has the form \(\langle S_{j}^{(s)}\rangle\), where \(0\leq j\leq n-1\) and \(1\leq s\leq n\)._
Proof.: By Lemma 4.10 we know that any semistable object in \(\mathbf{T}_{n}\) has the possible form \(S_{j}^{(t)}\) for some \(0\leq j\leq n-1\), where \(1\leq t<n\) or \(n|t\). According to the proof of Lemma 4.5, we obtain that \(\langle S_{j}^{(rn)}\rangle=\langle S_{j}^{(n)}\rangle\) for any \(r\geq 1\). Moreover, by Lemma 4.9, there exists a unique indecomposable semistable object of length \(n\). Then by Lemma 4.7, any two distinct semistable objects \(S_{j_{1}}^{(t_{1})}\) and \(S_{j_{2}}^{(t_{2})}\) with \(t_{i}\leq n\)\((i=1,2)\) have different phases. We are done.
Using the above results, we can classify all the finest stability data on any tube category via combinatorial method. In the following we state the classification result for the tube category \(\mathbf{T}_{3}\) as an example.
**Example 4.12**.: Up to \(\tau\)-actions, there are four equivalent classes of finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\mathbf{T}_{3}\) as follows:
1. \(\Phi=\{1,2,3,4,5\}\);
\begin{tabular}{|c|c|c|c|c|} \hline \(\Pi_{1}\) & \(\Pi_{2}\) & \(\Pi_{3}\) & \(\Pi_{4}\) & \(\Pi_{5}\) \\ \hline \(\langle S_{0}\rangle\) & \(\langle S_{2}\rangle\) & \(\langle S_{1}^{(3)}\rangle\) & \(\langle S_{1}^{(2)}\rangle\) & \(\langle S_{1}\rangle\) \\ \hline \(\langle S_{0}\rangle\) & \(\langle S_{1}^{(2)}\rangle\) & \(\langle S_{2}^{(3)}\rangle\) & \(\langle S_{2}\rangle\) & \(\langle S_{1}\rangle\) \\ \hline \end{tabular}
2. \(\Phi=\{1,2,3,4,5,6\}\);
\begin{tabular}{|c|c|c|c|c|c|} \hline \(\Pi_{1}\) & \(\Pi_{2}\) & \(\Pi_{3}\) & \(\Pi_{4}\) & \(\Pi_{5}\) & \(\Pi_{6}\) \\ \hline \(\langle S_{0}\rangle\) & \(\langle S_{1}^{(2)}\rangle\) & \(\langle S_{1}\rangle\) & \(\langle S_{2}^{(3)}\rangle\) & \(\langle S_{2}^{(2)}\rangle\) & \(\langle S_{2}\rangle\) \\ \hline \(\langle S_{0}\rangle\) & \(\langle S_{1}^{(2)}\rangle\) & \(\langle S_{2}^{(3)}\rangle\) & \(\langle S_{1}\rangle\) & \(\langle S_{2}^{(2)}\rangle\) & \(\langle S_{2}\rangle\) \\ \hline \end{tabular}
### Torsion pairs in tube categories
We have already obtained all possible semistable subcategories for finest stability data on the tube category \(\mathbf{T}_{n}\). This enables us to classify torsion pairs in \(\mathbf{T}_{n}\).
**Theorem 4.13**.: \((\mathcal{T},\mathcal{F})\) _is a torsion pair in \(\mathbf{T}_{n}\) if and only if one of the following holds (up to \(\tau\)-actions):_
1. \(\mathcal{T}\) _is a torsion class in_ \(\langle S_{1},S_{2},\cdots,S_{n-1}\rangle\)_, and_ \(\mathcal{F}=\mathcal{T}^{\perp}\)_;_
2. \(\mathcal{F}\) _is a torsionfree class in_ \(\langle S_{0},S_{1},\cdots,S_{n-2}\rangle\)_, and_ \(\mathcal{T}={}^{\perp}\mathcal{F}\)_._
Proof.: If \(\mathcal{T}\) is a torsion class in \(\langle S_{1},S_{2},\cdots,S_{n-1}\rangle\), then \(\mathcal{T}\) is closed under extensions and quotient objects in \(\langle S_{1},S_{2},\cdots,S_{n-1}\rangle\) and also in \(\mathbf{T}_{n}\). Hence \(\mathcal{T}\) is a torsion class in \(\mathbf{T}_{n}\) by [27, Lem. 1.1.3]. Then \((\mathcal{T},\mathcal{T}^{\perp})\) is a torsion pair in \(\mathbf{T}_{n}\).
Similarly, if \(\mathcal{F}\) is a torsionfree class in \(\langle S_{0},S_{1},\cdots,S_{n-2}\rangle\), then \(\mathcal{F}\) is closed under extensions and subobjects in \(\langle S_{0},S_{1},\cdots,S_{n-2}\rangle\) and also in \(\mathbf{T}_{n}\). Hence \(\mathcal{F}\) is a torsionfree class in \(\mathbf{T}_{n}\) by the dual of [27, Lem. 1.1.3]. Then \(({}^{\perp}\mathcal{F},\mathcal{F})\) is a torsion pair in \(\mathbf{T}_{n}\).
On the other hand, for any torsion torsion pair \((\mathcal{T},\mathcal{F})\) in \(\mathbf{T}_{n}\), by Propositions 3.5 and 4.6, there exists a finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) and a decomposition \(\Phi=\Phi_{-}\cup\Phi_{+}\), such that
\[\mathcal{T}=\langle\Pi_{\varphi}\mid\varphi\in\Phi_{+}\rangle,\quad\mathcal{F }=\langle\Pi_{\varphi}\mid\varphi\in\Phi_{-}\rangle.\]
By Lemma 4.9, we can assume, up to \(\tau\)-actions on \(\mathbf{T}_{n}\), \(S_{n-1}^{(n)}\) is the unique semistable object under \((\Phi,\{\Pi_{\varphi}\}_{y\in\Phi})\). Then \(S_{n-1}^{(n)}\in\mathcal{F}\) or \(S_{n-1}^{(n)}\in\mathcal{T}\). We consider the following two cases.
(1) If \(S_{n-1}^{(n)}\in\mathcal{F}\), then
\[\mathcal{T}={}^{\perp}\mathcal{F}\subseteq{}^{\perp}(S_{n-1}^{(n)})=\langle S _{1},S_{2},\cdots,S_{n-1}\rangle.\]
Since \(\mathcal{T}\) is a torsion class in \(\mathbf{T}_{n}\), it is closed under extensions and quotient objects. The same statements holds in the subcategory \(\langle S_{1},S_{2},\cdots,S_{n-1}\rangle\). Hence \(\mathcal{T}\) is a torsion class in \(\langle S_{1},S_{2},\cdots,S_{n-1}\rangle\). Obviously, by definition we have \(\mathcal{F}=\mathcal{T}^{\perp}\).
(2) If \(S_{n-1}^{(n)}\in\mathcal{T}\), then
\[\mathcal{F}=\mathcal{T}^{\perp}\subseteq(S_{n-1}^{(n)})^{\perp}=\langle S_{0},S_{1},\cdots,S_{n-2}\rangle.\]
Since \(\mathcal{F}\) is a torsionfree class in \(\mathbf{T}_{n}\), it is closed under extensions and subobjects. The same statements holds in the subcategory \(\langle S_{0},S_{1},\cdots,S_{n-2}\rangle\). Hence \(\mathcal{F}\) is a torsionfree class in \(\langle S_{0},S_{1},\cdots,S_{n-2}\rangle\). It follows that \(\mathcal{T}={}^{\perp}\mathcal{F}\) and we are done.
**Remark 4.14**.: In [5], Baur-Buan-Marsh classified the torsion pairs in \(\mathbf{T}_{n}\) via maximal rigid objects in the extension \(\overline{\mathbf{T}}_{n}\) of the tube category \(\mathbf{T}_{n}\). Here, \(\overline{\mathbf{T}}_{n}\) is the subcategory of Mod-\(\mathbf{k}C_{n}\) for cyclic quiver \(C_{n}\), whose objects are all filtered direct limits or filtered inverse limits of objects in \(\mathbf{T}_{n}\).
More precisely, there are two different types of torsion pairs: the ray type and the coray type, where the ray type corresponds to the maximal rigid object of Prufer type in \(\overline{\mathbf{T}}_{n}\); and the coray type corresponds to the maximal rigid object of adic type in \(\overline{\mathbf{T}}_{n}\). In fact, up to \(\tau\)-actions, the torsion pairs in \(\overline{\mathbf{T}}_{n}\) of ray type (_resp._ coray type) correspond to those in (1) (_resp. (2)) in Theorem 4.13.
As an explanation of Theorem 4.13, we list all the torsion pairs in the tube category \(\mathbf{T}_{3}\) as below.
**Example 4.15**.: Up to \(\tau\)-actions, the non-trivial torsion pairs \((\mathcal{T},\mathcal{F})\) in \(\mathbf{T}_{3}\) are classified as below:
## 5. Tame hereditary algebra
In this section we investigate the stability data for tame hereditary algebras. We show that each stability data can be refined to a finest one for the category of finitely generated modules over a tame hereditary algebra.
We recall from [28] and [32] for the definition and the main properties for tame hereditary algebra \(\Lambda=\mathbf{k}Q\), where \(Q\) is a tame quiver. The category \(\operatorname{mod}\operatorname{-}\Lambda\) of finitely generated modules over \(\Lambda\) is a hereditary abelian category. Moreover, there is a decomposition of indecomposable \(\Lambda\)-modules: \(\operatorname{ind}\left(\operatorname{mod}\operatorname{-}\Lambda\right)= \mathcal{P}\vee\mathcal{R}\vee\mathcal{I}\), where \(\mathcal{P}\) is the postprojective component, \(\mathcal{R}\) is the regular component, and \(\mathcal{I}\) is the preinjective component. The postprojective component \(\mathcal{P}\) consists of modules \(\tau^{-n}P_{i},i\in Q_{0},n\in\mathbb{Z}_{\geq 0}\), where \(P_{i}\) are indecomposable projective modules, and each indecomposable module \(M\) in \(\mathcal{P}\) is directing with \(\operatorname{End}\left(M\right)=\mathbf{k},\operatorname{Ext}^{i}\left(M,M \right)=0\) for all \(i\geq 1\); the regular component \(\mathcal{R}\) consists of a family \(\mathbf{T}=\{\Gamma(\mathbf{T}_{\mathbf{x}})\}_{x\in\mathbb{P}^{1}}\) of pairwise orthogonal stable tubes \(\Gamma(\mathbf{T}_{\mathbf{x}})\), in particular, all but finitely many of the tubes \(\Gamma(\mathbf{T}_{\mathbf{x}})\) in \(\mathcal{R}\) are homogeneous, and there are at most \(|Q_{0}|-2\) non-homogeneous tubes; and the preinjective component \(\mathcal{I}\) consists of modules \(\tau^{m}I_{i},i\in Q_{0},m\in\mathbb{Z}_{\geq 0}\), where \(I_{i}\) are indecomposable injective modules, and each indecomposable module \(N\) in \(\mathcal{I}\) is also directing with \(\operatorname{End}\left(N\right)=\mathbf{k},\operatorname{Ext}^{i}\left(N,N \right)=0\) for all \(i\geq 1\).
In the rest of this section we denote by \(\mathcal{A}=\operatorname{mod}\operatorname{-}\Lambda\), where \(\Lambda\) is a tame hereditary algebra.
**Proposition 5.1**.: _Each stability data on \(\mathcal{A}=\operatorname{mod}\operatorname{-}\Lambda\) can be refined to a finest one._
Proof.: Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}\). Since each indecomposable module \(M\) in the postprojective component \(\mathcal{P}\) is directing, the indecomposable modules in the subcategory \(\Pi_{\varphi}\cap\mathcal{P}\) can be ordered by a linearly ordered set \(\mathcal{P}_{\varphi}\) for any \(\varphi\in\Phi\), namely, \(\Pi_{\varphi}\cap\mathcal{P}=\{M_{\psi}\ |\ \psi\in\mathcal{P}_{\varphi}\}\), such that \(\operatorname{Hom}\left(M_{\psi^{\prime}},M_{\psi^{\prime\prime}}\right)=0\) for any \(\psi^{\prime}>\psi^{\prime\prime}\in\mathcal{P}_{\varphi}\).
Similarly, since each indecomposable module \(N\) in the preinjective component \(\mathcal{I}\) is directing, the indecomposable modules in the subcategory \(\Pi_{\varphi}\cap\mathcal{I}\) can be ordered by a linearly ordered set \(\mathcal{I}_{\varphi}\) for any \(\varphi\in\Phi\), namely, \(\Pi_{\varphi}\cap\mathcal{I}=\{N_{\psi}\ |\ \psi\in\mathcal{I}_{\varphi}\}\), such that \(\operatorname{Hom}\left(N_{\psi^{\prime}},N_{\psi^{\prime\prime}}\right)=0\) for any \(\psi^{\prime}>\psi^{\prime\prime}\in\mathcal{I}_{\varphi}\). Meanwhile, the subcategory \(\Pi_{\varphi}\cap\mathcal{R}\) of regular component \(\mathcal{R}\) can be ordered by a linearly ordered set \(\mathcal{R}_{\varphi}\) for any \(\varphi\in\Phi\), namely, \(\Pi_{\varphi}\cap\mathcal{R}=\{\Pi_{\varphi}\cap\Gamma(\mathbf{T}_{\mathbf{x} })\ |\ (\varphi,\mathbf{x})\in\mathcal{R}_{\varphi},\Pi_{\varphi}\cap(\Gamma(\mathbf{T}_{ \mathbf{x}})\neq 0\}\).
Put \(I_{\varphi}=\mathcal{P}_{\varphi}\cup\mathcal{R}_{\varphi}\cup\mathcal{I}_{\varphi}\). Then \(I_{\varphi}\) is a linearly ordered set with further relations \(\psi^{\prime}>\psi^{\prime\prime}>\psi^{\prime\prime\prime}\) for any \(\psi^{\prime}\in\mathcal{I}_{\varphi},\psi^{\prime\prime}\in\mathcal{R}_{\varphi}\) and \(\psi^{\prime\prime\prime}\in\mathcal{P}_{\varphi}\). Let \(\Psi=\cup_{\varphi\in\Phi}I_{\varphi}\), then \(\Psi\) is a linearly ordered set which keeps each \(I_{\varphi}\) as ordered subsets, and if \(\psi^{\prime}\in I_{\varphi^{\prime}},\psi^{\prime\prime}\in I_{\varphi^{ \prime\prime}}\) with \(\varphi^{\prime}>\varphi^{\prime\prime}\) in \(\Phi\), then \(\psi^{\prime}>\psi^{\prime\prime}\).
For any \(\varphi\in\Phi\) and \(\psi\in I_{\varphi}\), we define
\[P_{\psi}=\begin{cases}\operatorname{add}\,M_{\psi},&\text{if }\psi\in\mathcal{P}_{ \varphi};\\ \operatorname{add}\,N_{\psi},&\text{if }\psi\in\mathcal{I}_{\varphi};\\ \operatorname{add}\,\{\Pi_{\varphi}\cap\Gamma(\mathbf{T}_{\mathbf{x}})\},& \text{if }\psi=(\varphi,\mathbf{x})\in\mathcal{R}_{\varphi}.\end{cases}\]
By construction, \(P_{\psi}\) is extension closed, \(\operatorname{Hom}\left(P_{\psi^{\prime}},P_{\psi^{\prime\prime}}\right)=0\) for any \(\psi^{\prime}>\psi^{\prime\prime}\in\Psi\), and \(\Pi_{\varphi}=\cup_{\psi\in I_{\boldsymbol{\nu}}}P_{\psi}\). It is easy to see that \((\Psi,\{P_{\psi}\}_{\psi\in\Psi})\) is a stability data on \(\mathcal{A}\), which is finer than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\). Note that each subcategory \(P_{\psi}\) for any \(\psi\) in \(\mathcal{P}_{\varphi}\) or \(\mathcal{I}_{\varphi}\) contains a unique indecomposable direct summand. Moreover, for any \(\psi=(\varphi,\boldsymbol{x})\in\mathcal{R}_{\varphi}\), the semistable subcategory \(P_{\psi}=\text{add }\{\Pi_{\varphi}\cap\Gamma(\boldsymbol{\Upsilon}_{ \boldsymbol{x}})\}\) is a subcategory of a tube category \(\mathbf{T}_{\boldsymbol{x}}\). Then by similar arguments as in the proof of Proposition 4.6, we can make local refinement to obtain a finest stability data. This finishes the proof.
Consequently, we can classify torsion pairs for the category of finitely generated modules over any tame hereditary algebra via the stability data approach. In the following we provide a concrete example for tame hereditary algebra \(\widetilde{A}_{1}\).
**Example 5.2**.: Let \(Q:1\rightrightarrows 2\). Then \(\widetilde{A}_{1}{:=}\mathbf{k}Q\) is the minimal tame hereditary algebra. Denote by \(P_{i},I_{i},S_{i}\) (\(1\leq i\leq 2\)) the indecomposable projective, injective and simple \(\widetilde{A}_{1}\)-modules respectively. The postprojective component \(\mathcal{P}\) consists of modules \(\tau^{-n}P_{i}\) and the preinjective component \(\mathcal{I}\) consists of modules \(\tau^{n}I_{i}\), where \(i=1,2\) and \(n\in\mathbb{Z}_{\geq 0}\). For convenience, we denote by \(\mathcal{P}_{2k+1}{:=}\tau^{-k}P_{2}\), \(\mathcal{P}_{2k+2}:=\tau^{-k}P_{1}\), and \(\mathcal{I}_{2k+1}{:=}\tau^{k}I_{1}\), \(\mathcal{I}_{2k+2}:=\tau^{k}I_{2}\), for any \(k\in\mathbb{Z}_{\geq 0}\). The regular component \(\mathcal{R}\) consists of modules \(S_{\boldsymbol{x}}^{(d)},\boldsymbol{x}\in\mathbb{P}^{1},d\in\mathbb{Z}_{\geq 1}\) where each \(S_{\boldsymbol{x}}\) is a simple regular module.
For any finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{mod}\operatorname{-}\widetilde{A}_{1}\), the simple modules \(S_{1}\) and \(S_{2}\) are stable with \(\boldsymbol{\phi}(S_{1})\neq\boldsymbol{\phi}(S_{2})\) by Theorem 3.4. In fact, there are only two equivalent classes of finest stability data on \(\operatorname{mod}\operatorname{-}\widetilde{A}_{1}\) as below, according to \(\boldsymbol{\phi}(S_{1})>\boldsymbol{\phi}(S_{2})\) or \(\boldsymbol{\phi}(S_{1})<\boldsymbol{\phi}(S_{2})\) respectively.
1. \(\Phi=(0,\mathbb{Z}_{\geq 1})\cup\mathbb{P}^{1}\cup(1,\mathbb{Z}_{\geq 1})\) with \((0,n)<(0,n+1)<\boldsymbol{x}<(1,n+1)<(1,n)\) for any \(n\in\mathbb{Z}_{\geq 1},\boldsymbol{x}\in\mathbb{P}^{1}\), and there are no restrictions on the choice of ordering on the set \(\mathbb{P}^{1}\). For any \(k\in\mathbb{Z}_{\geq 1}\), \(\Pi_{(0,k)}=\langle\mathcal{P}_{k}\rangle\) and \(\Pi_{(1,k)}=\langle\mathcal{I}_{k}\rangle\); for any \(\boldsymbol{x}\in\mathbb{P}^{1}\), \(\Pi_{\boldsymbol{x}}=\langle S_{\boldsymbol{x}}\rangle\). In this case, each indecomposable module is semistable;
2. \(\Phi=\{1,2\}\) with \(1<2\), and \(\Pi_{1}=\langle S_{1}\rangle,\Pi_{2}=\langle S_{2}\rangle\). In this case, only \(S_{1},S_{2}\) are semistable indecomposable modules. For any other indecomposable module \(X\), assume \(X\) has dimension vector \(\underline{\underline{\mathbf{dim}}}\ X=(m,n)\), its HN-filtration is given by \[0\to S_{2}^{\oplus n}\to X\to S_{1}^{\oplus m}\to 0.\]
As a consequence of Propositions 5.1 and 3.5, we can obtain a classification of torsion pairs in \(\operatorname{mod}\operatorname{-}\widetilde{A}_{1}\) as below, where \(P\subseteq\mathbb{P}^{1}\) is a (possibly empty) subset of \(\mathbb{P}^{1}\) and \(n\in\mathbb{Z}_{\geq 1}\).
Recall that a subcategory \(\mathcal{T}\) is called _contravariantly finite_ in an abelian category \(\mathcal{A}\) if for each \(X\in\mathcal{A}\), there is a map \(f:T\to X\) with \(T\in\mathcal{T}\) such that \(\operatorname{Hom}\left(T^{\prime},T\right)\overset{\cdot f}{\longrightarrow} \operatorname{Hom}\left(T^{\prime},X\right)\) is surjective for all \(T^{\prime}\in\mathcal{T}\). Dually, a _covariantly finite_ subcategory is defined. A subcategory \(\mathcal{T}\) is called _functorially finite_ if it is both contravariantly finite and covariantly finite. We call a torsion pair \((\mathcal{T},\mathcal{F})\) functorially finite if the torsion class \(\mathcal{T}\) is functorially finite.
We remark that all the torsion pairs in the above table are functorially finite except for the first case. Moreover, for any functorially finite torsion pair, its torsion class \(\mathcal{T}\) is determined by a support \(\tau\)-tilting module, c.f. [1, Thm. 2.7]. More precisely, the support \(\tau\)-tilting module \(T=S_{i}\) if \(\mathcal{T}=\langle S_{i}\rangle\) for \(i=1,2\), and \(T=\mathcal{I}_{n}\oplus\mathcal{I}_{n-1}\) if \(\mathcal{T}=\langle\mathcal{I}_{m}\ |\ m\leq n\rangle\) with \(n\geq 2\); while \(T=\mathcal{P}_{n+1}\oplus\mathcal{P}_{n+2}\) if \(\mathcal{T}=\langle\mathcal{P}_{m},\mathcal{R},\mathcal{I}\ |\ m>n\rangle\).
## 6. Projective line
In this section we investigate the stability data for the category \(\operatorname{coh}\mathbb{P}^{1}\) of coherent sheaves over the projective line \(\mathbb{P}^{1}:=\mathbb{P}^{1}_{\mathbf{k}}\). We obtain classifications of finest stability data and torsion pairs on \(\operatorname{coh}\mathbb{P}^{1}\).
Recall from [18] and [22] that \(\operatorname{coh}\mathbb{P}^{1}\) is a (skeletally) small and Hom-finite hereditary abelian \(\mathbf{k}\)-linear category with Serre duality of the form
\[\operatorname{Ext}^{1}(X,Y)\cong\operatorname{DHom}\,(Y,\tau X),\]
where \(\operatorname{D}\,=\operatorname{Hom}_{\,\mathbf{k}}(-,\mathbf{k})\) and \(\tau\) is given by the grading shift with \((-2)\). The subcategories \(\operatorname{coh}_{0}\mathbb{P}^{1}\) of torsion sheaves and \(\operatorname{vect}\mathbb{P}^{1}\) of vector bundles form a split torsion pair \((\operatorname{coh}_{0}\mathbb{P}^{1},\operatorname{vect}\mathbb{P}^{1})\) in \(\operatorname{coh}\mathbb{P}^{1}\), namely, any coherent sheaf can be decomposed as a direct sum of a torsion sheaf and a vector bundle, and there are no nonzero homomorphisms from \(\operatorname{coh}_{0}\mathbb{P}^{1}\) to \(\operatorname{vect}\mathbb{P}^{1}\). The subcategory \(\operatorname{coh}_{0}\mathbb{P}^{1}\) splits into a coproduct of connected subcategories \(\coprod_{x\in\mathbb{P}^{1}}\mathbf{T}_{\mathbf{z}}\), where each \(\mathbf{T}_{\mathbf{z}}\) is generated by a simple sheaf \(S_{\mathbf{z}}\), and the associated AR-quivers \(\Gamma(\mathbf{T}_{\mathbf{z}})\) is a stable tube of rank one. Any indecomposable vector bundle in \(\operatorname{vect}\mathbb{P}^{1}\) is a line bundle, more precisely, it is of the form \(\mathcal{O}(n)\) for \(n\in\mathbb{Z}\). If \(n=0\), then \(\mathcal{O}:=\mathcal{O}(0)\) is the structure sheave of \(\mathbb{P}^{1}\). The homomorphism space \(\operatorname{Hom}\,(\mathcal{O}(m),\mathcal{O}(n))\) between two line bundles has dimension \(n-m+1\) if \(n\geq m\) and \(0\) otherwise. Moreover, if \(\mathcal{O}(n)\) is a line bundle and \(S_{\mathbf{z}}\) is a simple sheaf, then \(\operatorname{Hom}\,(\mathcal{O}(n),S_{\mathbf{z}})\cong\mathbf{k}\).
The following lemma shows that any indecomposable sheaf is semistable for \(\operatorname{coh}\mathbb{P}^{1}\).
**Lemma 6.1**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) be a stability data on \(\operatorname{coh}\mathbb{P}^{1}\), then any indecomposable sheaf is semistable._
Proof.: Obviously, any simple sheaf \(S_{\mathbf{z}}\) in \(\operatorname{coh}\mathbb{P}^{1}\) is stable since it has no non-trivial subsheaves. Then \(S_{\mathbf{z}}\in\Pi_{\varphi}\) for some \(\varphi\). It follows that the subcategory \(\langle S_{\mathbf{z}}\rangle\subseteq\Pi_{\varphi}\), hence any torsion sheaf \(S_{\mathbf{z}}^{(n)}\in\langle S_{\mathbf{z}}\rangle\) is semistable.
For any line bundle \(L\in\operatorname{coh}\mathbb{P}^{1}\), if it is not semistable, then its minimal semistable quotient sheaf is given by a torsion sheaf \(X\), and its maximal semistable subsheaf is given by another line bundle \(L^{\prime}\). Then \(\operatorname{Hom}\,(L^{\prime},X)\neq 0\) yields a contradiction. We are done.
There is a classical slope function \(\mu:\operatorname{ind}\,(\operatorname{coh}\mathbb{P}^{1})\to\mathbb{Z}\cup\{\infty\}\) on the category of coherent sheaves over \(\mathbb{P}^{1}\), satisfying \(\mu(\mathcal{O}(n))=n\) for any line bundle \(\mathcal{O}(n)\) and \(\mu(X)=\infty\) for any torsion sheaf \(X\). For any indecomposable sheaves \(X,Y\in\operatorname{coh}\mathbb{P}^{1}\) with \(\mu(X)\neq\mu(Y)\), it is well-known that
\[\operatorname{Hom}\,(X,Y)\neq 0\quad\text{if}\;\;\text{and}\;\;\text{only}\;\;\; \text{if}\quad\mu(X)<\mu(Y).\]
Denote by \(\overline{\mathbb{Z}}=\mathbb{Z}\cup\{\infty\}\), which is a linearly ordered set in the natrual way. Then the slope function gives a stability data \((\overline{\mathbb{Z}},\{P_{\psi}\}_{\psi\in\overline{\mathbb{Z}}})\), called _slope stability data_, where
\[P_{\psi}=\langle X\in\operatorname{ind}\,(\operatorname{coh}\mathbb{P}^{1}) \mid\mu(X)=\psi\rangle.\]
The following result shows that each finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\) can be obtained from the slope stability data by local refinements.
**Proposition 6.2**.: _Each finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\) is a refinement of the slope stability data \((\overline{\mathbb{Z}},\{P_{\psi}\}_{\psi\in\overline{\mathbb{Z}}})\)._
Proof.: Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) be a finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\). We only need to show that \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) is finer than \((\overline{\mathbb{Z}},\{P_{\psi}\}_{\psi\in\overline{\mathbb{Z}}})\).
For any non-zero indecomposable sheaves \(X,Y\in\Pi_{\varphi}\), we have \(\operatorname{Hom}\,(X,Y)\neq 0\neq\operatorname{Hom}\,(Y,X)\). It follows that \(\mu(X)=\mu(Y)\). This induces a well-defined map \(r:\Phi\to\overline{\mathbb{Z}}\), satisfying \(r(\varphi)=\mu(X)\) for any indecomposable \(X\in\Pi_{\varphi}\). Moreover, by Lemma 6.1, we know that any indecomposable sheaf is semistable under the stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\). It follows that \(r\) is surjective.
By definition, it remains to show the statements (1) and (2) in Definition 3.2 hold. In fact, for any \(\varphi_{1}>\varphi_{2}\in\Phi\) and any indecomposable \(X_{i}\in\Pi_{\varphi_{i}}\,(i=1,2)\), we have \(\operatorname{Hom}\,(X_{1},X_{2})=0\). Hence \(\mu(X_{1})\geq\mu(X_{2})\). That is, \(r(\varphi_{1})\geq r(\varphi_{2})\). On the other hand, for any \(\psi\in\overline{\mathbb{Z}}\),
\[P_{\psi}=\langle X\in\operatorname{ind}\;(\operatorname{coh}\mathbb{P}^{1})\mid \mu(X)=\psi\rangle=\langle X\mid X\in\Pi_{\varphi},r(\varphi)=\psi\rangle= \langle\Pi_{\varphi}\mid\varphi\in r^{-1}(\psi)\rangle.\]
Therefore, \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) is finer than \((\overline{\mathbb{Z}},\{P_{\psi}\}_{\psi\in\overline{\mathbb{Z}}})\). We are done.
The following result describes semistable subcategories for finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\).
**Proposition 6.3**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\). Then each semistable subcategory has the form \(\langle\mathcal{O}(n)\rangle\) for \(n\in\mathbb{Z}\), or \(\langle S_{\boldsymbol{x}}\rangle\) for some \(\boldsymbol{x}\in\mathbb{P}^{1}\). Further, the phases for the stable sheaves satisfy:_
1. \(\boldsymbol{\phi}(\mathcal{O}(n))<\boldsymbol{\phi}(\mathcal{O}(n+1))< \boldsymbol{\phi}(S_{\boldsymbol{x}}),\forall\ n\in\mathbb{Z},\boldsymbol{x} \in\mathbb{P}^{1}\)_;_
2. \(\boldsymbol{\phi}(S_{\boldsymbol{x}})\neq\boldsymbol{\phi}(S_{\boldsymbol{y} }),\forall\ \boldsymbol{x}\neq\boldsymbol{y}\in\mathbb{P}^{1}\)_, and there are no restrictions on the choice of ordering on the set_ \(\{\boldsymbol{\phi}(S_{\boldsymbol{x}})\ |\ \boldsymbol{x}\in\mathbb{P}^{1}\}\)_._
_Consequently, any stability data on \(\operatorname{coh}\mathbb{P}^{1}\) can be refined to a finest one._
Proof.: By Lemma 6.1, we know that any indecomposable sheaf is semistable in \(\operatorname{coh}\mathbb{P}^{1}\). Notice that \(\operatorname{Hom}\left(\mathcal{O}(n),\mathcal{O}(m)\right)\neq 0\) if and only if \(n\leq m\), and \(\operatorname{Hom}\left(\mathcal{O}(n),S_{\boldsymbol{x}}\right)\neq 0,\ \forall\ n \in\mathbb{Z},\boldsymbol{x}\in\mathbb{P}^{1}\). Moreover, there are no non-zero homomorphisms between torsion sheaves concentrated in distinct points. Then the result follows from Theorem 3.4 and Lemma 6.1 immediately.
Note that \(\mathbb{Z}\) is a linearly ordered set in the natural way, i.e., \(n<n+1\) for any \(n\in\mathbb{Z}\). Meanwhile, \(\mathbb{P}^{1}\) can be viewed as a linearly ordered set by fixing any choice of ordering on the points of the projective line. Then the union \(\mathbb{Z}\cup\mathbb{P}^{1}\) also forms a linearly ordered set by composing the relations \(n<\boldsymbol{x}\) for any \(n\in\mathbb{Z}\) and \(\boldsymbol{x}\in\mathbb{P}^{1}\). Note that there are infinitely many ordered relations on the set \(\mathbb{Z}\cup\mathbb{P}^{1}\), depending on the choices of ordering on the set \(\mathbb{P}^{1}\).
As a consequence of Proposition 6.3, we obtain the following classification result for finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\), parameterized by the linearly ordered set \(\mathbb{Z}\cup\mathbb{P}^{1}\).
**Proposition 6.4**.: _Any finest stability data on \(\operatorname{coh}\mathbb{P}^{1}\) has the form_
\[(\mathbb{Z}\cup\mathbb{P}^{1},\{\Pi_{\varphi}\}_{\varphi\in\mathbb{Z}\cup \mathbb{P}^{1}}),\]
_where \(\mathbb{Z}\cup\mathbb{P}^{1}\) is an arbitrary linearly ordered set defined as above, \(\Pi_{n}=\langle\mathcal{O}(n)\rangle\) for any \(n\in\mathbb{Z}\), and \(\Pi_{\boldsymbol{x}}=\langle S_{\boldsymbol{x}}\rangle\) for any \(\boldsymbol{x}\in\mathbb{P}^{1}\)._
Note that each stability data on \(\operatorname{coh}\mathbb{P}^{1}\) is coarser than a finest one. Then by Proposition 3.5, any torsion pair \((\mathcal{T},\mathcal{F})\) can be obtained from the finest stability data. As a consequence of Propositions 6.4 and 3.5, we can obtain a classification of torsion pairs in \(\operatorname{coh}\mathbb{P}^{1}\).
**Proposition 6.5**.: _The non-trivial torsion pairs \((\mathcal{T},\mathcal{F})\) in \(\operatorname{coh}\mathbb{P}^{1}\) are classified as below, where \(P\) is a non-empty subset of \(\mathbb{P}^{1}\) and \(n\) is an integer._
We remark that the torsion pair of the first case is not functorially finite, while it is functorially finite for the second case, and it is determined by the canonical tilting sheaf \(\mathcal{O}(n)\oplus\mathcal{O}(n+1)\).
## 7. Elliptic curves
In this section we investigate the stability data for the category \(\operatorname{coh}\mathbb{T}\) of coherent sheaves over a smooth elliptic curve \(\mathbb{T}\). We will classify finest stability data and torsion pairs for \(\operatorname{coh}\mathbb{T}\).
We recall from [3, 22, 23] for the main properties of the category \(\operatorname{coh}\mathbb{T}\) as below. Similar as \(\operatorname{coh}\mathbb{P}^{1}\), the category \(\operatorname{coh}\mathbb{T}\) is a (skeletally) small and Hom-finite hereditary abelian \(\mathbf{k}\)-linear category, satisfying the 1-Calabi-Yau property
\[\operatorname{Ext}^{1}(X,Y)\cong\operatorname{DHom}\left(Y,X\right),\]
\begin{table}
\begin{tabular}{|c|c|} \hline \(\mathcal{T}\) & \(\mathcal{F}\) \\ \hline \(\langle S_{\boldsymbol{x}}\ |\ \boldsymbol{x}\in P\subseteq\mathbb{P}^{1}\rangle\) & \(\langle\mathcal{O}(m),S_{\boldsymbol{x}}\ |\ m\in\mathbb{Z},\boldsymbol{x}\in\mathbb{P}^{1} \backslash P\rangle\) \\ \hline \(\langle\mathcal{O}(m),S_{\boldsymbol{x}}\ |\ m>n,\boldsymbol{x}\in\mathbb{P}^{1}\rangle\) & \(\langle\mathcal{O}(m)\ |\ m\leq n\rangle\) \\ \hline \end{tabular}
\end{table}
Table 5. Non-trivial torsion pairs \((\mathcal{T},\mathcal{F})\) in \(\operatorname{coh}\mathbb{P}^{1}\)
where \(\mathrm{D}\,=\mathrm{Hom}\,_{\mathbf{k}}(-,\mathbf{k})\). The subcategory \(\mathrm{coh}_{\mathbb{T}}\) of torsion sheaves splits into a coproduct of connected subcategories \(\coprod_{\mathbf{x}\in\mathbb{T}}\mathbf{T}_{\mathbf{x}}\), whose associated AR-quivers \(\Gamma(\mathbf{T}_{\mathbf{x}})\) are stable tubes of rank one generated by the simple sheaves \(S_{\mathbf{x}}\). The subcategory \(\mathcal{A}_{q}\) (\(q\in\mathbb{Q}\)) generated by all indecomposable vector bundles of slope \(q\) splits into a coproduct of connected tube subcategories \(\coprod_{\mathbf{x}\in\mathbb{T}}\mathcal{A}_{q,\mathbf{x}}\), whose associated AR-quivers \(\Gamma(\mathcal{A}_{q,\mathbf{x}})\) are stable tubes of rank one generated by the unique quasi-simple sheaves \(S_{q,\mathbf{x}}\). It turns out that \(\mathrm{coh}\,\mathbb{T}=\bigvee_{q\in\mathbb{Q}\cup\{\infty\}}\mathcal{A}_{q}\), where \(\mathcal{A}_{\infty}:=\mathrm{coh}_{\mathbb{0}}\mathbb{T}\). There exists a nonzero morphism from \(\mathcal{A}_{q}\) to \(\mathcal{A}_{r}\) if and only if \(q\leq r\). Moreover, if \(L\) is a line bundle and \(S_{\mathbf{x}}\) is a simple sheaf, then \(\mathrm{Hom}\,(L,S)\cong\mathbf{k}\).
The major difference between \(\mathrm{coh}\,\mathbb{T}\) and \(\mathrm{coh}\,\mathbb{P}^{1}\) are:
(1) the Grothendieck group \(\mathrm{K}_{0}(\mathbb{T})\) has infinite rank while \(\mathrm{K}_{0}(\mathbb{P}^{1})\) is free abelian of finite rank, thus permitting a close investigation of the Euler form, roots and radical vectors;
(2) all simple ordinary sheaves on \(\mathbb{P}^{1}\) do have the same class in \(\mathrm{K}_{0}(\mathbb{P}^{1})\), while simple sheaves concentrated at different points from \(\mathbb{T}\) possess distinct classes in \(\mathrm{K}_{0}(\mathbb{T})\);
(3) the line bundles of degree zero on \(\mathbb{T}\) form a one-parameter family, indexed by \(\mathbb{T}\), while \(\mathbb{P}^{1}\) has a unique line bundle of degree zero (i.e., the structure sheaf);
(4) there are no exceptional sheaves on \(\mathbb{T}\), while the study of exceptional sheaves on \(\mathbb{P}^{1}\) forms a major item.
The following lemma shows that any indecomposable sheaf is semistable for \(\mathrm{coh}\,\mathbb{T}\).
**Lemma 7.1**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) be a stability data on \(\mathrm{coh}\,\mathbb{T}\), then any indecomposable sheaf is semistable._
Proof.: Obviously, any simple sheaf \(S_{\mathbf{x}}\) in \(\mathrm{coh}\,\mathbb{T}\) is stable since it has no non-trivial subsheaves. Then \(S_{\mathbf{x}}\in\Pi_{\varphi}\) for some \(\varphi\). It follows that the subcategory \(\langle S_{\mathbf{x}}\rangle\subseteq\Pi_{\varphi}\), hence any torsion sheaf \(S_{\mathbf{x}}^{(n)}\in\langle S_{\mathbf{x}}\rangle\) is semistable.
For any quasi-simple vector bundle \(E\in\mathrm{coh}\,\mathbb{T}\) of slope \(q\), if it is not semistable, then its maximal semistable subsheaf is given by a sheaf \(E_{1}\) of slope \(q_{1}<q\), and its minimal semistable quotient sheaf is given by another sheaf \(E_{2}\) of slope \(q_{2}>q\). Then \(\mathrm{Hom}\,(E_{1},E_{2})\neq 0\) yields a contradiction. Hence any quasi-simple vector bundle is stable. In general, for any indecomposable vector bundle \(F\), we know that the top of \(F\) (=top\((F)\)) is quasi-simple, and \(F\) belongs to the subcategory \(\langle\mathrm{top}(F)\rangle\), whose AR-quiver is a homogeneous tube. Hence \(\mathrm{top}(F)\) is semistable implies \(F\) is semistable. We are done.
There is a classical slope function \(\mu:\mathrm{ind}\,(\mathrm{coh}\,\mathbb{T})\to\mathbb{Q}\cup\{\infty\}\) on the category of indecomposable coherent sheaves over \(\mathbb{T}\), satisfying \(\mu(E)=\mathrm{deg}(E)/\mathrm{rank}(E)\) for any sheaf \(E\), c.f. [3, Part 1]. Then \(\mu(E)\in\mathbb{Q}\) when \(E\) is a vector bundle, and \(\mu(X)=\infty\) for any torsion sheaf \(X\). Denote by \(\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}\), which is a linearly ordered set in the natrual way. Then the slope function gives a stability data \((\overline{\mathbb{Q}},\left\{P_{\psi}\right\}_{\psi\in\overline{\mathbb{Q}}})\), called _slope stability data_, where
\[P_{\psi}=\langle X\in\mathrm{ind}\,(\mathrm{coh}\,\mathbb{T})\mid\mu(X)=\psi\rangle.\]
By using the same proof as in Proposition 6.2, one obtains that
**Proposition 7.2**.: _Each finest stability data on \(\mathrm{coh}\,\mathbb{T}\) is a refinement of the slope stability data \((\overline{\mathbb{Q}},\left\{P_{\psi}\right\}_{\psi\in\overline{\mathbb{Q}}})\)._
The following result describes semistable subcategories for finest stability data on \(\mathrm{coh}\,\mathbb{T}\).
**Proposition 7.3**.: _Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) be a finest stability data on \(\mathrm{coh}\,\mathbb{T}\). Then each semistable subcategory has the form \(\langle S_{q,\mathbf{z}}\rangle\) for some \(q\in\overline{\mathbb{Q}},\mathbf{x}\in\mathbb{T}\). Further, the phases for the stable sheaves satisfy:_
1. \(\boldsymbol{\phi}(S_{p,\mathbf{z}})<\boldsymbol{\phi}(S_{q,\mathbf{y}});\quad \forall\ p<q,\mathbf{x},\mathbf{y}\in\mathbb{T};\)__
2. _there are no restrictions on the choice of ordering on the set_ \(\{\boldsymbol{\phi}(S_{p,\mathbf{z}}),\ \mathbf{z}\in\mathbb{T}\}\)_._
_Consequently, any stability data on \(\mathrm{coh}\,\mathbb{T}\) can be refined to a finest one._
Proof.: By Lemma 7.1, we know that any indecomposable sheaf is semistable in \(\mathrm{coh}\,\mathbb{T}\). Notice that for any \(p,q\in\overline{\mathbb{Q}},\ \mathbf{x},\mathbf{y}\in\mathbb{T}\), \(\mathrm{Hom}\,(S_{p,\mathbf{z}},\,S_{q,\mathbf{y}})\neq 0\) if and only if \(p<q\), or \((p,\mathbf{z})=(q,\mathbf{y})\). Then the result follows from Theorem 3.4 and Lemma 7.1 immediately.
Note that \(\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}\) is a linearly ordered set in the natural way, i.e., \(p<q<\infty\) for any \(p<q\in\mathbb{Q}\). Meanwhile, \(\mathbb{T}\) can be viewed as a linearly ordered set by fixing any choice of ordering on the points of the
elliptic curve \(\mathbb{T}\). Hence, for any \(q\in\overline{\mathbb{Q}}\), \(\{q\}\times\mathbb{T}\) becomes a linearly ordered set by fixing any choice of ordering on the points of \(\mathbb{T}\), and then the product \(\overline{\mathbb{Q}}\times\mathbb{T}=\bigcup_{q\in\overline{\mathbb{Q}}}(\{q \}\times\mathbb{T})\) also forms a linearly ordered set under the lexicographical order. There are infinitely many ordered relations on the set \(\overline{\mathbb{Q}}\times\mathbb{T}\), depending on the choices of ordering on the set \(\{q\}\times\mathbb{T}\) for \(q\in\overline{\mathbb{Q}}\).
As a consequence of Proposition 7.3, we obtain the following classification result for finest stability data on \(\operatorname{coh}\mathbb{T}\), parameterized by the linearly ordered set \(\overline{\mathbb{Q}}\times\mathbb{T}\).
**Proposition 7.4**.: _Any finest stability data on \(\operatorname{coh}\mathbb{T}\) has the form_
\[(\overline{\mathbb{Q}}\times\mathbb{T},\{\Pi_{(q,\boldsymbol{x})}\}_{(q, \boldsymbol{x})\in\overline{\mathbb{Q}}\times\mathbb{T}}),\]
_where \(\overline{\mathbb{Q}}\times\mathbb{T}\) is an arbitrary linearly ordered set defined as above, and \(\Pi_{(q,\boldsymbol{x})}=\langle S_{q,\boldsymbol{x}}\rangle\) for any \((q,\boldsymbol{x})\in\overline{\mathbb{Q}}\times\mathbb{T}\)._
Note that each stability data on \(\operatorname{coh}\mathbb{T}\) is coarser than a finest one. Then by Proposition 3.5, any torsion pair \((\mathcal{T},\mathcal{F})\) can be obtained from the finest stability data. As a consequence of Propositions 7.4 and 3.5, we can obtain a classification of torsion pairs in \(\operatorname{coh}\mathbb{T}\).
**Proposition 7.5**.: _The torsion pairs \((\mathcal{T},\mathcal{F})\) in \(\operatorname{coh}\mathbb{T}\) are classified as below, where \(q\in\overline{\mathbb{Q}}\) and \(P\subseteq\mathbb{T}\)._
## 8. Weighted projective lines
In this section we investigate the stability data for the category \(\operatorname{coh}\mathbb{X}\) of coherent sheaves over a weighted projective line \(\mathbb{X}\). We show that any stability data on \(\operatorname{coh}\mathbb{X}\) can be refined to a finest one when \(\mathbb{X}\) is of domestic type or tubular type. Moreover, we classify all the finest stability data on \(\operatorname{coh}\mathbb{X}\) for \(\mathbb{X}\) having weight type (2), as a by-product we obtain a classification of torsion pairs.
### Weighted projective lines
Following [16, 22], a _weighted projective line_\(\mathbb{X}=\mathbb{X}_{\mathbf{k}}\) over \(\mathbf{k}\) is given by a weight sequence \(\mathbf{p}=(p_{1},\ldots,p_{t})\) (called _weight type_) of positive integers, and a sequence \(\boldsymbol{\lambda}=(\lambda_{1},,\ldots,\lambda_{t})\) of pairwise distinct closed points in the projective line \(\mathbb{P}^{1}:=\mathbb{P}^{1}_{\mathbf{k}}\) which can be normalized as \(\lambda_{1}=\infty,\lambda_{2}=0,\lambda_{3}=1\). More precisely, let \(\mathbb{L}=\mathbb{L}(\mathbf{p})\) be the rank one abelian group (called _string group_) with generators \(\vec{x}_{1},\ldots,\vec{x}_{t}\) and the relations
\[p_{1}\vec{x}_{1}=\cdots=p_{t}\vec{x}_{t}=:\vec{c},\]
where \(\vec{c}\) is called the _canonical element_ of \(\mathbb{L}\). Each element \(\vec{x}\in\mathbb{L}\) has the _normal form_\(\vec{x}=\sum\limits_{i=1}^{t}l_{i}\vec{x}_{i}+l\vec{c}\) with \(0\leq l_{i}\leq p_{i}-1\) and \(l\in\mathbb{Z}\).
Denote by \(S\) the commutative algebra
\[S=S(\mathbf{p},\boldsymbol{\lambda})=\mathbf{k}[X_{1},\cdots,X_{t}]/I:= \mathbf{k}[x_{1},\ldots,x_{t}],\]
where \(I=(f_{3},\ldots,f_{t})\) is the ideal generated by \(f_{i}=X_{i}^{p_{i}}-X_{2}^{p_{2}}+\lambda_{i}X_{1}^{p_{1}}\) for \(3\leq i\leq t\). Then \(S\) is \(\mathbb{L}\)-graded by setting
\[\deg(x_{i})=\vec{x}_{i}\;\text{ for }1\leq i\leq t.\]
Finally, the weighted projective line associated with \(\mathbf{p}\) and \(\boldsymbol{\lambda}\) is defined to be
\[\mathbb{X}=\mathbb{X}_{\mathbf{k}}=\operatorname{Proj}^{\mathbb{L}}S,\]
the projective spectrum of \(\mathbb{L}\)-graded homogeneous prime ideals of \(S\).
Denote by \(p=\mathrm{l.c.m.}(p_{1},p_{2},\cdots,p_{t})\). There is a surjective group homomorphism \(\delta\colon\mathbb{L}\to\mathbb{Z}\) given by \(\delta(\vec{x}_{i})=\frac{p}{p_{i}}\) for \(1\leq i\leq t\). Recall that the _dualizing element_\(\vec{\omega}\) in \(\mathbb{L}\) is defined as \(\vec{\omega}=(t-2)\vec{c}-\sum_{i=1}^{t}\vec{x}_{i}\). Hence we have \(\delta(\vec{\omega})=p((t-2)-\sum_{i=1}^{t}\frac{1}{p_{i}})\). The the weighted projective line \(\mathbb{X}\) is called of _domestic, tubular or wild types_ provided that \(\delta(\vec{\omega})<0\), \(\delta(\vec{\omega})=0\) or \(\delta(\vec{\omega})>0\) respectively. More precisely, we have the following trichotomy for \(\mathbb{X}\) according to the weight sequence \(\mathbf{p}\) (up to permutation of weights):
1. domestic type: \((),(p),(p_{1},p_{2})\), \((2,2,p_{3})\), \((2,3,3)\), \((2,3,4)\) and \((2,3,5)\);
2. tubular type: \((2,2,2,2)\), \((3,3,3)\), \((4,4,2)\) and \((6,3,2)\);
3. wild type: all the other cases.
Here, \(()\) stands for the unweighted case, that is, the classical projective line.
The category of coherent sheaves on \(\mathbb{X}\) is defined to be the quotient category
\[\mathrm{coh}\,\mathbb{X}=\mathrm{mod}^{\mathbb{L}}S/\mathrm{mod}_{0}^{\mathbb{ L}}S,\]
where \(\mathrm{mod}^{\mathbb{L}}S\) is the category of finitely generated \(\mathbb{L}\)-graded \(S\)-modules, while \(\mathrm{mod}_{0}^{\mathbb{L}}S\) is the Serre subcategory of finite length \(\mathbb{L}\)-graded \(S\)-modules. For any \(\vec{x}\in\mathbb{L}\), the grading shift \(\vec{x}:E\mapsto E(\vec{x})\) gives an equivalence of \(\mathrm{coh}\,\mathbb{X}\). Moreover, \(\mathrm{coh}\,\mathbb{X}\) is a hereditary abelian category with Serre duality of the form
\[\mathrm{Ext}\,^{1}(X,Y)\cong\mathrm{D}\,\mathrm{Hom}\,(Y,X(\vec{\omega})).\]
This implies the existence of almost split sequences in \(\mathrm{coh}\,\mathbb{X}\) with the Auslander-Reiten translation \(\tau\) given by the grading shift with \(\vec{\omega}\).
It is known that \(\mathrm{coh}\,\mathbb{X}\) admits a splitting torsion pair \((\mathrm{coh}_{0}\mathbb{X},\mathrm{vect}\,\mathbb{X})\), where \(\mathrm{coh}_{0}\mathbb{X}\) and \(\mathrm{vect}\,\mathbb{X}\) are full subcategories of torsion sheaves and vector bundles, respectively. The subcategory \(\mathrm{coh}_{0}\mathbb{X}\) splits into a coproduct of connected tube subcategories \(\coprod_{x\in\mathbb{P}^{1}}\mathbf{T}_{\boldsymbol{x}}\), whose associated AR-quivers \(\Gamma(\mathbf{T}_{\boldsymbol{x}})\) are stable tubes of finite rank. The subcategory \(\mathrm{vect}\,\mathbb{X}\) contains line bundles, which have the forms \(\mathcal{O}(\vec{x})\) for \(\vec{x}\in\mathbb{L}\).
### Finest stability data on \(\mathrm{coh}\,\mathbb{X}\) of domestic or tubular type
In this subsection, we investigate the stability data on the category \(\mathrm{coh}\,\mathbb{X}\) of coherent sheaves for weighted projective lines of domestic or tubular type. We will show that each stability data on \(\mathrm{coh}\,\mathbb{X}\) can be refined to a finest one.
Recall that there is a slope function \(\mu:\mathrm{ind}\,(\mathrm{coh}\,\mathbb{X})\to\mathbb{Q}\cup\{\infty\}\) on the category of coherent sheaves over \(\mathbb{X}\), satisfying \(\mu(E)=\mathrm{deg}(E)/\mathrm{rank}(E)\) for any sheaf \(E\). Then \(\mu(E)\in\mathbb{Q}\) when \(E\) is a vector bundle, and \(\mu(X)=\infty\) for any torsion sheaf \(X\). Denote by \(\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}\), which is a linearly ordered set in the natural way. This slope function gives a stability data \((\overline{\mathbb{Q}},\{P_{\psi}\}_{\psi\in\overline{\mathbb{Q}}})\), called _slope stability data_, where
\[\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\},\quad P_{\psi}=\langle X\in \mathrm{ind}\,(\mathrm{coh}\,\mathbb{X})\mid\mu(X)=\psi\rangle.\]
By assumption, i.e., \(\mathbb{X}\) is of domestic or tubular type, each indecomposable sheaf is semistable. Moreover, \(\mathrm{Hom}\,(X,Y)\neq 0\) implies \(\mu(X)\leq\mu(Y)\) for any indecomposable sheaves \(X,Y\). This slope function gives a decomposition of the category \(\mathcal{A}=\mathrm{coh}\,\mathbb{X}=\mathrm{add}\{\mathcal{A}_{q}\mid q\in \overline{\mathbb{Q}}\}\), where \(\mathcal{A}_{q}=\langle X\in\mathrm{ind}\,(\mathrm{coh}\,\mathbb{X})\mid\mu( X)=q\rangle\).
Recall that each subcategory \(\mathcal{A}_{q}\) has one of the following forms:
1. \(\mathcal{A}_{q}\) is a semisimple category generated by finitely many orthogonal exceptional vector bundles, in case \(\mathbb{X}\) is domestic type and \(q\in\mathbb{Q}\);
2. \(\mathcal{A}_{q}\) is an abelian category consisting of a family of connected orthogonal tube subcategories parameterized by the projective line, in case \(q=\infty\) or \(\mathbb{X}\) is of tubular type.
The main result of this subsection is as follows.
**Proposition 8.1**.: _Each stability data on \(\mathrm{coh}\,\mathbb{X}\) can be refined to a finest one._
Proof.: Let \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) be a stability data on \(\mathcal{A}:=\mathrm{coh}\,\mathbb{X}\). We first claim that \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) can be refined to a finer one with each semistable category is contained in some \(\mathcal{A}_{q}:=\langle X\in\mathrm{ind}\,(\mathrm{coh}\,\mathbb{X})\mid\mu(X)=q\rangle\).
In fact, for any \(\varphi\in\Phi\), define \(I_{\varphi}=\{(\varphi,q)\mid q\in\overline{\mathbb{Q}},\ \Pi_{\varphi}\cap\mathcal{A}_{q}\neq 0\}\) and set \(\Psi=\cup_{\varphi\in\Phi}I_{\varphi}\). Then \(\Psi\) is a linearly ordered set under the lexicographical order, i.e., \((\varphi^{\prime},q^{\prime})>(\varphi^{\prime\prime},q^{\prime\prime})\) if and only if \(\varphi^{\prime}>\varphi^{\prime\prime}\), or \(\varphi^{\prime}=\varphi^{\prime\prime}\) and \(q^{\prime}>q^{\prime\prime}\). Set \(P_{\varphi,q}=\Pi_{\varphi}\cap\mathcal{A}_{q}\) for any \((\varphi,q)\in\Psi\). Then by Proposition 3.3, we obtain that \((\Psi,\{P_{\varphi,q}\}_{(\varphi,q)\in\Psi})\) is a finer stability data than \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\), with each \(P_{\varphi,q}\) contained in some \(\mathcal{A}_{q}\).
Recall that for weighted projective line of domestic or tubular type, the subcategory \(\mathcal{A}_{q}\) is generated by finitely many orthogonal exceptional vector bundles, or \(\mathcal{A}_{q}\) is a coproduct of connected orthogonal tube subcategories. Hence we can further make local refinement on \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) such that each semistable category \(\Pi_{\varphi}\) is generated by a unique exceptional vector bundle, or contained in a homogeneous tube, or contained in a non-homogeneous tube.
For the first two cases, the subcategory \(\Pi_{\varphi}\) satisfies \(\operatorname{Hom}\left(X,Y\right)\neq 0\neq\operatorname{Hom}\left(Y,X\right)\) for non-zero objects \(X,Y\in\Pi_{\varphi}\). For the third case, the semistable subcategory \(\Pi_{\varphi}\) is a subcategory of a tube category. Then by similar arguments as in the proof of Proposition 4.6, we can make local refinement to obtain a finest stability data. This finishes the proof.
### Semistable sheaves on \(\operatorname{coh}\mathbb{X}\) for weight type (2)
From now on, we focus on the weighted projective line \(\mathbb{X}\) of weight type (2). Recall that the group \(\mathbb{L}=\mathbb{L}(2)\) is the rank one abelian group with generators \(\vec{x}_{1},\vec{x}_{2}\) and the relations
\[2\vec{x}_{1}=\vec{x}_{2}:=\vec{c}.\]
Each element \(\vec{x}\in\mathbb{L}\) has the _normal form_\(\vec{x}=l\vec{c}\) or \(\vec{x}=\vec{x}_{1}+l\vec{c}\) for some \(l\in\mathbb{Z}\). In this case, each indecomposable bundle in \(\operatorname{vect}\mathbb{X}\) is a line bundle, hence has the form \(\mathcal{O}(\vec{x}),\vec{x}\in\mathbb{L}\).
The subcategory \(\operatorname{coh}_{0}\mathbb{X}(2)\) admits ordinary simple sheaves \(S_{\boldsymbol{x}}\) for each \(\boldsymbol{x}\in\mathbb{P}^{1}\setminus\{\infty\}\) and exceptional simple sheaves \(S_{1,0},S_{1,1}\). The ordinary simple sheaf \(S_{\boldsymbol{x}}\) is determined by the exact sequence
\[0\to\mathcal{O}\xrightarrow{X_{2}-\boldsymbol{x}X_{1}^{2}}\mathcal{O}(\vec{c} )\longrightarrow S_{\boldsymbol{x}}\to 0.\]
By contrast, multiplication by \(X_{1}\) leads to the exceptional simples \(S_{1,j}\) for \(j\in\mathbb{Z}/2\mathbb{Z}\):
\[0\to\mathcal{O}((j-1)\vec{x}_{1})\xrightarrow{X_{1}}\mathcal{O}(j\vec{x}_{1}) \to S_{1,j}\to 0.\]
From now on, we assume \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi\})\) is a finest stability data on \(\operatorname{coh}\mathbb{X}(2)\). Obviously, all the ordinary simple sheaves \(S_{\boldsymbol{x}}\) and exceptional simples \(S_{1,0},S_{1,1}\) in \(\operatorname{coh}\mathbb{X}(2)\) are stable since they have no non-trivial subsheaves. Note that \(S_{1,1}(\vec{c})=S_{1,0}\). Up to degree shift, we can assume
\[\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1}).\]
Then \(S_{1,1}^{(2n)}\) is also semistable for any \(n\geq 1\), and \(\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1}^{(2)})=\boldsymbol{\phi} (S_{1,1}^{(2n)})<\boldsymbol{\phi}(S_{1,1})\) since \(\langle S_{1,1}^{(2)}\rangle=\operatorname{add}\{S_{1,1}^{(2n)},n\geq 1\}\).
For any line bundle \(L\), we will denote by \(S_{i,L}\) the unique exceptional simple sheaf satisfying that \(\operatorname{Hom}\left(L,S_{i,L}\right)\neq 0\). For examples, \(S_{1,\mathcal{O}}=S_{1,0},S_{1,\mathcal{O}(\vec{x}_{1})}=S_{1,1}\).
**Lemma 8.2**.: _If a line bundle \(L\) is not semistable, then \(L(-\vec{x}_{1})\) is semistable and the HN-filtration of \(L\) is given by the exact sequence_
\[0\to L(-\vec{x}_{1})\to L\to S_{1,L}\to 0.\]
Proof.: Consider the HN-filtration of \(L\)
\[\begin{CD}\cdots L_{n-2}@>{}>{}>L_{n-1}@>{}>{}>L_{n}=L,\\ @V{}V{A_{n-1}}V\\ A_{n}\end{CD}\]
then the epimorphism \(L\twoheadrightarrow A_{n}\) implies \(A_{n}\) is a torsion sheaf. For any \(i\leq n-1\), \(\operatorname{Hom}\left(A_{i},A_{n}\right)=0\) implies \(\operatorname{Hom}\left(L_{n-1},A_{n}\right)=0\), so we have \(A_{n}=S_{1,0}\) or \(S_{1,1}\), and then \(L_{n-1}=L(-\vec{x}_{1})\). If \(L_{n-1}\) is not semistable, then \(L_{n-2}\neq 0\) and \(A_{n-1}=S_{1,1}\) or \(S_{1,0}\) by similar arguments as above. Note that \(A_{n-1}\neq A_{n}\). Hence \(\{A_{n-1},A_{n}\}=\{S_{1,0},S_{1,1}\}\). So \(\operatorname{Hom}\left(L_{n-2},S_{1,0}\right)=0=\operatorname{Hom}\left(L_{n-2 },S_{1,1}\right)\), a contradiction. Hence \(L_{n-1}\) is semistable. We are done.
**Lemma 8.3**.: _If \(L\) and \(L(\vec{x}_{1})\) are both semistable line bundles, then \(L(-\vec{x})\) are semistable for all \(\vec{x}\geq 0\)._
Proof.: Obviously, it suffices to show that \(L(-\vec{x}_{1})\) is semistable. Then the result follows by induction. For contradiction we assume \(L(-\vec{x}_{1})\) is not semistable. Then by Lemma 8.2, the HN-filtration of \(L(-\vec{x}_{1})\) is given by: \(0\to L(-\vec{c})\to L(-\vec{x}_{1})\to S_{1,L(-\vec{x}_{1})}\to 0\). But \(\operatorname{Hom}\left(L(\vec{x}_{1}),S_{1,L(-\vec{x}_{1})}\right)\neq 0\) implies \(\boldsymbol{\phi}(L(\vec{x}_{1}))<\boldsymbol{\phi}(S_{1,L(-\vec{x}_{1})})< \boldsymbol{\phi}(L(-\vec{c}))\), a contradiction to \(\operatorname{Hom}\left(L(-\vec{c}),L(\vec{x}_{1})\right)\neq 0\). We are done.
For any \(m\in\mathbb{Z}\), we denote by
\[\mathbb{L}_{m}:=\{k\vec{c}\ |\ k\in\mathbb{Z},k<m\}\cup\{\vec{x}_{1}+\mathbb{Z} \vec{c}\}\]
for the subset of \(\mathbb{L}\). Then semistable line bundles with respect to a finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) can be described as follows.
**Proposition 8.4**.: _For any finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{coh}\mathbb{X}(2)\), let \(\{\mathcal{O}(\vec{x})\ |\ \vec{x}\in X\}\) be the set of all the semistable line bundles. Then \(X=\vec{x}_{1}+\mathbb{Z}\vec{c},\ \mathbb{L},\) or \(\mathbb{L}_{m}\) for some \(m\in\mathbb{Z}\)._
Proof.: Recall that we have assumed \(\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1}).\) We claim that all \(\mathcal{O}(\vec{x}_{1}+k\vec{c})\) are semistable for \(k\in\mathbb{Z}\). In fact, assume there exists some \(\mathcal{O}(\vec{x}_{1}+k\vec{c})\) which is not semistable, then by Lemma 8.2, the HN-filtration of \(\mathcal{O}(\vec{x}_{1}+k\vec{c})\) is given by
\[0\to\mathcal{O}(k\vec{c})\to\mathcal{O}(\vec{x}_{1}+k\vec{c})\to S_{1,1}\to 0.\]
Hence \(\boldsymbol{\phi}(\mathcal{O}(k\vec{c}))>\boldsymbol{\phi}(S_{1,1})> \boldsymbol{\phi}(S_{1,0})\). Then \(\operatorname{Hom}\left(\mathcal{O}(k\vec{c}),S_{1,0}\right)\neq 0\) yields a contradiction. This proves the claim. It follows that \(\vec{x}_{1}+\mathbb{Z}\vec{c}\subseteq\,X\subseteq\,\mathbb{L}\).
Assume \(X\neq\vec{x}_{1}+\mathbb{Z}\vec{c}\) and \(X\neq\mathbb{L}\), then we need to show that \(X=\mathbb{L}_{m}\) for some \(m\in\mathbb{Z}\). Note that \(\mathbb{L}=\mathbb{Z}\vec{c}\cup\{\vec{x}_{1}+\mathbb{Z}\vec{c}\}\). Since \(X\neq\vec{x}_{1}+\mathbb{Z}\vec{c}\), there exists some \(k\in\mathbb{Z}\), such that \(\mathcal{O}(k\vec{c})\) is semistable. Since \(\mathcal{O}(k\vec{c}+\vec{x}_{1})\) is semistable, by Lemma 8.3 we obtain that \(\mathcal{O}(\vec{x})\) is semistable for any \(\vec{x}\leq k\vec{c}\). By assumption \(X\neq\mathbb{L}\), then there exists a minimal \(m\), such that \(\mathcal{O}(m\vec{c})\) is not semistable. It follows that \(\mathcal{O}((m-1)\vec{c})\) is semistable and \(\mathcal{O}(k\vec{c})\) is not semistable for any \(k\geq m\). That is, \(X=\mathbb{L}_{m}\). We are done.
### Finest stability data and torsion pairs on \(\operatorname{coh}\mathbb{X}(2)\)
Now we are in the position to give a classification of finest stability data on \(\operatorname{coh}\mathbb{X}(2)\). For this let's introduce some linearly ordered sets as below.
Note that \(\mathbb{L}\) is a linearly ordered set in the natural way, i.e., \(\vec{x}\leq\vec{y}\) if and only if \(\vec{y}-\vec{x}\geq 0\) in \(\mathbb{L}\). Then the subsets \(\mathbb{L}_{m}\left(m\in\mathbb{Z}\right)\) and \(\vec{x}_{1}+\mathbb{Z}\vec{c}\) inherit the linearly ordered relations of \(\mathbb{L}\) naturally. Set
\[\widetilde{\mathbb{X}}=\{(\infty,0),(\infty,\frac{1}{2}),(\infty,1),\boldsymbol {x}\ |\ \boldsymbol{x}\in\mathbb{P}^{1}\setminus\{\infty\}\}.\]
Then \(\widetilde{\mathbb{X}}\) can be viewed as a linearly ordered set by fixing any choice of ordering on the elements of \(\widetilde{\mathbb{X}}\) satisfying \((\infty,0)<(\infty,\frac{1}{2})<(\infty,1)\).
By taking union of two ordered sets and composing some new relations between them, we can obtain the following three family of linearly ordered sets, depending on the choices of ordering on the set \(\widetilde{\mathbb{X}}\):
* \(\mathbb{L}\cup\widetilde{\mathbb{X}}\): by composing \(\vec{x}<\boldsymbol{z}\) for any \(\vec{x}\in\mathbb{L}\) and \(\boldsymbol{z}\in\widetilde{\mathbb{X}}\).
* \(\{\vec{x}_{1}+\mathbb{Z}\vec{c}\}\cup\widetilde{\mathbb{X}}\): by composing \((\infty,0)<\vec{y}<\boldsymbol{z}\) for any \(\vec{y}\in\vec{x}_{1}+\mathbb{Z}\vec{c}\) and \(\boldsymbol{z}\in\widetilde{\mathbb{X}}\setminus\{(\infty,0)\}\).
* \(\mathbb{L}_{m}\cup\widetilde{\mathbb{X}}\): by composing \((m-1)\vec{c}<(\infty,0)<\vec{x}_{1}+(m-1)\vec{c}\), and \(\vec{y}<\boldsymbol{z}\) for any \(\vec{y}\in\mathbb{L}_{m}\) and \(\boldsymbol{z}\in\widetilde{\mathbb{X}}\setminus\{(\infty,0)\}\).
The main result of this subsection is as follows, which gives a classification of finest stability data of \(\operatorname{coh}\mathbb{X}(2)\) under the assumption \(\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1})\).
**Proposition 8.5**.: _Keep notations as above. For any \(\varphi\in\mathbb{L}\cup\widetilde{\mathbb{X}}\), define the subcategories \(\Pi_{\varphi}\subseteq\operatorname{coh}\mathbb{X}(2)\) as below:_
* \(\Pi_{\vec{x}}=\langle\mathcal{O}(\vec{x})\rangle,\vec{x}\in\mathbb{L}\)_;_
* \(\Pi_{(\infty,0)}=\langle S_{1,0}\rangle\)_;_ \(\quad\Pi_{(\infty,\frac{1}{2})}=\langle S_{1,1}^{(2)}\rangle\)_;_ \(\quad\Pi_{(\infty,1)}=\langle S_{1,1}\rangle\)_;_
* \(\Pi_{\boldsymbol{z}}=\langle\boldsymbol{S_{z}}\rangle,\boldsymbol{x}\in \mathbb{P}^{1}\setminus\{\infty\}\)_._
_Then there are only three types of the finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{coh}\mathbb{X}(2)\), where \(\Phi=\mathbb{L}\cup\widetilde{\mathbb{X}}\), \(\mathbb{L}_{m}\cup\widetilde{\mathbb{X}}\) or \(\{\vec{x}_{1}+\mathbb{Z}\vec{c}\}\cup\widetilde{\mathbb{X}}\) respectively._
Proof.: We first prove that \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is a finest stability data for \(\Phi=\{\vec{x}_{1}+\mathbb{Z}\vec{c}\}\cup\widetilde{\mathbb{X}}\). Indeed, it is easy to see that \(\operatorname{Hom}\left(\Pi_{\varphi^{\prime}},\Pi_{\varphi^{\prime\prime}} \right)=0\) for any \(\varphi^{\prime}>\varphi^{\prime\prime}\in\Phi\). Note that \(\langle S^{(2)}_{1,j}\rangle=\operatorname{add}\{S^{(2n)}_{1,j},n\geq 1\}\) for \(j=0,1\). Hence all the indecomposable sheaves which do not belong to any \(\Pi_{\varphi}\) are given by
* \(\mathcal{O}(k\vec{c}),\;k\in\mathbb{Z}\);
* \(S^{(2n+1)}_{1,1},\;n\geq 1;\quad S^{(2n+1)}_{1,0},\;n\geq 1;\quad S^{(2n+2)}_{1,0 },\;n\geq 0\).
They admit HN-filtrations of the following forms:
Hence \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is a stability data on \(\operatorname{coh}\mathbb{X}(2)\). Observe that each semistable subcategory is generated by an indecomposable sheaf. It is easy to see that \(\operatorname{Hom}\left(X,Y\right)\neq 0\neq\operatorname{Hom}\left(Y,X\right)\) for any \(\varphi\in\Phi\) and any non-zero \(X,Y\in\Pi_{\varphi}\). According to Theorem 3.4, the stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is finest.
Similarly, one can prove that \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) is a finest stability data on \(\operatorname{coh}\mathbb{X}(2)\) for \(\Phi=\mathbb{L}\cup\widetilde{\mathbb{X}}\) or \(\mathbb{L}_{m}\cup\widetilde{\mathbb{X}}\).
On the other hand, for any finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{coh}\mathbb{X}(2)\), we know that the simple sheaves are stable since they don't have non-trivial subsheaves. Up to degree shift, we can assume \(\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1})\). Hence \(S^{(2)}_{1,1}\) is semistable and \(S^{(2)}_{1,0}\) is not semistable. It follows that \(S^{(2n)}_{1,1}\) is semistable with \(\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S^{(2)}_{1,1})=\boldsymbol{\phi} (S^{(2n)}_{1,1})<\boldsymbol{\phi}(S^{(2)}_{1,1})\) and \(S^{(2n)}_{1,0}\) is not semistable for any \(n\geq 1\). According to Lemma 4.10, we know that \(S^{(2n+1)}_{1,j}\) is not semistable for any \(n\geq 1\) and \(j=0,1\). Therefore, the semistable sheaves in the subcategory \(\operatorname{coh}_{0}\mathbb{X}(2)\) are characterized by the linearly ordered set \(\widetilde{\mathbb{X}}\). Moreover, according to Proposition 8.4, the semistable line bundles are precisely characterized by one of the three linearly ordered sets \(\vec{x}_{1}+\mathbb{Z}\vec{c}\), \(\mathbb{L}\) or \(\mathbb{L}_{m}\left(m\in\mathbb{Z}\right)\). Note that \(\operatorname{Hom}(\Pi_{\varphi^{\prime}},\Pi_{\varphi^{\prime\prime}})=0\) for all \(\varphi^{\prime}>\varphi^{\prime\prime}\) in \(\Phi\). Then the finest stability data on \(\operatorname{coh}\mathbb{X}(2)\) are exhausted by the above three types.
This finishes the proof.
**Remark 8.6**.: For the classical projective line or an arbitrary smooth elliptic curve, any finest stability data on its category of coherent sheaves is a refinement of the slope stability data by Propositions 6.2 and 7.2, respectively. But this is not the case for weighted projective lines. In fact, the finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{coh}\mathbb{X}(2)\) for \(\Phi=\{\vec{x}_{1}+\mathbb{Z}\vec{c}\}\cup\widetilde{\mathbb{X}}\) is not finer than the slope stability data, since \(\boldsymbol{\phi}(S_{1,0})=(\infty,0)<\vec{x}_{1}+k\vec{c}=\boldsymbol{\phi}( \mathcal{O}(\vec{x}_{1}+k\vec{c}))\) for any \(k\in\mathbb{Z}\), but \(\mu(S_{1,0})>\mu(\mathcal{O}(\vec{x}_{1}+k\vec{c}))\).
As an application of Proposition 8.5, we can obtain a classification of torsion pairs in \(\operatorname{coh}\mathbb{X}(2)\).
**Proposition 8.7**.: _Up to degree shift, the non-trivial torsion pairs \((\mathcal{T},\mathcal{F})\) in \(\operatorname{coh}\mathbb{X}(2)\) are classified as below, where \(P\) is a non-empty subset of \(\mathbb{P}^{1}\) and \(Q\) is a (possibly empty) subset of \(\mathbb{P}^{1}\setminus\{\infty\}\)._
Proof.: For any finest stability data \((\Phi,\{\Pi_{\varphi}\}_{\varphi\in\Phi})\) on \(\operatorname{coh}\mathbb{X}(2)\), the simple sheaves \(S_{1,0}\) and \(S_{1,1}\) are stable. Up to degree shift, we can assume
\[\boldsymbol{\phi}(S_{1,0})<\boldsymbol{\phi}(S_{1,1}).\]
According to Corollary 2.8, the following subcategories
\[\Pi_{\geq\psi}=\langle\Pi_{\varphi}\mid\varphi\geq\psi\rangle, \quad\Pi_{<\psi}=\langle\Pi_{\varphi}\mid\varphi<\psi\rangle;\] \[\Pi_{>\psi}=\langle\Pi_{\varphi}\mid\varphi>\psi\rangle,\quad\Pi_ {\leq\psi}=\langle\Pi_{\varphi}\mid\varphi\leq\psi\rangle\]
give two torsion pairs \((\Pi_{\geq\psi},\Pi_{<\psi})\) and \((\Pi_{>\psi},\Pi_{\leq\psi})\) in \(\operatorname{coh}\mathbb{X}(2)\). These torsion pairs, besides the trivial ones, i.e., \((0,\operatorname{coh}\mathbb{X}(2))\) and \((\operatorname{coh}\mathbb{X}(2),0)\), have the forms in Table 7 due to Proposition 8.5 (up to degree shift), where \(P\) is a non-empty subset of \(\mathbb{P}^{1}\), and \(Q\) is a (possibly empty) subset of \(\mathbb{P}^{1}\setminus\{\infty\}\).
On the other hand, by Proposition 8.1, each stability data on \(\operatorname{coh}\mathbb{X}(2)\) can be refined to a finest one. Then by Proposition 3.5, the classification in Table 6 is complete.
**Remark 8.8**.: The torsion pairs in the category \(\operatorname{coh}\mathbb{X}\) of coherent sheaves for domestic and tubular weighted projective lines \(\mathbb{X}\) have been described in [33], via certain bounded t-structures \((\mathcal{D}^{\prime\leq 0},\mathcal{D}^{\prime\geq 0})\) with \(\mathcal{D}^{\leq-1}\subset\mathcal{D}^{\prime\leq 0}\subset\mathcal{D}^{ \leq 0}\) on the bounded derived category \(\mathcal{D}^{b}(\operatorname{coh}\mathbb{X})\), where \((\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})\) is the standard bounded t-structure on \(\mathcal{D}^{b}(\operatorname{coh}\mathbb{X})\) with heart \(\operatorname{coh}\mathbb{X}\). More precisely, for weighted projective line of weight type (2), there are two different types of non-trivial torsion pairs \((\mathcal{T},\mathcal{F})\):
(1) \((\mathcal{T},\mathcal{F})\) is a tilting torsion pair, corresponding to \((\mathcal{D}^{\prime\leq 0},\mathcal{D}^{\prime\geq 0})\) with finite length heart;
(2) \(\mathcal{T}\subset\operatorname{coh}_{0}\mathbb{X}(2)\) or \(\mathcal{F}\subset\operatorname{coh}_{0}\mathbb{X}(2)\), corresponding to \((\mathcal{D}^{\prime\leq 0},\mathcal{D}^{\prime\geq 0})\) not having finite length heart.
In fact, up to degree shift, the torsion pairs of the second type correspond to \(I,II,III\) or \(VI\), while the tilting torsion pairs correspond to \(IV\) or \(V\) in Table 7. Moreover, the tilting torsion pair IV is induced by the canonical tilting bundle \(\mathcal{O}\oplus\mathcal{O}(\vec{x}_{1})\oplus\mathcal{O}(\vec{c})\), and the tilting torsion pair V is induced by the tilting sheaf \(\mathcal{O}(\vec{x}_{1})\oplus\mathcal{O}(\vec{x}_{1}+\vec{c})\oplus S_{1,1}\).
**Acknowledgments.** This work was supported by the National Natural Science Foundation of China (Nos. 12271448, 11971398), the Natural Science Foundation of Fujian Province (No. 2022J01034), Natural Science Foundation of Xiamen (No. 3502Z20227184) and the Fundamental Research Funds for Central Universities of China (No. 20720220043).
|
2301.04582 | A Personalized Utterance Style (PUS) based Dialogue Strategy for
Efficient Service Requirement Elicitation | With the flourish of services on the Internet, a prerequisite for service
providers to precisely deliver services to their customers is to capture user
requirements comprehensively, accurately, and efficiently. This is called the
``Service Requirement Elicitation (SRE)'' task. Considering the amount of
customers is huge, it is an inefficient way for service providers to interact
with each user by face-to-face dialog. Therefore, to elicit user requirements
with the assistance of virtual intelligent assistants has become a mainstream
way. Since user requirements generally consist of different levels of details
and need to be satisfied by services from multiple domains, there is a huge
potential requirement space for SRE to explore to elicit complete requirements.
Considering that traditional dialogue system with static slots cannot be
directly applied to the SRE task, it is a challenge to design an efficient
dialogue strategy to guide users to express their complete and accurate
requirements in such a huge potential requirement space. Based on the
phenomenon that users tend to express requirements subjectively in a sequential
manner, we propose a Personalized Utterance Style (PUS) module to perceive the
personalized requirement expression habits, and then apply PUS to an dialogue
strategy to efficiently complete the SRE task. Specifically, the dialogue
strategy chooses suitable response actions for dynamically updating the
dialogue state. With the assistance of PUS extracted from dialogue history, the
system can shrink the search scope of potential requirement space. Experiment
results show that the dialogue strategy with PUS can elicit more accurate user
requirements with fewer dialogue rounds. | Demin Yu, Min Liu, Zhongjie Wang | 2023-01-07T04:10:33Z | http://arxiv.org/abs/2301.04582v1 | A Personalized Utterance Style (PUS) based Dialogue Strategy for Efficient Service Requirement Elicitation
###### Abstract
With the flourish of services on the Internet, a prerequisite for service providers to precisely deliver services to their customers is to capture user requirements comprehensively, accurately, and efficiently. This is called the "Service Requirement Elicitation (SRE)" task. Considering the amount of customers is huge, it is an inefficient way for service providers to interact with each user by face-to-face dialog. Therefore, to elicit user requirements with the assistance of virtual intelligent assistants has become a mainstream way. Since user requirements generally consist of different levels of details and need to be satisfied by services from multiple domains, there is a huge potential requirement space for SRE to explore to elicit complete requirements. Considering that traditional dialogue system with static slots cannot be directly applied to the SRE task, it is a challenge to design an efficient dialogue strategy to guide users to express their complete and accurate requirements in such a huge potential requirement space. Based on the phenomenon that users tend to express requirements subjectively in a sequential manner, we propose a Personalized Utterance Style (PUS) module to perceive the personalized requirement expression habits, and then apply PUS to an dialogue strategy to efficiently complete the SRE task. Specifically, the dialogue strategy chooses suitable response actions for dynamically updating the dialogue state. With the assistance of PUS extracted from dialogue history, the system can shrink the search scope of potential requirement space. Experiment results show that the dialogue strategy with PUS can elicit more accurate user requirements with fewer dialogue rounds.
Service Requirement Elicitation (SRE), Task-oriented dialogue, Personalized Utterance Style (PUS), Requirement space
## 1 Introduction
The Service Requirement Elicitation (SRE) has been becoming an important task in service oriented computing. With the development of computing servitization, more and more virtual services are deployed on the web. The variety of services that enriches life, also makes it difficult to choose appropriate services that meet user's personalized requirements. Without complete and accurate requirements, service providers cannot support requirements with customized services, thus leading to a mismatch between service supply and demand. The prerequisite for service providers is to capture diverse user requirements with efficient method, which is called the "Service Requirement Elicitation (SRE)" task. However it is an inefficient way for service providers to communicate with customers by face-to-face dialog considering the huge amount of customers. In this
case, it is becoming a trend to elicit user requirements with the assistance of intelligent dialogue systems [1; 2].
We adopt a special tree-like structure to model the complex requirement in dialogue. Unlike traditional dialogue tasks like flight checking or accommodation reservation, the dialog system for SRE task pays less attention to the entity recommendation but to capturing accurate needs of various domains [3]. In this way, the requirements in SRE task enable the decoupling of users and service providers. However, the requirements proposed by users in SRE are generally complex, hierarchical and transboundary. Most complex requirements can be divided into more finer-grained sub-requirements, and generally need to be satisfied by services from different domains. We introduce the Requirement Tree (\(R\_Tree\)) [2] to model complex requirements. From the perspective of dialog system, the huge amount of candidate requirements supported by multi-domain services and the dependencies among different levels of sub-requirements constitute a potential requirement space [3].
We propose **Personalized Utterance Style** (PUS)to describe requirement expression preference to assist requirement elicitation. Considering that users have personalized habits when expressing their requirements [4], the PUS characterizes users from three dimensions: Requirement Planning (RP) preference, Act State Transfer (AST) preference, and Requirement Attribute (RA) preference. The RP focuses on the tendency of following potential requirements. The AST aims to figure out features of dialogue actions in multiple rounds. The RA, like user profile, is used to record preference of constraints.
We regard our dialogue strategy as a requirement node searching and optimizing process in a huge potential requirement space. The goal of dialog system for SRE is to efficiently guide users to express requirements and generate the target \(R\_Tree\). As an example of requirement space shown in Fig. 1, there are dependencies between different levels of requirements distributed in multiple domains. Part of a traveler's requirements is to find a restaurant, and then for the restaurant, they want to learn about the local specialties. Once confirming the requirement about hotel, dialog system needs to guide user to express the following potential requirement, such as attraction or sub-requirements about hotel. To elicit complete and accurate service requirements in huge requirement space, we propose a PUS-based dialogue strategy, which uses \(R\_Tree\) to model dialog state and target dialog goal. Specifically, our strategy chooses suitable requirements for dynamically expanding the requirement tree of dialogue state with the assistance of PUS. The PUS can assist strategy generates suitable response dialog action with guiding requirement expression by narrowing the scope of potential requirement space. We apply our strategy to customized CrossWOZ dataset [5], and results show that the strategy which integrates utterance style can elicit more accurate and complete user requirements with fewer dialogue rounds and a higher success rate.
The main contributions of this work are as follows:
* We propose **Personalized Utterance Style** (PUS) to portray the requirements expression preference from the perspective of the whole personalized dialog context.
* We design a PUS-based dialog strategy to elicit complex requirements to generate dialog target \(R\_Tree\) by shrinking the search scope in the potential requirement space.
* We prove that the PUS-based dialog strategy can effectively elicit more accurate and complete user requirements and complete conversation in limited rounds. Furthermore, the introduced PUS feature can improve the performance of deep neutral network for dialogue.
Figure 1: The Process of Exploring Requirements in Potential Requirement Space.
## 2 Related Work
### Task-oriented Dialogue System for Service Requirement
Service requirements elicitation plays a decisive role in the service oriented computing [6]. Xiang et al. [7] propose to use ontology list to to narrow generic service requirements and capture services requirements. Hwang et al. [8] regard QoS as discrete random variables with probability mass functions to address the service selection problem. The goal of requirement elicitation task is to find out complete requirements as much detail as possible so that it could become easier to support downstream task [9]. However, these approaches strongly rely on domain data and manual rules, leading to bad interaction of users. In this paper, we consider the SRE task as a special dialogue task.
Conversation is a user-friendly method to elicit user requirements. With the development of dialogue technology in recent years, more and more task-oriented dialogue systems are applied to interact with users to provide specific services. There are many challenges for dialogue system to achieve specific tasks. First of all, the dialogue study has been blocked by the scale of data available. Zhu et al. [5] propose a multi-domain dialogue dataset with rich annotation within travel scenarios to advance dialogue system modeling. Secondly, depending on the different functions of dialogue systems, research usually focuses on dialogue state tracking (DST) and dialogue policy (DP) to improve dialogue effectiveness. Lee et al. [10] propose a universal and scalable model called SUMBT to track slots state by learning the relations between domain-slot-types and slot-values appearing in utterances. Takanobu et al. [11] propose a join policy optimization and reward estimation method to estimate the reward signal and infers the user goal in the dialog sessions. Zhang et al. [12] propose an three stage end-to-end dialogue model to handle the problem that multiple responses can be appropriate for the same dialog context. However, these dialogue systems with static slots of specific tasks and are limited to cover complete requirements and dialogue state, resulting in an inability to handle the user's complex requirements and continue the dialog.
### Personalized Features in Dialogue
Creating a virtual personal assistants (VPA) that can have realistic conversations with humans is one of the ultimate goals of Artificial Intelligence (AI). One of the challenges of VPA is to take the personalized information into account in order to achieve a customized conversation service for the user. Current research about user personalization are distributed in different stage of dialogue system. Qian et al. [13] propose the IMPChat that owns the consistent personality with the corresponding user by modeling implicit user profile. Zhang et al. [15] achieve the personalized response retrieval and generation based on similar user profile attributes. He et al. [17] propose to harness dialog historical information to adapt different scenarios and leverage the knowledge base and user profiles to achieve high quality conversational recommendation. We summarize the relevant works on personalized information for different dialogue tasks, as shown in Table 1. However, these studies have focused more on users' profile attributes for external entities, ignoring the users' preference of dialogue utterance from the whole context. Unlike the traditional user profile, the personalized utterance style (PUS) in this work aims to characterize the user's requirement expression preference during the conversation. Considering that language style or user profile can be distinguished or extracted from discrete utterances, the PUS requires the whole dialogue context to be analyzed.
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Task**} & **Related** & **Personalized** \\ & **Work** & **Information** \\ \hline Personalized & & Language Style \\ Dialogue & [13][14] & \& User Profiles \\ Consistency & & & User Profiles \& \\ \hline Personalized & & User Profiles \& \\ Response & [15][16] & Knowledge \\ Generation & & Template \\ \hline Personalized & & Dialogue History, \\ Recommendation & [17] & User Profiles \& \\ & & Knowledge \\ \hline \hline \end{tabular}
\end{table}
Table 1: Different task of personalized dialogue
## 3 Problem Formulation
### Requirement Model for SRE
We introduce the requirement tree (\(R\_Tree\)) [2] to model the complex structure of user requirements in SRE task. The \(R\_Tree\) can be considered as a tree structure consisting of Requirement Nodes. The definition of \(R\_Tree\) is shown in the following equation:
\[R\_Tree=(V,E) \tag{1}\]
Where \(V\) is the set of requirement nodes and \(E\) is the edges between requirements, which describes the dependency between two nodes. The edge \(e_{ij}\in E\) stands for the directed edge from \(v_{i}\) to \(v_{j}\), which means that \(v_{i}\) is the sub-requirement of \(v_{j}\). Each requirement node \(v_{i}\in V\) can be described as following equation:
\[v_{i}=(Req,Slots,Sub\_Reqs) \tag{2}\]
Where \(Req\) describes user's functional requirements, which is textual expression. \(Sub\_Reqs\) donates the dependencies of child nodes. For requirement node \(v_{i}\), the child nodes can be defined as \(Sub\_Reqs_{i}=\{v_{j}|\exists e_{ij}\in E\}\). \(Slots\) represents the non-functional constraints of user requirements. For each \(Slot\in Slots\), it is defined as equation 3.
\[Slot=(Slot\_Name,Slot\_Value) \tag{3}\]
Where \(Slot\_Name\) and \(Slot\_Value\) denote the specific attribute of constraint. The target \(R\_Tree\) in Fig. 2 shows an example of decomposing user requirements into an requirement tree.
As we introduce dialog system to elicit service requirements, the goal of system is to generate the target \(R\_Tree\) by multiple rounds of dialog. For every requirement expressed by user, dialog system construct details of requirement as an requirement node and update the \(R\_Tree\) based on dependencies among requirements.
### Framework of Dialogue System
Considering that completing specific tasks with the assistance of intelligent dialog system has become a mainstream way, we design a dialog system for SRE task based on generic structure of task-oriented dialog system. The overview of our dialogue system framework is shown in Fig. 2, which consists of four major modules: Natural Language Understanding (NLU), Natural Language Generation (NLG), Dialogue State Tracking (DST) and Dialogue Policy (DP) [18]. These modules form a message pipeline to handle user request and generate response. Different with general task-oriented dialogue system, we use requirement tree (\(R\_Tree\)) rather than static slots to model dialogue state. We abstract the semantic dialog action into the following triples:
\[<DA,Req,Slots> \tag{4}\]
where \(DA\) donates the Dialog Act, which can be interpreted as the atomic units of a conversation [19]. Similar to the concept in equation 3, the \(Slots\) in dialog action donates the different constraints of \(Req\).
During the conversation, the dialog policy uses the requirement discussed in the current context as a node pointer to locate the corresponding requirement node in the current dialog state. Based on the request dialogue action and the semantic slot information, relevant operations are performed on the requirement nodes. When the current requirement has been discussed completely, the dialog policy searches the potential requirement space and selects possible other requirements to continue the dialogue to achieve effective guidance. At this time, the requirement tree in the dialog state is also continuously expanded to maintain the current dialog context information. In order to confirm the user's current requirement and guide the expansion of the requirement more efficiently, the dialog policy will combine the user's requirement, the dialog state and the PUS to confirm the appropriate response action.
Figure 2: Overview of our Dialogue Framework
Note that how to tag user actions from user speech (NLU) or generate neutral language according to dialogue action (NLG) is not our main focus. We try to apply the existing accurate methods, like BERT1, to achieve these tasks and do not make enough innovative contributions.
Footnote 1: [https://github.com/thu-coai/ConvLab-2/tree/master/convlab2/nlu/jointBERT](https://github.com/thu-coai/ConvLab-2/tree/master/convlab2/nlu/jointBERT)
### Personalized Utterance Style Models
In order for the dialogue system to perceive requirements more accurately, the grasp of the user's dialogue habits is crucial in addition to the understanding of semantic information. Personalized utterance style can help to estimate the user's requirement expression trend and select targeted dialog strategy based on the user's dialog habits, so as to better guide the requirement expression. In this paper, we propose a **Personalized Utterance Style** (PUS) module as shown in Fig. 3, which mainly consists of three different parts: Act State Transfer (AST) Preference, Requirement Planning (RP) Preference and Requirement Attribute (RA) Preference.
#### 3.3.1 Act State Transfer Preference
Dialog Act (\(DA\)) represents the semantic motivation of current utterance. As an important part of the user requirement expression, learning user dialog state transfer preferences is an important task. We focus on modeling act state transfer habits from dialog acts sequence in dialogue history. The goal of this part is to learn \(DA\) states to generate system response act according to the user request act. The act state transfer preference, modeled as a state transfer graph, can be described as \(A=(Q,\Sigma,\delta)\), where \(Q\) donates the all act state trained from of DA sequence. The \(\Sigma\) donates the set of all dialog acts. The \(\delta\) donates the state transfer functions defined as follows:
\[DA_{out}=\delta(s,DA_{in}) \tag{5}\]
where \(DA\in\Sigma\) donates the input and output dialog acts, and the \(s\in Q\) donates the current state. According to related linguistic theory [20][21], we design a specific dialogue act category for the SRE task shown on Table. 2. Some cases in scenarios of SRE show that the DA system in Table 2 can basically cover the different utterances of dialogue.
#### 3.3.2 Requirement Planning Preference
The requirement planing model represents the user preference of requirement expression during the dialogue, which is able to guide the order of dialogue topic. The requirement planning preference can be modeled as Equation 6
\[P(\hat{g}_{j}|\{g_{j-N},...,g_{j-1}\}) \tag{6}\]
where \(g_{i}\) donates the \(i-th\) requirement which has been confirmed and \(\hat{g}_{j}\) donates the next potential requirement for conversation. The order of requirement expression is not unique. We assume that users with different PUS have different personalized habits to express all their requirements in unique order. These habits can provide dialogue system with effective tendency to plan the order of requirement confirmation. Most of requirement relationships are uncertain. Based on the \(R\_Tree\) model, there are dependencies between requirements, such as restaurant-Peking Duck, etc. The dependency is presented in \(R\_Tree\) that they are not in a topological order but a kind of parent-child node structure.
#### 3.3.3 Requirement Attribute Preference
Like the user profile [22], RA module is the process of understanding users by extracting their interests on different constraints. Based on RA preference, the system can guide users to express
Figure 3: The Overview of Personalized Utterance Style Modeling
their requirements in a more directional way during dialogue. We design a key-value structure to describe the RA preference. Specially we use text mining method [22] to extract constraints of different requirements from the dialogue corpus to generate or update the preferences.
The RA preference can be expressed as \(E=(U,P)\), where \(U\) describes the general information of the user. \(P\) represents the set of preferences for requirements attributes under different domains. The each identified requirement attribute preference \(p_{i}\) can be expressed as follows:
\[p_{i}=(Req,Name,Value,Freq) \tag{7}\]
\(Name\) and \(Value\) respectively donates the constraint name and value of preference of \(Req\). Here we divide the preference value into two types: Interval Value and Discrete Value. \(Freq\) represents the frequency of requirement constraint appeared in the last limited rounds of corpus. It can reflect the importance of user to different constraint attributes.
## 4 Methodology
### Personalized Utterance Style Mining
#### 4.1.1 Act State Transfer Preference
According to the \(DA\) categories proposed in Table 2, all utterances are tagged with dialogue acts so that all \(DA\)s in a dialogue about specific \(R\_Tree\) constitute a sequence of dialogue acts:
\[Seq_{DA}=\{Act^{1}_{usr},Act^{1}_{sys},...,Act^{N}_{usr},Act^{N}_{sys}\} \tag{8}\]
We consider that the \(Seq_{DA}\) contains the potential act state transfer (AST) tendencies. Users with different \(DA\) preference tend to choose different response \(DA\) when they in the same state of context. In this work, we use weighted finite state transducers (wFST) [23] to capture the latent AST preference. Unlike RNN, wFST can explicitly track the entire path it traversed, which gives additional symbolic constraints and information about the dialog history. A trained wFST serves as a scaffold for dialog history tracking. Meanwhile, wFST is more interpretable, as each state is explicitly represented by an action distribution which is easier for humans to interpret model decision. As part of PUS, the AST module can model the dialog acts states to generate response act according to request. Given all \(DA\) sequences of one user, we use the greedily node splitting algorithm[23] to train the initial wFST. For a trained wFST, we can understand the dialog state node meaning by checking input and output edges. Given the current dialogue act sequence \(\{Act^{1}_{usr},Act^{1}_{sys},...,Act^{t-1}_{usr}\}\), the last user dialogue act \(Act^{t-1}_{usr}\) is fed to wFST, and then it gives a probability density for the likelihood of transfer from the current state to each of the possible dialog acts. Compared to RNN, the wFST can not only return the current state embedding but also all information of the state it traversed since the start state.
#### 4.1.2 Requirement Planning Preference
Users with different PUS preference differ in the requirement expression order even within the same dialogue goal. Intuitively, given an \(R\_Tree\)
\begin{table}
\begin{tabular}{l l l l} \hline
**Rough Category** & **Meaning** & **Detailed Act** & **Example** \\ \hline \multirow{2}{*}{Statement} & \multirow{2}{*}{\begin{tabular}{l} Offer requirements, \\ entity or events \\ \end{tabular} } & State\_in & I want to find a five-star hotel. \\ & & State\_out & The hotel provides pick-up service. \\ \hline \multirow{3}{*}{Question} & \multirow{3}{*}{
\begin{tabular}{l} Offer different questions. \\ \end{tabular} } & Ques\_select & Do you prefer A or B? \\ & & Ques\_rec & How about traveling by bus? \\ & & Ques\_req & What do you need for the hotel? \\ \hline \multirow{2}{*}{Response} & Response to & Resp\_acc & This restaurant is a good choice. \\ & statements or questions. & Resp\_deny & I don’t think so. \\ & & Resp\_vag & I have no opinion. \\ \hline \multirow{2}{*}{General} & Verbalized acts & \multirow{2}{*}{General} & Thanks; Bye; \\ & or unknown acts. & & \\ \hline \end{tabular}
\end{table}
Table 2: The Dialogue Acts for Requirement Elicitation Scenarios
as the initial user requirements, the process of user requirement expression can be regarded as the node travelling for all requirement nodes of \(R\_Tree\), and there are different expression orders of requirement sequence. Given the sequence of requirements in the current context, dialogue system should continuously plan the next requirement to be identified, and the candidate requirements are chosen based on the potential patterns of requirement sequence in dialogue history. Therefore, we consider requirement planning as a simplified Sequence Recommendation (SeqRec) task [24].
The classical SeqRec tries to model the sequential user behaviors with items, mining the connection between user contexts, requirement, and goals for more accurate and customized recommendations. As a simplified version of the sequence recommendation task, we only need to focus on the recommendation of requirement sequences for different users, so we use Factorizing Personalized Markov Chains (FPMC)[24] to learn the latent preferences between users and requirement sequences. FPMC is designed by personalized transfer on incomplete Markov chains, which combines the features of both matrix decomposition and Markov chain recommendation models. Specifically, it combines the personalized set Markov Chains with the factorized transition cube results to capture user preferences over requirement-to-requirement transitions. Given all requirement sequences of all users, we use the S-BPR algorithm[24] to train a personalized transition cube to learn the requirement expression preference of different users. For the requirement planning task, FPMC can effectively solve the problems of user-requirement data coefficients and serialized local dependency modeling.
#### Requirement Attribute Preference
We mine requirement attribute preference from the user dialogue corpus. The essential task of preference mining is to effectively count the \(Freq\), which represents the frequency of the attribute that appears in the recent limited rounds of the user's corpus. Frequency can reflect the importance that users place on this constraint. For user's last \(N\) rounds of dialogue, the \(Freq\) domain of each preferred attribute is continuously updated as the conversation progresses, which also reflects that the user's attribute preferences are keeping changing. For each user's dialogue corpus, we analyze utterance of each round. For each utterance, the triples of dialog action, defined at equation 4, can be extracted with natural language understanding (NLU) model or manual rules. Since the scenarios in which entity attributes appear can be either active expressions of the user or assisted system recommendations, different scenarios for the appearance of entity attributes need to update \(Freq\) according to different strategies. We design the \(Freq\) update strategy as shown in Table 3. Users' preferences on requirement attributes can be reflected by their rejection or acceptance to different constraints. The \(Freq\) would upgrade when the constraint is approved. And the \(Freq\) would be decreased if user reject the constraint.
### Multi-Round dialogue for Requirement Elicitation
#### Requirement State Updating based on Requirement Pattern
In traditional task-oriented dialogue scenarios, dialogue state is usually updated by static slots, which is the basis for dialogue goals. These strategy will get impressive results in specific dialogue scenarios, such as fight enquiry. However, because
\begin{table}
\begin{tabular}{l l l} \hline Constraint scenarios & Strategy to update \(Freq\) & Example \\ \hline User initiated & \(Freq\) = \(Freq\) + 2 & User: I want to find a cheap hotel. \\ User Accept & \(Freq\) = \(Freq\) + 1 & Bot: Do you need a 5a attraction? \\ User: That’s great! & \\ & \(Freq\) = \(Freq\) -1 & Bot: Do you need a cheap hotel? \\ User: No, I prefer a luxury one. \\ \hline \end{tabular}
\end{table}
Table 3: Updating Strategy of \(Freq\)
of the complex requirements in SRE task, the strategy based on static-slot filling cannot play to their advantage. As for SRE task,the \(R\_Tree\) can effectively modeling complex requirements in multiple domains and won't be limited by fixed-slot in specific scenarios, which provides a foundation for dynamically requirement building.
In the real world, there will be lots of redundant requirement fragments appearing together on different \(R\_Tree\) in similar environment. As an example about travel in a food city shown in Fig. 4, most requirements of tourists will associate famous restaurant with the node of travel. These requirement fragments frequently appear together in user's requirement trees are called requirement patterns (\(RP\)) [2], which can be aggregated to form new coarse-grained requirements. These requirement patterns exist in different fields and can be used as the basic components to provide support for requirements mining. As for service provider, these requirement patterns do not need to be decomposed because there is already specific solution for those requirements [3].
Base on \(RP\), we can effectively address the domain-limited problem of dialogue state in dialogue and realize the dynamic expansion of the dialogue state. We use requirement pattern mining algorithm[2] to extract frequent requirement patterns from requirement trees in specific domain, and apply a RP-based requirement steering strategy to capture user requirements. Specifically, we store all \(RP\) from different domains in requirement pattern library (\(RPs\)) to provide necessary domain information. In dialogue process, once the basic requirement is captured, the \(RP\) in pattern library is matched according to requirement. Then we use static information in user portrait, such as external environment, time and place, and user attribute preferences to choose candidate \(RP\) that meet user requirements. Using \(RP\), system can choose different dialogue actions to perceive the possible expansion direction of current requirement and plan the direction of conversation topic. Finally, requirement reconstruction is completed with assistance of \(RP\). \(RP\) can be extracted from multiple fields in an offline state, and be used for real-time requirements adaptation using user profile. The dialogue topic guidance strategy based on \(RP\) can effectively realize adaptation process between multiple dialogue fields. As for strategy based on fixed slot filling, users can only choose to offer their requirements directly when there is no domain slot information. As the complexity of requirement increases, the unlimited fixed slot filling strategy will lead to rapid growth of dialogue rounds, leading to terrible dialogue experience.
Figure 4: The Example of Dialogue Strategy
#### Incorporating PUS into Requirement Elicitation Strategy
We design a dialogue strategy based on PUS to elicit user requirements more efficiently. The main target of PUS incorporation can be divided into two parts. Firstly, determine the next dialogue act response to the user request. Secondly, plan direction of dialogue topics with personalized preference. The overview of dialogue strategy with PUS is shown in Fig. 5.
For the entire requirement elicitation process of dialogue, we abstract the dialogue process into an inner-outer loop process according to requirement of \(R\_Tree\) and requirement constraints. Based on the Requirement Planning in PUS and \(RP\) matching, the dialogue strategy plans the order of requirement confirmation in the outer loop. Throughout the whole dialogue process, every requirement in \(R\_Tree\) would have different states for the dialogue system: potential requirement in space, candidate requirement for planning, requirement accepted, current requirement in context, requirement confirmed. All requirements before dialogue can be regarded as a huge potential requirement space, shown in Fig. 1. Then based on \(RP\) matching, the system would search some requirements as candidate requirements for requirement guidance. When user proactively proposes requirements or accepts the requirements from system recommendations, the requirement of requirements can be accepted. According to the requirement in the current context, there will be several rounds to discuss constraints about current requirement. When all constraints about current requirement have been confirmed, the system will choose another topic to continue the dialogue.
```
1:\(RPs\)
2:\(R\_Tree_{t}\)
3:\(R\_s_{c}\leftarrow\) Initial \(rp\) requirement extract from \(RPs\)
4:for\(len(Reqs_{c})>0\)do
5:\(Re\_Req\gets PUS.RP(R\_Tree_{c},RPs)\)
6:for\(not\)\(Topic\_Change(Context)\)do
7:\(<DA,Req_{cur},Slots>\leftarrow NLU()\)
8: Update \(R\_Tree_{c}\) State
9:\(Re\_DA\gets PUS.AST(DA,R\_Tree_{s})\)
10:\(Re\_Slots\leftarrow PUS.RA(RPs,Slots)\)
11: NLG(\(Re\_DA,Re\_Slots\))
12:endfor
13:\(Reqs_{c}\gets Update(RPs,Re\_Req,R\_Tree_{s})\)
14:endfor
15:\(R\_Tree_{t}\gets R\_Tree_{s}\)
16:return\(R\_Tree_{t}\)
```
**Algorithm 1** Algorithm of Requirement Elicitation Strategy
As for each round of dialogue, the dialogue system needs to determine the response action, described in equation 4, according to user state and dialogue context. We use the personalized act state transfer preference to predict the \(DA\) of response action. User profiles and constraints in \(RP\) can be equipped to generate suitable \(Slots\) of response. In the inner loop, the strategy will also judge whether the current requirement has changed according to user action and system state of \(R\_Tree\), so as to realize the transfer between inner and outer loops. The algorithm 1 describes the whole dialogue strategy for SRE. The \(Predict()\) function is to focus on the current system state and PUS to analyze user action preference, and generate optimal response action. The dialogue process is terminated when the requirement planning module no longer has candidate requirements or the user proposes to end the conversation. The dialogue example shown in Fig. 4 demonstrates the process of requirement elicitation of our dialogue strategy.
Figure 5: Dialogue Strategy with PUS
## 5 Experiments
### Experimental Setup
#### SRE Task Setup
The SRE task is oriented toward scenarios where dialogue system interacts with user and guides them to express complete requirements. For each conversation, user owns their requirements as dialogue goal, which can be regarded as a \(R\_Tree\). Then users express requirements or constraints of every requirement node through multiple dialog rounds in a personalized manner. Users can express their requirements either actively or passively. Then the dialogue system captures user's requirements and updates the target \(R\_Tree\). The conversation stops when all user requirements have been confirmed or the dialog round reaches the max number, which is set as 17 for the average dialogue corpus. Otherwise, the conversation is regarded as failed.
We re-implement the requirement-oriented dialogue strategy from Tian et al.[2], which also applies requirement tree to dialogue state and incorporates external knowledge into requirement discovery. To achieve personalized interaction with dialogue system, we use an Agenda-based user simulator [25] to simulate user behaviors. It uses a compact representation of the user goal of \(R\_Tree\) to guide dialog topic. Given a dialogue example with requirement tree, the simulator analyzes \(R\_Tree\) as dialogue goal. Then, it pushes different requirements and constraints into a stack structure in the order of their appearance in the original dialogue. At each turn, the simulator receives system dialogue actions, modifies its state, and outputs user dialogue actions according to specific hand-crafted rules. Both our pus-based dialogue strategy and requirement-based dialog strategy [2] interact with simulator to evaluate dialogue effectiveness.
#### Dataset Annotation
We use CrossWOZ2, a task-oriented conversation dataset, to extract the utterance style of different users.As for SRE task, dialogue system no longer does entity-oriented search actions, but focuses on understanding and guiding the user requirement expression. Therefore, we modify and delete question-answer pairs in original dataset that involve entity search with manual rules. Although the original dataset was not designed with personalized characteristics, the workers who played the USER role were encouraged to dialogue on their terms when collecting the data [5]. It is important to emphasize that the PUS preferences mentioned in this paper mainly refer to the workers' habits when expressing their requirements, rather than just attribute preferences in other user-profile related studies. For the SRE scenario, we use the dialogue act categories shown in Table 2 to re-annotate dialogue acts in CrossWOZ. We also use manual annotation to assist in labeling cases that cannot be handled by the rules if necessary.
Footnote 2: [https://github.com/thu-coni/CrossWOZ](https://github.com/thu-coni/CrossWOZ)
#### Evaluation Metrics
The number of dialogue rounds indicates efficiency of interaction during process wherein dialog system identifies user requirements. Typically, complex and redundant dialogues may result in a bad user experience. In other words, the fewer the dialog rounds, the better the user interaction. Thus, this study takes **Dialogue Round**, the average rounds of dialogues with the same scale requirements, as an evaluation index. In addition, we uses **Recall** and **F1** to evaluate results of generated \(R\_Tree\). Considering that every requirement nodes in target \(R\_Tree\) needs to be confirmed by user (requirement recommendation or statement), there would not be false requirement, and the Precision would always be 1. The PRF of \(R\_Tree\) can be calculated by following equations.
\[P_{tree}=\frac{R\_Tree_{init}\cap R\_Tree_{gen}}{R\_Tree_{gen}} \tag{9}\]
\[R_{tree}=\frac{R\_Tree_{init}\cap R\_Tree_{gen}}{R\_Tree_{init}} \tag{10}\]
\[F1_{tree}=2\times\frac{P_{tree}\times R_{tree}}{P_{tree}+R_{tree}} \tag{11}\]
Where \(R\_Tree_{init}\) indicates the initial \(R\_Tree\) of user requirements and \(R\_Tree_{gen}\) indicates the \(R\_Tree\) generated by dialogue system with conversation. Finally, we use **Success Rate** to indicate the effectiveness of dialogue system.
### Main Results
#### 5.2.1 Dynamic Requirement
Elicitation with PUS
For the SRE task, we use the re-annotated CrossWOZ dataset to complete dialogue experiments. The user simulator uses current dialogue state and different personalized weights to simulate different dialogue styles. Different weights can achieve different action selection tendencies according to dialogue records. Dialogue system updates dialogue state and determines response action for each user action request, depending on user's PUS and \(RP\). We apply our requirement-based dialogue strategy [2] and our pus-based dialogue strategy on dialogue simulation platform [26] and interact with dialog simulator for multi-turn dialogues.
After completing 4988 dialogue tests from 67 users, we obtained experiment results as shown in Fig. 6. Fig. 6(a) and Fig. 6(b) show accuracy of \(R\_Tree\) and dialogue success rate under different requirement scales. It is worth noting that accuracy of dialogue requirements includes not only requirement nodes, but also constraints of different requirements. Fig. 6(c) shows average number of rounds per dialogue for different requirement scales. The results show that the dialogue strategy incorporating pus can obtain more complete and accurate user requirements with fewer dialogue rounds, and can guarantee the success of dialogue under medium requirement scale. The main reason of difference is that system can take different dialog actions in specific context and search different directions in requirement space. According to requirement planning and action state selection in PUS, the dialogue system can select a more suitable direction with a higher probability. However, due to limitation of maximum number of rounds, the dialogue strategy that only relies on the \(RP\) can only attempt different search states in limited steps. The dialogue success rate is relatively low when there are too many potential requirements. Furthermore, since we set a large enough maximum number of dialog rounds, the dialog of strategy with PUS can confirm all requirements, indicating that the strategy can choose a more suitable response action with assistance of PUS.
#### 5.2.2 Ablations of Personalized Features
One of the advantages of SRE is that it enables decoupling of users and service providers, where the two parties do not need to interact but achieve to match the corresponding requirements with the assistance of \(R\_Tree\). While system response actions on real data are not necessarily the optimal requirement guidance strategy, the accuracy (PRF) of response actions can demonstrate the effectiveness of PUS model from dialogue history. We perform the PUS validity experiments of response action based on the Convlab framework [26]. Specifically, we
Figure 6: The Results of Requirement Elicitation
Figure 7: The Results of PUS Ablation
simulator and a dialogue policy incorporating the PUS model to interact in CrossWOZ. The dialogue policy is based on the rule policy in Convlab. We also choose worker id as the different user identifier. As a comparison, we extract action state transfer trends and requirement planning trends, called global tendency of dialog strategy, from all dialogue data. This global tendency contains the feature of generic response action. The experiment results are shown in Fig. 7.
The results show that the strategy with PUS has a significant improvement in the PRF of response action, which means the response action considering personalized features can get more effective selection in real scenarios. In addition, as the scale of requirement increases, dialogue strategy can rely more on the previous context information to get more effective response accuracy due to the distant contextual memory of utterance style model.
#### The Validity of PUS for Deep Network
It is necessary for deep learning models to have the ability to embed the user personalized feature, which is a potential preference hidden in user dialogue. Therefore, we designed a personalized feature extraction method, which can effectively vectorize the PUS model and integrate it into the deep learning model. The PUS can be embedded with following equation:
\[\begin{split} PUS_{embed}=& Concat(MLP(wFST(u,DA_{seq})), \\ & MASK(MLP(FPMC(u,Req_{seq}))))\end{split} \tag{12}\]
Taking DAMD[12] model as an example, which is an end-to-end task dialogue model. It tackles multi-domain response generation problem through proposed multi-action data augmentation framework. We try to incorporate PUS embedding into DAMD structure of action decision-making step. The whole structure is shown in the Fig. 8. Note that we ignore the text generation module of original DAMD because it is not necessary for SRE task.
The results in the Table 4 show that the DAMD model incorporating PUS features can improve the effect of Slot and Act prediction. However, the DAMD is an end-to-end dialog model and model whole process of dialog. We notice that the PUS only works on fine-grained utterance, while for overall dialogue indicators, such as dialogue target matching (Match), there is a large gap in performance. The possible reason is that we introduced PUS features in dialogue action recognition and decision-making module.
\begin{table}
\begin{tabular}{l c c} \hline \hline & **DAMD** & **DAMD (PUS)** \\ \hline
**Joint Goal (\%)** & 46.2 & **47.6** \\
**Slot Acc (\%)** & 96.3 & **96.4** \\
**Slot F1 (\%)** & 88.3 & **88.7** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results on DAMD with PUS
Figure 8: Structure of DAMD Net with PUS embedding
## 6 Conclusion
In this work, we focus on a human-machine dialogue system for service requirement elicitation task. We design a dialogue strategy base on the personalized utterance style(PUS) model and requirement pattern to effectively elicit user requirements. The dialogue strategy is able to dynamically expand the requirement tree and search target requirement based on cross-domain requirement patterns. We validated effectiveness of the dialogue strategy incorporating PUS model in a customized dataset, which can achieve the SRE task efficiently. As an essential part of service computing, efficient service requirement elicitation is the solid foundation for subsequent service provision. In the future, we hope to explore more factors of personalized utterance style in complex dialogue strategies to achieve a better experience for the service environment.
## 7 Conflict of Interest Statement
All authors declare that they have no conflicts of interest.
|
2302.12868 | Stiffness matrix method for modelling wave propagation in arbitrary
multilayers | Natural and engineered media usually involve combinations of solid, fluid and
porous layers, and accurate and stable modelling of wave propagation in such
complex multilayered media is fundamental to evaluating their properties with
wave-based methods. Here we present a general stiffness matrix method for
modelling waves in arbitrary multilayers. The method first formulates stiffness
matrices for individual layers based on the governing wave equations for fluids
and solids, and the Biot theory for porous materials. Then it utilises the
boundary conditions considered at layer interfaces to assemble the layer
matrices into a global system of equations, to obtain solutions for reflection
and transmission coefficients at any incidence. Its advantage over existing
methods is manifested by its unconditional computational stability, and its
validity is proved by experimental validations on single solid sheets, porous
layers, and porous-solid-porous battery electrodes. This establishes a powerful
theoretical platform that allows us to develop advanced wave-based methods to
quantitatively characterise properties of the layers, especially for layers of
porous materials. | Ming Huang, Frederic Cegla, Bo Lan | 2023-02-24T20:06:07Z | http://arxiv.org/abs/2302.12868v1 | # Stiffness matrix method for modelling wave propagation in arbitrary multilayers
###### Abstract
Natural and engineered media usually involve combinations of solid, fluid and porous layers, and accurate and stable modelling of wave propagation in such complex multilayered media is fundamental to evaluating their properties with wave-based methods. Here we present a general stiffness matrix method for modelling waves in arbitrary multilayers. The method first formulates stiffness matrices for individual layers based on the governing wave equations for fluids and solids, and the Biot theory for porous materials. Then it utilises the boundary conditions considered at layer interfaces to assemble the layer matrices into a global system of equations, to obtain solutions for reflection and transmission coefficients at any incidence. Its advantage over existing methods is manifested by its unconditional computational stability, and its validity is proved by experimental validations on single solid sheets, porous layers, and porous-solid-porous battery electrodes. This establishes a powerful theoretical platform that allows us to develop advanced wave-based methods to quantitatively characterise properties of the layers, especially for layers of porous materials.
## I Introduction
Multilayered media are ubiquitous in nature as well as in engineering structures, with examples spanning from minerals and the Earth's crust to composite laminates and electrochemical systems (such as batteries). The layers normally consist of different material types, commonly involving solids and oftentimes fluid and porous layers. The resulting structures are generally complex, with a representative case being the electrodes of lithium-ion batteries, with two fluid-saturated porous layers coated on a thin solid metal sheet.
Consequently, multilayered media typically exhibit unique structural and functional properties. Achieving and maintaining the properties relies strongly on non-destructive methods to evaluate them and to monitor their changes. Ultrasonic testing is frequently used for this purpose and has facilitated many application areas, such as the estimation of the thicknesses of thin layered sheets [1; 2], the inspection of composite laminates [3; 4], and the characterisation of the layered structures of lithium-ion batteries [5; 6]. Such evaluations generally utilise the information about the layers that the ultrasonic waves carry after interacting with them. Therefore, for an ultrasonic method to deliver optimal results, understanding the wave interactions with the layered media through physical models is essential.
Matrix formulations are most commonly used for such models with arbitrary numbers of layers. As the earliest formulation, the transfer matrix method [7; 8] relates the stresses and displacements at one interface of a layer to those at the other interface. With continuity at the interfaces considered, the method produces a matrix for the entire system by multiplying the matrices of individual layers. Solving the final matrix equation in different ways can deliver solutions for wave reflections and transmissions as well as guided waves in the system. However, it suffers from computational instability at large \(fd\) (\(f\) is frequency and \(d\) layer thickness) as inhomogeneous evanescent waves arise [9; 10]. To resolve this problem, a number of alternative formulations were proposed, and the global matrix method [11; 12] emerged as one of the preferred substitutes. Instead of using matrix multiplications, the global matrix method assembles the transfer matrices of all layers into a global system of equations. Its stability is achieved by eliminating the diverging exponential terms using different spatial origins for the partial waves [13; 14]. Another attractive approach to achieving computational stability at large \(fd\) is the stiffness matrix method [15; 16], which uses a stiffness matrix to link the stresses at the two interfaces of a layer to the respective displacements. This method can be implemented in a recursive form [15] similar to the matrix multiplication of the transfer matrix method, and can also be formulated in a global matrix form [16] in the same fashion as for transfer matrices. Both forms are unconditionally computationally stable. These matrix-based methods have received numerous applications in various fields, most notably in seismology [17], ocean acoustics [18], composites [19; 20] and guided ultrasonics [21; 22].
Although the matrix formulations focused mainly on solid layers and occasionally on fluid ones (e.g. [23; 24]), significant attention also centred on the development of matrix descriptions for porous layers. The formulations were mostly based on the Biot theory [25; 26; 27; 28] to describe the complex wave mechanics in fluid-saturated porous media. They all utilised transfer matrices to model the Biot waves in individual porous layers, but relied differently on matrix multiplication and global matrices to assemble layer matrices. The matrix multiplication method only applies to layered systems containing pure porous layers [29; 30] or alternating fluid/solid-porous layers [31], while the global matrix method is a more general model for arbitrarily stacked fluid, solid and porous layers [32; 33]. These developments have seen applications in, e.g., seismology [34] and sound-absorbing materials [33]. They are particularly useful for the inverse determination of important properties (porosity, tortu
osity etc.) for porous materials [33; 35], and compared to non-matrix based inversion studies that are limited to single/double-layered settings [36; 37; 38], they can deal with more complex cases with many layers of different types. However, the aforementioned instability problem arises, not only in the instability-prone matrix multiplication method but also to the supposedly-stable global matrix method. The problem in the former case occurs constantly at large \(fd\)[30], while that of the latter case, according to our analyses, arises less predictably at large incident angles of porous layers.
In this work, we present an intrinsically-stable stiffness matrix method for layered media with arbitrary numbers of fluid, solid and porous layers. The novelties and advantages are threefold. Firstly, the proposed method employs stiffness matrices to describe individual layers and uses global matrices to model assembled layers. Owing to the superior stability of both formulations, the proposed method exhibits intrinsic computational stability, and most importantly, it works exceptionally well for the cases that challenged existing methods. This allows us to reliably model highly-transmissible waves and guided modes that involve large wave angles in porous layers. Secondly, the proposed method is optimised to have simple expressions for both stiffness matrices and boundary conditions even for complex porous layers, thus enabling much easier computer implementation. Lastly, the proposed method is validated against experimental measurements to be working well for arbitrary single solid sheets, porous layers, and porous-solid-porous combinations. Based on the contributions, advanced ultrasonic techniques may be developed to characterise the properties of layered media, and an imperative application is to quantify the performance determinants of porous electrodes in lithium-ion batteries.
The paper is organised as follows. Section II provides a concise review of the well-established wave physics in different layer materials. Then Sec. III presents the proposed stiffness matrix method, demonstrating how the wave physics in individual layers are modelled by stiffness matrices and how the layer matrices are assembled into global matrices to obtain wave solutions. This is followed by experimental validations in Sec. IV to showcase the applicability of the method to complex layered media with a single solid/porous layer and multiple porous-solid-porous layers. Section V concludes this paper.
## II Wave physics in individual layers
We address a general problem of wave propagation in an arbitrary multilayer, as illustrated in Fig. 1. The medium contains \(n\) layers with infinite dimensions in the \(x\)- and \(y\)-directions, and layer \(i\) is defined by interfaces \(z_{i-1}\) and \(z_{i}\) in the \(z\)-direction with thickness \(d_{i}=z_{i}-z_{i-1}\). The layered system is bounded by half-spaces \(0\) and \(n+1\) on the two sides.
Individual layers in the system are each occupied by a fluid, elastic solid or fluid-saturated porous material. All three types of materials are treated as macroscopically isotropic and homogeneous. Wave propagation in individual layers is governed by different wave physics, depending on the nature of the layer material. The well-established governing equations in the three considered materials are reviewed in this section, and the stiffness matrix method will be formulated based upon them in the next.
### Fluid and solid layers
We start with fluid and solid layers that involve relatively simple wave physics. With linear elasticity assumed and body forces neglected, wave propagation in fluid and solid materials is governed by the wave equation [39]
\[\nabla\cdot\mathbf{\sigma}-\rho(\partial^{2}\mathbf{u}/\partial t^{2})=0, \tag{1}\]
where \(\mathbf{u}(\mathbf{x},t)\) and \(\mathbf{\sigma}(\mathbf{x},t)\) are the particle displacement field and the stress tensor, both as function of the position \(\mathbf{x}\) and time \(t\). \(\rho\) is the mass density of the material. \(\nabla\) denotes the vector differential operator, namely \(\nabla=[\partial/\partial x,\partial/\partial y,\partial/\partial z]^{\rm T}\). The stress tensor \(\mathbf{\sigma}\) is related to the strain tensor \(\mathbf{\varepsilon}\) by the generalised Hooke's law, given differently for fluid and solid materials by
\[p =K\varepsilon_{kk}, \tag{2}\] \[\sigma_{ij} =(K-2G/3)\varepsilon_{kk}\delta_{ij}+2G\varepsilon_{ij}, \tag{3}\]
Figure 1: Wave propagation in a multilayered medium with fluid, solid and porous layers. The medium has \(n\) layers and is bounded by half-spaces \(0\) and \(n+1\). An incident wave from \(0\) induces a reflected wave back into \(0\) and a wave transmitted through the layers to \(n+1\). Layer \(i\) is defined by the interfaces of \(z_{i-1}\) and \(z_{i}\) and the thickness of \(d_{i}=z_{i}-z_{i-1}\). Each layer is consisted of a fluid, solid or fluid-saturated porous material. There is one wave (longitudinal L) in fluid, two waves (longitudinal L and shear S) in solid, and three (fast L1 and slow L2 longitudinal, and shear) in a porous material.
where fluid has an omnidirectional stress \(p\) (also known as pressure) as a result of dilatation \(\varepsilon_{kk}\), while solid exhibits direction-dependent stress \(\sigma_{ij}\) (\(i,j\in\{x,y,z\}\)) due to dilatation \(\varepsilon_{kk}\) and shearing \(\varepsilon_{ij}\) (\(i\neq j\)) in the medium. Note that Einstein summation over the repeated index \(k\) from \(x\) to \(z\) is assumed for the dilatation \(\varepsilon_{kk}\). \(\delta_{ij}\) is the Kronecker delta. \(K\) and \(G\) are the bulk and shear moduli. The strain component \(\varepsilon_{ij}\) and dilatation \(\varepsilon_{kk}\) are related to the displacement field by
\[\varepsilon_{ij} =(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})/2, \tag{4}\] \[\varepsilon_{kk} =\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}=\nabla\cdot \mathbf{u}. \tag{5}\]
Substituting the above strain-displacement relation into the Hooke's law and then into Eq. 1 leads to an equation for the displacement field \(\mathbf{u}\). The equation can be further written for the scalar \(\varphi\) and vector \(\mathbf{H}\) potentials by using the Helmholtz decomposition [39; 40]
\[\mathbf{u}=\nabla\varphi+\nabla\times\mathbf{H}. \tag{6}\]
The two potentials describe respectively the longitudinal (dilatational) and shear (rotational) waves in the medium. For fluids, the vector potential \(\mathbf{H}\) vanishes due to the absence of shear waves, and solving the resulting wave equation for the scalar potential delivers a longitudinal wave solution with wave speed
\[c_{\mathrm{L}}=\sqrt{K/\rho}. \tag{7}\]
For solids, the wave equation is decoupled into two equations for the scalar and vector potentials respectively. The two equations give respectively the longitudinal and shear wave solutions, having the wave speeds of
\[c_{\mathrm{L}}=\sqrt{(K+4G/3)/\rho},\;c_{\mathrm{S}}=\sqrt{G/\rho}. \tag{8}\]
Note we have conveniently treated the two differently-polarised shear waves as a single wave mode because they have the the same speed in the considered isotropic solid. This applies to the porous media as discussed below.
### Fluid-saturated porous layers
Now we consider fluid-saturated porous layers. The wave physics are much more complicated in this case and are addressed by the widely-employed Biot theory [25; 26; 27] (or empirically by other models such as [41; 42]). Here, the solid frame is considered to be continuous, and the pores fully connected and saturated with fluid. The propagating wave is subjected to attenuation induced by scattering in the solid phase, and viscous and inertial dissipation in the fluid phase. When the wavelength is large compared to the average pore size, the propagating wave can be treated in a homogenised sense. The average displacement fields can then be characterised by \(\mathbf{u}^{\mathrm{s}}(\mathbf{x},t)\) and \(\mathbf{u}^{\mathrm{f}}(\mathbf{x},t)\) in the solid ('s') and fluid ('f') phases. The two wave fields are coupled and described by the wave equations [25; 43]
\[\nabla\cdot\mathbf{\sigma}^{\mathrm{s}}-\frac{\partial^{2}}{\partial t ^{2}}(\rho_{11}\mathbf{u}^{\mathrm{s}}+\rho_{12}\mathbf{u}^{\mathrm{f}}) \tag{9}\] \[-bF\frac{\partial}{\partial t}(\mathbf{u}^{\mathrm{s}}-\mathbf{u}^{ \mathrm{f}})=0,\] \[\nabla\cdot\mathbf{\sigma}^{\mathrm{f}}-\frac{\partial^{2}}{\partial t ^{2}}(\rho_{12}\mathbf{u}^{\mathrm{s}}+\rho_{22}\mathbf{u}^{\mathrm{f}})\] (10) \[-bF\frac{\partial}{\partial t}(\mathbf{u}^{\mathrm{f}}-\mathbf{u}^{ \mathrm{s}})=0,\]
where \(\rho_{11}\), \(\rho_{12}\) and \(\rho_{22}\) are the effective densities [25]
\[\rho_{12} =-(\alpha_{\infty}-1)\phi\rho_{\mathrm{f}}, \tag{11}\] \[\rho_{11} =(1-\phi)\rho_{\mathrm{s}}-\rho_{12},\] (12) \[\rho_{22} =\phi\rho_{\mathrm{f}}-\rho_{12}, \tag{13}\]
where \(\rho_{\mathrm{s}}\) and \(\rho_{\mathrm{f}}\) are the densities of the solid and fluid materials. \(\phi\) is the porosity and \(\alpha_{\infty}\) the tortuosity. The parameter \(b\) in the wave equations represents the viscous damping factor, given by [43; 35]
\[b=\eta\phi^{2}/k_{0} \tag{14}\]
with \(\eta\) being the dynamic viscosity of the fluid and \(k_{0}\) the permeability of the fluid through the porous medium. \(F\) is the viscous correction factor with a generalised form of [43; 25; 44]
\[F=\sqrt{1+iMf/(2f_{c})}, \tag{15}\]
which is dependent on the frequency \(f\). \(M\) is the shape factor, which is generally taken as unity. \(f_{c}\) is the viscous characteristic frequency, given by [44; 35]
\[f_{c}=\eta\phi/(2\pi\alpha_{\infty}\rho_{\mathrm{f}}k_{0}). \tag{16}\]
In Eqs. 9 and 10, the stress tensors \(\mathbf{\sigma}^{\mathrm{s}}\) and \(\mathbf{\sigma}^{\mathrm{f}}\) in the solid and fluid phases are related to the strain tensors by [43; 25]
\[\sigma^{\mathrm{s}}_{ij} =[(P-2N)\varepsilon^{\mathrm{s}}_{kk}+Q\varepsilon^{\mathrm{f}}_ {kk}]\delta_{ij}+2N\varepsilon^{\mathrm{s}}_{ij}, \tag{17}\] \[\sigma^{\mathrm{f}}_{ij} =[Q\varepsilon^{\mathrm{s}}_{kk}+R\varepsilon^{\mathrm{f}}_{kk}] \delta_{ij}, \tag{18}\]
with the strain tensors linked to the respective displacement fields in the solid and fluid by Eqs. 4 and 5. \(P\) and \(N\) are the effective longitudinal and shear moduli of the medium. \(R\) represents the pressure required for forcing a certain volume of the liquid into the medium whilst maintaining the total volume. \(Q\) signifies the coupling of volume change between the solid and liquid. These four elastic parameters are given by [27; 35]
\[P =K_{\mathrm{b}}+K_{\mathrm{f}}(1-\phi-K_{\mathrm{b}}/K_{\mathrm{s} })^{2}/\phi_{\mathrm{eff}}+4G_{\mathrm{b}}/3, \tag{19}\] \[N =G_{\mathrm{b}},\] (20) \[R =\phi^{2}K_{\mathrm{f}}/\phi_{\mathrm{eff}},\] (21) \[Q =\phi K_{\mathrm{f}}(1-\phi-K_{\mathrm{b}}/K_{\mathrm{s}})/\phi_{ \mathrm{eff}}, \tag{22}\]
where \(\phi_{\mathrm{eff}}=\phi+K_{\mathrm{f}}/K_{\mathrm{s}}(1-\phi-K_{\mathrm{b}}/K_ {\mathrm{s}})\) is an effective porosity of the fluid-saturated medium. \(K_{\mathrm{s}}\) and \(K_{\mathrm{f}}\) are
the bulk moduli of the solid and fluid materials, respectively. \(K_{\rm b}\) and \(G_{\rm b}\) are the in-vacuo bulk and shear moduli of the solid frame (namely, the porous solid after draining out the saturated fluid).
Using the Helmholtz decomposition, the wave equations in Eqs. 9 and 10 can be decoupled into two equations for longitudinal and shear waves, respectively [25]. The longitudinal wave equation delivers two solutions with the wave speeds of
\[c_{\rm L1}^{2} =\frac{2(PR-Q^{2})}{P\tilde{\rho}_{22}+R\tilde{\rho}_{11}-2Q\tilde {\rho}_{12}-\sqrt{\Delta}}, \tag{23}\] \[c_{\rm L2}^{2} =\frac{2(PR-Q^{2})}{P\tilde{\rho}_{22}+R\tilde{\rho}_{11}-2Q\tilde {\rho}_{12}+\sqrt{\Delta}}, \tag{24}\]
where
\[\tilde{\rho}_{12} =\rho_{12}+ibF/\omega, \tag{25}\] \[\tilde{\rho}_{11} =\rho_{11}-ibF/\omega,\] (26) \[\tilde{\rho}_{22} =\rho_{22}-ibF/\omega,\] (27) \[\Delta =(P\tilde{\rho}_{22}+R\tilde{\rho}_{11}-2Q\tilde{\rho}_{12})^{2}\] \[\qquad-4(PR-Q^{2})(\tilde{\rho}_{11}\tilde{\rho}_{22}-\tilde{\rho }_{12}^{2}).\]
The two longitudinal waves both involve coupled motion in the solid frame and the saturated fluid. The faster wave L1 propagates dominantly in the solid frame, with a speed slower than that of the solid and faster than the saturated fluid. The slower wave L2 travels predominantly in the fluid phase, having a speed slower than the fluid. The shear wave equation yields only one solution involving coupled motion between the solid and fluid, with the wave speed of
\[c_{\rm S}^{2}=N\tilde{\rho}_{22}/(\tilde{\rho}_{11}\tilde{\rho}_{22}-\tilde{ \rho}_{12}^{2}). \tag{29}\]
## III Stiffness matrix method
With the wave physics in individual layers discussed, here we address the propagation of waves in the entire layered system by formulating the stiffness matrix method.
### Stiffness matrix for the two interfaces of a layer
The formulation begins by establishing a stiffness matrix relation for a layer \(i\) by [15]
\[\begin{bmatrix}\mathbf{\sigma}_{i-1}\\ \mathbf{\sigma}_{i}\end{bmatrix}^{i}=\mathbf{K}^{i}\begin{bmatrix}\mathbf{u}_{i-1} \\ \mathbf{u}_{i}\end{bmatrix}^{i}, \tag{30}\]
which relates the stress vectors \(\mathbf{\sigma}\) on the two interfaces to the respective displacement vectors \(\mathbf{u}\) by the stiffness matrix \(\mathbf{K}\). Note that layer \(i\) is bounded by the interfaces \(i-1\) and \(i\) (see Fig. 1), and layers and interfaces are differently indicated by superscripts and subscripts throughout this paper wherever possible. The displacement and stress vectors each has \(m\) components that are representative of the \(m\) wave modes in the layer. As clarified in the preceding section, the fluid, solid and porous layers considered in this work have \(m=1\), \(2\) and \(3\) wave modes, respectively. Here we emphasise again that the two differently-polarised shear waves are treated as a single shear wave mode for solids and porous materials. We choose the displacement and stress vectors as
\[\text{Fluid}: \quad\mathbf{u}=\left[u_{z}\right]^{\rm T}, \mathbf{\sigma}=\left[p\right]^{\rm T}, \tag{31}\] \[\text{Solid}: \quad\mathbf{u}=\left[u_{z},u_{x}\right]^{\rm T}, \mathbf{\sigma}=\left[\sigma_{zz},\sigma_{xz}\right]^{\rm T},\] (32) \[\text{Porous}: \quad\mathbf{u}=\left[u_{z},u_{x}^{\rm s},\hat{u}_{z}\right]^{ \rm T}, \mathbf{\sigma}=\left[p,\sigma_{xz}^{\rm s},\hat{\sigma}_{zz}\right]^{ \rm T}, \tag{33}\]
where the dependencies on space \(\{x,y,z\}\) and time \(t\) are implied. For porous layers, the components \(u_{x}^{\rm s}\) and \(\sigma_{xz}^{\rm s}\) are for the solid frame; the other four components, however, contain the displacements and stresses of both the solid and fluid phases, given by
\[u_{z} =(1-\phi)u_{z}^{\rm s}+\phi u_{z}^{\rm f}, \tag{34}\] \[\hat{u}_{z} =u_{z}^{\rm s}-u_{z}^{\rm f},\] (35) \[p =\sigma_{zz}^{\rm s}+\sigma_{zz}^{\rm f},\] (36) \[\hat{\sigma}_{zz} =\sigma_{zz}^{\rm s}/(1-\phi)-\sigma_{zz}^{\rm f}/\phi. \tag{37}\]
which are chosen to ease the definition of boundary conditions in the next subsection; this will be discussed in detail below.
To obtain the displacement and stress vectors, we write the scalar \(\varphi\) and vector \(\mathbf{H}\) potentials in a layer as [30, 21]
\[\text{Fluid}: \quad\varphi=U_{\rm L},\,\mathbf{H}=\mathbf{0}, \tag{38}\] \[\text{Solid}: \quad\varphi=U_{\rm L},\,\mathbf{H}=\left[0,U_{\rm S},0\right]^{ \rm T},\] (39) \[\text{Porous}: \quad\varphi^{\rm s}=U_{\rm L1}+U_{\rm L2},\,\mathbf{H}^{\rm s}= \left[0,U_{\rm S},0\right]^{\rm T},\] (40) \[\quad\varphi^{\rm f}=\mu_{\rm L1}U_{\rm L1}+\mu_{\rm L2}U_{\rm L2},\,\mathbf{H}^{\rm s}=\mu_{\rm S}\mathbf{H}^{\rm s}, \tag{41}\]
with
\[U_{i}=(a_{i}^{+}e^{ik_{iz}z}+a_{i}^{-}e^{-ik_{iz}z})e^{i(k_{x}x-\omega t)}, \tag{42}\]
where \(i\in\{\rm L,L1,L2,S\}\). Here each scalar potential corresponds to a longitudinal wave mode and each vector potential to a shear mode. Without loss of generality, the nonzero components in the vector potentials are assumed to be in the \(y\) direction, meaning that the particle motion of the shear waves lies in the \(x\)-\(z\) plane. Each wave mode travels in both the forward ('+') and backward ('-') directions, and these two waves are represented by the two terms in Eq. 42 with amplitudes \(a^{+}\) and \(a^{-}\). In addition, it is implied in the potentials that the incident wave has a wavenumber component of \(k_{x}\) in the transverse \(x\)-direction of the layered system. According to the Snell's law, all propagating waves in all layers have the same transverse component of wavenumber. As a result, the common term \(e^{i(k_{x}x-\omega t)}\) in Eq. 42 is invariant,
and the wavenumber component in the \(z\)-direction for a wave \(i\) (\(i\in\{\mathrm{L},\mathrm{L}1,\mathrm{L}2,\mathrm{S}\}\)) is given by
\[k_{iz}=\sqrt{k_{i}^{2}-k_{x}^{2}}, \tag{43}\]
where the total wavenumber \(k_{i}\) is related to the wave speed \(c_{i}\) by \(k_{i}=\omega/c_{i}\), with \(\omega=2\pi f\) being the angular frequency. Also, we have to emphasise that the potentials are given separately for the solid and fluid phases in a porous medium, in the same fashion as for the displacement and stress fields in Sec. II. The potentials for the two phases are related by the ratios \(\mu_{i}\) due to the coupling of the wave fields between the two phases. The ratios can be obtained from the physical relations in Sec. II, given by [35]
\[\mu_{\mathrm{L}1} =(\tilde{\rho}_{11}-P/c_{\mathrm{L}1}^{2})/(Q/c_{\mathrm{L}1}^{2} -\tilde{\rho}_{12}), \tag{44}\] \[\mu_{\mathrm{L}2} =(\tilde{\rho}_{11}-P/c_{\mathrm{L}2}^{2})/(Q/c_{\mathrm{L}2}^{2 }-\tilde{\rho}_{12}),\] (45) \[\mu_{\mathrm{S}} =-\tilde{\rho}_{12}/\tilde{\rho}_{22}. \tag{46}\]
In the potentials, the \(z\)-coordinate origin of each wave is defined to be at its entry to the layer. Therefore, forward propagating waves (L+, L1+, L2+, S+) in a layer \(i\) have their origin at \(z_{i-1}\) and backward propagating waves (L-, L1-, L2-, S-) have their origin at \(z_{i}\). Such selection ensures that every single exponential term in the potentials is normalised to unity on its entry interface and decays towards the exit interface. This essentially eliminates the numerical overflow of the exponential terms as the waves become inhomogeneous, thus impeding numerical instability [15; 21].
Based on the potentials, the components of the displacement vector \(\mathbf{u}\) in Eqs. 31-33 are obtained using the Helmholtz decomposition (Eq. 6). For a given layer \(i\), the displacement vector can then be evaluated at its two interfaces \(z_{i-1}\) and \(z_{i}\), leading to a matrix-form result
\[\begin{bmatrix}\mathbf{u}_{i-1}\\ \mathbf{u}_{i}\end{bmatrix}^{i}=\begin{bmatrix}\mathbf{D}^{+}&\mathbf{D}^{-} \mathbf{E}\\ \mathbf{D}^{+}\mathbf{E}&\mathbf{D}^{-}\end{bmatrix}^{i}\begin{bmatrix}\mathbf{ A}^{+}\\ \mathbf{A}^{-}\end{bmatrix}^{i}=\mathbf{D}^{i}\mathbf{A}^{i}, \tag{47}\]
where a common term \(ie^{i(k_{x}x-\omega t)}\) is implied. \(\mathbf{D}\) is the displacement matrix, and the expressions of its \(m\times m\) sub-matrices \(\mathbf{D}^{+}\) and \(\mathbf{D}^{-}\) are provided in Table 1 for the three material types. \(\mathbf{E}\) is a \(m\times m\) diagonal matrix, with the diagonal being \(\begin{bmatrix}e^{ik_{\mathrm{L},d}}\end{bmatrix}^{\mathrm{T}}\), \(\begin{bmatrix}e^{ik_{\mathrm{L},d}},e^{ik_{\mathrm{S},d}}\end{bmatrix}^{ \mathrm{T}}\) and \(\begin{bmatrix}e^{ik_{\mathrm{L},d}},e^{ik_{\mathrm{L},2,d}},e^{ik_{\mathrm{S}, d}}\end{bmatrix}^{\mathrm{T}}\) for fluid, solid and porous materials. \(\mathbf{A}\) is the amplitude vector, and its sub-vectors are \(\mathbf{A}^{\pm}=\begin{bmatrix}a_{\mathrm{L}}^{\pm}\end{bmatrix}^{\mathrm{T}}\), \(\begin{bmatrix}a_{\mathrm{L}}^{\pm},a_{\mathrm{S}}^{\pm}\end{bmatrix}^{ \mathrm{T}}\) and \(\begin{bmatrix}a_{\mathrm{L}1}^{\pm},a_{\mathrm{L}2}^{\pm},a_{\mathrm{S}}^{\pm} \end{bmatrix}^{\mathrm{T}}\) for the three types of materials.
The components of the stress vector \(\mathbf{\sigma}\) in Eqs. 31-33 are obtained using the respective strain-displacement relations and the Hooke's laws in Sec. II, yielding
\[\begin{bmatrix}\mathbf{\sigma}_{i-1}\\ \mathbf{\sigma}_{i}\end{bmatrix}^{i}=\begin{bmatrix}\mathbf{S}^{+}&\mathbf{S}^{-} \mathbf{E}\\ \mathbf{S}^{+}\mathbf{E}&\mathbf{S}^{-}\end{bmatrix}^{i}\begin{bmatrix} \mathbf{A}^{+}\\ \mathbf{A}^{-}\end{bmatrix}^{i}=\mathbf{S}^{i}\mathbf{A}^{i}, \tag{48}\]
with \(ie^{i(k_{x}x-\omega t)}\) implied. \(\mathbf{S}\) is the stress matrix and its \(m\times m\) sub-matrices \(\mathbf{S}^{+}\) and \(\mathbf{S}^{-}\) are listed in Table 1 for the three material types. The diagonal matrix \(\mathbf{E}\) and the amplitude vector \(\mathbf{A}\) are the same as those for the displacement vector in Eq. 47.
Equations 47 and 48 relate the displacement and stress vectors on the layer interfaces to the amplitude vector. By obtaining the amplitude vector \(\mathbf{A}^{i}\) from Eq. 47 and substituting into Eq. 48, we arrive at the stiffness matrix relation in Eq. 30, with the stiffness matrix given by
\[\mathbf{K}^{i} =\begin{bmatrix}\mathbf{K}_{11}^{i}&\mathbf{K}_{12}^{i}\\ \mathbf{K}_{21}^{i}&\mathbf{K}_{22}^{i}\end{bmatrix}=\mathbf{S}^{i}\left( \mathbf{D}^{i}\right)^{-1} \tag{49}\] \[=\begin{bmatrix}\mathbf{S}^{+}&\mathbf{S}^{-}\mathbf{E}\\ \mathbf{S}^{+}\mathbf{E}&\mathbf{S}^{-}\end{bmatrix}^{i}\left(\begin{bmatrix} \mathbf{D}^{+}&\mathbf{D}^{-}\mathbf{E}\\ \mathbf{D}^{+}\mathbf{E}&\mathbf{D}^{-}\end{bmatrix}^{i}\right)^{-1},\]
where \(\mathbf{K}_{pq}^{i}\left(p,q\in\{1,2\}\right)\) are \(m\times m\) sub-matrices.
### Boundary conditions across the interface of two layers
After establishing the stiffness relation for the two interfaces of a layer, now we consider the wave interaction across the interface of two neighbouring layers. The interaction is defined by the boundary conditions, which can be expressed in matrix form for the displacement and stress vectors as
\[\mathbf{B}_{i}^{i}\mathbf{u}_{i}^{i}+\mathbf{B}_{i}^{i+1}\mathbf{u}_{i}^{i+ 1} =\mathbf{0}, \tag{50}\] \[\mathbf{C}_{i}^{i}\mathbf{\sigma}_{i}^{i}+\mathbf{C}_{i}^{i+1}\mathbf{ \sigma}_{i}^{i+1}=\mathbf{0}, \tag{51}\]
\begin{table}
\begin{tabular}{l l} Layer & Matrix \\ \hline \multirow{2}{*}{Fluid} & \(\mathbf{D}^{+}=\begin{bmatrix}k_{Lx}\end{bmatrix}\), & \(\mathbf{D}^{-}=\begin{bmatrix}-k_{Lx}\end{bmatrix}\) \\ & \(\mathbf{S}^{+}=\begin{bmatrix}\mu\omega^{2}\end{bmatrix}\), & \(\mathbf{S}^{-}=\begin{bmatrix}\mu\omega^{2}\end{bmatrix}^{2}\) \\ \hline \multirow{5}{*}{Solid} & \(\mathbf{D}^{+}=\begin{bmatrix}k_{Lx}&k_{x}\\ k_{x}&-k_{Sz}\end{bmatrix}\), & \(\mathbf{D}^{-}=\begin{bmatrix}-k_{Lx}&k_{x}\\ k_{x}&k_{Sz}\end{bmatrix}\) \\ & \(r_{i}=k_{z}^{2}(Q+\mu_{i}R)/N\), \(s_{i}=[\phi q_{i}+(\phi-1)r_{i}]/[\phi(\phi-1)]\) for \(i\in\{\mathrm{L}1,\mathrm{L}2,\mathrm{S}\}\). \\ \end{tabular}
\end{table}
Table 1: Sub-matrices of the displacement \(\mathbf{D}\) and stress \(\mathbf{S}\) matrices for different types of layer materials. For the porous case, \(h_{i}=\mu_{i}-1\), \(g_{i}=1+\phi h_{i}\), \(q_{i}=k_{i}^{2}(P+\mu_{i}Q)/N-2k_{x}^{2}\), \(r_{i}=k_{z}^{2}(Q+\mu_{i}R)/N\), \(s_{i}=[\phi q_{i}+(\phi-1)r_{i}]/[\phi(\phi-1)]\) for \(i\in\{\mathrm{L}1,\mathrm{L}2,\mathrm{S}\}\).
across the interface \(i\) (subscript) between layers \(i\) and \(i+1\) (superscript); see Fig. 1 for the numbering of layers and interfaces. \(\mathbf{B}\) and \(\mathbf{C}\) are the boundary matrices for the displacement and stress vectors.
The boundary matrices vary depending on the material types of the two neighbouring layers. When the two layers are made of the same material, the displacements and stresses of the two layers need to be continuous across the interface, resulting in the boundary matrices
\[\mathbf{B}_{i}^{i}=-\mathbf{B}_{i}^{i+1}=\mathbf{C}_{i}^{i}=-\mathbf{C}_{i}^{i +1}=\mathbf{I}_{m\times m}, \tag{52}\]
where \(\mathbf{I}_{m\times m}\) represents the identity matrix of size \(m\).
When the two layers have different material types, they involve different numbers of wave modes \(m\), leading to different numbers of components in the displacement and stress vectors. In this case, the boundary matrices are complicated by the fact that there are not only continuity conditions but also Dirichlet conditions for the displacement and/or stress components. Across a fluid-solid interface for example, beside the two continuity conditions \(u_{z}^{\mathrm{f}}=u_{z}^{\mathrm{s}}\) and \(p^{\mathrm{f}}=\sigma_{zz}^{\mathrm{s}}\), an extra Dirichlet condition exists for the shear stress of the solid layer (namely \(\sigma_{xz}^{\mathrm{s}}=0\)) due to the lack of shear stresses in the neighbouring fluid. These three conditions translate to the boundary matrices in Table 2, alongside those for other interface types.
Before proceeding, we shall revisit the choice of the displacement and stress components for porous layers in Eqs. 34-37. As aforementioned, the choice makes it straightforward to define boundary conditions, specifically: (1) Equating \(u_{z}\) in Eq. 34 to the \(u_{z}\) of a neighbouring layer prescribes the conservation of fluid and solid volume through a fluid- or solid-porous interface. (2) Applying the Dirichlet condition to \(\hat{u}_{z}\) in Eq. 35 ensures no mass is lost over a solid-porous interface. (3) The equality of \(p\) in Eq. 36 to the \(p\) or \(\sigma_{zz}\) of a neighbouring fluid or solid layer satisfies the continuity of normal stress across the interface. (4) The use of the Dirichlet condition on \(\hat{\sigma}_{zz}\) in Eq. 37 additionally guarantees the continuity of fluid pressure across a fluid-porous interface. These aspects deliver very simple boundary matrices as given in Table 2 despite the complex wave physics in porous layers.
### Global matrix for the calculation of reflection and transmission coefficients
Substituting the stiffness matrix relation in Eq. 30 into the stress boundary condition in Eq. 51, we have
\[\mathbf{C}_{i}^{\mathrm{f}}(\mathbf{K}_{21}^{i}\mathbf{u}_{1-1}^{i}+\mathbf{K }_{22}^{i}\mathbf{u}_{i}^{i})+\mathbf{C}_{i}^{i+1}(\mathbf{K}_{11}^{i+1} \mathbf{u}_{1}^{i+1}+\mathbf{K}_{12}^{i+1}\mathbf{u}_{i+1}^{i+1})=\mathbf{0}, \tag{53}\]
which gives the following result when combined with the displacement boundary condition in Eq. 50 as
\[\begin{bmatrix}\mathbf{0}&\mathbf{B}_{i}^{i}&\mathbf{B}_{i}^{i+1}&\mathbf{0} \\ \mathbf{C}_{i}^{\mathrm{f}}\mathbf{K}_{21}^{i}&\mathbf{C}_{i}^{\mathrm{f}} \mathbf{K}_{22}^{i}&\mathbf{C}_{i}^{\mathrm{f}}\mathbf{K}_{11}^{i+1}&\mathbf{ C}_{i}^{i+1}\mathbf{K}_{12}^{i+1}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{i-1}^{i}\\ \mathbf{u}_{i}^{i+1}\\ \mathbf{u}_{i+1}^{i+1}\end{bmatrix}=\mathbf{0}. \tag{54}\]
With this procedure applied to all interfaces of a layered system, a global matrix for the interfacial displacements can be obtained. We note that a recursive algorithm can be used instead if all layers belong to the same material type [15], leading to an assembled stiffness matrix with the same dimension as that of a single stiffness matrix. This recursive method, however, is not considered here because we attempt to address a general case with arbitrarily stacked fluid, solid and porous layers.
The reflection and transmission coefficients for a layered system can be calculated upon substituting the wave conditions in the two half-spaces \(0\) and \(n+1\) (see Fig. 1) into the global matrix. Without loss of generality, the two half-spaces are considered to be occupied by the same fluid material, which is the case for air- and water-coupled configurations as commonly used for the testing of layered structures. In the half-space \(0\), a monochromatic plane wave is impinged on the \(z=0\) surface at an incident angle of \(\theta\). Thus, the incident wave has the wavenumber components of
\[k_{x}=k_{\mathrm{L}}^{0}\sin\theta,\,k_{\mathrm{L}z}^{0}=k_{\mathrm{L}}^{0} \cos\theta, \tag{55}\]
where \(k_{\mathrm{L}}^{0}=\omega/c_{\mathrm{L}}^{0}\) is the wavenumber in the half-space \(0\). As used throughout this paper, the \(x\)-direction wavenumber component \(k_{x}\) is a common term shared by all waves in the entire structure, which is prescribed by the Snell's law.
As the incident wave comes into the layered medium, part of the wave is reflected back into the half-space \(0\), and the rest travels through the layers and is transmitted into the half-space \(n+1\). Let us assume the incident wave to have a unit amplitude, then the amplitudes of the reflected and transmitted waves equal to the reflection \(R\) and transmission \(T\) coefficients of the system. In this case, the amplitude vectors for the half-spaces \(0\) and \(n+1\) are \(\mathbf{A}^{0}=\left[1,R\right]^{\mathrm{T}}\) and \(\mathbf{A}^{n+1}=\left[T,0\right]^{\mathrm{T}}\) (note the absence of backward travelling wave in \(n+1\)). To avoid having an origin at \(-\infty\) in the half-space \(0\), we place the \(z\)-coordinate origin of both forward and backward propagating waves at the interface \(0\). Then from Eqs. 47 and
\begin{table}
\begin{tabular}{l l} Interface & Matrix \\ \hline Fluid-solid & \(\mathbf{B}^{\mathrm{f}}=\mathbf{I}_{1\times 1},\,\,\,\mathbf{B}^{ \mathrm{s}}=-\left[1\,\,\,\,0\right]\), \\ & \(\mathbf{C}^{\mathrm{f}}=\left[1\,\,\,\,0\right]^{\mathrm{T}},\,\,\,\mathbf{C}^{ \mathrm{s}}=-\mathbf{I}_{2\times 2}\) \\ \hline Fluid-porous & \(\mathbf{B}^{\mathrm{f}}=\mathbf{I}_{1\times 1},\,\,\,\mathbf{B}^{ \mathrm{p}}=-\left[1\,\,\,\,0\,\,0\right]\), \\ & \(\mathbf{C}^{\mathrm{f}}=\left[1\,\,\,\,0\,\,0\right]^{\mathrm{T}},\,\,\,\mathbf{C}^{ \mathrm{p}}=-\mathbf{I}_{3\times 3}\) \\ \hline Solid-porous & \(\mathbf{B}^{\mathrm{s}}=\left[\begin{matrix}1&0&0\\ 0&1&0\end{matrix}\right]^{\mathrm{T}},\,\,\,\mathbf{B}^{ \mathrm{p}}=-\mathbf{I}_{3\times 3}\), \\ & \(\mathbf{C}^{\mathrm{s}}=\mathbf{I}_{2\times 2},\,\,\,\mathbf{C}^{ \mathrm{p}}=-\left[\begin{matrix}1&0&0\\ 0&1&0\end{matrix}\right]\) \\ \end{tabular}
\end{table}
Table 2: Boundary matrices \(\mathbf{B}\) and \(\mathbf{C}\) for the displacement and stress vectors across the interface of two layers. The superscripts f, s and p indicate respectively fluid, solid and porous layers.
48, we obtain the displacement and stress vectors on the interface \(0\) as
\[\mathbf{u}_{0}^{0}=\left[k_{\mathrm{L}z}^{0}(1-R)\right]^{\mathrm{T}},\,\mathbf{ \sigma}_{0}^{0}=\left[i\rho\omega^{2}(1+R)\right]^{\mathrm{T}}. \tag{56}\]
Similarly, the displacement and stress vectors on the interface \(n+1\) can be determined as
\[\mathbf{u}_{n}^{n+1}=\left[k_{\mathrm{L}z}^{0}T\right]^{\mathrm{T}},\,\mathbf{ \sigma}_{n}^{n+1}=\left[i\rho\omega^{2}T\right]^{\mathrm{T}}, \tag{57}\]
where \(k_{\mathrm{L}z}^{n+1}=k_{\mathrm{L}z}^{0}\) is considered. Subsequently, the stiffness matrix relations for the two half-spaces can be obtained from Equations 56 and 57 as
\[\mathbf{\sigma}_{0}^{0}=\mathbf{K}^{0}\mathbf{u}_{0}^{0},\,\mathbf{\sigma}_{n}^{n+1}= \mathbf{K}^{n+1}\mathbf{u}_{n}^{n+1}, \tag{58}\]
with
\[\mathbf{K}^{0}=\left[\frac{i\omega Z^{0}}{\cos\theta}\frac{1+R}{1-R}\right],\, \mathbf{K}^{n+1}=\left[\frac{i\omega Z^{0}}{\cos\theta}\right], \tag{59}\]
where \(Z^{0}=\rho^{0}c_{\mathrm{L}}^{0}\) is the acoustic impedance in both half-spaces.
Incorporating Eq. 58 into the global matrix, we have
\[\begin{bmatrix}\mathbf{B}_{0}^{0}&\mathbf{B}_{0}^{1}&\mathbf{0}&\mathbf{0}& \cdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{C}_{0}^{0}\mathbf{K}^{0}&\mathbf{C}_{0}^{1}\mathbf{K}_{11}^{1}& \mathbf{C}_{0}^{0}\mathbf{K}_{12}^{1}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{B}_{1}^{1}&\mathbf{B}_{2}^{1}&\mathbf{0}& \cdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}_{1}^{1}\mathbf{K}_{21}^{1}&\mathbf{C}_{1}^{1}\mathbf{K }_{22}^{1}&\mathbf{C}_{1}^{2}\mathbf{K}_{11}^{2}&\mathbf{C}_{1}^{2}\mathbf{K }_{12}^{2}&\cdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}& \mathbf{B}_{n}^{n}&\mathbf{B}_{n}^{n+1}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{C}_{n}^{ n}\mathbf{K}_{21}^{n}&\mathbf{C}_{n}^{n}\mathbf{K}_{22}^{n}&\mathbf{C}_{n}^{n+1} \mathbf{K}^{n+1}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{0}^{0}\\ \mathbf{u}_{0}^{0}\\ \mathbf{u}_{1}^{1}\\ \mathbf{u}_{1}^{1}\\ \mathbf{u}_{2}^{1}\\ \vdots\\ \mathbf{u}_{n}^{n}\\ \mathbf{u}_{n}^{n+1}\end{bmatrix}=\mathbf{V}\mathbf{U}=\mathbf{0}, \tag{60}\]
where the global matrix \(\mathbf{V}\) is a square matrix with an even size, and it has only one unknown variable of \(R\). For the equation to have nontrivial solutions, the determinant of \(\mathbf{V}\) must vanish. We can observe from \(\mathbf{V}\) that its first column has only two nonzero elements on the first and second rows, so its determinant can be expressed as
\[\det\mathbf{V}=\det\mathbf{V}_{1,1}-\left(\frac{i\omega Z^{0}}{\cos\theta} \frac{1+R}{1-R}\right)\det\mathbf{V}_{2,1}=0, \tag{61}\]
where the values of the two nonzero elements have been incorporated. \(\mathbf{V}_{i,j}\) is the submatrix formed by removing the \(i\)-th row and \(j\)-th column of \(\mathbf{V}\). Equation 61 produces the result for the reflection coefficient as
\[R=\frac{Z^{\mathrm{eff}}-Z^{0}}{Z^{\mathrm{eff}}+Z^{0}}, \tag{62}\]
with
\[Z^{\mathrm{eff}}=\frac{\cos\theta}{i\omega}\frac{\det\mathbf{V}_{1,1}}{\det \mathbf{V}_{2,1}}. \tag{63}\]
To calculate the transmission coefficient \(T\), we consider the displacement relation between the two half-spaces, obtained from Eqs. 56 and 57 as \(T\mathbf{u}_{0}^{0}+(R-1)\mathbf{u}_{n}^{n+1}=\mathbf{0}\). Replacing the first row of \(\mathbf{V}\) in Eq. 60 with this relation and evaluating the determinant of the resulting matrix, we arrive at the expression for the transmission coefficient
\[T=(R-1)\frac{\det\mathbf{V}_{1,\mathrm{end}}}{\det\mathbf{V}_{1,1}}, \tag{64}\]
where 'end' represents the last column of \(\mathbf{V}\).
The calculated \(R\) and \(T\) are complex numbers, carrying both amplitude and phase information of the reflected and transmitted waves. We point out that we can readily obtain the three acoustic indicators commonly used in sound-absorbing applications, with surface impedance given by Eq. 63, absorption coefficient by \(1-|R|^{2}\) and transmission loss by \(-10\log|T^{2}|\)[33]. Though it will not be discussed in detail here, we emphasise that above formulations can be easily adapted to other boundary conditions, such as those bounded by an impervious hard wall on one side and those by solids on both sides.
### Relation to transfer matrix method and stability
The stiffness matrix method is closely related to the well-established transfer matrix method [30; 32; 33]. To demonstrate this relationship, we reorganise the displacements and stresses in Eqs. 47 and 48 to the two layer interfaces, yielding
\[\begin{bmatrix}\mathbf{u}_{i-1}\\ \mathbf{\sigma}_{i-1}\end{bmatrix}^{i} =\begin{bmatrix}\mathbf{D}^{+}&\mathbf{D}^{-}\mathbf{E}\\ \mathbf{S}^{+}&\mathbf{S}^{-}\mathbf{E}\end{bmatrix}^{i}\begin{bmatrix}\mathbf{ A}^{+}\\ \mathbf{A}^{-}\end{bmatrix}^{i}, \tag{65}\] \[\begin{bmatrix}\mathbf{u}_{i}\\ \mathbf{\sigma}_{i}\end{bmatrix}^{i} =\begin{bmatrix}\mathbf{D}^{+}\mathbf{E}&\mathbf{D}^{-}\\ \mathbf{S}^{+}\mathbf{E}&\mathbf{S}^{-}\end{bmatrix}^{i}\begin{bmatrix}\mathbf{ A}^{+}\\ \mathbf{A}^{-}\end{bmatrix}^{i}. \tag{66}\]
Inverting Eq. 66 and substituting the resulting amplitude vector into Eq. 65 leads to a transfer matrix relation
\[\begin{bmatrix}\mathbf{u}_{i-1}\\ \boldsymbol{\sigma}_{i-1}\end{bmatrix}^{i} =\begin{bmatrix}\mathbf{D}^{+}&\mathbf{D}^{-}\mathbf{E}\\ \mathbf{S}^{+}&\mathbf{S}^{-}\mathbf{E}\end{bmatrix}^{i}\left(\begin{bmatrix} \mathbf{D}^{+}\mathbf{E}&\mathbf{D}^{-}\\ \mathbf{S}^{+}\mathbf{E}&\mathbf{S}^{-}\end{bmatrix}^{i}\right)^{-1}\begin{bmatrix} \mathbf{u}_{i}\\ \boldsymbol{\sigma}_{i}\end{bmatrix}^{i} \tag{67}\] \[=\mathbf{T}^{i}\begin{bmatrix}\mathbf{u}_{i}\\ \boldsymbol{\sigma}_{i}\end{bmatrix}^{i},\]
which relates the displacements and stresses on the interface \(i-1\) to those on the interface \(i\) of the layer \(i\). The elements of the transfer matrix \(\mathbf{T}\) can be expressed by those of the stiffness matrix \(\mathbf{K}\) (and vice versa) [15]; the expressions are not provided here because they are numerically unstable. Using the boundary conditions in Eqs. 50 and 51, the transfer matrix relation is assembled into a global matrix form in order to solve the wave reflection and transmission problem in the layered system in the same fashion as presented above [32, 33].
As mentioned, the stiffness matrix formulation brings the benefit of intrinsic stability. To compare it with transfer matrix method, we have run a wide variety of calculation cases with the layered system containing different numbers of layers of different material types. The stiffness matrix method is found to be numerically stable in all cases; while the transfer matrix method delivers practically identical results when numerically-stable, it does suffer from instability under certain conditions, even when assembled in the global matrix configuration. This might seem contradicting to the discussions in [21] and in the Introduction - the transfer matrix method is supposed to be free from instability in global matrix formulation. This is found to be indeed true for the cases involving fluid and/or solid layers only. However, instability does arise for porous layers, potentially due to the existence of slow waves (slower speed than waves in fluid). A prominent example case is provided in Figure 2 for a porous layer, showing unstable blow-ups at high frequencies beyond the first critical angle. Another unstable case arises at the \(f\to 0\) limit where instability leads to a non-unity transmission coefficient (supposed to be unity because the layered structure is transparent to the transmitting wave). This latter case is less important so is not plotted here. Our stiffness matrix method does not suffer from these issues and is thus more advantageous.
## IV Results and experimental validation
Having fully established the stiffness matrix method, we present wave propagation results predicted by it (referred to as 'theory') and compare them with experimental measurements for a range of layered media. We employ the experimental setup sketched in Fig. 3(a). The sample is placed between two ultrasonic transducers, with the source transducer generating wave into the coupling fluid and the receiver transducer recording the wave that travelled through the fluid-sample-fluid path. The sample is mounted vertically onto a stepper motor-driven rotary stage and the angle \(\theta\) between the incident wave and the sample surface is thus automatically controlled. The two transducers are each fixed onto a kinematic mount, through which they are adjusted to have their axes aligned and their active surfaces parallel to the sample surface at the normal incidence of \(\theta=0^{\circ}\).
We consider three samples of gradually increasing complexity as illustrated in Fig. 3(b). To experimentally measure the transmission coefficient of each sample, a reference signal \(u_{\mathrm{r}}(t)\) is first acquired when the sample is absent, and then the transmitted signal \(u_{\mathrm{t}}(t)\) through the sample is recorded at a desired incident angle \(\theta\). Example reference signal and transmitted signal through the solid steel sample at \(22.5^{\circ}\) are provided in Fig. 3(c), which are acquired using a pair of \(0.5\) MHz transducers. The two signals are subsequently transformed into the frequency domain to obtain the spectra \(U_{\mathrm{r}}(f)\) and \(U_{\mathrm{t}}(f)\); see Fig. 3(d) and (e) for the amplitude and phase spectra of the example signals. The frequency-dependent transmission coefficient at \(\theta\) is calculated by \(T(f)=U_{\mathrm{t}}(f)/U_{\mathrm{r}}(f)e^{-ik_{\mathrm{f}}d\cos\theta}\), with the exponential term accounting for the propagation across the sample thickness \(d\) in the coupling fluid (with wave number \(k_{\mathrm{f}}\cos\theta\) in the propagation direction) when the sample is absent. The transmission coefficient is mostly small especially in an air-coupled setting, and the transmitted signal is thus very weak. To reach a signal-to-noise ratio of around \(30\) dB for the transmitted signal, the voltage of the excitation (5-cycle Hann-windowed toneburst) to the source transducer is maintained at around \(100\) V and the
Figure 2: Comparison of the transfer matrix and stiffness matrix methods for wave reflection and transmission in a porous layer at an incident angle of \(\theta=40^{\circ}\) (beyond the critical angle of the fast longitudinal wave). The porous layer is made of sintered glass beads saturated with water and the entire layer is bounded by water [35], and this calculation case will be studied in detail in Fig. 5. (a) and (c) Amplitudes of the reflection and transmission coefficients, \(|R|\) and \(|T|\), obtained from the two methods. (b) and (d) Respective zoomed-in plots, highlighting the unstable blow-ups of the transfer matrix method.
recorded signal by the receiver transducer is pre-amplified by 40 dB. In addition, each signal recording is the average of 256 signal firings, as a commonly-employed technique to suppress electronic noises.
### Single solid layer
We start with a simple case of a solid layer submerged in air at 20 \({}^{\circ}\)C room temperature. The properties of the sample is given in Fig. 4(a), and those of air are \(K=1.4\times 10^{5}\) Pa, \(\rho=1.3\) kg/m\({}^{3}\) and \(\eta=1.8\times 10^{-5}\) Pa\(\cdot\)s. This case is an important first step for understanding wave propagation in multilayered media, such as battery electrodes that have a thin solid layer in the middle as discussed later.
The theoretical transmission coefficient amplitude \(|T|\) is provided in Fig. 4(b) as a function of frequency \(f\) and incident angle \(\theta\). Since the amplitude spans over a few orders of magnitude, a logarithmic colour scale is used in the plot for better visualisation. At low frequencies where the wavelength is substantially larger than the sample thickness, the transmission coefficient in all directions approaches unity, meaning that nearly all energy is transmitted through the layer as if the layer is not present. The transmission coefficient is mostly very small at other frequencies, but it tends to reach unity again at large incident angles beyond the two critical angles of \(3.5^{\circ}\) and \(6.2^{\circ}\). This large-angle, highly-transmissible region arises due to the excitation of the fundamental anti-symmetric guided wave (A\({}_{0}\) mode) in the thin layer as detailed in our prior work [45]. Physically, the wave resonates in the layer at the presence of the A\({}_{0}\) mode, and it causes the majority of the energy to travel through the layer to the other side.
The theoretical prediction is corroborated by the experimental results in Fig. 4(c) and (d) that are collected using three pairs of air-coupled transducers with the frequencies of 0.5, 0.7 and 1.0 MHz. The two figure plots display respectively the angle dependence of the transmission coefficient amplitude at \(f=0.5\) and 0.7 MHz and the frequency dependence at \(\theta=22.5^{\circ}\) and \(43.5^{\circ}\). Both plots show very good agreement between the theory and the experiment. Prominently, in comparison to the experiment, the theory has
Figure 3: Experimental setup, samples and example signals. (a) Experimental setup with a transducer generating wave into a layered sample and another transducer receiving the transmitted wave. The sample is mounted onto a rotary stage to adjust the angle of incidence. (b) Three validation samples, with the first being a solid layer, the second a porous layer and the third a porous-solid-porous layer. (c) Example reference signal recorded when the sample is absent and transmitted signal when the wave has a \(22.5^{\circ}\) incident angle to the \(50\,\mathrm{\SIUnitSymbolMicro m}\) steel sample. Each signal is the average of 256 recordings acquired using a pair of 0.5 MHz transducers. (d) Amplitude spectra of the two example signals. (e) Respective phase spectra.
Figure 4: Wave transmission through a solid layer. (a) The properties of the thin steel sample with a thickness of \(50\,\mathrm{\SIUnitSymbolMicro m}\). (b) The amplitude of theoretically predicted transmission coefficient as a function of frequency \(f\) and incident angle \(\theta\). (c) Comparison of theoretical and experimental results at the frequencies of 0.5 and 0.7 MHz. (d) Similar comparison at the incident angles of \(22.5^{\circ}\) and \(43.5^{\circ}\).
fine details at, and around, the highly-transmissible A\({}_{0}\) peaks. We have also carried out experiments on 250 and 500 \(\mathrm{\SIUnitSymbolMicro m}\) thick samples and the results show similar theory-experiment agreement and transmission characteristics.
### Single porous layer
Going further, we analyse wave transmission through a porous layer of the same granular microstructure as active battery electrodes. We consider a well-studied porous material made of sintered glass beads [46, 47, 48, 35] as its microstructure characteristics are highly controllable. We use the parameters and experimental results of the sample S3 by Jocker et al. [35]. The sample has the parameters in Fig. 5(a) and is submerged in water of \(K=2.2\) GPa, \(\rho=1000\) kg/m\({}^{3}\) and \(\eta=0.001\) Pa\(\cdot\)s. Figure 5(b) gives the wave speed and attenuation coefficient of the three Biot waves in the porous medium, namely the fast and slow longitudinal waves and the shear wave. As affected by the porous network (porosity, tortuosity, permeability etc), the two faster waves travel more slowly than in pure glass (5850 and 3250 m/s [43]) and the slow wave is even slower than in the fluid (1483 m/s). Also, the three waves exhibit different levels of attenuation, increasing around an order of magnitude each from the fast longitudinal through the shear to the slow longitudinal waves. The three waves show distinctive wave speed and attenuation characteristics in the low-frequency viscous and high-frequency inertial regimes separated by the viscous characteristic frequency \(f_{\mathrm{c}}=0.013\) MHz. In the viscous regime, the slow wave demonstrates more pronounced wave speed dispersion but smaller increase of attenuation with frequency than the other two waves. By contrast, all three waves tend to have constant speeds and the same power dependence of attenuation on frequency in the inertial regime.
Our theoretical prediction of the transmission coefficient through the sample is given in Fig. 5(c) and (d), showing respectively the amplitude and phase maps against frequency \(f\) and incident angle \(\theta\). The transmission is small beyond the critical angle of the fast longitudinal wave, which is annotated as the solid line in the maps. Below this critical angle, the transmission coefficient exhibits a cyclic behaviour, which is particularly obvious as we look along the horizontal line at the normal incidence of \(\theta=0^{\circ}\) in the amplitude map. This cyclic behaviour arises because of the resonances of the three waves, predominately the fast longitudinal wave, reverberating between the two boundaries of the porous layer. The resonances also cause small phase fluctuations as can be seen in the phase map.
Figure 5(c) and (d) compare our theoretical prediction with experimental results [35] at the incident angles of \(\theta=0^{\circ}\) and \(\theta=18^{\circ}\), respectively. In both cases, the theory agrees remarkably well with the experimental points in both amplitude and phase, building our confidence in using the theory to describe multilayered porous electrodes in what follows.
### Porous-solid-porous anode
Now we examine wave transmission through a porous-solid-porous battery anode. We focus on the most-widely anode material with two 50 \(\mathrm{\SIUnitSymbolMicro m}\) active layers coated on both sides of a 10 \(\mathrm{\SIUnitSymbolMicro m}\) copper film. The active layer has a granular porous microstructure with graphite particles joined together by binder materials. The active material has been well characterised using advanced techniques [49, 50, 51] and the relevant parameters are summarised in Fig. 6(a). The three Biot waves in the active material are detailed in Fig. 6(b), showing similar speed and attenuation profiles to those in sintered glass beads in
Figure 5: Wave transmission through a porous layer. (a) Properties of the porous material of sintered glass beads saturated with water [35]. (b) Wave speed (left) and attenuation coefficient (right) of the fast and slow longitudinal waves and the shear wave in the porous material. The vertical dash-dotted line represents the viscous characteristic frequency \(f_{\mathrm{c}}\). (c) and (d) Amplitude and phase maps of theoretically predicted transmission coefficient against frequency \(f\) and incident angle \(\theta\). The solid line in the maps denotes the first critical angle for the fast longitudinal wave. (e) Comparison of amplitude (top) and phase (bottom) between the theoretical and experimental results at the normal incidence of \(\theta=0^{\circ}\). (f) Similar comparison at the incident angle of \(\theta=18^{\circ}\). The experimental points in (e) and (f) are taken from Jocker et al. [35].
Fig. 5(b). The use of airborne ultrasound in this case and the micron-level pore structure lead the slow longitudinal wave to have exceptionally small speed and excessively large attenuation. Thus, in comparison to the other two waves, the slow wave has minimal contribution to the transmitted wave. In addition, the large viscous characteristic frequency of \(f_{\mathrm{c}}=9.75\) MHz means that we have to focus in the viscous regime because the inertial regime can barely be reached with an air-coupled setup (generally limited to \(\leq 5\) MHz).
Figure 6(c) and (d) display respectively the theoretical amplitude and phase maps of the transmission coefficient through the anode. As in a single solid or porous layer, the wave perceives the anode as transparent irrespective of the incident angle in the low-frequency, long-wavelength range. At higher frequencies, the wave tunnels through the anode when the frequency and incident angle are suitably combined to excite guided wave modes in individual layers or in the whole anode, evidenced also by the \(\pi\) phase jump in the phase map. The highly-transmissible tunneling is reminiscent of the above single solid layer but shows many more complex features due to the multilayered nature of the anode.
The theoretical results are evaluated against experimental measurements in Fig. 6(e) and (f). The theory matches very well with the experiment, particularly in the transmission amplitude, from both the angle- and frequency-dependent results. The agreement is less satisfactory for the phase results in Fig. 6(e) because of the difficulty in obtaining accurate phase information from noisy experimental signals. Overall, the evaluation results highlight the very good applicability of the theory to describe wave transmission through complex multilayered electrodes involving micron-level porous structures.
## V Summary and Outlook
In summary, we have developed a general stiffness matrix method for modelling wave propagation in multilayered media with arbitrary numbers and combinations of fluid, solid and porous layers. With individual layers described by stiffness matrices, and layer interfaces defined by boundary conditions, the proposed method assembles a global system of equations to solve the reflection and transmission of oblique waves in the layered system. The method is intrinsically and unconditionally stable and is thus more advantageous over existing transfer matrix-based methods, which have been demonstrated to show instability in certain configurations. Experimental validations have proved the validity of the proposed method over a range of layered cases, ranging from single solid and porous layers to a porous-solid-porous anode.
The presented work is a powerful modelling tool for developing new methods to quantify the properties of layered media. A particularly exciting possibility is the quantification of the porous parameters (e.g. porosity, tortuosity and permeability) in Li-ion battery electrodes, which we have already analysed in this paper tentatively. These electrode parameters are key performance determinants of the batteries, hence non-destructive evaluation capabilities, potentially offered by wave-based methods, could bring considerable benefits via closed-loop control and quality assurance during battery manufacturing.
###### Acknowledgements.
We acknowledge Prof Kirill Horoshenkov of Sheffield University for insightful discussions on poroelasticity, Antonio De Sanctis of Imperial College and Peiyao Huang of Cambridge University for assistance in experimental setup. B.L. gratefully thanks the generous support from the Imperial College Research Fellowship Scheme, and M.H. appreciates the funding by the Imperial College Non-Destructive Evaluation Group.
Figure 6: Wave transmission through a porous-solid-porous anode. (a) Properties of the multilayered anode [49; 50; 51] in an air-coupled setting. (b) Wave speed (left) and attenuation coefficient (right) of the three Biot waves in the graphite anode coating material, with the viscous characteristic frequency \(f_{\mathrm{c}}\) annotated as the vertical dash-dotted line. (c) and (d) Amplitude and phase maps of theoretically predicted transmission coefficient against frequency \(f\) and incident angle \(\theta\). (e) Angular dependence of amplitude (top) and phase (bottom) at 0.5 MHz, comparing the theoretical and experimental results. (f) Frequency dependence at the normal incidence of \(\theta=0^{\circ}\). |
2307.11567 | CortexMorph: fast cortical thickness estimation via diffeomorphic
registration using VoxelMorph | The thickness of the cortical band is linked to various neurological and
psychiatric conditions, and is often estimated through surface-based methods
such as Freesurfer in MRI studies. The DiReCT method, which calculates cortical
thickness using a diffeomorphic deformation of the gray-white matter interface
towards the pial surface, offers an alternative to surface-based methods.
Recent studies using a synthetic cortical thickness phantom have demonstrated
that the combination of DiReCT and deep-learning-based segmentation is more
sensitive to subvoxel cortical thinning than Freesurfer.
While anatomical segmentation of a T1-weighted image now takes seconds,
existing implementations of DiReCT rely on iterative image registration methods
which can take up to an hour per volume. On the other hand, learning-based
deformable image registration methods like VoxelMorph have been shown to be
faster than classical methods while improving registration accuracy. This paper
proposes CortexMorph, a new method that employs unsupervised deep learning to
directly regress the deformation field needed for DiReCT. By combining
CortexMorph with a deep-learning-based segmentation model, it is possible to
estimate region-wise thickness in seconds from a T1-weighted image, while
maintaining the ability to detect cortical atrophy. We validate this claim on
the OASIS-3 dataset and the synthetic cortical thickness phantom of Rusak et
al. | Richard McKinley, Christian Rummel | 2023-07-21T13:18:43Z | http://arxiv.org/abs/2307.11567v1 | # CortexMorph: fast cortical thickness estimation via diffeomorphic registration using VoxelMorph
###### Abstract
The thickness of the cortical band is linked to various neurological and psychiatric conditions, and is often estimated through surface-based methods such as Freesurfer in MRI studies. The DiReCT method, which calculates cortical thickness using a diffeomorphic deformation of the gray-white matter interface towards the pial surface, offers an alternative to surface-based methods. Recent studies using a synthetic cortical thickness phantom have demonstrated that the combination of DiReCT and deep-learning-based segmentation is more sensitive to subvoxel cortical thinning than Freesurfer.
While anatomical segmentation of a T1-weighted image now takes seconds, existing implementations of DiReCT rely on iterative image registration methods which can take up to an hour per volume. On the other hand, learning-based deformable image registration methods like VoxelMorph have been shown to be faster than classical methods while improving registration accuracy. This paper proposes CortexMorph, a new method that employs unsupervised deep learning to directly regress the deformation field needed for DiReCT. By combining CortexMorph with a deep-learning-based segmentation model, it is possible to estimate region-wise thickness in seconds from a T1-weighted image, while maintaining the ability to detect cortical atrophy. We validate this claim on the OASIS-3 dataset and the synthetic cortical thickness phantom of Rusak et al.
Keywords:MRI Morphometry cortical thickness Unsupervised image registration Deep learning
## 1 Introduction
Cortical thickness (CTh) is a crucial biomarker of various neurological and psychiatric disorders, making it a primary focus in neuroimaging research. The cortex, a thin ribbon of grey matter at the outer surface of the cerebrum, plays a vital role in cognitive, sensory, and motor functions, and its thickness has been linked to a wide range of neurological and psychiatric conditions, including Alzheimer's disease, multiple sclerosis, schizophrenia, and depression, among others. Structural magnetic resonance imaging (MRI) is the primary modality
used to investigate CTh, and numerous computational methods have been developed to estimate this thickness on the sub-millimeter scale. Among these, surface-based methods like Freesurfer [5, 6] have been widely used, but they are computationally intensive, making them less feasible for clinical applications. Optimizations based on Deep Learning have brought the running time for a modified Freesurfer pipeline down to one hour. [7] The DiReCT method [4] offers an alternative to surface-based morphometry methods, calculating CTh via a diffeomorphic deformation of the gray-white matter interface (GWI) towards the pial surface (the outer edge of the cortical band). The ANTs package of neuroimaging tools provides an implementation of DiReCT via the function KellyKapowski: for readablility we refer below to KellyKapowski with its default parameters as ANTs-DiReCT. The ANTs cortical thickness pipeline uses ANTs-DiReCT together with a three-class segmentation (grey matter, white matter, cerebrospinal fluid) provided by the Atropos segmentation method, taking between 4 and 15 hours depending on the settings and available hardware [1, 15]. A more recent version of ANTs provides a deep-learning based alternative to Atropos, giving comparable results to ANTs but accelerating the overall pipeline to approximately one hour, such that now the running time is dominated by the time needed to run ANTs-DiReCT [16]. Meanwhile, Rebsamen et al. have shown that applying DiReCT to the output of a deep-learning-based segmentation model trained on Freesurfer segmentations (rather than Atropos) yields a CTh method which agrees strongly with Freesurfer, while having improved repeatability on repeated scans [12]. Subsequently, a digital phantom using GAN-generated scans with simulated cortical atrophy showed that the method of Rebsamen et al. is more sensitive to cortical thinning than Freesurfer [13].
The long running time of methods for determining CTh remains a barrier to application in clinical routine: a running time of one hour, while a substantial improvement over Freesurfer and ANTs cortical thickness, is still far beyond the real-time processing desirable for on-demand cortical morphometry in clinical applications. In terms of both the speed and performance, VoxelMorph and related models are known to outperform classical deformable registration methods, suggesting that a DiReCT-style CTh algorithm based on unsupervised registration models may enable faster CTh estimation. [3, 2, 18] In this paper, we demonstrate that a VoxelMorph style model can be trained to produce a diffeomorphism taking the GWI to the pial surface, and that this model can be used to perform DiReCT-style CTh estimation in seconds. We trained the model on 320 segmentations derived from the IXI and ADNI datasets, and demonstrate excellent agreement with ANTs-DiReCT on the OASIS-3 dataset. Our model also shows improved performance on the digital CTh phantom of Rusak et al.[13]
## 2 Methods
### DiReCT Cortical Thickness estimation
The estimation of CTh using the DiReCT method [4] proceeds as follows: first a (partial volume) segmentation of the cortical white matter (WM) and cortical
grey matter (GM) is obtained. Second, a forward deformation field \(\phi\) mapping the white-matter (WM) image towards the WM+GM image is computed. This forward deformation field should be a diffeomorphism, in order that the deformation field is invertible and the topology of the inferred pial surface is the same as the GWI. Third, the diffeormorphism is inverted to obtain the reverse the deformation field, taking the pial surface towards the GWI. Finally, the CTh is determined by computing the magnitude of the reverse field at the GWI: specifically, at each voxel of WM adjacent to the GM. In ANTs-DiReCT, the forward transform (from WM to WM+GM) is calculated by a modified greedy algorithm, in which the WM surface is propagated iteratively in the direction of the surface normal until it reaches the outer GM surface or a predefined spatial prior maximum is reached. The approximate inverse field is then determined by numerical means using kernel based splines (as implemented in ITK).
The absence of a reliable gold-standard ground truth for CTh makes comparisons between methods difficult. This situation has recently been improved by the publication of a synthetic cortical atrophy phantom: a dataset generated using a GAN conditioned on subvoxel segmentations, consisting of 20 synthetic subjects with 19 induced sub-voxel atrophy levels per subject (ten evenly spaced atrophy levels from 0 to 0.1mm, and a further nine evenly spaced atrophy levels from 0.1mm to 1mm). [13] The purpose of this digital phantom is to explore the ability of CTh algorithms to resolve subtle changes of CTh. The paper of Rusak et al. analyzed the performance of several CTh methods on this dataset, finding that the DL+DiReCT method [12] (which combines a deep network trained on Freesurfer annotations with ANTs-DiReCT) was the most sensitive to cortical atrophy and had the best agreement with the synthetically induced thinning.
Figure 1: End-to-end unsupervised architecture for DiReCT: velocity field \(z\) is regressed from WM and WM+GM segmentations, using a Unet. This velocity field is then integrated by seven scaling and squaring layers (\(\int\)) to yield forward and reverse deformation fields \(\phi_{z}\) and \(\phi_{-z}\), which are used to deform the input images in spatial transformer (ST) blocks. Components of the loss function are marked in orange.
### CortexMorph: VoxelMorph for DiReCT
The original VoxelMorph architecture, introduced in [3], utilized a Unet architecture to directly regress a displacement field from a fixed brain image and a moving brain image. Application of a spatial transform layer allows the moving image to be transformed to the space of the fixed image, and compared using a differentiable similarity metric such as mean squared error or cross-correlation. Since the spatial transformation is also a differentiable operation, the network can be trained end-to-end. Later adaptations of the concept employed a regression of a stationary _velocity field_, with the deformation field being calculated via an integration layer: the principal advantage of this formulation is that integrating through a velocity field yields a _diffeomorphism_. [2] Since diffeomorphic registration is required in the DiReCT method, we adopt this velocity-field form of VoxelMorph for our purposes.
The setup of our VoxelMorph architecture, CortexMorph, is detailed in Figure 1. The two inputs to the network are a partial volume segmentation of white matter (WM), and a partial volume segmentation of grey matter plus white matter (WM+GM). These are fed as entries into a Unet, the output of which is a velocity field \(z\), which is then integrated using 7 steps of scaling and squaring to yield a displacement field \(\phi_{z}\). This displacement field is then applied to the WM image to yield the deformed white matter volume \(\mathrm{WM}\circ\phi_{z}\). By integrating \(-z\) we obtain the reverse deformation field \(\phi_{-z}\), which is applied to the WM+GM image to obtain a deformed volume \((\mathrm{WM}+\mathrm{GM})\circ\phi_{-z}\). This simplifies the DiReCT method substantially: instead of needing to perform a numerical inversion of the deformation field, the reverse deformation field can be calculated directly. The deformed volumes are then compared using a loss function \(\mathcal{L}\) to their non-deformed counterparts: both directions of deformation are weighted equally in the final objective function. To encourage smoothness, a discrete approximation of the squared gradient magnitude of the velocity field \(\mathcal{L}_{\mathrm{smooth}}\) is added to the loss as a regularizer. [2] As a result, our loss has the following form
\[\mathcal{L}(\mathrm{WM},(\mathrm{WM}+\mathrm{GM})\circ\phi_{-z})+\mathcal{L}( \mathrm{WM}+\mathrm{GM},\mathrm{WM}\circ\phi_{z})+\lambda\mathcal{L}_{ \mathrm{smooth}}(z) \tag{1}\]
### Data and WM/GM segmentation
Training data and validation for our VoxelMorph model was derived from two publicly available sources: images from 200 randomly selected elderly individuals from the ADNI dataset [10] and images from 200 randomly selected healthy adults from the IXI dataset (brain-development.org/ixi-dataset). From each of these datasets, 160 images were randomly chosen to serve as training data, yielding in total 320 training cases and 80 validation cases. For testing our pipeline, we use two sources different from the training/validation data: the well-known OASIS-3 dataset (2,643 scans of 1,038 subjects, acquired over \(>10\) years on three different Siemens scanners), and the CTh phantom of Rusak et al. [13, 14]
For WM/GM segmentation, we employed the DeepSCAN model [12, 11], which is available as part of DL+DiReCT ([https://github.com/SCAN-NRA](https://github.com/SCAN-NRA) D/DL-DiReCT), since this is already known to give high-quality CTh results when combined with ANTs-DiReCT. This model takes as input a T1-weighted image, performs resampling and skull-stripping if necessary (provided by HDBET [9]) and produces a partial volume segmentation \(P_{w}\) of the white matter and \(P_{g}\) of the cortex (the necessary inputs to the DiReCT algorithm) with 1mm isovoxel resolution. It also produces a cortical parcellation in the same space (necessary to calculate region-wise CTh measures). We applied this model to the training data, validation data, and the 400 synthetic MRI cases of the CTh phantom, both to produce ANTs-DiReCT CTh measurements and also as an input to our VoxelMorph models.
### Training and model selection
Our network was implemented and trained in Pytorch (1.13.1). We utilized a standard Unet (derived from the nnUnet framework [8]) with 3 pooling steps and a feature depth of 24 features at each resolution. The spatial transformer/squaring and scaling layers/gradient magnitude loss were incorporated from the official VoxelMorph repository. For the loss function \(\mathcal{L}\) we tested both L1 loss and mean squared error (MSE). We tested values of the smoothness parameter lambda between 0 and 0.05. The models were trained with the Adam optimizer, with a fixed learning rate of \(10^{-3}\) and weight decay \(10^{-5}\). Patches of size \(128^{3}\) were used as training data in batches of size 2.
The training regime was fully unsupervised with respect to cortical thickness: neither the deformation fields yielded by ANTs-DiReCT nor the CTh results computed from those deformation fields were used in the objective function. Since we are interested in replacing the iterative implementation of DiReCT with a deep learning counterpart, we used the 80 validation examples for model selection, selecting the model which showed best agreement in mean global CTh with the results of ANTs-DiReCT. The metric for agreement chosen is intra-class correlation coefficient, specifically ICC(2,1) (the proportion of variation explained by the individual in a random effects model, assuming equal means of the two CTh measurement techniques), since this method is sensitive to both absolute agreement and relative consistency of the measured quantity. ICC was calculated using the python package Pingouin. [17]
### Testing
The VoxelMorph model which agreed best with ANTs-DiReCT on the validation set was applied to segmentations of the OASIS-3 dataset, to confirm whether model selection on a small set of validation data would induce good agreement with ANTs-DiReCT on a much larger test set (metric, ICC(2,1)) and to the synthetic CTh phantom of Rusak et al, to determine whether the VoxelMorph model is able to distinguish subvoxel changes in CTh (metric, coefficient of determination (\(R^{2}\))).
## 3 Results
The best performing model on the validation set (in terms of agreement with DiReCT) was the model trained with MSE loss and a \(\lambda\) of 0.02. When used to measure mean global CTh, this model scored an ICC(2,1) of 0.91 (95% confidence interval [0.9, 0.92]) versus the mean global CTh yielded by ANTs-DiReCT on the OASIS-3 dataset. For comparison, on the same dataset the ICC between Freesurfer and the ANTs-DiReCT method was 0.50 ([95% confidence interval -0.08, 0.8]). A breakdown of the ICC by cortical subregion can be seen in Figure 2: these range from good agreement (entorhinal right, ICC = 0.87) to poor (caudalanteriorcingulate right, ICC=0.26), depending on the region. However, ICC(2,1) is a measure of absolute agreement, as well as correlation: all regional Pearson correlation coefficients lie in a range [0.64-0.90] (see supplementary material for a region-wise plot of the Pearson correlation coefficients).
Performance of this model on the CTh digital phantom can be seen in Figure 3: agreement with the induced level of atrophy is high (metric: Coefficient of Determination between the induced and the measured level of atrophy, across all 20 synthetic subjects) in both the wide range of atrophy (up to 1mm) and the fine-grained narrower range of atrophy (up to 0.1mm), suggesting that the VoxelMorph model is able to resolve small changes in CTh.
Calculating regional CTh took between 2.5s and 6.4s per subject (mean, 4.3s, standard deviation 0.71s) (Nvidia A6000 GP, Intel Xeon(R) W-11955M CPU).
Figure 2: Region-wise performance of CortexMorph: ICC(2,1) of mean region-wise cortical thickness between CortexMorph and ANTs-DiReCT, using the segmentations generated by DeepSCAN on the OASIS-3 dataset.
Figure 3: Performance of ANTs-DiReCT and CortexMorph on the CTh phantom of Rusak et al, based on segmentations derived from DeepSCAN. Above: performance on the whole synthetic dataset, comprising twenty synthetic individuals, each with a baseline scan and 19 ’follow-up’ images with induced levels of uniform cortical atrophy. Measured atrophy is defined as the difference between the mean CTh as measured on the synthetic baseline scan and the mean CTh measured on the synthetic follow-up, averaged across the whole cortex. Below: The same data, but focused only on the range [0-0.1mm] of induced atrophy. \(R^{2}\) denotes the coefficient of determination between the induced and measured atrophy levels.
## 4 Conclusion
Our experiments suggest that the classical, iterative approach to cortical thickness estimation by diffeomorphic registration can be replaced with a VoxelMorph network, with \(\sim\) 800 fold reduction in the time needed to calculate CTh from a partial volume segmentation of the cortical grey and white matter. Since such segmentations can also be obtained in a small number of seconds using a CNN or other deep neural network, we have demonstrated for the first time reliable CTh estimation running on a timeframe of seconds. This level of acceleration offers increased feasibility to evaluate CTh in the clinical setting. It would also enable the application of ensemble methods to provide multiple thickness measures for an individual: given an ensemble of, say, 15 segmentation methods, a plausible distribution of CTh values could be reported for each cortical subregion within one minute: this would allow better determination of the presence of cortical atrophy in an individual than is provided by point estimates. We are currently investigating the prospect of leveraging the velocity field to enable fast calculation of other morphometric labels such as grey-white matter contrast and cortical curvature: these too could be calculated with error bars via ensembling.
This work allows the fast calculation of diffeomorphisms for DiReCT on the GPU. We did not consider the possibility of directly implementing/accelerating the classical DiReCT algorithm on a GPU in this work. Elements of the ANTs-DiReCT pipeline implement multithreading, yielding for example a 20 minute runtime with 4 threads: however, since some parts of the pipeline cannot be parallelized it is unlikely that iterative methods can approach the speed of direct regression by CNN.
Given the lack of a gold standard ground truth for CTh, it is necessary when studying a new definition of CTh to compare to an existing silver standard method: this would typically be Freesurfer, but recent results suggest that this may not be the optimal method when studying small differences in CTh. [13] We have focused on comparison to the DL+DiReCT method for this study, since the results of this model on the CTh phantom are already reported and represent the state-of-the-art. For this reason, it made sense to use the outputs of the underlying CNN as inputs to our pipeline. However, the method we describe is general and could be applied to any highly performing segmentation method. Similarly, while we performed model selection to optimize agreement with the CTh values produced by Rebsamen et al, this optimization could easily be tuned to instead optimize agreement with Freesurfer. Alternatively, we could abandon agreement and instead select models based on consistency (given by a different variant of ICC) or Pearson correlation with a baseline model: this could lead to models which deviate from the baseline model but are better able to capture differences between patients or cohorts.
## Acknowledgements
This work was supported by a Freenovation grant from the Novartis Forschungsstiftung, and by the Swiss National Science Foundation (SNSF) under grant number 204593 (ScanOMetrics).
|
2305.04075 | PointCMP: Contrastive Mask Prediction for Self-supervised Learning on
Point Cloud Videos | Self-supervised learning can extract representations of good quality from
solely unlabeled data, which is appealing for point cloud videos due to their
high labelling cost. In this paper, we propose a contrastive mask prediction
(PointCMP) framework for self-supervised learning on point cloud videos.
Specifically, our PointCMP employs a two-branch structure to achieve
simultaneous learning of both local and global spatio-temporal information. On
top of this two-branch structure, a mutual similarity based augmentation module
is developed to synthesize hard samples at the feature level. By masking
dominant tokens and erasing principal channels, we generate hard samples to
facilitate learning representations with better discrimination and
generalization performance. Extensive experiments show that our PointCMP
achieves the state-of-the-art performance on benchmark datasets and outperforms
existing full-supervised counterparts. Transfer learning results demonstrate
the superiority of the learned representations across different datasets and
tasks. | Zhiqiang Shen, Xiaoxiao Sheng, Longguang Wang, Yulan Guo, Qiong Liu, Xi Zhou | 2023-05-06T15:47:48Z | http://arxiv.org/abs/2305.04075v1 | # PointCMP: Contrastive Mask Prediction for Self-supervised Learning
###### Abstract
Self-supervised learning can extract representations of good quality from solely unlabeled data, which is appealing for point cloud videos due to their high labelling cost. In this paper, we propose a contrastive mask prediction (PointCMP) framework for self-supervised learning on point cloud videos. Specifically, our PointCMP employs a two-branch structure to achieve simultaneous learning of both local and global spatio-temporal information. On top of this two-branch structure, a mutual similarity based augmentation module is developed to synthesize hard samples at the feature level. By masking dominant tokens and erasing principal channels, we generate hard samples to facilitate learning representations with better discrimination and generalization performance. Extensive experiments show that our PointCMP achieves the state-of-the-art performance on benchmark datasets and outperforms existing full-supervised counterparts. Transfer learning results demonstrate the superiority of the learned representations across different datasets and tasks.
## 1 Introduction
Recently, LiDARs have become increasingly popular in numerous real-world applications to perceive 3D environments, such as autonomous vehicles and robots. Point clouds acquired by LiDARs can provide rich geometric information and facilitate the machine to achieve 3D perception. Early works focus on parsing the real world from static point clouds [9, 24, 64], while recent researches pay more attention to understanding point cloud videos [14, 16, 54, 55]. Since annotating point clouds is highly time and labor consuming [1, 57], learning from point cloud videos in a self-supervised manner draws increasing interest. Although contrastive learning and mask prediction paradigms [6, 19, 21, 22, 58, 63] have shown the effectiveness of self-supervised learning on images or static point clouds, these methods cannot be directly extended to point cloud videos due to the following three challenges:
**(i) Multiple-Granularity Information Matters.** The contrastive learning paradigm [3, 4, 6, 19, 22, 63] usually focuses on extracting global semantic information based on instance-level augmentations. In contrast, the mask prediction paradigm [2, 21, 23, 58, 2] pays more attention to modeling local structures while ignoring global semantics. However, since fine-grained understanding of point cloud videos requires not only local spatio-temporal features but also global dynamics [16, 55], existing paradigms cannot be directly adopted.
**(ii) Sample Generation.** The contrastive learning paradigm is conducted by pulling positive samples while pushing negative ones [3, 6, 8, 19, 22, 63, 8], and the mask prediction paradigm learns representations by modeling the visible parts to infer the masked ones [2, 21, 39, 49, 58, 62].
Figure 1: A comparison between (a) contrastive learning paradigm, (b) mask prediction paradigm, and (c) our method.
Both paradigms rely heavily on the augmented samples at the input level. Further, as demonstrated in several works [18, 25, 26, 42], self-supervised learning can significantly benefit from proper hard samples. However, the spatial disorder, temporal misalignment, and uneven information density distribution impose huge challenges on hard sample generation for point cloud videos at the input level.
**(iii) Leakage of Location Information.** The mask prediction paradigm usually learns to reconstruct masked raw signals by modeling visible ones [2, 21, 39, 49, 58, 62]. For images, the contents are decoupled from the spatial position such that positional encoding is provided as cues to predict masked regions. However, for point clouds with only \(xyz\)-coordinates, positional encoding may be used as shortcuts to infer the masked points without capturing geometric information [30, 39].
In this paper, we propose a contrastive mask prediction framework for self-supervised learning on point cloud videos, termed as PointCMP. To address challenge (i), our PointCMP integrates the learning of both local and global spatio-temporal features into a unified two-branch structure, and simultaneously conducts self-supervised learning at different granularities (Fig. 1(c)). For challenge (ii), we introduce a mutual similarity based augmentation module to generate hard masked samples and negative samples at the feature level. To handle challenge (iii), instead of directly regressing the coordinates of masked points, token-level contrastive learning is conducted between the predicted tokens and their target embeddings to mitigate information leakage.
Our contributions are summarized as follows:
* We develop a unified self-supervised learning framework for point cloud videos, namely PointCMP. Our PointCMP integrates the learning of multiple-granularity spatio-temporal features into a unified framework using parallel local and global branches.
* We propose a mutual similarity based augmentation module to generate hard masked samples and negative samples by masking dominant tokens and principal channels. These feature-level augmented samples facilitate better exploitation of local and global information in a point cloud video.
* Extensive experiments and ablation studies on several benchmark datasets demonstrate the efficacy of our PointCMP on point cloud video understanding.
## 2 Related Work
In this section, we first briefly review two mainstream self-supervised learning frameworks. Then, we present recent advances for point cloud video understanding.
### Contrastive Learning
Contrastive learning has greatly promoted the development of self-supervised learning [3, 4, 6, 7, 8, 19, 22, 52, 63]. Usually, semantically consistent sample pairs are separately encoded by an asymmetric siamese network, and then contrastive loss aligns them to facilitate the encoder to learn discriminative representations [50, 51, 45]. For contrastive learning on images, data augmentation has been widely investigated to generate positive and negative samples to improve the discriminability of representations [6, 18, 32, 40].
Recently, contrastive learning has also been studied on static point clouds. Specifically, Xie _et al_. [57] used random geometric transformations to generate two views of a point cloud and associated matched point pairs in these two views using contrastive loss. Zhang _et al_. [66] constructed two augmented versions of a point cloud and used their global features to setup an instance discrimination task for pre-training.
### Mask Prediction
Mask prediction has demonstrated its effectiveness in numerous computer vision tasks and draws increasing interest [2, 21, 23, 49, 58]. Bao _et al_. [2] proposed a BERT-style framework [27] to predict token identities of masked patches based on visible ones. Then, Zhou _et al_. [68] developed an online tokenizer for better image BERT pre-training. Later, Feichtenhofer _et al_. [17] and Tong _et al_. [46] introduced mask prediction to videos and obtained representations rich in local details by inferring masked spatio-temporal tubes.
Recently, several efforts have been made to extend the mask prediction paradigm to point clouds. Specifically, Yu _et al_. [62] proposed PointBERT and introduced a masked point modeling task for point cloud pre-training. Pang _et al_. [39] proposed Point-MAE to reconstruct masked point coordinates using high-level latent features learned from unmasked ones. Liu _et al_. [30] proposed to use binary point classification as a pretext task for point cloud masked autoencoding.
Most existing contrastive learning and mask prediction methods rely on input-level augmentation to conduct self-supervised learning on static point clouds. Nevertheless, it is intractable to directly extend these methods to point cloud videos as more complicated augmentation operations are required to cover the additional temporal dimension. To remedy this, we propose to synthesize samples at the feature level based on mutual similarities, which enables reasonable sample generation without considering the unstructured data formats of point cloud videos.
### Point Cloud Video Understanding
Spatial disorder and temporal misalignment make point cloud videos more challenging to be parsed using a neu
ral network than structured data like images. To leverage the advanced techniques developed for structured data, previous methods transform point clouds into a sequence of bird's eye views [33], voxels [10, 61], and pillar grids [60]. However, these transformations inevitably lead to the loss of geometric details. Recently, more attention has been paid to learning directly on raw points using attention-based models [13, 14, 54, 55], convolution-based models [15, 16, 31], and hand-crafted temporal descriptors [53, 67]. Specifically, Fan [15] proposed a spatio-temporal decoupled encoder, which alternately performs spatial and temporal convolution to model raw point sequences hierarchically. Then, they further developed P4Transformer [13] that utilizes point spatio-temporal tubes to aggregate local neighborhoods into tokens.
Despite the huge success of self-supervised learning methods in video understanding [5, 20, 29, 41, 44, 48, 56, 65], self-supervised point cloud video understanding is still under-investigated. Recently, Wang [47] designed a pretext task, namely recurrent order prediction (ROP), to predict the temporal order of shuffled point cloud segments for self-supervised learning. However, this method can only capture clip-level temporal structures and cannot exploit finer spatio-temporal details. To parse a point cloud video, it is important for a self-supervised method to capture both spatio-temporal local structures and global semantics. To this end, we develop a unified PointCMP framework that can enable networks to simultaneously learn information with different granularities.
## 3 Method
The architecture of our PointCMP is illustrated in Fig. 2. Given a point cloud video, it is first uniformly divided into \(L\) segments. Then, these segments are fed to an encoder to produce tokens \(\mathbf{Z}\in\mathbb{R}^{L\times N\times C}\) by aggregating local spatio-temporal information, where \(N\) means the token number aggregated from each segment and \(C\) is the number of channels. Meanwhile, a global token \(\mathbf{Z}_{global}\in\mathbb{R}^{C}\) with global semantics is also obtained following [15]. Next, \(\mathbf{Z}_{global}\) and \(\mathbf{Z}\) are passed to a mutual similarity based augmentation module for online sample generation. Afterwards, a local contrastive learning branch and a global contrastive learning branch are employed to capture multi-granularity information.
### Mutual Similarity based Augmentation
Hard samples have been demonstrated to be critical to the performance of self-supervised learning [18, 26, 42]. However, it is challenging to generate hard samples for orderless and unstructured point cloud videos at the input level. To address this issue, we introduce a mutual similarity based augmentation module to synthesize hard samples at the feature level.
**Hard Masked Samples.** Our intuition is that reconstruction is easier when tokens sharing higher similarities with the global token are visible. Therefore, we are motivated to mask these tokens to synthesize hard masked samples. Specifically, the similarity \(\mathbf{s}^{i}\) between the \(i\)-th token \(\mathbf{z}^{i}\) and the global token \(\mathbf{Z}_{global}\) is calculated as:
\[\mathbf{s}^{i}=\frac{\mathbf{z}^{i}}{\|\mathbf{z}^{i}\|_{2}}\cdot\frac{\mathbf{Z}_{global}}{\| \mathbf{Z}_{global}\|_{2}}. \tag{1}\]
Then, the top 40% tokens with the highest similarities are selected as dominant ones. Note that, point patches corresponding to adjacent tokens usually share overlapped regions [62]. That is, token-level masking may introduce shortcuts for mask prediction. To remedy this, segment-level masking is adopted as different segments are isolated. Specifically, \(L_{m}\) segments with the most dominant tokens are selected with all tokens (\(\mathbb{R}^{L_{m}\times N\times C}\)) being masked. By masking these tokens that share high similarity with the global token, the difficulty of mask prediction is largely increased. It is demonstrated in Sec. 4.6 that our hard masked
Figure 2: An overview of our PointCMP.
samples can facilitate the encoder to achieve much higher accuracy.
**Hard Negative Samples.** Our major motivation is that different channels contain information of various importance, and the channels with higher correlation with the global token are more discriminative. Consequently, we synthesize hard negative samples by erasing these channels. Specifically, the correlation of the \(c\)-th channel in the \(i\)-th token \(\mathbf{s}_{c}^{i}\) is calculated as:
\[\mathbf{s}_{c}^{i}=\frac{\mathbf{z}_{c}^{i}}{\|\mathbf{z}^{i}\|_{2}}\cdot\frac{\mathbf{z}_{c}^{ global}}{\|\mathbf{Z}_{global}\|_{2}}, \tag{2}\]
where \(\mathbf{z}_{c}^{global}\) is the \(c\)-th channel of the global token. Then, we rank \(\mathbf{s}_{c}^{i}\) to obtain the order of each channel \(\mathbf{o}_{c}^{i}\), and sum up the resultant ranks across all tokens:
\[\mathbf{A}_{c}=\sum_{i=1}^{L\times N}\mathbf{o}_{c}^{i}, \tag{3}\]
Next, the top 20% channels are selected as principal channels and erased to produce hard negative samples.
### Local Contrastive Learning Branch
In the local branch, we first generate positional embedding for each token by feeding its spatio-temporal coordinate \((x,y,z,t)\) to a linear layer. Then, these positional embeddings are summed with their tokens and fed to a regressor to predict masked tokens using the context and position cues. Next, the predicted tokens \(\mathbf{Z}_{pre}\in\mathbb{R}^{L_{m}\times N\times C}\) are passed to a spatio-temporal matching module, as shown in Fig 3. Specifically, \(\mathbf{Z}_{pre}\) is pooled to obtain a global representation \(\mathbb{R}^{L_{m}\times C}\), which is then added to \(\mathbf{Z}_{pre}\). Afterwards, the resultant token is fed into a decoder to predict their position \(\mathbf{P}_{pre}\in\mathbb{R}^{L_{m}\times N\times 3}\). Here, a three-layer Transformer [13] is adopted as the regressor and FoldingNet [59] is used as the decoder.
As discussed in Sec. 1, the positional embeddings may lead to leakage of location information when inferring the coordinates for masked points. To remedy this, we adopt a contrastive loss to associate the representations of predicted tokens \(\mathbf{Z}_{pre}\) and corresponding groundtruth tokens \(\mathbf{Z}_{gt}\) learned by the encoder. Specifically, the tokens located at \(\mathbf{P}_{gt}\) are obtained through trilinear interpolation by querying \(\mathbf{Z}_{pre}\) located at \(\mathbf{P}_{pre}\), resulting in \(\mathbf{\hat{Z}}_{pre}\). For the \(i\)-th token \(\mathbf{z}_{i}\in\mathbf{\hat{Z}}_{pre}\), the corresponding token in \(\mathbf{Z}_{gt}\) is adopted as the positive sample \(\mathbf{z}_{+}\). Meanwhile, other tokens are regarded as negative samples. This avoids directly using the token position correspondence to construct sample pairs. The InfoNCE loss [37] is used for training:
\[\mathcal{L}_{\mathbf{z}_{i}}\!=\!-\!\log\frac{\exp\left(\mathbf{z}_{i}^{T}\mathbf{z}_{+}/ \tau\right)}{\exp\left(\mathbf{z}_{i}^{T}\mathbf{z}_{+}/\tau\right)+\sum_{\mathbf{z}_{j} \in\Phi}\exp\left(\mathbf{z}_{i}^{T}\mathbf{z}_{j}/\tau\right)}, \tag{4}\]
where \(\tau\) is a temperature parameter and \(\Phi\) is a negative sample set. Through token-level contrastive learning, the encoder can alleviate the shortcuts of positional encoding to capture fine-grained local information.
### Global Contrastive Learning Branch
In the global branch, we focus on learning discriminative representations at the video level. We take the global token \(\mathbf{Z}_{global}\) as the query \(\mathbf{v}_{i}\), and the resultant tokens produced by the regressor are pooled to obtain the positive sample \(\mathbf{v}_{+}\), as shown in Fig. 2. Meanwhile, the hard negative sample \(\mathbf{v}_{-}\) in addition with samples from other videos in batch \(\mathcal{B}\) are passed to a max-pooling layer, resulting in negative samples. Then, all samples are projected into the latent space and the InfoNCE loss [37] is adopted for training:
\[\mathcal{L}_{\mathbf{v}_{i}}\!=\!-\!\log\frac{\exp\left(\mathbf{v}_{i}^{T}\mathbf{v}_{+}/ \tau\right)}{\exp\left(\mathbf{v}_{i}^{T}\mathbf{v}_{-}/\tau\right)\!+\!\sum_{\mathbf{v}_{j }\in\{\mathcal{B}\cup\mathbf{v}_{+}\}}^{i\neq j}\exp\left(\mathbf{v}_{i}^{T}\mathbf{v}_{j} /\tau\right)}. \tag{5}\]
Overall, the total loss of our PointCMP is defined as:
\[\mathcal{L}_{total}=\mathcal{L}_{NCE}^{local}+\mathcal{L}_{NCE}^{global}, \tag{6}\]
where \(\mathcal{L}_{NCE}^{local}\) refers to the InfoNCE loss in the local branch (Eq. 4), and \(\mathcal{L}_{NCE}^{global}\) refers to the InfoNCE loss in the global branch (Eq. 5). With both loss terms, our PointCMP can simultaneously learn both local and global information.
## 4 Experiments
In this section, we first present the datasets and implementation details used in the experiments. Then, we compare our PointCMP to previous methods under four
Figure 3: Network Architecture of our spatio-temporal matching module.
widely used protocols, including end-to-end fine-tuning, linear probing, semi-supervised learning, and transfer learning. Finally, we conduct ablation studies to demonstrate the effectiveness of our method.
### Datasets and Implementation Details
**Datasets.** We conduct experiments on 3D action recognition and 3D gesture recognition tasks. Four benchmark datasets are employed, including NTU-RGBD [43], MSRAction-3D [28], NvGesture [36], and SHREC'17 [11].
* **NTU-RGBD**. The NTU-RGBD dataset consists of 56,880 videos with 60 action categories performed by 40 subjects. Following the cross-subject setting of [43], this dataset is split into 40,320 training videos and 16,560 test videos.
* **MSRAction-3D**. The MSRAction-3D dataset contains 567 videos with 23k frames. It consists of 20 fine-grained action categories performed by 10 subjects. Following [15], this dataset is split into 270 training videos and 297 test videos.
* **NvGesture**. The NvGesture dataset is comprised of 1532 videos with 25 gesture classes. Following [35], this dataset is split into 1050 training videos and 482 test videos.
* **SHREC'17**. The SHREC'17 dataset consists of 2800 videos in 28 gestures. Following [11], 1960 videos are used as the training set and 840 videos are adopted as the test data.
**Pre-training Details.** During pre-training, 16 frames were sampled as a clip from each point cloud video, with 1024 points being selected for each frame. The frame sampling stride was set to 2 and 1 on NTU-RGBD and MSRAction-3D, respectively. Then, we divided each clip into 4 segments and random scaling was utilized for data augmentation. Our model was pre-trained for 200 epochs with a batch size of 80, and linear warmup was utilized for the first 5 epochs. The initial learning rate was set to 0.0003 with a cosine decay strategy. The spatial search radius was initially set to 0.5/0.1 on NTU-RGBD/MSRAction-3D and the number of neighbors for the ball query was set to 9. The temperature parameter \(\tau\) was set to 0.01/0.1 in the local/global InfoNCE loss term.
### End-to-end Fine-tuning
We first evaluate our representations by fine-tuning the pre-trained encoder with a linear classifier in a supervised manner. The MSRAction-3D dataset was used for both pre-training and fine-tuning. During fine-tuning, 2048 points were selected for each frame and the pre-trained model was trained for 35 epochs with a batch size of 24. The initial learning rate was set to 0.015 with a cosine decay strategy. Following [15], the initial spatial search radius was set to 0.5 and the number of neighbors for the ball query was set to 9. Quantitative results are presented in Table 1.
As we can see, our PointCMP introduces significant accuracy improvements over the baseline trained in a fully supervised manner. Especially, the accuracy achieved using 8/12 frames is improved from 83.50%/87.88% to 89.56%/91.58%. This shows that our PointCMP can learn beneficial knowledge from point cloud videos in a self-supervised manner, which contributes to higher accuracy after fine-tuning.
### Linear Probing
We then conduct experiments to validate the effectiveness of our PointCMP via linear probing. The MSRAction-3D dataset was used for both pre-training and linear probing. Specifically, the pre-trained encoder is frozen and an additional linear classifier is added for supervised training. The experimental settings are the same as Sec. 4.2.
From Table 1, we can see that the pre-trained encoder using PointCMP outperforms the fully supervised baseline
\begin{table}
\begin{tabular}{c|l|c c c c c} \hline \hline \multicolumn{2}{c|}{**Methods**} & \multicolumn{6}{c}{**\#Frames**} \\ \cline{2-7} & 4 & 8 & 12 & 16 & 24 \\ \hline \multirow{10}{*}{Supervised Learning} & MeteorNet [31] & 78.11 & 81.14 & 86.53 & 88.21 & 88.50 \\ & Kinet [67] & 79.80 & 83.84 & 88.53 & 91.92 & 93.27 \\ & PST\({}^{2}\)[54] & 81.14 & 86.53 & 88.55 & 89.22 & - \\ & PPTr [55] & 80.97 & 84.02 & 89.89 & 90.31 & 92.33 \\ & P4Transformer [13] & 80.13 & 83.17 & 87.54 & 89.56 & 90.94 \\ & PST-Transformer [14] & 81.14 & 83.97 & 88.15 & 91.98 & 93.73 \\ & PSTNet [15] & 81.14 & 83.50 & 87.88 & 89.90 & 91.20 \\ & PSTNet++ [16] & 81.53 & 83.50 & 88.15 & 90.24 & 92.68 \\ \hline End-to-end Fine-tuning & **PSTNet + PointCMP** & **84.02** & **89.56** & **91.58** & **92.26** & **93.27** \\ Linear Probing & **PSTNet + PointCMP** & 78.11 & 88.55 & 90.24 & 91.92 & 92.93 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Action recognition accuracy (%) on MSRAction-3D.
even under the linear probing setting. Our method surpasses the baseline under most frame settings with notable margins (e.g., 88.55%/90.24% vs. 83.50%/87.88% under 8/12 frames). This clearly demonstrates the high quality of the representations learned by PointCMP.
### Semi-supervised Learning
We also conduct experiments to evaluate our PointCMP under the setting of semi-supervised learning. The cross-subject training set of NTU-RGBD was used for pre-training. Specifically, we used only 50% training set of NTU-RGBD to fine-tune the pre-trained encoder in a supervised manner. Following [15], the initial spatial search radius was set to 0.1, the number of neighbors for the ball query was set to 9, and 2048 points were samples for each frame. The model was fine-tuned for 50 epochs with a batch size of 24. The initial learning rate was set to 0.015 with a cosine decay strategy.
Table 2 compares the quantitative results produced by our PointCMP and previous fully supervised approaches. Averaged accuracy over 3 experiments is reported for our method. It can be observed that our PointCMP achieves comparable performance to the fully supervised baseline even with only 50% data (88.5% vs. 90.5%). This further demonstrates the superiority of the representations learned by our PointCMP.
### Transfer Learning
To evaluate the generalization performance of our PointCMP, we conduct experiments by transferring pre-trained encoder to other datasets or tasks. Specifically, the encoder was first pre-trained on the cross-subject training set of NTU-RGBD, and then fine-tuned with an additional MLP head on MSRAction-3D, NvGesture, and SHREC'17.
**Transfer to MSRAction-3D.** We first fine-tuned the pre-trained encoder on MSRAction-3D following the experimental settings in Sec. 4.2. We compare our PointCMP with ROP [47] in Table 3. Note that, since the official code for ROP is unavailable, we report its performance on 4D MinkNet [10] and MeteorNet [31] for comparison. Although PSTNet uses only points as input, our PointCMP facilitates this baseline to surpass ROP by over 2% accuracy. In addition, our PointCMP introduces more significant accuracy improvements as compared to ROP (5.03% vs. 4.26%).
**Transfer to NvGesture and SHREC'17.** The encoder was further transferred from action recognition to gesture recognition through fine-tuning on NvGesture and SHREC'17. Specifically, the pre-trained model was fine-tuned for 100 epochs with a batch size of 16. The initial learning rate was set to 0.01 with a cosine decay strategy. During fine-tuning, 32 frames were utilized with 512/256 points sampled for each frame on NvGesture/SHREC'17. We compare our fine-tuned models to previous supervised state-of-the-art methods in Table 4. As we can see, after fine-tuning for 100 epochs, our PointCMP facilitates PSTNet to produce very competitive accuracy. In addition, our PointCMP also allows for faster convergence such that more significant improvements are achieved after fine-tuning for only 35 epochs (_e.g._, 78.9% vs. 84.0% on NvGesture). This also shows the superior generalization capability cross different tasks of the representations learned by our PointCMP.
branch, respectively. Then, model A3 is introduced by removing the mutual similarity based augmentation module. Quantitative results are presented in Table 5.
As we can see, with only local or global branch, the performance of model A1 and A2 are limited (89.22% and 49.49%). This is because, both local and global information contribute to the recognition of point cloud videos. When these two branches are combined, complementary information can be exploited such that better accuracy is achieved by model A3 (89.76%). However, without the mutual similarity based augmentation module, model A3 still suffers an accuracy drop of 2.16% as compared to A4. This further validates the effectiveness of our mutual similarity based augmentation module.
**Hard Masked Samples.** The masking strategy contributes to the quality of hard masked samples and plays a critical role in the local branch of our PointCMP. Consequently, we conduct experiments to study different masking strategies and compare their results in Table 6.
As we can see, segment-wise masking strategy significantly outperforms token-wise masking strategy under different masking ratios. As compared to token-wise strategy, segment-wise strategy can better avoid the leakage of information caused by overlapped point patches, which facilitates the network to better exploit local structures in a point cloud video. Moreover, similarity-based masks introduce notable performance gains on segment-wise strategy, with accuracy being improved from 90.81%/88.15% to 91.92%/90.24%. This demonstrates the effectiveness of our hard masked samples.
We further visualize the points corresponding to dominant tokens with high similarity to the global token in Fig. 4. As we can see, tokens corresponding to moving body parts (e.g., arms in Fig. 4(c)) are highlighted, which is consistent with our intuition. This demonstrates the feasibility of our mutual similarity based augmentation to synthesize reasonable hard samples. With these discriminative regions being masked, the encoder is encouraged to leverage more context for mask prediction, with representations of higher quality being learned.
**Hard Negative samples.** To demonstrate the effectiveness of hard negative samples in the global branch of our PointCMP, model C1 is introduced by excluding hard samples during training. That is, only samples in other videos are employed as negatives. Furthermore, we conduct experiments to study different channel erasing strategies. Quantitative results are presented in Table 7.
It can be observed that model C1 suffers an accuracy drop of 1.40% as compared to C3 when hard negative samples are excluded. Using random channel erasing to generate hard negative samples, model C2 improves C1 with accuracy being increased from 90.52% to 91.29%. With our mutual similarity based augmentation module, hard negative samples of higher quality can be synthesized such that better performance can be achieved. This validates the
\begin{table}
\begin{tabular}{l l|c|c c c c} \hline \hline \multicolumn{2}{c|}{**Granularity**} & \multicolumn{2}{c|}{**Mask**} & \multicolumn{4}{c}{**Masking Ratio**} \\ \cline{3-7} & & & 25\% & 50\% & 75\% & 90\% \\ \hline B1 & Token & Random & 71.72 & 71.72 & 76.77 & 78.11 \\ B2 & Token & Similarity-based & 70.03 & 81.82 & 84.18 & 88.55 \\ \hline B3 & Segment & Random & 90.81 & 88.15 & 79.80 & - \\ B4 (Ours) & Segment & Similarity-based & **91.92** & **90.24** & **86.53** & - \\ \hline \hline \end{tabular}
\end{table}
Table 6: Ablation studies on hard masked samples.
\begin{table}
\begin{tabular}{l c|c|c} \hline \hline \multicolumn{2}{c|}{**Hard Sample**} & \multicolumn{1}{c|}{**Strategy**} & \multicolumn{1}{c}{**Accuracy (\%)**} \\ \hline C1 & \(\bigtimes\) & - & 90.52 \\ C2 & ✓ & Random & 91.29 \\ C3 (Ours) & ✓ & Similarity-based & **91.92** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Ablation studies on hard negative samples.
\begin{table}
\begin{tabular}{l c c|c} \hline \hline \multicolumn{2}{c|}{**Hard Sample**} & \multicolumn{1}{c|}{**Strategy**} & \multicolumn{1}{c}{**Accuracy (\%)**} \\ \hline D1 & Local & \(\bigtimes\) & 86.20 \\ D2 & Local & ✓ & 89.22 \\ \hline D3 & Local \& Global & \(\bigtimes\) & 90.24 \\ D4 (Ours) & Local \& Global & ✓ & **91.92** \\ \hline \hline \end{tabular}
\end{table}
Table 8: Ablation studies on the spatio-temporal matching module.
Figure 4: Visualization of hard masked samples. Points corresponding to dominant tokens are marked in green.
\begin{table}
\begin{tabular}{l c c c|c} \hline \hline & **\begin{tabular}{c} **Local** \\ **Branch** \\ \end{tabular} & **\begin{tabular}{c} **Global** \\ **Branch** \\ \end{tabular} & **
\begin{tabular}{c} **Similarity-based** \\ **Augmentation** \\ \end{tabular} & **Accuracy (\%)** \\ \hline A1 & ✓ & & & 89.22 \\ A2 & & ✓ & & 49.49 \\ A3 & ✓ & ✓ & & 89.76 \\ A4 (Ours) & ✓ & ✓ & ✓ & **91.92** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Ablation studies on architecture designs.
effectiveness of the hard negatives generated by principal channel erasing.
Following [18], we visualize cosine similarity of representations learned from different sample pairs in Fig. 5 to study the importance of our hard negative samples. A model pre-trained on MSRAction-3D without using hard negatives is utilized for analyses. As shown in Fig. 5(a), the similarities of positive pairs are close to 1 with an average value of 0.875. On the contrary, negative pairs are gathered around 0 with an average value of 0.013. For our hard negatives, their average similarity score is increased to 0.315, which means these samples remain difficult for the pre-trained encoder if they are not included for training. This further shows the necessity of our hard negatives.
**Spatio-temporal Matching Module.** In the local branch of our PointCMP, a spatio-temporal matching module is adopted to conduct local contrastive learning. To study its effectiveness, we first developed model D2 with only the local branch. Then, we introduced model D1 and D3 by removing this matching module from D2 and D4, respectively. Quantitative results are presented in Table 8. As we can see, the spatio-temporal matching module facilitates D4 to produce an accuracy improvement of 1.68% and introduces a more significant improvement of 3.02% to D2. We further visualize the evolution of the local contrastive loss (i.e., \(\mathcal{L}_{NCE}^{local}\) in Eq. 6) in Fig. 7. Without the spatio-temporal matching module, the loss decreases rapidly to near 0 and the networks cannot be further optimized. This is because the leakage of location information is leveraged by the network as shortcuts without capturing geometric information. In contrast, our matching module alleviates positional information leakage and increases the hardness of learning to help the network ultimately achieve higher accuracy (Table 8).
**Representation Visualization.** We further visualize the feature distributions using t-SNE to demonstrate the effectiveness of our PointCMP. With only global branch as many previous methods do, the learned representations have blurred boundaries between different categories with limited discriminative capability, as shown in Fig. 6(a). In contrast, the representations extracted using our PointCMP can better exploit both global and local information with clearer boundaries between different categories, as shown in Fig. 6(b). This clearly demonstrates the high discrimination of the representations learned by our method.
## 5 Conclusion
In this paper, we develop a self-supervised learning framework termed PointCMP for point cloud videos. Our PointCMP unifies the complementary advantages of contrastive learning and mask prediction paradigms to simultaneously learn both global and local spatio-temporal features at different granularities. To promote the training of PointCMP, we propose a mutual similarity based augmentation module to generate hard masked and negative samples at the feature level. Experiments on benchmark datasets show that our PointCMP achieves state-of-the-art performance on both action and gesture recognition tasks.
**Acknowledgments.** This work was partially supported by the National Natural Science Foundation of China (No. U20A20185, 61972435), the Guangdong Basic and Applied Basic Research Foundation (2022B1515020103), and Shanghai Science and Technology Innovation Action Plan (21DZ203700).
Figure 5: Visualization of cosine similarity histograms between the representations of query samples and their paired (a) positive samples, (b) negative samples, and (c) hard negative samples generated by channel erasing.
Figure 6: The t-SNE visualization of representation distributions on MSRAction-3D (a) after pre-training only with global contrastive learning and (b) after pre-training using our PointCMP.
Figure 7: Evolution of local contrastive learning loss during pre-training on MSRAction-3D. |
2304.13620 | ChartSumm: A Comprehensive Benchmark for Automatic Chart Summarization
of Long and Short Summaries | Automatic chart to text summarization is an effective tool for the visually
impaired people along with providing precise insights of tabular data in
natural language to the user. A large and well-structured dataset is always a
key part for data driven models. In this paper, we propose ChartSumm: a
large-scale benchmark dataset consisting of a total of 84,363 charts along with
their metadata and descriptions covering a wide range of topics and chart types
to generate short and long summaries. Extensive experiments with strong
baseline models show that even though these models generate fluent and
informative summaries by achieving decent scores in various automatic
evaluation metrics, they often face issues like suffering from hallucination,
missing out important data points, in addition to incorrect explanation of
complex trends in the charts. We also investigated the potential of expanding
ChartSumm to other languages using automated translation tools. These make our
dataset a challenging benchmark for future research. | Raian Rahman, Rizvi Hasan, Abdullah Al Farhad, Md Tahmid Rahman Laskar, Md. Hamjajul Ashmafee, Abu Raihan Mostofa Kamal | 2023-04-26T15:25:24Z | http://arxiv.org/abs/2304.13620v3 | # ChartSumm: A Comprehensive Benchmark for Automatic Chart Summarization of Long and Short Summaries
###### Abstract
Automatic chart to text summarization is an effective tool for the visually impaired people along with providing precise insights of tabular data in natural language to the user. A large and well-structured dataset is always a key part for data driven models. In this paper, we propose ChartSumm: a large-scale benchmark dataset consisting of a total of 84,363 charts along with their metadata and descriptions covering a wide range of topics and chart types to generate short and long summaries. Extensive experiments with strong baseline models show that even though these models generate fluent and informative summaries by achieving decent scores in various automatic evaluation metrics, they often face issues like suffering from hallucination, missing out important data points, in addition to incorrect explanation of complex trends in the charts. We also investigated the potential of expanding ChartSumm to other languages using automated translation tools. These make our dataset a challenging benchmark for future research.
## 1 Introduction
Automatic chart summarization is a task where the goal is to describe important data points and trends in a chart in natural language. Chart summaries are helpful to better interpret the chart, making it useful for the visually impaired people as well as to improve the performance of different information retrieval algorithms (Obeid and Hoque, 2020; Carenini et al., 2013; Li et al., 2013).
Scarcity of large scale well defined datasets with chart image, metadata and well described summaries is a major challenge in automatic chart summarization. To our best knowledge, there are only four datasets (Obeid and Hoque, 2020; Zhu et al., 2021; Hsu et al., 2021; Kanthara et al., 2022) available for the chart to text summarization task, making this task a low resource problem. Among these datasets, three of them (Obeid and Hoque, 2020; Zhu et al., 2021; Kanthara et al., 2022) contain chart images with metadata and well defined summaries while the SciCAP dataset (Hsu et al., 2021) only contains chart images and captions.
In this work, we address the scarcity of public datasets in the automatic chart summarization task. We propose "**ChartSumm**", a large scale dataset for chart to text summarization comprising of \(84,363\) chart images with corresponding chart metadata and summaries. (see Figure 1 for an example). We also propose two test sets based on the summary length. In this paper, our major contributions are summarized below:
(i) Proposing a new benchmark dataset for the automatic chart summarization task. To our best knowledge, our ChartSumm dataset is currently the largest dataset proposed for this task. Meanwhile, we also introduce two different test sets to separately compare the performance on generating short and long summaries.
(ii) Conducting a series of experiments using strong baselines to demonstrate how models trained on our dataset have better generalization capability than other existing datasets. In addition, we identify the limitations of the state-of-the-art models in our proposed dataset. Furthermore, we also explore the scope of expanding our dataset
Figure 1: An example chart-summary pair from our proposed dataset.
to other languages through translation and evaluate the performance in a human-annotated test set in the Bengali language. To our best knowledge, this is the first work that investigated the Chart Summarization task in any languages except English. The dataset and codes are available at [https://github.com/pranonrahman/ChartSumm](https://github.com/pranonrahman/ChartSumm).
## 2 Related Work
Existing Chart-To-Text summarization systems generate summaries from either the chart image Hsu et al. (2021) or the chart metadata Gong et al. (2019); Obeid and Hoque (2020); Kanthara et al. (2022). Before the advent of deep learning, most early work utilized a two stage approach that applied content selection using different statistical tools in the first step followed by generating summaries using pre-defined templates Reiter (2007); Zhu et al. (2021). However, predefined template-based architectures frequently lack generality and fail to capture complex trends in data. In recent years, deep learning-based techniques have gained significant attention Gong et al. (2019); Obeid and Hoque (2020); Gajbhiye and Lopes (2021); Zhu et al. (2021); Hsu et al. (2020); Zhou et al. (2021); Dadhich et al. (2021); Sreevalsan-Nair et al. (2021); Luo et al. (2021); Chai et al. (2021); Kanthara et al. (2022) due to their superior performance over the template-based approaches. Nonetheless, due to the lack of Chart-To-Text summarization datasets, not only that the models proposed for this task require improvements, the generalized effectiveness of these models is also yet to be investigated.
Among the four benchmark Chart-To-Text summarization datasets that are publicly available, the Chart2Text Obeid and Hoque (2020) summarization dataset is the first dataset proposed for this task that includes 8,305 samples collected from the Statista1. However, the size of this dataset is quite small and so effective data-driven methods cannot be trained on top of that. Later, the SciCAP Hsu et al. (2021) data was proposed for the chart captioning task from chart images. Thus, it is not suitable for methods that can only generate summary from metadata. The recently proposed AutoChart Zhu et al. (2021) dataset is based on some predetermined templates and so this dataset does not contain much variance in the chart descriptions. More recently, Kanthara et al., Kanthara et al. (2022) proposed the Chart-To-Text dataset that consists of chart images, with their corresponding metadata and human written descriptions. Though this dataset is currently the largest dataset available for this task containing 44,085 charts that were collected from the Statista website and the Pew website, our proposed ChartSumm dataset is almost double in size than the Chart-To-Text dataset.
Footnote 1: [https://www.statista.com/](https://www.statista.com/)
## 3 The ChartSumm Dataset
This section describes how we compile a large-scale dataset consisting of \(84,363\) examples from the Knoema2 and the Statista website for the chart to text summarization task along with their analysis.
Footnote 2: [https://knoema.com/atlas](https://knoema.com/atlas)
### Dataset Construction
**Knoema:** It is a statistical service-based online platform that contains the economic indicator of more than 200 countries. Knoema provides a short description for each statistic generated by its digital data assistant named Yodatai3 that summarizes basic information about datasets. To construct our dataset, we first crawl over 1,10,000 statistics from Knoema. Then, we filter out the statistics where the source of data is not publicly available, resulting in 43,179 publicly available statistics. Afterward, we collect the chart metadata and their corresponding short descriptive captions. Since the statistics in Knoema are shown with respect to the year, we
\begin{table}
\begin{tabular}{l l l l l l} \hline
**Dataset Name** & **Task** & **Data Source** & **Fernodality** & **Summary Type** & **Example Count** \\ \hline SciCAP Jing et al. (2021) & Image - Test caption & Scientific, Pogoon & Class Image & Short region from scientific, pogoon & 270,000 \\ \hline Chart2Text Obeid and Hoque (2020) & Table - Test description & Station & Machine Image & Densight image & 8,305 \\ \hline AutoCart Zhu et al. (2021) & Table - Test description & Different politis-sponsing & Machine & Automatic template generated summary & 23,543 \\ \hline Chart-To-Text (Kanthara et al., 2022) & Table - Test description & Station & Channel Image & Densight image & Densight image \\ \hline Chart-To-Text (Kanthara et al., 2022) & 1. Table - Test description & Station & Channel Image & Densight image & 44,085 \\ \hline ChartSumm (ours) & Table - Test description & Known & Decision & & \\ \hline \end{tabular}
\end{table}
Table 1: Comparison between existing datasets and our proposed dataset.
classify each chart as a simple line chart. The title and caption of the chart are then tokenized while we remove white spaces and newlines using stemming. We also normalize the numerical entities.
**Statista:** It is also an online platform where statistics on a wide range of topics are published along with a short human-written description of the statistics. Topics in Statista include economics, marketing, industry, and opinion research. For dataset creation, at first, we crawl over \(750,000\) available pages in Statista research to collect a list of \(41,184\) publicly available charts along with summaries and chart metadata. Then we classify the data into simple and complex charts depending on the number of columns in the chart. Similar to the Knoema dataset, we also apply tokenization and stemming. Since many examples in the Statista dataset did not contain the x_label, we apply the following heuristic rules to automatically classify the x_labels as Year, Month, Day, Quarter, Country, City, and Continent:
* **Year:** If all x values were integers less than 2050 and greater than 1800 we set the x label to "year".
* **Month:** If all x values were names of months, we set the x label to "month".
* **Day:** If all x values were names of days(saturday, sunday...), we set the x label to "month".
* **Quarter:** If most x values were Q1/Q2/Q3/Q4 and optionally followed by an integer(year) we set x label as "quarter".
* **Country:** If more than 30% x values contained values from the list of all countries collected from Wikipedia, we labeled the x label as "country"
* **City:** If more than 30% x values contained values from the list of cities collected from World City Database, we labeled the x label as "city"
* **Area:** If the x labels contained the names of general areas like continent names, sub continent names, etc, we set the x label as "area".
* **NER:** We also used named entity recognition to identify some other named types such as companies, social medias etc.
For charts where the x_labels could not be automatically determined, the value for the x_label is set to "x_label".
To classify the charts into different types (bar/line/pie), we use ChartReader4. The charts are then divided into simple and complex categories. We need to find **missing x labels** since some of the scrapped data have missing x labels. So we manually identify them using the following methods.
Footnote 4: [https://github.com/Cvrane/ChartReader](https://github.com/Cvrane/ChartReader)
### Dataset Analysis
In this section, we analyze our proposed ChartSumm dataset. At first, we compare our ChartSumm dataset with some existing datasets in Table 1. We find the both Chart2Text [10] and Chart-To-Text [13] collected their data from a source called Statista in which the summaries are bit descriptive and longer in length. Whereas our dataset contains both long and short summaries.
Note that our proposed dataset contains line charts, bar charts, and pie charts. For Statista. bar chart is the most common type (\(64.70\%\) in statistica) followed by line chart (\(33.76\%\) in statistica) followed by pie chart (\(1.54\%\) in statistica). For knoema, all charts are line charts. In Figure 2, we show the topic distribution of our dataset. For topic modeling, we perform Latent Dirichlet allocation (LDA) [1] by creating a topic per document and a word per topic model, both of which are based on Dirichlet distributions. We find that our proposed ChartSumm dataset covers a large spectrum of topics including Economy & politics (\(21.60\%\)), Society & Science (\(13.03\%\)), Internet & Media (\(11.43\%\)), Public life & Health (\(10.42\%\)), Sports & Entertainment (\(9.14\%\)), Consumer Goods (\(7.71\%\)), Retail & Trade (\(5.35\%\)), Education (\(5.32\%\)), etc.
Figure 2: Topic distribution of ChartSumm
In Table 2, we show the average cell counts for the tables, as well as character and token counts for summaries and titles, respectively. We find that in terms of average number of tokens, the summaries of simple and complex Statista charts were about \(35\%\) and \(59.9\%\) longer than the Knoema charts, respectively. The data obtained from each source was classified into train, validation, and test sets, following an 80:10:10 split ratio. We show the number of samples in our training, validation, and test sets in table 4. Since, one source of our dataset is Statista that is similar to the Chart-To-Text [14] dataset, we measure the overlaps between the data samples from Statista in our ChartSumm dataset and the Chart-To-Text dataset. For similarity measurement, we first tokenize the captions and then calculate the percentage of matched tokens. We assume two samples are exactly similar when the similarity is greater than \(90\%\). Table 3 shows that only \(5,338\) captions in our dataset overlaps with the samples from Statista in Chart-To-Text.
## 4 Experiments
In this section, we present the baseline models that we utilize to benchmark the performance in our proposed dataset, followed by the fine-tuning process, the evaluation metrics, and the experimental results.
### Baselines:
We use T5-Base [17] and BART [11] as our baselines due to their effectiveness in Chart-To-Text tasks [14]. T5 is a large pre-trained language model trained on multiple sequence-to-sequence tasks. BART is a sequence-to-sequence model pre-trained for the language modeling task using the denoising autoencoder architecture. We fine-tune three variants of BART: (i) BART-Base, (ii) BART-Large-CNN, and (iii) BART-Large-XSUM. We implement all models using HuggingFace [13]. Below we describe our model fine-tuning process.
### Fine-tuning Process
We used chart metadata (title, corresponding data table, labels) to fine-tune all four of our pre-trained baseline models. We flattened the table by rows and concatenated it with the caption of the table separated by a separator token. To mimic the pre-training process of T5, we added the prefix: "Summarize chart: " before each example. Figure 3 shows the fine-tuning stages.
All four baseline models were fine-tuned for \(3\) epochs with a batch size of \(8\). The initial learning rate during our fine-tuning was \(1e-6\). We used AdamW [12, 13] as our optimizer and cross-entropy as the loss function. We used Google Colab5 for our experiments.
Footnote 5: [https://colab.research.google.com/](https://colab.research.google.com/)
### Evaluation Metrics:
We use five evaluation metrics in our automated evaluation: (i) BLEU [20]: it uses n-gram overlaps between reference text and machine-generated text to determine similarity score, (ii) BLEURT [14]: it evaluates how fluent the candidate is and how well it transfers the reference's meaning (we utilize BLEURT base-128 for our evaluation), (iii) Perplexity: it is a measurement that quantifies how well a probability model predicts a sample (we utilized pre-trained GPT-2 [17] to measure perplexity), (iv) CIDEr: [2] it uses n-grams
\begin{table}
\begin{tabular}{|l|l|l|l|l|l|} \hline
**Train-k** & **Train-s** & **Valid-k** & **Valid-s** & **Test-k** & **Test-s** & **Total** \\ \hline
34,503 & 32,985 & 4,338 & 4,101 & 4,338 & 4,098 & **84,363** \\ \hline \end{tabular}
\end{table}
Table 4: Split distribution of ChartSumm.
\begin{table}
\begin{tabular}{|l|l|l|l|l|l|} \hline & **ChartSumm** & **Chart-To-Text** & **Overlaps** \\ \hline Simple & 33067 & 27868 & 4144 \\ \hline Complex & 8338 & 6943 & 1194 \\ \hline Total & 41405 & 34811 & 5338 \\ \hline \end{tabular}
\end{table}
Table 3: Overlaps between Chartsumm and Chart-To-Text Statista samples.
Figure 3: Fine-tuning process of baseline models
\begin{table}
\begin{tabular}{|l|l|l|l|l|} \hline & **ChartSumm** & **Chart-To-Text** & **Overlaps** \\ \hline Simple & 33067 & 27868 & 4144 \\ \hline Complex & 8338 & 6943 & 1194 \\ \hline Total & 41405 & 34811 & 5338 \\ \hline \end{tabular}
\end{table}
Table 2: Dataset analysis of ChartSumm.
overlaps and calculates average cosine similarity between the candidate sentence and the reference sentences, to capture the grammatical qualities with richer semantics, (v): Content Selection (CS): it measures how closely the generated text matches the reference documents (Wiseman et al., 2017).
### Experimental Results:
To evaluate the performance, we use three test sets: (i) _test-k_: denotes ChartSumm test set compiled from Knoema where the summaries are precise and well structured, (ii) _test-s_: denotes ChartSumm test set compiled from Statista where the summaries are descriptive and have a longer length, and (iii) _Chart
\begin{table}
\begin{tabular}{p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}} \hline \hline
**Gold:** Between 1992 and 2018, Georgia renewable surface for remained stable at around 62.1 billion cubic meters per year. & **T5-Base:** In 2018, renewable surface for Georgia was 62.1 million cubic meters per year. & **Comments:** Model predicts factually incorrect information and captures incorrect trend. & **Comments:** Model fails to generate informative summary. \\ \hline \hline \end{tabular}
\end{table}
Table 6: Examples of generated summaries containing error. Red indicates factually incorrect output, orange indicates failure to capture trend and violet indicates hallucination error
\begin{table}
\begin{tabular}{p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}|p{113.8pt}} \hline \hline
**Fine-based on** & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} & \multicolumn{3}{p{113.8pt}}{**Model (v)**} \\ \hline \multirow{4}{*}{**Categories**} & **Task** & **100** & **15.99** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** & **15.98** \\ & **T5-Base** & 11.3 & **1.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.3 & **1.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** \\ & **T5-Base** & 11.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.99** & **15.
_To-Text (Statista)_: the Statista version of the test set of Chart-To-Text (Kantharaj et al., 2022). We run experiments using the full training sets of both ChartSumm and Chart-To-Text, with additional experiments with the training subsets of Statista and Knoema from ChartSumm.
Our experimental results are shown in Table 5. In terms of BLEURT, CIDER, and CS metrics, we observe that BART-Large models (Bart-Large-CNN and Bart-Large-XSUM) outperform other models in all three test sets, while T5 performs the best in terms of PPL in all test sets. Furthermore, we observe that models fine-tuned on ChartSumm-K always perform better than other models in test sets containing data from Knoema only (except the PPL metric). We again observe similar trends in terms of models fine-tuned on ChartSumm-S. More importantly, all models fine-tuned on our ChartSumm-S dataset outperform the baselines that are fine-tuned on Chart-To-Text even in the Chart-To-Text test set, showing the effectiveness of our proposed dataset. Meanwhile, models fine-tuned on Chart-To-Text performs very poorly in ChartSumm-K (about 82.48% lower in terms of best performing T5 model), indicating that the Chart-To-Text dataset is not generalizable to generate summaries that are shorter and precise. To further investigate the performance, we do some error analyses in the following section.
### Error Analysis and Challenges:
For error analysis, we randomly sampled \(100\) instances with their summaries generated by different baseline models.
We notice that in many cases, even though the generated summary is fluent and readable, it contains factually incorrect information and predicted wrong trend in data (see the first example in Table 6. We also notice that models sometimes fail to generate informative summaries while also failing to predict anything about data (see the second example in Table 6). In both Bart-Large models, we also find some cases where the models generate information that is fully irrelevant to the chart (i.e., the Hallucination effect (Gong et al., 2019; Obeid and Hoque, 2020; Wiseman et al., 2021)) (see the third example in Table 6).
### Evaluation of ChartSumm in Other Languages
To assess the potential for expanding the use of ChartSumm to other languages, we undertook a study where we translated ChartSumm into Bengali and fine-tuned a pre-trained mT5 (Xue et al., 2021) model to evaluate its performance. To our best knowledge, this is the first study that has explored the task of summarizing charts in languages other than English.
**Translation:** To translate the training and validation sets of ChartSumm into Bengali, we utilized NLLB (Costa-jussa et al., 2022), which is a state-of-the-art neural machine translation model. To ensure proper evaluation, the test data was translated into Bengali with the assistance of human annotators who are undergraduate students and have proficiency in both English and Bengali.
**Baseline:** In our study, we employed mT5 (Xue et al., 2021) as a baseline model due to its efficacy in text summarization tasks in Bengali. We fine-tuned a variant of mT5 which was pre-trained on multilingual XL-SUM (Hasan et al., 2021). Our baseline model was fine-tuned for 4 epochs using a batch size of 8, with an initial learning rate of 0.000001. The AdamW optimizer was used for the fine-tuning process and cross-entropy was utilized as the loss function.
**Evaluation:** In our study, we conducted automatic evaluation of the text summarization models using the BLEU, which measures the degree of similarity between the reference and the machine-generated text using n-gram overlaps. However, we were unable to employ other evaluation metrics such as BLEURT and CIDEr for the Bengali language, as they require language-specific models that are not available for Bengali. Therefore, we only used BLEU as an evaluation metric for our Bengali text summarization experiment. We present the outcomes of our experiments in Ta
\begin{table}
\begin{tabular}{l|c c c|c c c} \hline \multirow{2}{*}{**Fine-tuned On**} & \multicolumn{3}{c|}{**Test-k**} & \multicolumn{3}{c}{**Test-s**} \\ \cline{2-7} & **BLEU-1** & **BLEU-2** & **BLEU-N** & **BLEU-1** & **BLEU-2** & **BLEU-N** \\ \hline Full dataset & 17.07 & 6.41 & 3.22 & 28.45 & 12.28 & 7.69 \\ \hline Knoema & **17.80** & **6.62** & **3.79** & 12.02 & 3.60 & 1.25 \\ \hline Statista & 15.44 & 5.02 & 1.73 & **28.93** & **12.63** & **7.97** \\ \hline \end{tabular}
\end{table}
Table 7: Experimental results on different test sets of ChartSumm translated to Bengali.
ble 7. It is evident from the results that models fine-tuned on ChartSumm-s and ChartSumm-k performed better on their corresponding test sets in comparison to the scenario when evaluated on the combined test set. From this, we can see that the model can perform better in the respective test sets even with machine-generated translations. This opens up the possibility to investigate the performance of ChartSumm in other languages through automatic machine translations. Meanwhile, performance evaluation via human-annotated training data in other languages is also something worth investigating in the future.
## 5 Conclusion
In this work, we present a new large scale benchmark dataset for the automatic chart summarization task to address the low resource problem in such tasks. Our proposed dataset is almost double in size than the existing largest dataset available for this task. Thus, the proposed ChartSumm dataset will serve as a strong benchmark for researchers in this relatively new area of natural language generation. We utilize three BART models and one T5 model as baselines and conduct extensive experiments using various evaluation metrics to identify the challenges in this task. Experimental results showed that models fine-tuned with our proposed ChartSumm dataset could achieve better domain generalization than other existing benchmark datasets. We also explored the possibility of extending ChartSumm to other languages through automatic machine translation. In the future, we would like to extend ChartSumm to a multilingual dataset to address the scarcity of well-formatted datasets in other low-resource languages. We will also study how to incorporate query relevance (Laskar et al., 2022, 2020a, 2020a, 2022; Kantharaj et al., 2022a; Hoque et al., 2022; Laskar et al., 2020c), and entity recognition (Laskar et al., 2022c, 20, 20, 20) capabilities in this task.
|
2306.15428 | Evaluating The Impact Of Species Specialisation On Ecological Network
Robustness Using Analytic Methods | Ecological networks describe the interactions between different species,
informing us of how they rely on one another for food, pollination and
survival. If a species in an ecosystem is under threat of extinction, it can
affect other species in the system and possibly result in their secondary
extinction as well. Understanding how (primary) extinctions cause secondary
extinctions on ecological networks has been considered previously using
computational methods. However, these methods do not provide an explanation for
the properties which make ecological networks robust, and can be
computationally expensive. We develop a new analytic model for predicting
secondary extinctions which requires no non-deterministic computational
simulation. Our model can predict secondary extinctions when primary
extinctions occur at random or due to some targeting based on the number of
links per species or risk of extinction, and can be applied to an ecological
network of any number of layers. Using our model, we consider how false
positives and negatives in network data affect predictions for network
robustness. We have also extended the model to predict scenarios in which
secondary extinctions occur once species lose a certain percentage of
interaction strength, and to model the loss of interactions as opposed to just
species extinction. From our model, it is possible to derive new analytic
results such as how ecological networks are most robust when secondary species
degree variance is minimised. Additionally, we show that both specialisation
and generalisation in distribution of interaction strength can be advantageous
for network robustness, depending upon the extinction scenario being
considered. | Chris Jones, Damaris Zurell, Karoline Wiesner | 2023-06-27T12:38:35Z | http://arxiv.org/abs/2306.15428v2 | Evaluating The Impact Of Species Specialisation On Ecological Network Robustness Using Analytic Methods
###### Abstract
Ecological networks describe the interactions between different species, informing us of how they rely on one another for food, pollination and survival. If a species in an ecosystem is under threat of extinction, it can affect other species in the system and possibly result in their secondary extinction as well. Understanding how (primary) extinctions cause secondary extinctions on ecological networks has been considered previously using computational methods. However, these methods do not provide an explanation for the properties which make ecological networks robust, and can be computationally expensive. We develop a new analytic model for predicting secondary extinctions which requires no non-deterministic computational simulation. Our model can predict secondary extinctions when primary extinctions occur at random or due to some targeting based on the number of links per species or risk of extinction, and can be applied to an ecological network of any number of layers. Using our model, we consider how false positives and negatives in network data affect predictions for network robustness. We have also extended the model to predict scenarios in which secondary extinctions occur once species lose a certain percentage of interaction strength, and to model the loss of interactions as opposed to just species extinction. From our model, it is possible to derive new analytic results such as how ecological networks are most robust when secondary species degree variance is minimised. Additionally, we show that both specialisation and generalisation in distribution of interaction strength can be advantageous for network robustness, depending upon the extinction scenario being considered.
## 1 Introduction
No species exists in isolation, depending upon interactions with other species to feed, reproduce or maintain a stable population [1, 2]. Modelling the interactions between species is therefore of great importance in ecology, and one approach to this problem is to model interactions as an ecological network [3, 4]. Ecosystems are increasingly threatened by the effects of climate change [5, 6] which can cause sudden and widespread extinction events. Therefore, it is useful to model extinctions on ecological networks in order to understand the possible knock-on effects of species extinctions, as this may help to identify methods for conserving or reinforcing ecosystems in the future [7, 8, 9].
Species extinctions on ecological networks have been extensively studied in the past 20 years, with simplistic topological models providing predictions for the impact of extinctions under scenarios including: extinctions which occur at random or with some ordering [10], extinctions on networks made up of numerous trophic levels [11], and extinctions which occur due to a loss of interaction strength over a certain threshold [12].
The models used are not the only possible approach to understanding the robustness of ecological networks. Other models consider the size of the largest component in the interaction network [13, 14], or take a more dynamical approach as is the case with Bayesian network models [15, 16]. Here we restrict ourselves to what we refer to as simplistic topological network models, which originate from Memmott et al. [10]. In these models, we are concerned with the point at which a given species goes extinct due to losing either a certain number of neighbours or a certain amount of interaction strength.
Previous work on simplistic topological network models have been largely computational, where extinctions are simulated in order to obtain predictions. Limited analytic work has been done to predict the robustness of ecological networks which are either maximally or minimally nested [17], but there is no existing analytic framework which can predict the robustness of any given simple ecological network. In the following, we develop such a model, which improves upon computational methods by providing an insight into the properties that make ecological networks robust, and by cutting computational cost.
We start by considering the same scenario put forward by Memott et al. [10], where a bipartite mutualistic network (such as a plant pollinator network) undergoes extinctions on one trophic level, with species on the other trophic level experiencing secondary extinctions if they lose some or all of their neighbours. Secondary extinctions may be predicted for random or targeted primary extinctions, and secondary extinctions are predictable on networks with more than two trophic layers [11]. Our model may also be used to predict the effects of errors in network data, where interactions are erroneously included or excluded. The model is then developed further, taking into account the variable interaction strengths of neighbouring species, where a species will go extinct if it loses a certain amount of interaction strength, as considered by Schleuning et al. [12]. We also consider the scenario in which species go extinct gradually, modelled by the loss of interaction strength as opposed to entire species.
Having developed an analytic model for these scenarios, we can determine the topological properties which make ecological networks robust. Previously, the roles of nestedness and specialisation have been debated as possible sources of robustness [10, 17, 18]. We use our model to demonstrate that, when interaction strength is irrelevant, a network is most robust against random extinctions when the variance of its secondary species degree distribution is minimised. When interaction strength is included, we show that if secondary species' interaction strength is maximally specialised then network robustness is constant regardless of network degree distribution or extinction sensitivity. If interaction strength is maximally generalised, networks with high degree secondary species have robustness that is solely dependent upon extinction sensitivity. As a result, high specialisation makes a network more robust if it is highly sensitive to interaction loss, and high generalisation is better for robustness if interaction loss sensitivity is low.
## 2 Introducing the Analytic Framework
In the model of Memmott et al. [10], species in one trophic level (e.g. pollinators) undergo extinctions, and this impacts species in an adjacent, secondary trophic level (e.g. plants). If species in the secondary level lose all of their neighbours, they suffer a secondary extinction. Primary extinctions may occur at random or according to some ordering, such as highest to lowest degree, where the degree of a species is the number of interactions/links/edges it has. We can plot the proportion of secondary species which survive against the proportion of primary extinctions in order to visualise how robust a given ecological network is against extinction, and an example of such a "robustness curve" is shown in Figure 1.
The area under the robustness curve in Figure 1**(b)** may be calculated in order to give a single metric for ecological network robustness, and this is given by [17]
\[R=\frac{1}{N_{p}}\sum_{\varphi=0}^{N_{p}}Pr(\text{survive}|\varphi), \tag{1}\]
where \(N_{p}\) is the total number of primary species, \(\varphi\) is the number of primary species which have gone extinct at a given point, and \(Pr(\text{survive}|\varphi)\) is the average probability of a randomly chosen secondary species surviving after some \(\varphi\) primary species have been removed. Previously, calculations of the robustness curve and the robustness value \(R\) have been done computationally, with some analytic results being derived for extreme cases [17]. As we show in the following, it is in fact possible to analytically predict the robustness curve of any given simple ecological network for a variety of extinction scenarios.
Let us consider some species \(A\) in the secondary trophic level, which initially has degree \(k_{A}\) and therefore \(k_{A}\) unique neighbours in the primary level. If species in the primary level go extinct at random, we want to know the probability that species \(A\) has degree \(k_{A}-j\) (i.e. \(j\) extinct neighbours) after some \(\varphi\) number of primary species have gone extinct. If there are \(N_{p}\) primary species, then there are \(\binom{N_{p}}{\varphi}\) different possible combinations of primary species extinctions. We then need to find how many of those combinations include
\(j\) neighbours of \(A\). There are \(\binom{k_{A}}{j}\) possible combinations for removing \(j\) neighbours of \(A\), and therefore there are \(\binom{N_{p}-k_{A}}{\varphi-j}\) possible combinations for removing \(\varphi-j\) species which are not neighbours of \(A\), so long as \(\varphi\geq j\). Multiplying \(\binom{k_{A}}{j}\) by \(\binom{N_{p}-k_{A}}{\varphi-j}\) gives us the total number of combinations of length \(\varphi\) which include \(j\) neighbours of \(A\), and so we may write the probability of \(A\) having degree \(k_{A}-j\) once \(\varphi\) primary species are extinct as
\[Pr(k_{A}^{\prime}=k_{A}-j|\varphi)=\begin{cases}\frac{\binom{k_{A}}{j}\binom{N _{p}-k_{A}}{\varphi-j}}{\binom{N_{p}}{\varphi}}&\text{if }\varphi\geq j,\\ 0&\text{otherwise},\end{cases} \tag{2}\]
where \(k_{A}^{\prime}\) refers to \(A\)'s actual degree value once \(\varphi\) primary species have gone extinct. This is the hyper-geometric distribution, which describes a process of sampling without replacement where each sample may pass (a neighbour of \(A\) is removed) or fail (a non-neighbouring primary species is removed). If we specify that species \(A\) goes extinct once its degree is \(k_{A}^{\prime}=k_{A}-i_{k}\) or below (i.e. it has lost at least \(i_{k}\) neighbours), then the disconnection probability for secondary species \(A\) once some \(\varphi\) primary species are extinct is
\[Pr(A\text{ extinct}|\varphi)=\sum_{j=i_{k}}^{k_{A}}Pr(k_{A}^{\prime}=k_{A}-j| \varphi), \tag{3}\]
Since extinction probability is only dependent upon the initial degree of a given secondary species species, the total number of primary species in the network and the number of primary species removed, we may extend this to all secondary species of initial degree \(k\). Consequently, the average secondary extinction probability over the entire network is
\[Pr(\text{extinct}|\varphi)=\sum_{k=0}p(k)\sum_{j=i_{k}}^{k}Pr(k^{\prime}=k-j| \varphi), \tag{4}\]
where \(p(k)\) is the probability of some randomly chosen secondary species having an initial degree of \(k\). Given that \(Pr(\text{survive}|\varphi)\) is simply \(1-Pr(\text{extinct}|\varphi)\), we can rewrite the expression for robustness \(R\) from Equation 1 as
\[R=1-\frac{1}{N_{p}}\sum_{\varphi=0}^{N_{p}}Pr(\text{extinct}|\varphi). \tag{5}\]
Figure 1: **(a)** An example plant pollinator network and **(b)** its associated robustness curve. For this network, pollinators are treated as primary species, and plants as secondary species. Random primary extinctions are simulated repeatedly and the proportion of surviving secondary species is recorded in order to generate the robustness curve.
Here we note that this analytic model is considerably computationally cheaper than brute force simulation. With an efficient implementation, calculating \(Pr(\text{extinct}|\varphi)\) analytically takes \(O(p)\) time, where \(p\) is the number of unique non-zero entries in the secondary species degree distribution. By contrast, estimating \(Pr(\text{extinct}|\varphi)\) computationally once takes \(O(N_{s})\) time, where \(N_{s}\) is the number of secondary species and \(N_{s}\geq p\). In practice, it is often necessary to run several thousand simulations in order to produce an accurate estimate of \(Pr(\text{extinct}|\varphi)\), and so our analytic approach is substantially computationally cheaper than the brute force method.
In Figure 2**(a)**, we demonstrate the results of our model by comparing the analytically predicted robustness curve for an ecological network against the average curve obtained computationally when all neighbours must be removed for extinction to occur (i.e. \(i_{k}=k\)). The ecological data used is from a study of plant pollinator networks in Japan by Kato [19]. We can see that the computationally obtained curve converges to our predicted curve as the number of simulations increases, indicating that our method accurately predicts the average robustness curve.
We also compare the absolute curve divergence between predicted and simulated curves for an increasing number of simulations in Figure 2**(b)**. The absolute curve divergence \(D\) is given by
\[D=\frac{\sum_{\varphi=0}^{N_{p}}|Pr(\text{survive}|\varphi)_{predict}-Pr(\text {survive}|\varphi)_{sim}|}{N_{p}}, \tag{6}\]
where \(Pr(\text{survive}|\varphi)_{predict}\) and \(Pr(\text{survive}|\varphi)_{sim}\) are the predicted and average simulated secondary species survival probabilities respectively. In Figure 2 we can see that the computationally generated result converges towards our prediction in the limit of a large number of simulations.
## 3 Minimum Variance Maximises Robustness Against Random Extinctions
Using this analytic model, we can prove that an ecological network is more robust when the secondary species degree distribution's variance is minimised, under the condition that secondary species average degree is held constant. In other words, the network is most robust if all secondary species have the same number of links to primary species as one another (or are as close to equal as possible). Let us consider the scenario in which a secondary species only goes extinct if it loses all of its primary neighbours. Now let us take two secondary species \(A\) and \(B\), where \(k_{A}<k_{B}-1\), so if we were to remove an edge from \(B\) and add it to \(A\) they would be closer together in degree and \(A\) would not have more edges than \(B\). How would rewiring an edge like
Figure 2: **(a)** Analytically predicted and computationally simulated robustness curves of a real world network from a study by Kato [19] and **(b)** the absolute curve divergence between the analytic and simulated curve as computational simulations increase, plotted on a log-log scale.
this affect their extinction probabilities, and by extension, the network's robustness? For a given number of extinct primary neighbours \(\varphi\) we can write the change in average extinction probability for the network as
\[\Delta P =Pr(\text{extinct}|\varphi)_{\text{rewired}}-Pr(\text{extinct}| \varphi)_{\text{initial}}\] \[=Pr(k_{A}^{\prime}+1=0|\varphi)-Pr(k_{A}^{\prime}=0|\varphi)+Pr(k_ {B}^{\prime}-1=0|\varphi)-Pr(k_{B}^{\prime}=0|\varphi)\] \[=\frac{1}{\binom{N_{p}}{\varphi}}\Bigg{[}\binom{N_{p}-(k_{A}+1)} {\varphi-(k_{A}+1)}-\binom{N_{p}-k_{A}}{\varphi-k_{A}}+\binom{N_{p}-(k_{B}-1)} {\varphi-(k_{B}-1)}-\binom{N_{p}-k_{B}}{\varphi-k_{B}}\Bigg{]}. \tag{7}\]
This may be simplified using the identity \(\binom{x}{y}=\binom{x-1}{y-1}+\binom{x-1}{y}\) to give
\[\Delta P=\frac{1}{\binom{N_{p}}{\varphi}}\Bigg{[}\binom{N_{p}-k_{B}}{\varphi- (k_{B}-1)}-\binom{N_{p}-(k_{A}+1)}{\varphi-k_{A}}\Bigg{]}, \tag{8}\]
and if we expand the terms for \(\binom{N_{p}-k_{B}}{\varphi-(k_{B}-1)}\) and \(\binom{N_{p}-(k_{A}+1)}{\varphi-k_{A}}\) into their factorial forms, we can rearrange to give
\[\Delta P=\frac{(N_{p}-k_{B})!}{\binom{N}{\varphi}(\varphi-k_{A})!(N-\varphi-1 )!}\Bigg{[}\prod_{k=k_{A}+1}^{k_{B}-1}(\varphi-(k-1))-\prod_{k=k_{A}+1}^{k_{B} -1}(N_{p}-k)\Bigg{]}. \tag{9}\]
Since \(\prod_{k=k_{A}+1}^{k_{B}-1}(\varphi-(k-1))\leq\prod_{k=k_{A}+1}^{k_{B}-1}(N_{ p}-k)\) when \(\varphi<N\) and \(k_{A}<k_{B}-1\), we know that the change in disconnection probability \(\Delta P\leq 0\) under the same conditions. Therefore, the disconnection probability must decrease or remain the same under the rewiring procedure, and so robustness must increase or remain the same. Additionally, we can prove that this rewiring will always reduce the variance of the secondary species degree distribution. The change in variance \(\Delta\text{Var}\) is given by
\[\Delta\text{Var} =E(k^{2})_{\text{rewired}}-E(k^{2})_{\text{initial}}\] \[=\frac{1}{N_{s}}\Big{[}(k_{A}+1)^{2}-k_{A}^{2}+(k_{B}-1)^{2}+k_{ B}^{2}\Big{]}\] \[=\frac{2}{N_{s}}(k_{A}-k_{B}+1), \tag{10}\]
where \(N_{s}\) is the number of secondary species in the network. From this, we know \(\Delta\text{Var}<0\) when \(k_{A}<k_{B}-1\), so variance always decreases for rewiring an edge from species \(B\) to species \(A\) given that initially \(k_{A}<k_{B}-1\). This proves that, for a secondary species degree distribution of fixed degree, lower degree distribution variance entails higher robustness, and vice versa.
This result tells us exactly the structural properties that make secondary species in ecological networks robust against random primary species extinctions, namely equally distributed interactions. Previous research has indicated this before [17], however, this was not conclusively proven, nor was the relationship between robustness and secondary species degree variance established. In the work of Burgos et al. [17], robustness is related to nestedness, which is the propensity of primary species to interact with secondary species which other, higher degree primary species also interact with. Nestedness affects the degree distributions of both primary and secondary species, however, we know from our model that for robustness against random primary extinctions, the primary species degree distribution is irrelevant. Therefore, for random primary extinctions, nestedness is not necessarily an indicator of secondary species robustness.
To illustrate the relation between degree variance and robustness, we provide a series of example networks in Figure 3, each with equal numbers of interactions and primary and secondary species, but different robustness and secondary degree distribution variance. We also show the correlation between robustness and secondary degree distribution variance for a network undergoing edge rewiring. The edge rewiring procedure starts on a network where a single secondary species is connected to all primary species, with all other secondary species having one primary neighbour, and one by one edges are swapped from the highest to lowest degree secondary species until secondary species degree variance is minimised.
We can see clearly that the highest variance network (i) has lowest robustness, and the lowest variance network (iv) has the highest robustness. Additionally, networks (ii) and (iii) have the same variance and robustness values as one another, as the only difference between them is the degree distribution of primary species. The difference in primary species degree distribution means that they are not considered to have the same nestedness as one another, but they are equally robust, demonstrating the fact that nestedness and robustness are not necessarily related.
## 4 Targeted Species Extinctions
The model demonstrated in the preceding sections only predicts robustness when primary species are removed at random, but it is also possible to adjust the model to predict robustness when primary species are removed in descending or ascending degree order. This means that species with many links (descending order) or few links (ascending order) go extinct first. In these scenarios, primary species are effectively sorted into some \(n\) different groups based on degree value. All species within a group are removed in a random order before moving onto the next group which is higher or lower in degree value, depending on the scenario. If we consider some secondary species, it will have some \(k_{l}\) neighbours in a given primary species group where all primary species have degree \(l\). As before, we set some threshold number \(i_{k}\) of neighbouring species which must be lost before a given secondary species of degree \(k\) goes extinct. We can then say that the secondary species will go extinct as we remove primary species from some group \(d\) if it satisfies the conditions \(\sum_{l=d}^{n}k_{l}\geq i_{k}\) and \(\sum_{l=d+1}^{n}k_{l}<i_{k}\) for descending degree order removal, and \(\sum_{l=0}^{d}k_{l}\geq i_{k}\) and \(\sum_{l=0}^{d-1}k_{l}<i_{k}\) for ascending degree order removal. We can therefore write the probability of a some secondary species \(A\) going extinct when primary species are removed in descending degree order as
\[Pr(\text{A extinct}|\varphi)=\begin{cases}0&\text{if }\sum_{l=d}^{n}k_{l}<i_{k} \text{ or }\varphi_{d}<j_{d},\\ \frac{(k_{d})}{(\varphi_{d}^{n})}\frac{(\varphi_{d}-k_{d})}{(\varphi_{d}^{n} )}&\text{if }\sum_{l=d}^{n}k_{l}\geq i_{k}\text{ and }\sum_{l=d+1}^{n}k_{l}<i_{k}\text{ and }\varphi_{d}\geq j_{d},\\ 1&\text{if }\sum_{l=d+1}^{n}k_{l}\geq i_{k},\end{cases} \tag{11}\]
where \(N_{d}\) is the number of primary species in group \(d\), \(\varphi_{d}\) is the number of primary species removed from group \(d\) and \(j_{d}=i_{k}-\sum_{l=d+1}^{n}k_{l}\). Similarly, for primary species removal in ascending degree order we have
Figure 3: **(a)** Example networks with different second neighbour degree variances and **(b)** network robustness plotted against second neighbour degree variance. Network (i) exhibits the highest variance and lowest robustness, networks (ii) and (iii) have the same variance and robustness even though they have different primary species degree distributions, and network (iv) has the lowest variance and highest robustness.
\[Pr(\text{A extinct}|\varphi)=\begin{cases}0&\text{if }\sum_{l=0}^{d-1}k_{l}<i_{k} \text{ or }\varphi_{d}<j_{d},\\ \frac{\binom{k_{d}}{j_{d}}\binom{N_{d}-k_{d}}{j_{d}-j_{d}}}{\binom{N_{d}}{j_{d} }}&\text{if }\sum_{l=0}^{d}k_{l}\geq i_{k}\text{ and }\sum_{l=0}^{d-1}k_{l}<i_{k}\text{ and }\varphi_{d}\geq j_{d},\\ 1&\text{if }\sum_{l=0}^{d-1}k_{l}\geq i_{k},\end{cases} \tag{12}\]
where \(j_{k}=i_{k}-\sum_{l=0}^{d-1}k_{l}\). The extinction probabilities \(Pr(\text{A extinct}|\varphi)\) from Equations 11 and 12 can be averaged over the degree distribution of the network in a manner similar to Equation 4, which allows us to predict the average extinction probability for any number of removed primary species. Therefore, we can predict the robustness curves for descending and ascending degree removal of primary species, and an example is given in Figure 4.
As before, our analytic model can successfully predict the average extinction probabilities for secondary species as primary species are removed. The scenarios in which primary species are removed in descending and ascending degree order have been referred to as the "worst" and "best" case scenarios respectively. However, our analytic model suggests that from the perspective of robustness, this may not exactly be the case. Under descending degree order removal, a secondary species' extinction probability only depends upon how many lowest degree neighbours it has, and for ascending degree order removal it depends upon the number of highest degree neighbours. In Figure 5, we provide an example ecosystem for which descending degree order removal gives higher network robustness than ascending degree removal.
Figure 4: Analytically predicted robustness curves for targeted primary species removal. Predictions are made on the plant pollinator network from a study by Kato [19]. The blue curve is for removal of primary species in ascending degree order (lowest degree first) and the orange curve is for removal of primary species in descending degree order (highest degree first).
Figure 5: **(a)** Example ecological network and **(b)** targeted removal robustness curves. Removing low degree primary species (pollinators) first gives a lower robustness than removing high degree primary species.
While this is a specifically constructed example, it demonstrates that finding the true worst or best case scenario for secondary extinctions is not necessarily a case of removing primary species in descending or ascending degree order respectively. As such, a possible future line of enquiry is to try and establish the true worst and best case scenarios for species extinction on any given network.
Thusfar in this section we have considered species extinctions which are targeted based on degree value, but this is only one possible extinction ordering. Recent research has examined extinction scenarios in which species are lost according to their extinction risk as assessed by the IUCN (International Union for Conservation of Nature) Red List [20]. Species are ranked in the Red List from Critically Endangered to Least Concern, and in work by Lamperty and Brosi [20] frugivore species in a seed dispersal network are removed from highest to lowest extinction risk. Since species only belong to one of these extinction risk categories, it is necessary to simulate extinctions from each risk category in descending risk order, with the order of extinctions within each group randomised. This framework fits well with our targeted species extinction model.
Using data from Bello et al. [21], we can replicate the results of Lamperty and Brosi [20], predicting the survival of plant species in a seed dispersal network as frugivore species are lost in descending extinction risk order. We calculate extinction probability in the same way as Equation 11, except instead of our primary species groups being organised by degree value, they are now organised by extinction risk. In Figure 6, we show simulated and predicted plant species survival probabilities as frugivore species are removed in descending extinction risk order, with plant species going extinct once they have lost all of their frugivore neighbours.
This demonstrates the fact that our framework for targeted species extinctions may be extended to any ordering of primary species loss where primary species are sorted into groups which go extinct in some order, but extinction within groups occurs at random.
## 5 Multi-layer Ecosystems
So far, we have only demonstrated our model for bipartite systems, i.e. those which include only two groups that interact with one another. However, real world ecosystems can exist on several distinct layers, for example, predators may feed on pollinators which in turn pollinate plants. Another extension for our model is to predict species extinction in a group of species not directly adjacent to the group undergoing primary extinction. This is predictable analytically, but only for the scenario in which a species must lose all of its neighbours in order to go extinct.
Scenarios such as this have previously been considered by Pocock et al. [11], using computational methods. Here we construct an example network to demonstrate robustness predictions on multi-layer networks. Let us consider a system of plants, pollinators and predators, where predators feed on pollinators, who in turn feed on plants. We want to know the probability of a predator going extinct after a certain number of plant extinctions. For some predator species A, the species will go extinct if all of the pollinator species it is
Figure 6: Analytically predicted and computationally simulated robustness curves of a real world seed dispersal networks, where primary frugivore species go extinct according to their IUCN extinction risk. This replicates results from Lamperty and Brosi [20] using data from Bello et al. [21].
connected to go extinct, which only occurs once all of their plant species neighbours go extinct. Therefore, the extinction probability of predator species A is simply dependent upon the \(u_{A}\) initial number of unique plant species to which it is connected via its pollinator species neighbours. Therefore, we can treat the predator species in this system as our secondary species and the plant species as primary species.
Similar to Equation 2, the extinction probability for a secondary species A once some \(\varphi\) number of primary species have been removed is
\[Pr(u_{A}^{\prime}=0|\varphi)=\begin{cases}\frac{\binom{N_{p}-u_{A}}{\varphi-u_{ A}^{\prime}}}{\binom{N_{p}}{\varphi}}&\text{if }\varphi\geq u_{A},\\ 0&\text{otherwise},\end{cases} \tag{13}\]
As before, we can average Equation 13 over the distribution of secondary species connected to \(u\) unique primary species to predict secondary extinctions as primary species are removed. The analytically predicted and computationally simulated robustness curves for this are given in Figure 7.
While this expands the scope of our analytic model beyond simply two layer ecosystems, it is important to note that thusfar we can only model extinctions on multi-layer systems if species go extinct after losing all of their neighbours. Therefore, we cannot consider as many different extinction scenarios on multi-layer networks than we can for bipartite networks.
## 6 False Positives and Negatives in Network Data
Beyond simply predicting robustness, we may also be interested in how predictions of robustness are affected by errors in network data, as ecological data can be error prone [22, 23]. For example, networks may vary across environmental gradients [24] or may constitute metawebs inferred from proxies [25, 26]. One may be interested in how the robustness of networks change as false edges are added in (false positives) or true edges are removed (false negatives). In the simplest case, let us consider the random addition and removal of edges. Since robustness against random primary species removal only depends upon the degree distribution of secondary species, we can analytically predict how robustness will change as edges are randomly added or removed by modelling the changes to the secondary degree distribution.
For random edge addition and removal, we can define recursive formulae which describe how the secondary degree distribution will change. For random edge addition, the recursive formula which describes the probability of randomly choosing a secondary species with degree \(k\) after some \(t\) edges have been added is
Figure 7: **(a)** Example three layer network of plants (primary species), pollinators and predators (secondary species), and **(b)** the associated analytically predicted and computationally simulated robustness curves.
\[p(k)_{t}=p(k)_{t-1}-p(k)_{t-1}\frac{N_{p}-k}{N_{p}N_{s}-(E+t-1)} \tag{14}\] \[+p(k-1)_{t-1}\frac{N_{p}-(k-1)}{N_{p}N_{s}-(E+t-1)},\]
where \(E\) is the total number of edges in the network before any additional edges have been added. Note that \(N_{p}N_{s}-(E+t-1)\) is the total possible number of edges which could be added to the network once \(t-1\) edges have been added.
For random edge removal, the recursive formula for the probability of choosing a secondary species with degree \(k\) after some \(v\) edges have been removed is
\[p(k)_{v}=p(k)_{v-1}-p(k)_{v-1}\frac{k}{E-(v-1)}+p(k+1)_{v-1}\frac{k+1}{E-(v-1)}. \tag{15}\]
In order to simulate the presence of false positives or negatives in network data, we can apply these recursive formulae a certain number of times in order to adjust the degree distribution, and then assess the impact of false positives and negatives by predicting robustness for either scenario.
In Figure 8, we show the robustness curve for a plant pollinator network undergoing random extinctions where \(i_{k}=k\), comparing the original predicted curve against the predicted curves for including false positives and negatives.
In terms of robustness, false positives have a more significant impact than false negatives in small quantities. The network we use to generate the data shown in Figure 8 has 1125 edges, and we see that up to a change in edges \(\Delta E=500\) (roughly 44% the original number of edges) false positives increase robustness more than false negatives decrease it. For false positives, as \(R\to 1\), additional edges contribute less and less to robustness and for false negatives, as \(\Delta E\to E\), \(R\to 0\), so we see a larger change in robustness due to false negatives for large numbers of errors. We verify similar results on a dataset of several networks, and details of this data from 18 real world plant pollinator networks is given in the Supplementary Materials. For this dataset, we find that if we measure robustness where 20% of the original number of edges are added/removed for false positives/negatives, then the net change in robustness is always positive, i.e. false positives always increase robustness more than false negatives decrease it.
If we assume that ecological network data gathering in the real world is reasonably accurate, i.e. unlikely to over/under record interactions by more than 20%, then we would expect false positives to introduce more error into calculations of robustness than false negatives. Particular care is needed for robustness analyses
Figure 8: **(a)** Analytically predicted robustness curves for a real world ecological network from a study by Kato [19] for the original network, the network with 200 edges randomly added and the network with 200 edges randomly removed. The original network has 1125 edges. **(b)** Shows how network robustness changes as edges are added or removed, and **(c)** shows the difference between the robustness of networks with errors and the original networks as edges are added or removed.
based on metaewbs of potential trophic interactions for which the false positive and false negative rates are difficult to ascertain [25, 26].
While this result indicates that false positives have more impact than false negatives, it only provides one perspective for how these errors may be introduced into network data. One future avenue of enquiry is to establish the likely sources of errors and model those, as opposed to modelling errors randomly.
## 7 Species Specialisation and Generalisation
In previous sections, we have only considered networks in which interactions are weighted equally, i.e. each one of a secondary species' interactions is as important for its survival. However, on real ecological networks, a secondary species may interact more with one primary species than another, and this has an impact on a secondary species survivability [27]. We can specify a certain percentage of total interaction strength that a species must lose before it goes extinct, an approach used before before by Schleuning et al. [12]. We can update our extinction probability for some secondary species A to
\[Pr(A\text{ extinct}|\varphi)=\sum_{j=1}^{k_{A}}Pr(k^{\prime}_{A}=k_{A}-j|\varphi )Pr(\sum_{0}^{j}W\geq i_{A}), \tag{16}\]
where \(W\) is a random variable representing some randomly chosen weight corresponding to the interaction strength with a neighbour of \(A\), \(i_{A}\) is the interaction strength threshold for \(A\) that must be removed before \(A\) goes extinct, and \(Pr(\sum_{0}^{j}W\geq i_{A})\) is the probability of choosing some \(j\) weighted interactions which exceed the threshold \(i_{A}\). We can get the value of \(i_{A}\) from specifying some ratio of interaction strength that must be lost for secondary extinction to occur as the sensitivity threshold \(T\), and calculating \(i_{A}=\lceil Tk_{A}\rceil\). The notation of \(\lceil x\rceil\) refers to \(x\) being rounded up to the nearest integer. The extinction probability \(Pr(A\text{ extinct}|\varphi)\) may be averaged over all species then over values of \(\varphi\) to obtain a robustness value for the network.
There is no closed form solution for \(Pr(\sum_{0}^{j}W\geq i_{A})\) since the weights \(w_{z}\) do not necessarily follow a particular distribution. It is instead necessary to estimate \(Pr(\sum_{0}^{j}W\geq i_{A})\) in some way. The brute force method is to randomly sample \(j\) weights using a Monte Carlo method, however, this must be repeated many times in order to give an accurate estimate, and is subject to statistical fluctuations.
Instead, we have developed a deterministic sampling method, where a species' weights \(w_{z}\) are arranged in size order and assigned a variable \(y_{z}\), which takes values of 0 or 1. For \(j\) removals there will be \(j\) values of \(y_{z}=1\), with the rest equal to 0. We can express the sequence of weights as a sequence of 0 or 1 \(y_{z}\) values, giving us a binary number. If weights were ordered as powers of 2, i.e. \(w_{z}=2^{k_{A}-z}\), then we could find the \(n^{th}\) binary sequence of \(y_{z}\) values with \(j\) values equal to 1 above which all \(\sum_{z=0}^{k_{A}}w_{z}y_{z}\geq i_{A}\) and below which all \(\sum_{z=0}^{k_{A}}w_{z}y_{z}<i_{A}\), allowing us to calculate \(Pr(\sum_{0}^{j}W\geq i_{A})\) exactly. However, weights are not typically ordered as powers of two, so finding an exact result this way is rarely possible. Instead we can order binary sequences of \(y_{z}\) values and sample from these orderings at some specified "depth" in order to estimate \(Pr(\sum_{0}^{j}W\geq i_{A})\), where depth effectively determines how many samples are taken. This sampling method is fully deterministic, so for a given sequence of weights and a specified depth, we always return the same estimate for \(Pr(\sum_{0}^{j}W\geq i_{A})\). Further details about how this algorithm operates are given in the Supplementary Materials.
Using our deterministic sampling method, we can provide quasi-analytic predictions for secondary species survival on networks undergoing random primary extinctions, where interaction strength is weighted unevenly and secondary extinctions occur after the loss of a certain percentage of interaction strength. Example predictions are given in Figure 9, alongside results showing how our deterministic estimate becomes increasingly accurate with greater depth, and a comparison between the time taken to estimate \(Pr(\sum_{0}^{j}W\geq i_{A})\) and prediction accuracy for our deterministic sampling method and for a brute force Monte Carlo method.
From this, we can see that as the depth of the deterministic estimation increases, we get diminishing returns in terms of prediction accuracy, and that it is more computationally efficient to use the deterministic estimation method as opposed to Monte Carlo simulation in order to get the same level of prediction accuracy.
Having developed an analytic framework for secondary species extinctions when interaction strength is weighted unevenly, we can examine some extreme scenarios of interaction strength weighting. One property
of interest in ecological networks is specialisation [28], where specialist species tend to interact with a small number of species very strongly, and generalist species tend to interact with many species evenly. Given an ecological network with a set number of primary and secondary species, and a set distribution of interactions, we can examine the most specialist interaction weighting and the most generalist interaction weighting.
In the most specialist case, each secondary species weights one of its interactions at close to 100% of its interaction strength, and all others close to 0%. Therefore, a given secondary species \(A\) only goes extinct when it loses the neighbour with which it shares almost all interaction strength. If a neighbour of \(A\) goes extinct, the probability of losing the heavily weighted neighbour is simply \(\frac{1}{k_{A}}\), so \(Pr(\sum_{0}^{j}W\geq i_{A})=\frac{j}{k_{A}}\). This give an extinction probability for some species \(A\) of
\[Pr(A\text{ extinct}|\varphi) =\sum_{j=1}^{k_{A}}Pr(k_{A}^{\prime}=k_{A}-j|\varphi)\frac{j}{k_{ A}},\] \[=\frac{\mathbb{E}(k_{A}^{\prime})}{k_{A}}\] \[=\frac{\varphi}{N_{p}}, \tag{17}\]
which we derive from the fact that \(\mathbb{E}(k_{A}^{\prime})=k_{A}\frac{\varphi}{N_{p}}\) since \(Pr(k_{A}^{\prime}=k_{A}-j|\varphi)\) describes the hypergeometric distribution. This results in a robustness value of \(R=0.5\), regardless of the secondary degree distribution, number of primary species or threshold.
For the most generalist case, each secondary species weights all of its interactions evenly, which means \(Pr(\sum_{0}^{j}W\geq i_{A})=0\) when \(j<i_{k}\), and \(Pr(\sum_{0}^{j}W\geq i_{A})=1\) when \(j\geq i_{k}\). Therefore, \(Pr(A\text{ extinct}|\varphi)=\sum_{j=i_{k}}^{k_{A}}Pr(k_{A}^{\prime}=k_{A}-j|\varphi)\), the same as Equation 3.
Given these results, when is it more advantageous for a network to be highly specialist or highly generalist in terms of robustness? Let us consider some secondary species \(A\) which is connected to all primary species in its network, i.e. \(k_{A}=N_{p}\). Therefore, the extinction probability for \(A\) is given by
Figure 9: **(a)** Analytically predicted and computationally simulated robustness curves for a real world network from a study by Kato [19] where unevenly weighted interaction strength is taken into account and extinctions occur over a specified threshold \(T\) of interaction strength loss. Analytic predictions are given for threshold values of \(70\%,50\%\) and \(30\%\), and computationally simulated curve is given for 50%. **(b)** Comparison between depth of the estimation for \(Pr(\sum_{0}^{j}W\geq i_{A})\) and divergence between analytically predicted robustness curve and the simulated curve averaged over 5000 iterations. **(c)** Divergence between the predicted robustness curve and the 5000 iteration simulation curve compared against the time taken, with data for both the deterministic estimation method and the Monte Carlo method.
\[Pr(A\text{ extinct}|\varphi) =\sum_{j=i_{k}}^{k_{A}}\begin{cases}\frac{\binom{N_{p}}{j}\binom{o}{o-j }}{N_{p}}&\text{if }\varphi\geq j,\\ 0&\text{otherwise},\end{cases}\] \[=\begin{cases}1&\text{if }\varphi\geq i_{k},\\ 0&\text{otherwise},\end{cases} \tag{18}\]
If all secondary species in a network have \(k=N_{p}\), then when they are maximally generalist, the network robustness is \(R=\frac{i_{b}}{N_{p}}\). Therefore, such a network is more robust when secondary species are maximally generalist if more than \(50\%\) of interaction strength must be lost before a secondary species goes extinct, i.e. when the sensitivity threshold \(T>0.5\). Conversely, the network is more robust when secondary species are maximally specialist if less than \(50\%\) of interaction strength must be lost to make secondary species go extinct, so \(T<0.5\). To illustrate this, we provide robustness curves in Figure 10 of **(a)** a single secondary species with \(k_{A}=N_{p}\), and of **(b)** an entire real world network.
From these results, we know that either extreme of species specialisation can be advantageous from the perspective of maximising network robustness, depending upon the sensitivity of the network, i.e. the proportion of interaction strength that must be lost for secondary species to go extinct. However, we see in Figure 10**(b)** that it is not strictly the case on real networks that maximum generalisation is always better for robustness than maximum specialisation when \(T>0.5\), as the maximum generalist curve when \(T=0.5\) gives \(R=0.471\). This is due to the fact that secondary species typically have \(k<N_{p}\) on real networks. Additionally, we note that the robustness values from the maximum generalist and maximum specialist interaction weightings do not necessarily give the maximum and minimum robustness values for a given threshold. Nevertheless, these results are still indicative of the fact that species generalisation and specialisation can both improve network robustness in different contexts, and so we might expect that in the real world, a network that has developed to be highly generalist is less sensitive to interaction loss than a network which has developed in order to be highly specialist.
## 8 Interaction Loss
The models we have considered are only concerned with the loss of primary species as a whole, where a primary species is removed at each "step" in the extinction process. Considering the loss of entire species at a time can skew our understanding of network robustness, for example a plant animal network with more animal than plants will appear more robust against primary extinctions of animal than against primary
Figure 10: **(a)** Shows species survival for maximal generalists and maximal specialists at various sensitivity thresholds \(T\) when \(k_{A}=N_{p}\). **(b)** Gives robustness curves for maximal generalisation and maximal specialisation on a real world network from a study by Kato [19] at various sensitivity thresholds \(T\).
extinctions of plants [12]. However, is this a realistic understanding of how species go extinct? There may be more animal species than plant species, but what if there is a very large population of each plant species and a small population of each animal species? Extinctions may be experienced more gradually, where a species' population dies off over time rather than all at once [29]. This process can be modelled by examining the loss of interactions as opposed to the loss of species, which in network terms entails considering edge removal as opposed to node removal.
If interaction strength between species is represented as integer values, then we can treat each unit of interaction strength as an edge, so secondary species have degree values equal to the sum of their interaction strength with other species. We then have \(E\) "edges" (i.e. total interaction strength on the network), and we remove some \(\varphi\) units of interaction strength. For a given secondary species \(A\), after removing some \(\varphi\) interaction strength the probability that it has lost some interaction strength \(j\) is
\[Pr(k_{A}^{\prime}=k_{A}-j|\varphi)=\begin{cases}\frac{\binom{k_{A}}{j}\binom{E- k_{A}}{e-j}}{\binom{k_{A}}{\varphi}}&\text{if $\varphi\geq j$},\\ 0&\text{otherwise},\end{cases} \tag{19}\]
where, as before in the case of Equation 2, \(k_{A}\) is the initial degree/total interaction strength of \(A\), and \(k_{A}^{\prime}\) is the degree/total interaction strength of \(A\) after removing \(\varphi\) interaction strength. From this, it is straightforward to predict secondary species' survival probability and network robustness using a similar logic as in Section 2. Predictions for interaction loss on an ecological network are given in Figure 11.
These predictions of interaction loss give different robustness values than predictions of species extinctions. For example, for species extinctions on a real network [19] when true interaction strength values are used (as shown in Figure 9**(a)**) and \(T=0.7\), we have \(R_{species}=0.619\). By contrast, for interaction loss on the same network when \(T=0.7\), we have \(R_{inter}=0.653\). Therefore, modelling secondary species extinctions as an outcome of interaction loss as opposed to primary species extinctions gives a different perspective on network robustness, allowing one to identify networks which are fragile against primary species loss but robust against interaction loss, and vice versa. A similar logic to that presented in Section 2 may be followed in order to show that a network with a set number of primary species, secondary species and total interaction strength is maximally robust against interaction loss when the variance in interaction strength per secondary species is minimised.
## 9 Discussion
In conclusion, we have successfully extended the robustness framework of Memmott et al. [10] such that we may make predictions of ecological network robustness analytically. For random extinctions, we have shown that networks with low second species degree variance are highly robust. We are also able to predict secondary
Figure 11: Robustness curves for interaction loss on a real world network from a study by Kato [19] at varying sensitivity thresholds \(T\).
extinctions as primary species go extinct according to some degree or extinction risk based targeting, and we can predict secondary extinctions on ecological networks with more than two layers. Additionally, we can model the influence of random false positives and negatives in network data on robustness, finding that in small quantities false positives have a greater impact than false negatives on network robustness.
Our model is also capable of predicting the robustness of networks where interaction strength is weighted unevenly between different secondary species' neighbours, and species go extinct once a certain proportion of interaction strength has been lost. We have given results for robustness when interaction strength is equally distributed (maximally generalist), and when interaction strength is shared solely with one neighbouring species (maximally specialist). From this, we know that maximal generalisation and maximal specialisation can both produce a more robust network, depending on the proportion of interaction strength that must be lost before secondary species extinction. Finally, we have demonstrated the fact that it is also possible to model interaction strength loss as opposed to simply species extinction, representing a more "gradual" extinction scenario.
These results represent a substantial advancement in analytic understanding of ecological network robustness. However, there are still many open questions. We can predict the average secondary species extinction probability for a given number of primary extinctions, but we may also want to analytically predict the possible error in robustness curves by finding the standard deviation in secondary species extinction probability for a given number of primary extinctions. Additionally, we may want to establish the true worst and best case scenarios for secondary extinctions, as these have not been definitively identified. For errors (i.e. false positives and negatives) in network data, our current results examine errors which occur at random, but this may not be the case in the real world. Errors may occur due to some specific reason or dynamic, and identifying what this is may allow us to better mathematically model data errors and their influence.
Beyond these possible improvements, it is also important to acknowledge that in recent years, ecologists have considered properties of ecological networks which affect robustness and go beyond simpler models of species extinction. For example, ontogenetic niche shifts, where species change their diets when they undergo changes such as growing from a larvae to an adult, can affect the structure and robustness of interaction networks [30]. Another consideration is how interactions can be "rewired" after species extinctions [12, 31], which to predict analytically would likely require combinatoric methods for sampling with fuzzy replacement [32]. These examples highlight the fact that there is still considerable room for analytic models of ecological network robustness to develop, and there are ongoing areas of research in both ecological networks and combinatorics which may complement one another well, so it may be useful for there to be a greater dialogue between these fields in future.
## Funding
C.J. was supported by Engineering and Physical Sciences Research Council Doctoral Training Partnership funding (EP/R513179/1). We thank the Research Focus Data-centric Sciences of the University of Potsdam for financial support.
## Acknowledgements
We thank Jane Memmott for providing feedback on a summary version of this paper.
## Conflict of Interest Statement
There authors have no conflict of interest to declare.
## Data Availability
The code used in generating the results for this paper may be found online at [https://github.com/cj14373/eco-analytic.git](https://github.com/cj14373/eco-analytic.git)
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Supplementary Materials for Evaluating The Impact Of Species Specialisation On Ecological Network Robustness Using Analytic Methods
Chris Jones \(\copyright\)\({}^{1}\), Damaris Zurell \(\copyright\)\({}^{2}\), and Karoline Wiesner \(\copyright\)\({}^{3}\)
\({}^{1}\)School of Mathematics, University of Bristol, Bristol, UK
\({}^{2}\)Institute of Biochemistry and Biology, University of Potsdam, Potsdam, Germany
\({}^{3}\)Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
## False Positive and Negative Robustness Data
In this section, we provide data for 18 different plant pollinator networks where we have measured their robustness against pollinator extinction (\(R_{real}\)) as well as their robustness when they have 20% randomly added false positive edges (\(R_{pos}\)) and 20% randomly removed false negative edges (\(R_{neg}\)). To determine whether false positives or false negatives have a larger impact on network robustness, we measure the net change in network robustness. This is calculated as \(\frac{R_{pos}+R_{neg}-2R_{real}}{R_{real}}\), so this measures net change relative to \(R_{real}\). If the net change is positive, this means that false positives change robustness more than false negatives, and if the net change is negative, then false negatives have more impact than false positives. This data relates to results stated in Section 6.
In Table 1, we can see that for all networks the net change in robustness is positive. This indicates the
\begin{table}
\begin{tabular}{|l|c|c|c|c|c|} \hline Network Name & \(R_{real}\) & \(R_{pos}\) & \(R_{neg}\) & \(\frac{R_{pos}+R_{neg}-2R_{real}}{R_{real}}\) & References \\ \hline MPL004 & 0.8246 & 0.8945 & 0.8035 & 0.0591 & [1] \\ \hline MPL005 & 0.7770 & 0.8509 & 0.7602 & 0.0736 & [2] \\ \hline MPL009 & 0.8545 & 0.8925 & 0.8313 & 0.0174 & [3] \\ \hline MPL015 & 0.9180 & 0.9999 & 0.9033 & 0.0733 & [4] \\ \hline MPL016 & 0.8760 & 0.9217 & 0.8625 & 0.0368 & [5] \\ \hline MPL021 & 0.7600 & 0.8644 & 0.7554 & 0.1312 & [6] \\ \hline MPL028 & 0.8089 & 0.8673 & 0.7965 & 0.0568 & [7] \\ \hline MPL029 & 0.8142 & 0.8556 & 0.7894 & 0.0204 & [7] \\ \hline MPL043 & 0.8335 & 0.8792 & 0.8163 & 0.0343 & [8] \\ \hline MPL044 & 0.7680 & 0.8514 & 0.7589 & 0.0969 & [9] \\ \hline MPL047 & 0.8827 & 0.9371 & 0.8683 & 0.0454 & [10] \\ \hline MPL048 & 0.8583 & 0.9316 & 0.8514 & 0.0775 & [10] \\ \hline MPL049 & 0.8846 & 0.9250 & 0.8704 & 0.0297 & [11] \\ \hline MPL054 & 0.7473 & 0.8146 & 0.7303 & 0.0674 & [12] \\ \hline MPL055 & 0.7457 & 0.8143 & 0.7343 & 0.0768 & [13] \\ \hline MPL056 & 0.8125 & 0.8694 & 0.7960 & 0.0496 & [14] \\ \hline MPL057 & 0.8020 & 0.8945 & 0.7906 & 0.1011 & [15] \\ \hline MPL062 & 0.9695 & 1.0000 & 0.9628 & 0.0245 & [16] \\ \hline \end{tabular}
\end{table}
Table 1: Data table of 18 real world plant pollinator networks, where datasets were accessed from [17]. Data regarding network names, real robustness, robustness values with 20% false positives and negatives, net changes in robustness and references are included.
fact that false positives have a greater impact on network robustness than false negatives. The average net change is \(5.95\%\), and the median net change is \(5.80\%\).
## Deterministically Estimating the Weight Sum
Here we present the details behind our algorithm for estimating values of \(Pr(\sum_{0}^{j}W\geq i_{A})\), as discussed in Section 7. Given some weight vector \(\mathbf{w}\) of length \(k_{A}\) with elements \(w_{z}\) for some species \(A\), we want to know how many combinations of \(j\) elements of \(\mathbf{w}\) sum up to \(\geq i_{A}\), the extinction threshold for species \(A\). We may express our choices of weights with \(\mathbf{y}\), a vector with elements \(y_{z}\) which can take values of \(0\) or \(1\), and \(\sum_{z=0}^{k_{A}}y_{z}=j\). Values of \(\mathbf{y}\) correspond to whether or not an element of \(\mathbf{w}\) has been chosen, so if \(y_{z}=1\) then the \(z^{th}\) element of \(\mathbf{w}\) has been chosen. We may then calculate \(\sum_{0}^{j}W\) as \(\sum_{z=0}^{k_{A}}w_{z}y_{z}=\mathbf{w}\cdot\mathbf{y}\). There are a total of \(\binom{k_{A}}{j}\) possible combinations of weights, so it is computationally impractical to calculate all possible combinations of weights above small values of \(k_{A}\) and \(j\). Instead, it is necessary to estimate the number of weight combinations which meet or exceed the threshold \(i_{A}\) in order to calculate \(Pr(\sum_{0}^{j}W\geq i_{A})\).
As mentioned in the main text, it is possible to express \(\mathbf{y}\) as a binary number and provide an ordering of possible values of \(\mathbf{y}\) such that, were \(w_{z}\) values powers of \(2\) ordered in descending value (i.e. \(w_{z}=2^{k_{A}-z}\)), it would always be possible to identify the \(n^{th}\) ordering of \(y\) above which \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\) and below which all \(\mathbf{w}\cdot\mathbf{y}<i_{A}\). However, this is not necessarily a realistic sequence of \(w_{z}\) weights. In Table 2, we provide two example \(\mathbf{w}\) vectors of length \(k_{A}=4\), where weights are in descending size order. In the \(\mathbf{y}\) vector column, we give each possible ordering of \(\mathbf{y}\) for which \(\sum_{z=0}^{k_{A}}y_{z}=2\). The \(\mathbf{y}\) vectors are themselves ordered such that they are in ascending size order for the binary numbers the correspond to (e.g. \(\mathbf{y}=(0,1,0,1)\) corresponds to the binary number \(0101\), which is \(5\) in base \(10\)).
We can see that when \(\mathbf{w}=(8,4,2,1)\), the values of \(\mathbf{w}\cdot\mathbf{y}\) are in ascending size order, and so it is straightforward to identify the ordering of \(\mathbf{y}\) above which \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\) and below which \(\mathbf{w}\cdot\mathbf{y}<i_{A}\) for any given \(i_{A}\). However, when \(\mathbf{w}=(3,3,2,1)\) the values of \(\mathbf{w}\cdot\mathbf{y}\) are not ordered, and so if we were to set \(i_{A}=5\) for example, we cannot identify an ordering of \(\mathbf{y}\) above which \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\) and below which \(\mathbf{w}\cdot\mathbf{y}<i_{A}\).
While in this example it is straightforward to explicitly enumerate all possible orderings of \(\mathbf{y}\), this is impractical for for larger \(k_{A}\) and \(j\) values. Therefore, we must consider how to estimate the number of orderings of \(\mathbf{y}\) for which \(\mathbf{w}\cdot\mathbf{y}\). If we order values of \(\mathbf{w}\) in descending size order, then it is possible to establish certain groups of \(\mathbf{y}\) vectors which will always give \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\) or \(\mathbf{w}\cdot\mathbf{y}<i_{A}\). For example, let us consider the \(\mathbf{w}=(3,3,2,1)\) and \(\sum_{z=0}^{k_{A}}y_{z}=2\) scenario from Table 2. If we set the threshold \(i_{A}=4\), then we know that all \(\mathbf{y}\) vectors of the forms \((1,...)\) and \((0,1,...)\) must give \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\), and all \(\mathbf{y}\) vectors of the form \((0,0,...)\) must give \(\mathbf{w}\cdot\mathbf{y}<i_{A}\). Therefore, for this example we would only need to sample \(3\) orderings of \(\mathbf{y}\) in order to accurately estimate the value of \(Pr(\sum_{0}^{j}W\geq i_{A})\), as opposed to having to sample all \(6\) orderings.
We can extend this idea in order to develop an estimation algorithm. Given some \(\mathbf{w}\) weights arranged in descending size order, we specify some prefix of \(\mathbf{y}\) (referred to as \(\mathbf{y}^{p}\)) which gives the first \(z=1,2,...,x\) values of \(\mathbf{y}\). We then create a "bracket" for the prefix \(\mathbf{y}^{p}\), finding \(\mathbf{y}^{p}_{upper}\) and \(\mathbf{y}^{p}_{lower}\) which are the highest and lowest orderings of \(\mathbf{y}\) with prefix \(\mathbf{y}^{p}\) respectively. If \(\mathbf{y}^{p}_{upper}\) and \(\mathbf{y}^{p}_{lower}\) satisfy \(\mathbf{w}\cdot\mathbf{y}^{p}\geq i_{A}\), then we count all of the orderings in the bracket towards our estimation of \(Pr(\sum_{0}^{j}W\geq i_{A})\). If \(\mathbf{y}^{p}_{upper}\) and \(\mathbf{y}^{p}_{lower}\) fulfil \(\mathbf{w}\cdot\mathbf{y}^{p}<i_{A}\), then we simply discount the prefix. If \(\mathbf{w}\cdot\mathbf{y}^{p}_{upper}\geq i_{A}\) and \(\mathbf{w}\cdot\mathbf{y}^{p}_{lower}<i_{A}\), then we lengthen the prefix and repeat. We can specify some maximum depth \(d\) of prefix, where \(\sum_{z=0}^{k_{A}}y^{p}_{z}\leq d\), i.e. \(d\) is the maximum
\begin{table}
\begin{tabular}{|c|c|c|} \hline & \(\mathbf{w}\cdot\mathbf{y}\) & \(\mathbf{w}\cdot\mathbf{y}\) \\ \(\mathbf{y}\) & \(\mathbf{w}=(8,4,2,1)\) & \(\mathbf{w}=(3,3,2,1)\) \\ \hline (0,0,1,1) & 3 & 3 \\ \hline (0,1,0,1) & 5 & 4 \\ \hline (0,1,1,0) & 6 & 5 \\ \hline (1,0,0,1) & 9 & 4 \\ \hline (1,0,1,0) & 10 & 5 \\ \hline (1,1,0,0) & 12 & 6 \\ \hline \end{tabular}
\end{table}
Table 2: Ordering of \(\mathbf{y}\) vector with corresponding weight sum values for different \(\mathbf{w}\) vectors.
number of \(y_{z}=1\) values permitted in the prefix. This allows us to perform the estimation while specifying an endpoint, preventing us from searching through too many orderings of \(\mathbf{y}^{p}\) and conserving computational resources.
If we reach the maximum prefix depth such that \(\sum_{z=0}^{k_{A}}y_{z}^{p}=d\) and we still have \(\mathbf{w}\cdot\mathbf{y}_{upper}^{p}\geq i_{A}\) and \(\mathbf{w}\cdot\mathbf{y}_{lower}^{p}<i_{A}\), then we estimate the number of orderings in the bracket which give \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\). If we set \(\mathbf{w}\cdot\mathbf{y}_{upper}^{p}=U\) and \(\mathbf{w}\cdot\mathbf{y}_{lower}^{p}=L\), and the number of orderings in the bracket \(b\) as \(s_{b}\), then we estimate the number of orderings \(n_{b}\) in the bracket for which \(\mathbf{w}\cdot\mathbf{y}\geq i_{A}\) as
\[n_{b}\approx\Big{\lfloor}\frac{U-i_{A}+1}{U-L}s_{b}\Big{\rfloor}, \tag{1}\]
which is accurate if the values of \(\mathbf{w}\cdot\mathbf{y}\) are equally spaced in the bracket. We express this algorithm as Algorithm 1 in pseudocode form below.
```
1:while\(\sum_{z=0}^{k_{A}}y_{z}^{p}>0\) and \(\sum_{z=0}^{k_{A}}y_{z}^{p}\leq j\) and \(k_{A}-\text{length}(\mathbf{y}^{p})\geq j-\sum_{z=0}^{k_{A}}y_{z}^{p}\)do
2: Specify some prefix \(\mathbf{y}^{p}\) and calculate corresponding \(U\) and \(L\) values
3:if\(U\geq i_{A}\) and \(L\geq i_{A}\)then
4: Record \(n_{b}\) = \(s_{b}\)
5: Remove all values from \(\mathbf{y}^{p}\) after and including the last non-zero value of \(\mathbf{y}^{p}\)
6: Append (0,1) to \(\mathbf{y}^{p}\)
7:elseif\(U<i_{A}\) and \(L<i_{A}\)then
8:if\(\sum_{z=0}^{k_{A}}y_{z}^{p}\geq 2\)then
9: Remove all values from \(\mathbf{y}^{p}\) after and including the second to last non-zero value of \(\mathbf{y}^{p}\)
10: Append (0,1) to \(\mathbf{y}^{p}\)
11:elseif\(\sum_{z=0}^{k_{A}}y_{z}^{p}<2\)then
12: Remove all values from \(\mathbf{y}^{p}\)
13:endif
14:elseif\(U\geq i_{A}\) and \(L<i_{A}\)then
15:if\(\sum_{z=0}^{k_{A}}y_{z}^{p}\leq d\)then
16: Append (1) to \(\mathbf{y}^{p}\)
17:elseif\(\sum_{z=0}^{k_{A}}y_{z}^{p}>d\)then
18: Record \(n_{b}=\Big{\lfloor}\frac{U-i_{A}+1}{U-L}s_{b}\Big{\rfloor}\)
19: Remove all values from \(\mathbf{y}^{p}\) after and including the last non-zero value of \(\mathbf{y}^{p}\)
20: Append (0,1) to \(\mathbf{y}^{p}\)
21:endif
22:endif
23:endwhile
24:return\(Pr(\sum_{0}^{j}W\geq i_{A})=\frac{\sum_{b}n_{b}}{\binom{k_{A}}{j}}\)
```
**Algorithm 1** Deterministic Weight Sum Algorithm
In order to illustrate the importance of the depth parameter in providing an accurate result for \(Pr(\sum_{0}^{j}W\geq i_{A})\), we provide an example in the following. Let us consider a system where \(\mathbf{w}=(5,5,3,2,1)\), \(j=3\) and \(i_{A}=9\). In Figure 1, we provide a dendrogram representation of the different possible \(\mathbf{y}^{p}\) prefixes, sorted vertically by depth (i.e. value of \(\sum_{z=0}^{k_{A}}y_{z}^{p}\)) and horizontally by binary number value represented by \(\mathbf{y}\).
For the prefix \(\mathbf{y}^{p}=(1)\), we get \(\mathbf{y}_{upper}^{p}=(1,1,1,0,0)\) and \(\mathbf{y}_{lower}^{p}=(1,0,0,1,1)\), resulting in \(U=13\) and \(L=7\) respectively. For this prefix at depth 1, \(U>i_{A}\) and \(L<i_{A}\). If we set the maximum depth \(d=1\), then we need to estimate \(n_{b}\) as \(\Big{\lfloor}\frac{U-i_{A}+1}{U-L}s_{b}\Big{\rfloor}=4\), whereas the real value for \(n_{b}\) is actually 5. If we run our algorithm with the conditions \(\mathbf{w}=(5,5,3,2,1)\), \(j=3\), \(i_{A}=9\) and \(d=1\), we get an estimation of \(Pr(\sum_{0}^{j}W\geq i_{A})=0.5\), whereas the true probability is 0.7. Once we increase the depth to \(d=2\), then we get and estimation of \(Pr(\sum_{0}^{j}W\geq i_{A})=0.7\), but this comes at the cost of evaluating more prefixes and therefore greater computational expense. As we show in the main text, this method is more accurate than Monte Carlo methods when run for comparable lengths of time. Additionally, it is deterministic, so for a given set of intial parameters it will always return the same estimate of \(Pr(\sum_{0}^{j}W\geq i_{A})\). |
2305.14266 | Refraction laws for two-dimensional plasmons | Despite numerous applications of two-dimensional plasmons for electromagnetic
energy manipulation at the nanoscale, their quantitative refraction and
reflection laws (analogs of Fresnel formulas in optics) have not yet been
established. This fact can be traced down to the strong non-locality of
equations governing the 2d plasmon propagation. Here, we tackle this difficulty
by direct solution of plasmon scattering problem with Wiener-Hopf technique. We
obtain the reflection and transmission coefficients for 2d plasmons at the
discontinuity of 2d conductivity at arbitrary incidence angle, for both gated
and non-gated 2d systems. At a certain incidence angle, the absolute
reflectivity has a pronounced dip reaching zero for gated plasmons. The dip is
associated with wave passage causing no dynamic charge accumulation at the
boundary. For all incidence angles, the reflection has a non-trivial phase
different from zero and $\pi$. | Dmitry Svintsov, Georgy Alymov | 2023-05-23T17:16:53Z | http://arxiv.org/abs/2305.14266v1 | # Refraction laws for two-dimensional plasmons
###### Abstract
Despite numerous applications of two-dimensional plasmons for electromagnetic energy manipulation at the nanoscale, their quantitative refraction and reflection laws (analogs of Fresnel formulas in optics) have not yet been established. This fact can be traced down to the strong non-locality of equations governing the 2d plasmon propagation. Here, we tackle this difficulty by direct solution of plasmon scattering problem with Wiener-Hopf technique. We obtain the reflection and transmission coefficients for 2d plasmons at the discontinuity of 2d conductivity at arbitrary incidence angle, for both gated and non-gated 2d systems. At a certain incidence angle, the absolute reflectivity has a pronounced dip reaching zero for gated plasmons. The dip is associated with wave passage causing no dynamic charge accumulation at the boundary. For all incidence angles, the reflection has a non-trivial phase different from zero and \(\pi\).
Quantitative laws of wave reflection from a boundary between dissimilar media play a fundamental role in physics. In electrodynamics, such relations are known as Fresnel's formulas[1] and represent an indispensible tool for design of any optical element, be it a cavity, a polarizer, or anti-reflection coating. Similar laws can be found in acoustics of gases, liquids and solids[2]. In quantum mechanics, the problem of reflection and transmission at a potential step is a primary tool to demonstrate the wave-like nature of elementary particles.
Electromagnetically thin conductive media, be it 2d materials, quantum wells, or inversion layers in semiconductors, support a special type of electromagnetic waves known as two-dimensional plasmons[3; 4; 5]. At realistic densities of charge carriers, they can be confined by \(\sim 10^{2}\) times compared to free-space electromagnetic wavelength in vacuum[6]. This fact motivates their application for compact light detectors[7; 8; 9] and sources[10; 11], as well as for observation of zero-point electromagnetic fluctuation phenomena at the macro-scale[12].
Given the above motivation, it is surprising that quantitative Fresnel-type laws of 2d plasmon reflection haven't been yet derived. The complexity of such derivation stems from strong non-locality of dynamic equations governing wave propagation. As a result of non-locality, wave-like solutions break down at the interface of two conductive media. A conventional scheme of reflectance and transmittance derivation based on matching conditions fails. A number of works dealt with approximate reflection laws for 2d plasmons using numerical techniques[13; 14; 15; 16] and simulators[17], yet exact expressions for reflectance and transmittance have not been obtained.
Here, we resolve this complexity by a direct solution of scattering integral equation in piecewise-uniform 2d medium with the Wiener-Hopf technique. It is widely applied to diffraction problems at semi-infinite interfaces (wedges[18], waveguide terminations[19] and others). It was used many years ago in the problem of surface wave reflection at the normal incidence between metals with dissimilar surface impedances[20] and, quite recently, to 2d plasmon incident normally to the boundary between two regions of graphene with different conductivity[21]. A problem of inclined incidence is more complicated due the presence of two non-trivial field components, wherein two coupled integral equations are formed[22]. The Wiener-Hopf method is generally inapplicable in these situations[23]. The latter complexity is resolved in the quasi-static approximation. Such approximation is successfully used to describe the spectrum of edge magnetoplasmons[24] and similar waves[25; 26; 27].
We obtain a full analytical solution for reflection and transmission of 2d plasmon at the interface between 2d systems with different conductivities at arbitrary incidence angle \(\alpha\). In addition to universal total internal reflection, we find a certain angle \(\alpha^{*}\) at which the reflection is minimized. The reflection falls completely to zero if the wave propagates in the presence of ground plane (gated plasmon). This phenomenon may look similar to Brewster effect in optics, but has a different origin. At this angle \(\alpha^{*}\), the incident and transmitted waves cause no accumulation of charge at the interface, hence, no physical reason for reflection appears. In the case of non-gated plasmon, the reflection coefficient has a non-trivial phase shift which becomes large in the case of gliding incidence.
We proceed to the solution of the scattering problem for 2d plasmons schematically shown in Fig. 1 a. A plasma wave is incident from the left 2d section with conductivity \(\sigma_{L}\) at the boundary with right 2d section with conductivity \(\sigma_{R}\) at angle \(\alpha\), causing a reflected (r) and transmitted (t) waves. All wave characteristics (potential \(\varphi\), current density j) are harmonically varying in time as \(e^{-i\omega t}\), this time-dependent term will be skipped. The frequency dependence of conductivity \(\sigma(\omega)\) can be arbitrary and not limited to Drude model. The only requirement is that \(\sigma\) has a large positive imaginary part, such that transverse-magnetic 2d plasmons are well-defined.
The governing equation for electric potential \(\Phi\left(\mathrm{r}\right)\) in the 2d plane can be presented symbolically as:
\[\Phi\left(\mathrm{r}\right)=\mathcal{L}\left[\Phi\right], \tag{1}\]
where \(\mathcal{L}\left[...\right]\) is the integro-differential linear operator linking the potential created by charges in 2DES to the non
uniform field producing these charges:
\[\mathcal{L}\left[f\right]=\frac{1}{i\omega}\int d^{2}\mathrm{r}^{\prime}G\left( \mathrm{r}-\mathrm{r}^{\prime}\right)\nabla_{r^{\prime}}\left[\sigma\left( \mathrm{r}^{\prime}\right)\nabla_{r^{\prime}}f\left(\mathrm{r}^{\prime}\right)\right] \tag{2}\]
Above, \(G\left(\mathrm{r}\right)=\left|\mathrm{r}\right|^{-1}-\left|\mathrm{r}^{2}+4d^ {2}\right|^{-1/2}\) is the Green's function of the electrostatic problem, \(d\) is the distance to the screening plate (gate), \(\sigma\left(\mathrm{r}\right)\) is the distribution of 2d conductivity \(\sigma\left(\mathrm{r}\right)=\sigma_{L}\theta\left(-x\right)+\sigma_{R} \theta\left(x\right)\).
To solve the scattering problem, we split the full potential into the incident and scattered fields \(\Phi=\varphi_{i}+\varphi\). We choose the incident field as a semi-bounded plasma wave \(\varphi_{i}\left(\mathrm{r}\right)=\varphi_{i}\exp\left(iq_{i}x+iq_{y}y\right) \theta\left(-x\right)\). After such decomposition, the governing equation for scattered fields \(\varphi\) takes the form:
\[\varphi\left(\mathrm{r}\right)=\left\{\mathcal{L}\left[\varphi_{i}\right]- \varphi_{i}\left(\mathrm{r}\right)\right\}+\mathcal{L}\left[\varphi\right]. \tag{3}\]
We recognize that the term in curly brackets is equivalent to 'external source' creating the scattered field. From now on, the solution of scattering problem for 2d plasmons will be not much different from the solution of half-plane diffraction problems under external free-space illumination that have been studied extensively [18; 28; 29; 30].
We apply two subsequent Fourier transforms to Eq. 3. The first one with respect to the \(y\)-coordinate is trivial. The emerging wave vector \(q_{y}\) will be considered as an independent variable of the problem, the conserved \(y\)-component of plasmon momentum. Further on, we split the scattered potential into the 'left' and 'right' functions \(\varphi_{q_{y}}\left(x\right)=\varphi_{L}\left(x\right)\theta\left(-x\right)+ \varphi_{R}\left(x\right)\theta\left(x\right)\), and apply the second Fourier transform \(F\left[\varphi\right]\left(q_{x}\right)=\int\limits_{-\infty}^{+\infty}\varphi \left(x\right)e^{-iq_{x}x}dx\) with respect to \(x\)-coordinate. This leads us to fully Fourier-transformed scattering problem:
\[\varepsilon_{L}\left(q_{x}\right)\left[\varphi_{i}\left(q_{x} \right)+\varphi_{L}\left(q_{x}\right)\right]+\varepsilon_{R}\left(q_{x}\right) \varphi_{R}\left(q_{x}\right)=\\ \frac{q_{x}}{\omega}G\left(q\right)\left[\sigma_{R}-\sigma_{L} \right]\varphi_{0}, \tag{4}\]
where we have introduced the effective 2d dielectric functions of the left and right media
\[\varepsilon_{\alpha}\left(q_{x}\right)=1+\frac{i\sigma_{\alpha}}{\omega}q^{ 2}G\left(q\right),\qquad\alpha=\left\{L,\,R\right\} \tag{5}\]
and the Fourier-transformed Green's function of the electrostatic problem \(G\left(q\right)=2\pi q^{-1}(1-e^{-2qd})\), \(q=[q_{x}^{2}+q_{y}^{2}]^{1/2}\). The term on the right-hand side containing the value or real-space potential at the boundary \(\varphi_{q_{y}}(0)\equiv\varphi_{0}\) has emerged due to discontinuous electric field at the boundary.
Solution of (4) is based on inspection of analytic properties of emerging functions in the plane of complex \(q_{x}\)-variable and is given in Supplementary material, section I. The main property is that two functions \(F_{+}\left(q_{x}\right)\) and \(F_{-}\left(q_{x}\right)\) being analytic in the upper and lower half-planes and identical in a stripe \(\left|\mathrm{Im}\,q_{x}\right|<\delta\) should be equal to a polynomial of complex \(q_{x}\). This polynomial degenerates to zero if we require finiteness of potentials at infinity. Such splitting of Eq. (4) is quite straightforward if we know the decomposition of dielectric function \(\varepsilon_{\alpha}\left(q_{x}\right)=\varepsilon_{\alpha}^{+}\left(q_{x} \right)\varepsilon_{\alpha}^{-}\left(q_{x}\right)\) into the functions analytic and free of zeros in the upper (+) and lower (-) half-planes. In any case, it can be achieved with general formula:
\[\varepsilon_{\alpha}^{\pm}(q_{x})=\exp\left\{\pm\frac{1}{2\pi i}\int_{-\infty }^{+\infty}\frac{\ln\varepsilon_{\alpha}(u)du}{u-q_{x}\pm i0^{+}}\right\}, \tag{6}\]
while alternative approaches and semi-analytical formulas can also be available (see Supplementary Material, section II).
The result of splitting for scattering equation (4) reads as
\[\left[M_{+}\left(q_{x}\right)-M_{+}\left(q_{i}\right)\right]\varphi _{i}\left(q_{x}\right)+M_{+}\left(q_{x}\right)\varphi_{L}\left(q_{x}\right)-i \frac{\varphi_{0}}{2}L_{+}\left(q_{x}\right)= \tag{7}\] \[-M_{+}\left(q_{i}\right)\varphi_{i}\left(q_{x}\right)-M_{-}\left(q _{x}\right)\varphi_{R}\left(q_{x}\right)+i\frac{\varphi_{0}}{2}L_{-}\left(q_{x }\right), \tag{8}\]
\[M_{+}\left(q_{x}\right)=\frac{\varepsilon_{L}^{+}\left(q_{x} \right)}{\varepsilon_{R}^{+}\left(q_{x}\right)},M_{-}\left(q_{x}\right)=\frac{ \varepsilon_{R}^{+}\left(q_{x}\right)}{\varepsilon_{-}^{+}\left(q_{x}\right)}, \tag{9}\] \[L_{\pm}\left(q_{x}\right)=\pm\frac{M_{\pm}\left(q_{x} \right)}{q_{x}\pm iq_{y}}\pm\frac{M_{\pm}\left(q_{x}\right)-M_{\pm}\left(\pm iq _{y}\right)}{q_{x}\mp iq_{y}}\mp\frac{M_{\mp}\left(\mp iq_{y}\right)}{q_{x}\pm iq _{y}}. \tag{10}\]
Figure 1: (A) Schematic of the problem: a 2d plasma wave (red) is incident from the left on the conductivity step from \(\sigma_{L}\) to \(\sigma_{R}\), causing a reflected wave (green) and a transmitted wave (blue). Incidence angle is \(\alpha\), refraction angle is \(\alpha^{\prime}\) (B) Analytic structure of the Fourier-transformed scattering equation. The 2d dielectric functions \(\varepsilon_{L/R}(q_{x})\) have branch cuts starting at \(\pm iq_{y}\) and running to \(\pm i\infty\). They have simple zeros at wave vectors of the incident, transmitted and reflected waves, \(q_{x}=\left\{q_{i},\,q_{i},\,q_{r}\right\}\). The ’incident’ zero is compensated by the pole of Fourier-transformed incident potential (shown with hollow circle).
The left- and right-hand sides of such equation are now analytic in the upper and lower half-planes, respectively. They are identical in the stripe \(\left|\mathrm{Im}\,q_{x}\right|<\mathrm{Im}\,q_{i}\), where \(\mathrm{Im}\,q_{i}\) is the decay constant of the incident wave (can approach zero in the final result). Hence, both sides are zero identically, which yields the solution for scattering problem in the Fourier space:
\[\varphi_{L}\left(q_{x}\right)=M_{+}^{-1}(q_{x})\times\\ \left\{i\frac{\varphi_{0}}{2}L_{+}\left(q_{x}\right)-\left[M_{+} \left(q_{x}\right)-M_{+}\left(q_{i}\right)\right]\varphi_{i}\left(q_{x}\right)\right\} \tag{10}\]
\[\varphi_{R}\left(q_{x}\right)=M_{-}^{-1}(q_{x})\times\\ \left\{i\frac{\varphi_{0}}{2}L_{-}\left(q_{x}\right)-M_{+}\left(q _{i}\right)\varphi_{i}\left(q_{x}\right)\right\}. \tag{11}\]
A remaining problem is to link the real-space potential at \(x=0\), \(\varphi_{0}\), to that in the incident wave, \(\varphi_{i}\). This is achieved by evaluating the inverse transform \(\varphi_{0}=\pi^{-1}\lim_{x\to 0^{-}}\int_{-\infty}^{+\infty}\varphi_{L} \left(q_{x}\right)e^{-iq_{x}x}dq_{x}\) and solving a simple self-consistency system. This leads to
\[\varphi_{0}=2\varphi_{i}\frac{M_{+}\left(q_{i}\right)}{M_{+}\left(iq_{y} \right)+M_{-}\left(-iq_{y}\right)} \tag{12}\]
and completes the formal solution.
The real-space profiles of fields \(\varphi_{L}(x)\) and \(\varphi_{R}(x)\) are evaluated by inverse Fourier transforms of (10) and (10). The spatial structure of the real fields becomes transparent if we evaluate the transforms along the loops in \(q_{x}\)-plane shown in Fig. 1 b. The branch-cut contributions to \(\varphi(x)\) would correspond to evanescent fields near the edge with non-propagating nature. The residues at the poles \(q_{x}=q_{r}\) and \(q_{x}=q_{t}\) would yield the amplitudes or transmitted and reflected plasmons, respectively:
\[r=\frac{M_{+}\left(q_{i}\right)}{\left.\partial M_{+}/\partial q _{x}\right|_{q_{x}=q_{r}}}\left\{\frac{1}{q_{r}-q_{i}}-\frac{q_{r}}{q_{r}^{2} +q_{y}^{2}}-\frac{iq_{y}}{q_{r}^{2}+q_{y}^{2}}\frac{M_{+}^{2}\left(iq_{y} \right)-1}{M_{+}^{2}\left(iq_{y}\right)+1}\right\}, \tag{13}\] \[t=\frac{1}{\left.\partial M_{-}/\partial q_{x}\right|_{q_{x}=q_ {t}}}\left\{\frac{M_{+}\left(q_{i}\right)}{q_{t}-q_{i}}-M_{+}\left(q_{t} \right)\left[\frac{2q_{t}}{q_{t}^{2}+q_{y}^{2}}-\frac{iq_{y}}{q_{t}^{2}+q_{y} ^{2}}\frac{M_{-}^{2}\left(-iq_{y}\right)-1}{M_{-}^{2}\left(-iq_{y}\right)+1} \right]\right\} \tag{14}\]
Equations 13 and 14 represent the central results of this paper. To check its correctness, we note that for small separation between gate and 2DES \(qd\ll 1\), the dielectric function \(\varepsilon_{\alpha}^{G}\left(q_{x}\right)\) has a very simple analytic structure. Namely \(\varepsilon_{\alpha}^{G}\left(q_{x}\right)=1-\left(q_{x}^{2}+q_{y}^{2}\right) /q_{\mathrm{p}\alpha}^{2}\), where \(q_{p\alpha}^{2}=i\omega/4\pi d\sigma_{\alpha}\) is the absolute value of plasmon wave vector. The factorization of such dielectric function is immediately achieved:
\[\varepsilon_{\alpha}^{G}\left(q_{x}\right)=\frac{\sqrt{q_{p\alpha}^{2}-q_{y}^ {2}}-q_{x}}{q_{p\alpha}}\frac{\sqrt{q_{p\alpha}^{2}-q_{y}^{2}}+q_{x}}{q_{p \alpha}} \tag{15}\]
Introducing (15) into (13), we get very simple refraction laws for gated plasmons:
\[r^{G}=\frac{q_{pL}^{2}\sqrt{q_{pR}^{2}-q_{y}^{2}}-q_{pR}^{2}\sqrt{q_{pL}^{2}- q_{y}^{2}}}{q_{pL}^{2}\sqrt{q_{pR}^{2}-q_{y}^{2}}+q_{pR}^{2}\sqrt{q_{pL}^{2}- q_{y}^{2}}}. \tag{16}\]
The same result could be obtained in a simpler fashion, just by matching the potential and current across the boundary. This approach is correct in the case of strong screening, where electrostatics becomes local. Hence, coincidence of Wiener-Hopf result with wave matching result in the gated case serves as a check for this complex method. On the other hand, for non-gated plasmons the matching approach does not work, and we have to deal with full Wiener-Hopf expressions for reflection and transmission (13) and (14).
The computed reflection coefficient, according to Eq. 13, is shown in Figs. 2 and 3, for both the absolute value and phase. Expressing the parameters of left and right 2DES sections through the respective absolute values of plasmon wave vector \(q_{\mathrm{pL}}\) and \(q_{\mathrm{pR}}\), we can present the result in unified fashion for non-gated and gated plasmons. These are shown with solid and dashed lines, respectively. It is possible to show that absolute reflectance falls down to zero for gated plasmons at the incidence angle \(\alpha^{*}\) satisfying the Brewster-type condition
\[\tan\alpha^{*}=\frac{q_{pR}}{q_{pL}}. \tag{17}\]
Above, we have used the Snell's law \(q_{pL}\sin\alpha=q_{pR}\sin\alpha^{\prime}\) which is valid for arbitrary waves. For non-gated plasmons, the reflectance (13) has a dip not reaching zero, but becoming more pronounced for smaller 'contrast' of left and right sections. A similar dip was observed in electromagnetic simulations [17].
It is tempting to associate a dip in transmission with Brewster effect, similarly to conventional optics. In that case, the reflected wave vector should be co-directional with light-induced dipole moment in the medium # 2, but the dipole intensity in such direction turns to zero. Such explanation does not apply in the case of 2d plasmons, which have electric vector parallel to the propagation direction. The dipole emission intensity is thus always non-zero for directions of 2d plasmon reflection. Finally, the concept of canonical dipole radiation does
not apply to 2d plasmons treated in the non-retarded approximation, \(c\rightarrow\infty\).
A careful analysis shows that induced dipoles at the boundary of two 2DES sections do not appear at all at the non-reflection angle \(\alpha^{*}\). More precisely, the magnitudes of surface currents \(\mathbf{j}\) in the incident and transmitted wave are fine-tuned to cause no linear charge accumulation at the boundary. As a result, no physical stimulus appears for the reflection. To prove this viewpoint, we rewrite the no-reflection conduction (17) via conductivity and wave vector
\[\sigma_{L}q_{x}=\sigma_{R}q_{x}^{\prime}. \tag{18}\]
This is precisely the condition of current continuity \(j_{x}=-\sigma(x)\varphi^{\prime}(x)\) between incident and transmitted wave, from which the absence of dynamic charge immediately follows.
Interestingly, the reflection dip for non-gated plasmons never reaches zero. This fact is linked to the scattered fields having the non-propagating spatial structure, evanescent waves. As a result, some charge accumulation in the boundary layer should appear for non-gated plasmons, even despite the fulfillment of current matching condition (18). The latter condition applies only to the plane-wave part of the solution, and does not include evanescent fields.
The second important property of non-gated plasmon reflection, illustrated in Fig. 2 b, is the non-trivial reflection phase shift. It is different from zero and \(\pi\), and grows monotonically with increasing the angle of incidence. The case of gated plasmons, shown in the same figure with dashed lines, demonstrates a simpler behavior. The phase shift changes here stepwise between zero and \(\pi\) at the non-reflection angle \(\alpha^{*}\). This situation is analogous to the reflection of \(p\)-polarized waves near the
Figure 3: Computed reflectances \(|r|\) and reflection phases \(\mathrm{arg}r\) for a two-dimensional plasmon incident from medium with low conductivity to the medium with high conductivity (\(\mathrm{Im}\sigma_{L}<\mathrm{Im}\sigma_{R}\), \(q_{p,L}>q_{p,R}\)). Solid lines represent the result for non-gated plasmons, while dashed lines correspond to gated 2d plasmons.
Figure 2: Computed reflectances \(|r|\) and reflection phases \(\mathrm{arg}r\) for a two-dimensional plasmon incident from medium with high conductivity to the medium with low conductivity (\(\mathrm{Im}\sigma_{L}>\mathrm{Im}\sigma_{R}\), \(q_{p,L}<q_{p,R}\)). Solid lines represent the result for non-gated plasmons, while dashed lines correspond to gated 2d plasmons.
Brewster angle, though the origin of non-reflectance here is completely different.
Minimization of reflection for non-gated plasmons, and its full absence for gated plasmons, persists also for incidence from medium with low conductivity to the medium with high conductivity. As illustrated in Fig. 3 plotted for this case, the reflectance dip occurs at angles slightly below the angle of total internal reflection \(\alpha_{\mathrm{irr}}\). Remarkably, the variation of both amplitude and phase of reflection is very abrupt between \(\alpha^{*}\) and \(\alpha_{\mathrm{irr}}\). These abrupt variations would result in strong Goos-Hanchen shifts for the reflected waves [31]. In our particular 2d setup, Goos-Hanchen shift can be interpreted as excitation of leaky inter-edge plasmons [32]. From practical viewpoint, abrupt phase variations can be used for sensing applications, wherein the identified object modifies the properties of 2d conductivity [33].
The method for obtaining reflection and transmission used here can be extended to include non-local conduction effects, such as electron drift [34; 35] and viscosity [36]. It can also be applied to scattering of 2d plasmons at the boundary between gated and non-gated domains [37; 38; 39]. The obtained reflection and transmission coefficients can be used as building blocks for design of more complex structures, such as 2d plasmonic crystals with alternating doping [40; 41; 11] and 2d plasmonic waveguides. Possible applications of such structures currently lie in the fields of ultra long-wavelength radiation detection, emission, and modulation.
This work was supported by the Russian Science Foundation (Grant No. 21-72-10163).
|
2305.15178 | Uncertainty Voting Ensemble for Imbalanced Deep Regression | Data imbalance is ubiquitous when applying machine learning to real-world
problems, particularly regression problems. If training data are imbalanced,
the learning is dominated by the densely covered regions of the target
distribution and the learned regressor tends to exhibit poor performance in
sparsely covered regions. Beyond standard measures like oversampling or
reweighting, there are two main approaches to handling learning from imbalanced
data. For regression, recent work leverages the continuity of the distribution,
while for classification, the trend has been to use ensemble methods, allowing
some members to specialize in predictions for sparser regions. In our method,
named UVOTE, we integrate recent advances in probabilistic deep learning with
an ensemble approach for imbalanced regression. We replace traditional
regression losses with negative log-likelihood, which also predicts sample-wise
aleatoric uncertainty. Our experiments show that this loss function handles
imbalance better. Additionally, we use the predicted aleatoric uncertainty
values to fuse the predictions of different expert models in the ensemble,
eliminating the need for a separate aggregation module. We compare our method
with existing alternatives on multiple public benchmarks and show that UVOTE
consistently outperforms the prior art, while at the same time producing
better-calibrated uncertainty estimates. Our code is available at
link-upon-publication. | Yuchang Jiang, Vivien Sainte Fare Garnot, Konrad Schindler, Jan Dirk Wegner | 2023-05-24T14:12:21Z | http://arxiv.org/abs/2305.15178v3 | # Mixture of Experts with Uncertainty Voting for Imbalanced Deep Regression Problems
###### Abstract
Data imbalance is ubiquitous when applying machine learning to real-world problems, particularly regression problems. If training data are imbalanced, the learning is dominated by the densely covered regions of the target distribution, consequently, the learned regressor tends to exhibit poor performance in sparsely covered regions. Beyond standard measures like over-sampling or re-weighting, there are two main directions to handle learning from imbalanced data. For regression, recent work relies on the continuity of the distribution; whereas for classification there has been a trend to employ mixture-of-expert models and let some ensemble members specialize in predictions for the sparser regions. Here, we adapt the mixture-of-experts approach to the regression setting. A main question when using this approach is how to fuse the predictions from multiple experts into one output. Drawing inspiration from recent work on probabilistic deep learning, we propose to base the fusion on the aleatoric uncertainties of individual experts, thus obviating the need for a separate aggregation module. In our method, dubbed MOUV, each expert predicts not only an output value but also its uncertainty, which in turn serves as a statistically motivated criterion to rely on the right experts. We compare our method with existing alternatives on multiple public benchmarks and show that MOUV consistently outperforms the prior art, while at the same time producing better calibrated uncertainty estimates. Our code is available at _link-upon-publication_.
## 1 Introduction
Data imbalance is the norm, rather than the exception in real-world machine learning applications, and in regression tasks, in particular. Outside the realm of carefully curated research datasets, the distribution of the target values is typically non-uniform. Some parts of the distribution are covered by training examples much more densely than others, and as a result, machine learning models tend to be biased towards those well-represented regions and perform poorly in under-represented ones He and Garcia (2009). What is more, these sparse regions of the distribution are often important. In several applications, the prediction results matter specifically for rare, unusual conditions like extreme wind speeds in meteorology Maskey et al. (2020), or particularly high biomass in vegetation mapping Lang et al. (2022). Therefore, addressing the imbalance problem is an active area of machine learning research.
Traditional attempts to mitigate the impact of imbalance rely either on over-sampling rare data samples or on re-weighting the loss function to increase the cost of prediction errors at rare samples He and Garcia (2009). More recently, several authors have revisited the issue in the context of deep learning, typically through variants of the _mixture of experts_ framework. An ensemble of "expert" models is trained in such a way that they can each attend to a different part of the distribution. Then
their predictions are aggregated to obtain the final inference. The challenge in such methods consists in ensuring complementarity between the different experts and designing an aggregation method that synthesizes the predictions of individual ensemble members according to their relevance. A naive solution is to use the ensemble average, but this risks giving too much weight to predictions that are irrelevant to the specific data point. More elaborate solutions tune the aggregation weights in an unsupervised fashion (Zhang et al., 2021). Once optimized, these weights are still fixed and subject to the same limitation. It has also been proposed to use dynamic weights obtained from a _sample-level_ voting module that is trained with an independent objective (Wang et al., 2020). All works mentioned so far focus on classification problems. _Imbalanced regression_, on the other hand, has been studied a lot less and has only recently started to gain attention, especially since the publication of a suitable benchmark (Yang et al., 2021). The prevalent idea so far has been to exploit the continuity of regression functions, either by smoothing the features and labels (Yang et al., 2021) or by regularizers that encourage similar latent features at similar (continuous) labels (Gong et al., 2022). On the contrary, the mixture-of-experts idea has barely been explored in the context of imbalanced regression, despite the fact that model ensembles are common for deep regression (Lang et al., 2022; Yeo et al., 2021; Becker et al., 2023).
Here, we introduce a mixture of experts model with uncertainty voting (MOUV) for deep imbalanced regression. We adopt the expert ensemble framework for regression and propose a principled and straightforward way to dynamically aggregate the predictions. Rather than adding an empirically designed or learned voting module, we leverage the fact that uncertainty estimation techniques for deep regression (Kendall and Gal, 2017) inherently compute statistically meaningful weighting coefficients. Specifically, we use the estimated aleatoric uncertainties of individual experts to combine their predictions. To achieve this with a low computational overhead, we follow recent literature (Zhou et al., 2020) and construct a light ensemble, consisting of a shared encoder backbone and separate decoding branches for different experts. We experimentally evaluate our approach against other methods for deep imbalanced regression on a diverse set of tasks, including age regression, meteorological prediction, and text similarity prediction. We show that MOUV sets a new state-of-the-art performance on three out of four tasks and performs competitively on the last. Importantly, while MOUV improves overall performance, the gains are most significant for rare output values that are under-represented in the training data. In a large part, these gains can be attributed to the uncertainty-based aggregation. As an additional benefit, the uncertainties predicted by MOUV are better calibrated and, therefore, more informative for downstream tasks that rely on the regression output. blowing this approach, we integrate uncertainty estimates from experts who specialize in different data distributions to mitigate the impact of imbalanced data on regression.
To summarize, our contributions are:
* We introduce MOUV, a novel, efficient end-to-end method for imbalanced regression.
* We show empirically on four different datasets that MOUV advances the state of the art in terms of both the regression errors and the associated uncertainty estimates.
* To the best of our knowledge, MOUV is the first deep imbalanced regression method that combines the mixture-of-experts scheme with ideas from probabilistic deep learning.
## 2 Related Work
Imbalanced RegressionAs imbalanced regression receives less attention than imbalanced classification, early works usually use methods originally proposed for imbalanced classification. For example, the Synthetic Minority Oversampling Technique (SMOTE) introduced in Chawla et al. (2002) can create synthetic samples to relieve the imbalance in classification, which is also applied in regression problems Branco et al. (2017). Similarly, Lang et al. (2022) follows the class-balanced loss idea to add frequency-based weights to the loss function, so the model pays more attention to minority samples. More recently, Yang et al. (2021) introduces a public benchmark for imbalanced regression and investigates the continuity nature of regression problems with label and feature smoothing techniques. This public benchmark has encouraged more work focusing on the imbalanced regression. Then Gong et al. (2022) proposes a ranking-based regularization method to utilize the continuity property of the regression targets and enhance representation learning in the imbalanced regression.
Imbalanced ClassificationMost works studying imbalanced dataset focus on classification tasks. Early works include re-weighting Cui et al. (2019), re-sampling, Chawla et al. (2002) and data augmentation Zhang et al. (2017). A recent direction in imbalanced classification is the mixture-of-expert idea. Wang et al. (2020) proposes a two-stage method: in the first stage, they optimize three experts as three branches and use Kullback-Leibler divergence in the loss function to encourage expert diversity; in the second stage, they aggregate experts by training binary classifiers as dynamic expert assignment modules. Zhang et al. (2021) enforces different distribution for each expert explicitly to ensure the diversity of experts and utilizes a self-supervised training method at the test stage to combine experts for the final output. Although Zhang et al. (2021) requires no additional training for expert aggregation, the learnt weights are fixed at the _dataset-level_ instead of _sample-level_. Our work adapts the mixture-of-expert idea to regression task and we propose a new uncertainty-based expert aggregation mechanism, which requires no additional training and combines experts based on per-sample weights.
Application of Uncertainty Estimationprobabilistic deep learning methods like Kendall and Gal (2017) estimate both the mean target value and the uncertainty, which helps the model interpretation. Recent works further use uncertainty to achieve stronger predictions instead of solely producing uncertainty as a nice-to-have output. Yeo et al. (2021) utilizes a ensemble of probabilistic deep learning models to increase robustness to domain shifts. Each member of the ensemble is uniquely perturbed during training, and the aggregated ensemble prediction via the corresponding uncertainty achieves a more robust prediction against image corruption. Similarly, Becker et al. (2023) combines the predictions based on uncertainty to generate the country-wide map of forest structure variables, which is more robust to clouds. Here we also use estimated uncertainty to aggregate the knowledge of experts but our method is different from the previous methods in two aspects. First, we use multiple branches trained with different losses instead of a simple ensemble of models with different initializations, which is more computationally efficient. The second difference is the source of diversity among experts. Instead of relying on the randomness or the specified middle domain, we encourage different data distributions for each expert to achieve diverse predictions.
## 3 Method
Figure 1 shows a schematic overview of MOUV, the following paragraphs describe its components. MOUV consists of joint training of \(M\) different regression experts. Each expert predicts a sample-dependent aleatoric uncertainty, and that uncertainty is used to combine the predictions.
We consider a generic univariate regression dataset \(\mathcal{D}=\{(x_{n},y_{n}),n\in\llbracket 1,N\rrbracket\}\) of size \(N\), with \(x_{n}\) the input tensors, and \(y_{n}\) the corresponding scalar target values. We define \(B\) equally spaced bins across the target range and approximate the frequency distribution of the data by counting the number of data points per bin, \(\mathbf{f}=(f_{1},\cdots,f_{B})\).
Figure 1: **Overview of MOUV. A shared backbone encodes the input \(x\) into a representation \(z\). A mixture of \(M\) different experts use this shared representation to make their predictions. Each expert predicts a regression value \(\hat{y}\) as well as the uncertainty \(\hat{s}\) of that prediction. At inference time, we use the prediction of the most certain expert \(m_{0}\).**
Multi-headed architectureInstead of training \(M\) independent models, we follow recent literature Zhou et al. (2020) and design a multi-headed architecture with a shared backbone encoder and \(M\) regression heads that act as different experts. This design has the advantage that it is computationally lightweight and lets all experts rely on a common representation. The shared backbone encoder \(F_{trunk}\) can be selected according to the task at hand and maps each input point \(x_{n}\) to an embedding \(z_{n}\). The latter is processed by \(M\) different regression heads \(G_{m}\) that each output their individual expert prediction.
Aleatoric uncertainty predictionEach expert \(m\) makes two predictions: the target value \(\hat{y}_{n}^{m}\) and its associated aleatoric uncertainty \(\hat{s}_{n}^{m}\). Following Yeo et al. (2021), we train these predictions by minimizing the negative log-likelihood of the Laplace distribution:
\[\hat{y}_{n}^{m},\hat{s}_{n}^{m} =G_{m}(z_{n})\,, \tag{1}\] \[\mathcal{L}_{NLL}^{m} =\frac{1}{N}\sum_{n=1}^{N}w_{n}^{m}(exp(-\hat{s}_{n}^{m})\|y_{n} -\hat{y}_{n}^{m}\|_{1}+\hat{s}_{n}^{m})\,. \tag{2}\]
For numeric stability, we optimize \(\hat{s}_{n}\), the logarithm of the scale parameter in the Laplace distribution.
Joint training of diverse expertsEach expert \(m\) is trained with a different weighting of the samples \(w_{n}^{m}\), so as to achieve diversity and to make experts focus on different parts of the target distribution. The weights for expert \(m\) are defined as:
\[w_{n}^{m} =\left(\frac{1}{f_{b(n)}}\right)^{p_{m}}\,,\] (3) with \[p_{m} =\frac{m}{M-1},m\in\{0,...,M-1\}\, \tag{4}\]
with \(b(n)\) the bin in which sample \(n\) falls. Parameter \(p_{m}\) controls how strongly an expert concentrates on samples from sparse regions of the input distribution, with larger \(p\) corresponding to stronger rebalancing: when \(p=0\), the expert treats each sample equally; when \(p=1\), the expert employs inverse-frequency weighting and fully compensates density variations in the input. Different settings of \(p\) are complementary: unweighted standard regression learns the correct frequency prior and gives all data points the same influence on the latent representation; whereas inverse frequency weighting ensures that the model is not dominated by the dense regions of the distribution and fails to learn about the sparse ones. Intermediate versions between those extremes, like the popular inverse-squareroot weighting Lang et al. (2022), attempt to find a compromise. Together, the ensemble of experts strikes a balance by offering solutions according to several different weighting schemes and picking the least uncertain one on a case-by-case basis.
Dynamic learningFor representation learning, it is arguably more correct to assign samples equal weight. It is not obvious why the feature extractor that transforms raw data into a latent representation should to a large degree depend on the properties of rare, potentially not overly representative samples. Inspired by Zhou et al. (2020), we employ a dynamic learning strategy that initially focuses on the latent encoding and gradually phases in the remaining experts that have unequal weighting schemes:
\[\mathcal{L} =\alpha\mathcal{L}_{NLL}^{0}+(1-\alpha)\sum_{m=1}^{M-1}\mathcal{L }_{NLL}^{m}\, \alpha =1-(\frac{T}{T_{max}})^{2}\,, \tag{6}\]
where \(T\) is the current epoch number, and \(T_{max}\) is the maximum epoch number. \(\mathcal{L}_{NLL}^{0}\) is the loss for the expert with \(m=0\), which treats all samples equally. \(\alpha\) balances representation learning against mitigating the data imbalance.
Uncertainty-based expert aggregationDuring inference time, the predictions from multiple experts are combined based on the estimated uncertainty. One natural solution would be to weight the predictions using the inverse uncertainties. However, we obtain better experimental performance with selecting the output with the lowest predicted uncertainty:
\[\hat{y}_{n}=\hat{y}_{n}^{m_{0}}\,\ \ \hat{s}_{n}=\hat{s}_{n}^{m_{0}}\ \ \ \eqref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eqeq:eq:eqeq:eqeq:eq:eqeq:eqeq:eq:eqeq:eq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eqeq:eq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeq:eqeqeq:eqeqeq:eqeq:eqeqeqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeq:eqeq:eqeqeq:eqeqeq:eqeqeq:eqeq:eqeqeq:eqeqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeqeqeq:eqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeq:eqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeq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The baselines methods are to some degree complementary and can be combined. We therefore also test combinations of them as. e.g., LDS+FDS is a reweighting of features and loss terms, and RankSim is an additional regularizer that can be combined with different loss functions. For **AgeDB**, **IMDB-WIKI**, and **STS-B** datasets, the performance metrics for the baselines are taken from Yang et al. (2021); Gong et al. (2022). The **Wind** dataset is not included in that benchmark, so we run the baselines, using the official, public implementations of Yang et al. (2021); Gong et al. (2022).
MetricsFollowing Yang et al. (2021), we report the Mean Absolute Error (MAE) to evaluate regression performance on the **AgeDB**, **IMDB-WIKI**, and **Wind** datasets. Some work also reports Geometric Mean (GM), but in practice that metric strongly correlates with the MAE. For completeness we report it in the supplementary material. To be comparable, we follow Cer et al. (2017) for **STS-B** and report the Pearson correlation coefficient, expressed as a percentage (\(P\%\)). We also report the Mean Squared Error (MSE) of **STS-B** in the supplementary material. We report these metrics on the complete test set (All), as well as separately for different data density regimes. To that end the test data are binned into a frequency distribution. Bins with >100 samples form the many-shot regime (denotes _many_ in the tables), bins with 20 to 100 samples form the medium-shot (_med._) regime, and bins with <20 samples are the few-shot (_few_) regime Yang et al. (2021). Similar to other studies, we use bins of size \(1\) on **AgeDB**, **IMDB-WIKI**, and **Wind**, and of size \(0.1\) on **STS-B**. We report the Uncertainty Calibration Error (UCE) to evaluate the quality of the predicted uncertainties.
Implementation detailsAll codes to conduct the experiments is implemented in Pytorch Paszke et al. (2019). We use ResNet-50 as backbone for **AgeDB** and **IMDB-WIKI**, and ResNet-18 for **Wind**. For **STS-B**, we use BiLSTM+GloVe as the baseline, following Wang et al. (2018). The number of experts in MOUV (\(M\)) is tuned by training different instances and selecting the best one based on validation set performance. This gives \(M=2\) for **AgeDB** and **Wind**, and, \(M=3\) for **IMDB-WIKI** and **STS-B**. For further details about model training, see the supplementary material.
### Imbalanced Regression Experiment
Comparison to state-of-the-artWe report the numerical results of our experiments in Table 1. In terms of overall performance, MOUV outperforms all existing approaches on **AgeDB**, **IMDB-WIKI**, and **Wind** and is a close second on the **STS-B** dataset. On **AgeDB**, **IMDB-WIKI**, and **Wind**, our work also achieves the best performance on the _medium-shot_ and _few-shot_ regions of the distribution. The gain in few-shot performance compared to the Vanilla model ranges from \(43\%\) on **AgeDB** to \(20\%\) on **IMDB-WIKI**. The margin w.r.t. the closest competitor ranges from \(21\%\) on **AgeDB** to \(3\%\) on **Wind**. At the same time, MOUV still reaches the second-best performance in data-rich region (_many_), highlighting that it indeed leverages the predictions of different experts to respond to imbalanced datasets with marked density variations. On the **STS-B** dataset, MOUV achieves the second-highest Pearson correlation overall, as well as in the _medium-density_ regime, and the highest one for the _many-shot_ regime. On the contrary, in the _few-shot_ setting it does not shine, although it still outperforms not only the Vanilla model, but also standard baselines like RRT, LDS+FDS, and INV. For this particular dataset, RankSim clearly appears to be a more suitable strategy. We speculate that this may be due to the irregular distribution and the subjectively defined, non-metric target signal, for which it is challenging to obtain well-calibrated uncertainties. It appears that an approach based on pairwise feature similarity like RankSim, rather than metric uncertainty, is better suited for such data.
Figure 3: Per-expert and aggregated MAE on **IMDB-WIKI**. The uncertainty-based aggregation of MOUV nearly matches the performance of the best available expert on each subset of the test data.
In summary, MOUV sets a new state of the art for three of the four datasets, while on **STS-B** it is close second overall, but struggles in the _few-shot_ regime. MOUV is very flexible and can be readily adapted to different tasks and instantiated with different encoder and decoder architectures. It comes at a marginal computational cost. For instance, when using ResNet50 as backbone, each expert only increases the parameter count by 0.01%.
Successful expert aggregationAs visually illustrated in Figure 3 and numerically supported in Sec. 4.4, the uncertainty voting component of MOUV is able to dynamically select the expert that makes the best prediction, in the majority of cases: the prediction quality of the ensemble comes close to the one of an oracle that always picks the right expert, across all data regimes.
Uncertainty predictionTo assess the reliability of the predicted uncertainty itself, we calculate the uncertainty calibration metric in Table 2. Unsurprisingly, we observe the same pattern as for the actual regression targets: uncertainty is more difficult to estimate in the few-shot regime, i.e., in areas of low sample density. MOUV outperforms the vanilla network, trained with negative log-likelihood (NLL) loss, and the largest gains occur in the few-shot regime. e.g., the uncertainty calibration error (UCE) for samples from few-shot regions drops by 41% on **AgeDB** and by 37% on **Wind**.
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline & \multicolumn{3}{c}{**AgeDB \(\downarrow\)**} & \multicolumn{3}{c}{**IMDB-WIKI \(\downarrow\)**} \\ \cline{2-9} & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** \\ \cline{2-9} Vanilla & 7.77 & 6.62 & 9.55 & 13.67 & 8.06 & 7.23 & 15.12 & 26.33 \\ +RankSim & 7.13 & 6.51 & 8.17 & 10.12 & 7.72 & 6.93 & 14.48 & 25.38 \\ \cline{2-9} RRT & 7.74 & 6.98 & 8.79 & 11.99 & 7.81 & 7.07 & 14.06 & 25.13 \\ +LDS,FDS & 7.66 & 6.99 & 8.60 & 11.32 & 7.65 & 7.06 & 12.41 & 23.51 \\ +RankSim & 7.11 & 6.53 & 8.00 & 10.04 & 7.55 & 6.83 & 13.47 & 24.72 \\ +LDS,FDS+RankSim & 7.13 & 6.54 & 8.07 & 10.12 & 7.37 & **6.80** & 11.80 & 23.11 \\ \cline{2-9} SQINV & 7.81 & 7.16 & 8.80 & 11.20 & 7.87 & 7.24 & 12.44 & 22.76 \\ +LDS,FDS & 7.55 & 7.01 & 8.24 & 10.79 & 7.78 & 7.20 & 12.61 & 22.19 \\ +RankSim & 6.91 & **6.34** & 7.79 & 9.89 & 7.42 & 6.84 & 12.12 & 22.13 \\ +LDS,FDS+RankSim & 7.03 & 6.54 & 7.68 & 9.92 & 7.69 & 7.13 & 12.30 & 21.43 \\ \cline{2-9}
**MOUV** & **6.82** & 6.55 & **7.37** & **7.80** & **7.36** & 6.81 & **11.78** & **20.96** \\ \cline{2-9} & & \multicolumn{3}{c}{**Wind \(\downarrow\)**} & \multicolumn{3}{c}{**STS-B \(\uparrow\)**} \\ \cline{2-9} & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** \\ \cline{2-9} Vanilla & 7.48 & 7.38 & 13.10 & 21.42 & 74.2 & 72.0 & 62.7 & 75.2 \\ +RankSim & 7.43 & 7.33 & 12.49 & 20.50 & 76.8 & 71.0 & **72.9** & 85.2 \\ \cline{2-9} RRT & 7.51 & 7.39 & 13.67 & 22.79 & 74.5 & 72.4 & 62.3 & 75.4 \\ +LDS,FDS & 7.52 & 7.40 & 13.64 & 22.35 & 76.0 & 73.8 & 65.2 & 76.7 \\ +RankSim & 7.44 & 7.34 & 12.73 & 21.03 & **77.1** & 72.2 & 68.3 & **86.1** \\ +LDS,FDS+RankSim & 7.45 & 7.35 & 12.75 & 20.93 & 76.6 & 71.7 & 68.0 & 85.5 \\ \cline{2-9} SQINV/INV & 7.90 & 7.82 & 11.97 & 20.26 & 72.8 & 70.3 & 62.5 & 73.2 \\ +LDS,FDS & 7.75 & 7.68 & 11.98 & 15.87 & 76.0 & 74.0 & 65.2 & 76.6 \\ +RankSim & 7.79 & 7.71 & 12.22 & 20.07 & 69.9 & 65.2 & 60.1 & 76.0 \\ +LDS,FDS+RankSim & 7.71 & 7.63 & 12.16 & 16.70 & 75.8 & 70.6 & 69.0 & 82.7 \\ \cline{2-9}
**MOUV \(\kappa\)** & **7.30** & **7.23** & **11.09** & **15.43** & 77.0 & **74.3** & 70.0 & 77.9 \\ \hline \hline \end{tabular}
\end{table}
Table 1: **Main experiment. We report the regression performance (MAE\(\downarrow\) for AgeDB, IMDB-WIKI, Wind datasets, Pearson correlation (%)\(\uparrow\) for STS-B). For each column, the best results are in bold and the second best results are underlined.**
### Ablation Study
We investigate the contribution of different design choices in our method by training the following variants on the same benchmark data:
* **NLL**: The Vanilla architecture, but trained with negative log likelihood loss (NLL), instead of a standard L1 or L2 loss.
* **2-branch**, **3-branch**: The multi-head setup of our model without uncertainty estimation, corresponding to a naive model ensemble. We train both a two-branch (2-branch) and a three-branch (3-branch) version.
* **avg-vote**: This approach only differs from MOUV in that it combines the expert predictions by averaging, rather than based on the estimated uncertainty.
* **inv\(\sigma\)-vote**: Here, instead of selecting the lowest-uncertainty prediction in MOUV, we compute the weighted average of all experts' predictions based on invert uncertainty.
* **oracle-vote**: As an upper bound for the performance of MOUV, we also report the performance it would achieve if it had access to an oracle that selects the best expert for each data point (instead of using the predicted uncertainty).
Probabilistic trainingRunning a single-head regression network, but replacing the standard regression loss (Vanilla) with the NLL loss already leads to an increase in overall performance across all four datasets. On three of the four datasets, performance in the _few-shot_ regime also improves by
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c} \hline \hline & \multicolumn{4}{c}{**AgeDB \(\downarrow\)**} & \multicolumn{4}{c}{**IMDB-WIK \(\downarrow\)**} & \multicolumn{4}{c}{**Wind \(\downarrow\)**} \\ & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** \\ \cline{2-13} NLL & 1.76 & 1.05 & **2.66** & 6.01 & 2.36 & 1.83 & 7.37 & 17.71 & 6.68 & 6.59 & 11.52 & 20.95 & 0.68 & 0.65 & 0.77 & 0.71 \\
**MOUV** & **1.08** & **0.72** & 2.69 & **3.54** & **1.94** & **1.37** & **6.57** & **15.74** & **6.49** & **6.44** & **9.45** & **13.20** & **0.64** & **0.62** & **0.71** & **0.69** \\ \hline \hline \end{tabular}
\end{table}
Table 2: **Uncertainty calibration**. UCE of MOUV vs. Negative log-likelihood (NLL).
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c} \hline \hline & \multicolumn{4}{c}{**AgeDB \(\downarrow\)**} & \multicolumn{4}{c}{**IMDB-WIK \(\downarrow\)**} & \multicolumn{4}{c}{**IMDB-WIK \(\downarrow\)**} \\ & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** \\ \cline{2-13} Vanilla & 7.77 & 6.62 & 9.55 & 13.67 & 8.06 & 7.23 & 15.12 & 26.33 \\ \cline{2-13} NLL & 7.05 & **6.24** & 8.11 & 11.80 & 7.86 & 7.86 & 7.27 & 12.69 & 22.70 & 7.86 & 7.27 & 12.69 & 22.70 \\
3-branch & 7.80 & 7.19 & 8.93 & 10.44 & 7.61 & 7.03 & 12.21 & 22.46 & **7.34** & **6.77** & 12.06 & 21.30 \\ avg-vote & **6.80** & 6.38 & 7.58 & 8.66 & 6.86 & 6.86 & 6.31 & 11.29 & 20.59 & **7.36** & **6.77** & 12.01 & 21.23 \\ oracle-vote & 6.13 & 5.76 & 6.85 & 7.59 & 7.59 & 7.36 & 6.81 & **11.78** & **20.96** & **7.90** & **77.9** & **7.36** & 6.81 & **11.78** & **20.96** \\ \cline{2-13}
**MOUV** & 6.82 & 6.55 & **7.37** & **7.80** & & & & & & & & & & & \\ \cline{2-13} & \multicolumn{4}{c}{**Wind \(\downarrow\)**} & \multicolumn{4}{c}{**Few**} & \multicolumn{4}{c}{**STS-B \(\uparrow\)**} \\ & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** & **All** & **Many** & **Med.** & **Few** \\ \cline{2-13} Vanilla & 7.48 & 7.38 & 13.10 & 21.42 & 74.2 & 72.0 & 62.7 & 75.2 & 76.2 & 73.5 & 68.7 & 76.8 \\ NLL & 7.36 & 7.26 & 12.74 & 22.67 & 76.0 & 73.4 & 68.2 & 74.0 & 76.0 & 73.4 & 68.2 & 74.0 \\
2-branch \(\kappa\) & 7.56 & 7.49 & 11.15 & 17.83 & 75.9 & 72.5 & **71.8** & 74.5 & 75.9 & 72.5 & **71.8** & 74.5 \\
3-branch \(\kappa\) & **7.30** & 7.23 & 11.18 & 15.90 & 76.9 & 74.2 & 70.3 & 77.8 & 76.9 & 74.2 & 70.3 & 77.8 \\ inv\(\sigma\)-vote \(\kappa\) & **7.30** & **7.22** & 11.17 & 15.88 & 76.9 & 74.2 & 70.3 & 77.8 & 77.1 & 74.4 & 70.7 & 77.9 \\
**oracle-vote \(\kappa\)** & **7.22** & 7.15 & 10.91 & 15.33 & 77.1 & 74.4 & 70.7 & 77.9 & 77.9 & 77.
\(1-2\)pts. In other words, a probabilistic training objective by itself already mitigates the imbalance problem to some degree.
Multi-head structureAlso a multi-head architecture with specialized heads generally boosts overall performance, even without the probabilistic loss. The two and three-branch models (2-branch and 3-branch) improve performance primarily in the _few-shot_ regions of the distribution, by \(\approx 3\)pt MAE on **AgeDB**, **IMDB-WIKI**, and **Wind**, and by \(\approx 1\)pt \(P\%\) on **STS-B**. However, these models tend to suffer in high-density regions: _many-shot_ performance is lower on three dataset for the 2-branch model and on two datasets for the 3-branch model.
Uncertainty votingIn addition to improving the overall performance, the NLL training objective we use in MOUV allows us to select the best prediction based on aleatoric uncertainty. Compared to the 2-branch and 3-branch models, this brings a more significant improvement in the _few-shot_ regime, without sacrificing performance in _many-shot_ regions. For instance, MOUV reduces the error on **AgeDB**, by \(5.87\)pt in the _few-shot_ region, compared to only \(1.87\)pt and \(3.23\)pt reductions with the NLL and 3-branch models, respectively. At the same time, MOUV still outperforms the Vanilla model in the _many-shot_ regime. This highlights how uncertainty-based voting helps to select the correct expert at inference time, which also becomes apparent when comparing MOUV against the avg-vote model. While overall performance is similar, uncertainty voting excels in the _few-shot_ regions and consistently beats average voting. Average voting only seems beneficial in the _many-shot_ regions, where it brings the benefit of a traditional model ensemble. The performance of the invert uncertainty weighting approach (in\(\sigma\)-vote) also consistently underperforms in the _medium-shot_ and _few-shot_ regions. This suggests that in data-scarce parts of the distribution, selecting the best expert leads to better performance than ensembling. Lastly, a comparison of MOUV with oracle-voting bound demonstrates that the proposed uncertainty-based expert selection achieves near-perfect decisions in the medium- and low-density regions. While better uncertainty calibration would certainly help to further improve the results, it seems that the per-expert predictions, rather than the mechanism to combine them, are the current bottleneck.
Kernel density estimation.As a last ablation we employ KDE instead of histogram binning to estimate the sample density. Table 4 compares the performance of our method with KDE to the one with simple binning, across all datasets. On **AgeDB** and **IMDB-WIKI**, the MAE differences are marginal (\(<0.5\)pt) while they are more noticeable on datasets with irregular distribution, especially in the _few-shot_ regions. In particular, the MAE drops by 3.61pt in _few-shot_ regions of the **Wind** dataset when using KDE in MOUV. The result suggests that the more sophisticated approach to density estimation benefits datasets with irregular distributions, while making little difference for datasets with already relatively smooth distributions. We also re-trained the SQINV and RRT methods on **Wind** with KDE, but did not observe any performance improvement.
## 5 Conclusion
We have proposed MOUV, a simple and effective method for deep imbalanced regression. Our method, which can be understood as an ensemble over variably rebalanced regressors, can be freely combined with different encoder backbones and comes with negligible computational overhead. To our knowledge, it is also the first approach to deep imbalanced regression that combines the mixture-of-experts concept with ideas from probabilistic deep learning and uncertainty estimation. In experiments on four different datasets, MOUV reaches the best overall performance and sets a new state of the art for three of them. Importantly, our method decreases the prediction error particularly
\begin{table}
\begin{tabular}{l c c c c} \hline \hline & **All** & **Many** & **Med.** & **Few** \\ \hline
**AgeDB**\(\downarrow\) & +0.21 & +0.43 & -0.37 & -0.24 \\
**IMDB-WIKI**\(\downarrow\) & +0.03 & +0.07 & -0.37 & 0 \\
**Wind**\(\downarrow\) & 0 & +0.01 & -0.45 & -3.61 \\
**STS-B**\(\uparrow\) & +0.4 & +0.6 & -0.9 & +2.1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: **Impact of KDE**. Relative performance variation when switching from histogram-based frequencies to KDE with MOUV. Improvements with KDE are underlined.
in under-represented, low-density regions, while maintaining excellent performance in high-density regions. By construction, MOUV provides well-calibrated predictive uncertainties along with the target values, which enhance interpretability of the results and aid downstream tasks.
## Broader Impact
Our work can have a positive impact across multiple domains and industries. In healthcare, our methodology can enhance the accuracy of predictive models, leading to improved diagnoses and treatment recommendations, especially on rare cases. It also contributes to fairness and bias mitigation in machine learning, by promoting equitable decision-making processes and mitigating the bias of deep learning models towards over-represented parts of the population.
|
2310.18832 | Responsible AI (RAI) Games and Ensembles | Several recent works have studied the societal effects of AI; these include
issues such as fairness, robustness, and safety. In many of these objectives, a
learner seeks to minimize its worst-case loss over a set of predefined
distributions (known as uncertainty sets), with usual examples being perturbed
versions of the empirical distribution. In other words, aforementioned problems
can be written as min-max problems over these uncertainty sets. In this work,
we provide a general framework for studying these problems, which we refer to
as Responsible AI (RAI) games. We provide two classes of algorithms for solving
these games: (a) game-play based algorithms, and (b) greedy stagewise
estimation algorithms. The former class is motivated by online learning and
game theory, whereas the latter class is motivated by the classical statistical
literature on boosting, and regression. We empirically demonstrate the
applicability and competitive performance of our techniques for solving several
RAI problems, particularly around subpopulation shift. | Yash Gupta, Runtian Zhai, Arun Suggala, Pradeep Ravikumar | 2023-10-28T22:17:30Z | http://arxiv.org/abs/2310.18832v1 | # Responsible AI (RAI) Games and Ensembles
###### Abstract
Several recent works have studied the societal effects of AI; these include issues such as fairness, robustness, and safety. In many of these objectives, a learner seeks to minimize its worst-case loss over a set of predefined distributions (known as uncertainty sets), with usual examples being perturbed versions of the empirical distribution. In other words, aforementioned problems can be written as min-max problems over these uncertainty sets. In this work, we provide a general framework for studying these problems, which we refer to as _Responsible AI (RAI) games_. We provide two classes of algorithms for solving these games: (a) game-play based algorithms, and (b) greedy stagewise estimation algorithms. The former class is motivated by online learning and game theory, whereas the latter class is motivated by the classical statistical literature on boosting, and regression. We empirically demonstrate the applicability and competitive performance of our techniques for solving several RAI problems, particularly around subpopulation shift.
+
Footnote †: The relevant code for this work can be found at [https://github.com/yashgupta-7/rai-games](https://github.com/yashgupta-7/rai-games)
## 1 Introduction
In recent years, AI is increasingly being used in high-stakes decision-making contexts such as hiring, criminal justice, and healthcare. Given the impact these decisions can have on people's lives, it is important to ensure these AI systems have beneficial social effects. An emerging line of work has attempted to formalize such desiderata ranging over ethics, fairness, train-time robustness, test-time or adversarial robustness, and safety, among others. Each of these forms rich sub-fields with disparate desiderata, which are sometimes collated under the umbrella of "responsible AI". Many organizations are increasingly advocating the use of responsible AI models (Microsoft, 2021; Google, 2020).
But how do we do so when the majority of recent work around these problems is fragmented and usually focuses on optimizing one of these aspects at a time (DRO (Namkoong and Duchi, 2017; Duchi and Namkoong, 2018), GDRO (Sagawa et al., 2019), CVaR (Zhai et al., 2021), Distribution Shift (Hashimoto et al., 2018; Zhai et al., 2021))? Indeed optimizing for just one of these aspects has even been shown to exhibit adverse effects on the other aspects (Roh et al., 2020). To address this, we study a general framework that is broadly applicable across many of the settings above, and which we refer to as _Responsible AI (RAI)_ games. Our starting point is the recent understanding of a unifying theme in many of these disparate problems, that a learner seeks to minimize its worst-case loss over a set of predefined distributions. For example, in fairness, we seek to perform well on all sub-groups in the data. In robustness, we aim to design models that are robust to perturbations of the training data or the test distribution. This allows us to set up a zero-sum game between a learner that aims to learn a responsible model and an adversary that aims to prevent the learner from doing so. In
the general RAI game setting, this is a computationally intractable game that need not even have a Nash equilibrium. To address this computational issue, we study a relaxation of the single predictor RAI game, which we term the _ensemble_ RAI game, which can also be motivated as a linearization of the original RAI game.
We note that our framework encompasses not only the responsible AI settings but also the setting of classical boosting. Drawing upon the insights from boosting, we provide boosting-based algorithms for solving responsible AI games. We provide convergence guarantees of our algorithms by relying on the connections between boosting and online convex optimization, two-player gameplay (Arora et al., 2012; McMahan, 2011; Bubeck, 2011). We also conduct empirical analyses to demonstrate the convergence and utility of our proposed algorithms. Interestingly, the algorithms allow for _plug-and-play_ convenience, with changes in the RAI settings requiring only simple changes to the algorithms. More importantly, we could consider _intersections_ of different responsible AI considerations, which in turn can simply be incorporated into our algorithms. Finally, we also study the population risks of our algorithms in certain important settings. We show a surprising result that for the case of binary classification with the \(0/1\) loss, the optimal predictor for a large class of RAI games is the same as the Bayes optimal predictor, thus generalizing an emerging line of results demonstrating this for certain specific games (Hu et al., 2018). Under such settings, solving the RAI game could nonetheless be helpful in finite sample settings (as also demonstrated in our experiments) since the RAI game serves to encode desiderata satisfied by the Bayes optimal classifier.
## 2 Problem Setup and Background
We consider the standard supervised prediction setting, with input random variable \(X\in\mathcal{X}\subseteq\mathbb{R}^{d}\), output random variable \(Y\in\mathcal{Y}\), and samples \(S=\{(x_{i},y_{i})\}_{i=1}^{n}\) drawn from a distribution \(P_{\text{data}}\) over \(\mathcal{X}\times\mathcal{Y}\). Let \(\widehat{P}_{\text{data}}\) denote the empirical distribution over the samples. We also have a set \(H\) of hypothesis functions \(h:\mathcal{X}\mapsto\mathcal{Y}\) from which we wish to learn the best predictor. We evaluate the goodness of a predictor via a loss function \(\ell:\mathcal{Y}\times\mathcal{Y}\mapsto\mathbb{R}\), which yields the empirical risk: \(\widehat{R}(h)=\mathbb{E}_{\widehat{P}_{\text{data}}}\ell(h(x),y)\quad\text{ where}\quad\mathbb{E}_{\widehat{P}_{\text{data}}}(f(x,y))=\frac{1}{n}\sum_{i=1}^{n}f(x_{i},y_{i})\). Apart from having low expected risk, most settings require \(h\) to have certain properties, for example, robustness to distribution shift, fairness w.r.t subpopulations, superior tail performance, resistance to adversarial attacks, robustness in the presence of outliers, etc. We cast all these subproblems into an umbrella term "Responsible AI". Each of these properties has been studied extensively in recent works, albeit individually. In this work, we attempt to provide a general framework to study these problems.
### Related Work
We draw our unified framework from seminal works over the past decade by responsible AI researchers on devising non-expected risk objectives, _particularly min-max problems_, to ensure ML models are responsible. These have resulted in a multitude of different objectives (even for a single responsible AI desideratum such as fairness), and also multiple different sub-communities (so that fairness and multiple disparate robustness communities are relatively fractured), many (if not all) of which we combine within a single umbrella. There is emerging work on relating worst-case performance to invariance (Buhlmann, 2018); in other words, we might be able to get approximate group invariance via minimizing an appropriately constructed worst-group risk and vice-versa.
**RAI aspects as constraints.** Many prior works have enforced robustness as a constrained optimization (Shafieezadeh-Abadeh et al., 2015; Gao and Kleywegt, 2022; Namkoong and Duchi, 2016; Ben-Tal et al., 2011). There have also been few prior works enforcing fairness constraints (Mandal et al., 2020). To the best of our knowledge, there exists minimal prior work focusing on multiple desiderata at once in this regard.
**Multi Objective Optimization.** Several works have considered a multi-objective view of ensuring fairness in classifiers (Martinez et al., 2020; Oneto et al., 2018). If used for multiple RAI objectives, there is usually overhead in choosing a model that achieves a good trade-off between various losses. Also, it is difficult to guarantee that the solution is robust to any of the involved aspects. Our framework guarantees a certain level of performance on each of the RAI aspects under consideration.
**Distribution shift.**(Koh et al., 2021) classifies distribution shift problems into two categories: Domain generalization, and subpopulation shift. In this work, we focus on the subpopulation shift problem, where the target distribution is absolutely continuous to the source distribution. It has two
main applications: fairness [Hashimoto et al., 2018, Hu et al., 2018, Sagawa et al., 2019, Zhai et al., 2021b] and long-tail learning (_i.e._ learning on class-imbalanced datasets) [Cao et al., 2019, Menon et al., 2021, Kini et al., 2021].
**Distributionally Robust Optimization (DRO).** In DRO one aims to study classifiers that are robust to deviations of the data distribution. DRO has been studied under various uncertainty sets including \(f\)-divergence based uncertainty sets [Namkoong and Duchi, 2017, Duchi and Namkoong, 2018, Sagawa et al., 2019], Wasserstein uncertainty sets [Sinha et al., 2017, Gao et al., 2022], Maximum Mean Discrepancy uncertainty sets [Staib and Jegelka, 2019], more general uncertainty sets in the RKHS space [Zhu et al., 2020]. [Li et al., 2021a] evaluate model performance under worst-case subpopulations. Owing to its importance, several recent works have provided efficient algorithms for solving the DRO objective [Namkoong and Duchi, 2016, Qi et al., 2020, Kumar et al., 2023, Jin et al., 2021]. However, a lot of these techniques are specific to particular perturbation sets and are not directly applicable to the more general framework we consider in our work. Furthermore, in our work, we aim to learn an ensemble of models instead of a single model.
**Boosting.** Classical boosting aims to improve the performance of a weak learner by combining multiple weak classifiers to produce a strong classifier [Breiman, 1999, Friedman et al., 2000, Friedman, 2001, Freund and Schapire, 1995, Freund et al., 1996, Mason et al., 2000]. Over the years, a number of practical algorithms have been introduced such as AdaBoost [Schapire, 1999], LPBoost [Demiriz et al., 2002], gradient boosting [Mason et al., 1999], XGBoost [Chen and Guestrin, 2016], boosting for adversarial robustness [Zhang et al., 2022], [Meunier et al., 2021], [Balcan et al., 2023], and holistic robustness [Bennouna and Parys, 2022]. The algorithms we develop for RAI games are inspired by these algorithms.
**Fairness.** There are a number of fairness notions for algorithmic fairness, ranging from individual fairness [Dwork et al., 2012, Zemel et al., 2013], group fairness [Hardt et al., 2016a, Zafar et al., 2017], counterfactual fairness [Kusner et al., 2017], Rawlsian max-min fairness [Rawls, 2020, Hashimoto et al., 2018] and others [Barocas et al., 2017, Chouldechova and Roth, 2018, Mehrabi et al., 2021]. Our framework includes the popular notion of minimax group fairness. It doesn't capture other notions of group fairness such as Demographic Parity, Equality of Odds, Equality of Opportunity.
**Population RAI Risks.** Several recent works have studied properties of the population risks arising in various responsible AI scenarios. Hu et al. [2018] showed that the minimizer of population DRO risk (under general \(f\)-divergences) is the classical Bayes optimal classifier. Li et al. [2021b], Duchi and Namkoong [2018], Sinha et al. [2017] provided generalization guarantees for DRO risk under various divergence families ranging from \(f\)-divergences to Wasserstein perturbations.
## 3 RAI Games
In many cases, we do not wish to compute an unweighted average over training samples; due to reasons of noise, tail risk, robustness, and fairness, among many other "responsible AI" considerations.
**Definition 1** (RAI Risks): _Given a set of samples \(\{(x_{i},y_{i})\}_{i=1}^{n}\), we define the class of empirical RAI risks (for Responsible AI risks) as: \(\widehat{R}_{W_{n}}(h)=\sup_{w\in W_{n}}\mathbb{E}_{w}(h(x),y)\), where \(W_{n}\subseteq\Delta_{n},\) is some set of sample weights (a.k.a uncertainty set), and \(\mathbb{E}_{w}(f(x,y))=\sum_{i=1}^{n}w_{i}f(x_{i},y_{i}).\)_
Various choices of \(W_{n}\) give rise to various RAI risks. Table 1 presents examples of RAI risks that are popular in ML. Interestingly, classical problems such as boosting are special cases of RAI risks. In this work, we rely on this connection to design boosting-inspired algorithms for minimizing RAI risks. More choices for \(W_{n}\) can be obtained by combining the one's specified in Table 1 using union, intersection, convex-combination operations. For example, if one wants models that are fair to certain pre-specified groups, and at the same-time achieve good tail-risk, then one could choose \(W_{n}\) to be the intersection of Group-DRO and \(\alpha\)-CVaR uncertainty sets.
Given the empirical RAI risk \(\widehat{R}_{W_{n}}(h)\) of a hypothesis, and set of hypotheses \(H\), we naturally wish to obtain the hypothesis that minimizes the empirical RAI risk: \(\min_{h\in H}\widehat{R}_{W_{n}}(h)\). This can be seen as solving a zero-sum game.
**Definition 2** (RAI Games): _Given a set of hypothesis \(H\), and a RAI sample weight set \(W_{n}\), the class of RAI games is given as: \(\min_{h\in H}\max_{w\in W_{n}}\mathbb{E}_{w}(h(x),y).\)_
We thus study RAI Games for the special cases above and for an arbitrary constraint set \(W_{n}\).
## 4 Ensemble RAI Games
In this section, we begin our discussion about ensembles. In general, a statistical caveat with Definition 2 is that good worst-case performance over the sample weight set \(W_{n}\) is generally harder, and for a simpler set of hypotheses \(H\), there may not exist \(h\in H\) that can achieve such good worst-case performance. Thus it is natural to consider deterministic ensemble models over \(H\), which effectively gives us more powerful hypothesis classes. Let us first define RAI risk for such classifiers.
**Definition 3** (**Deterministic Ensemble**): _Consider the problem of classification, where \(\mathcal{Y}\) is a discrete set. Given a hypothesis class \(H\), a deterministic ensemble is specified by some distribution \(Q\in\Delta_{H}\), and is given by: \(h_{\text{der};Q}(x)=\arg\max_{y\in\mathcal{Y}}\mathbb{E}_{h\sim Q}\mathbb{I}[h (x)=y]\). Correspondingly, we can write the deterministic ensemble RAI risk as \(\widehat{R}_{W_{n}}(h_{\text{der};Q}(x))=\max_{w\in W_{n}}\mathbb{E}_{w}\ell( h_{\text{der};Q}(x),y)\)._
We discuss alternative definitions of deterministic ensembles in the Appendix. This admits a class of deterministic RAI games:
**Definition 4** (**Deterministic Ensemble RAI Games**): _Given a set of hypothesis \(H\), a RAI sample weight set \(W_{n}\), the class of RAI games for deterministic ensembles over \(H\) is given as:_
\[\min_{Q\in\Delta_{H}}\max_{w\in W_{n}}\mathbb{E}_{w}\ell(h_{\text{der};Q}(x), y).\]
However, the aforementioned game is computationally less amenable because of the non-smooth nature of de-randomized predictions. Moreover, there are some broader challenges with RAI games given by Definitions 2 and 4. Firstly, they need not have a Nash Equilibrium (NE), and in general, their min-max and max-min game values need not coincide. This poses challenges in solving the games efficiently. Next, in some cases, directly optimizing over the worst-case performance might not even be useful. For instance, [11, 12] show the pessimistic result that for classification tasks where when models are evaluated by the zero-one loss, ERM achieves the lowest possible DRO loss defined by some \(f\)-divergence or the \(\alpha\)-CVaR loss, given that the model is deterministic. To this end, we consider the following randomized ensemble:
**Definition 5** (**Randomized Ensemble**): _Given a hypothesis class \(H\), a randomized ensemble is specified by some distribution \(Q\in\Delta_{H}\), and is given by: \(\mathbb{P}[h_{\text{rand};Q}(x)=y]=\mathbb{E}_{h\sim Q}\mathbb{I}[h(x)=y]\). Similarly, we can define its corresponding randomized ensemble RAI risk: \(\widehat{R}_{\text{rand};W_{n}}(Q)=\max_{w\in W_{n}}\mathbb{E}_{h\sim Q} \mathbb{E}_{w}\ell(h(x),y)\)._
We can then also define the class of ensemble RAI games:
**Definition 6** (**Randomized Ensemble RAI Games**): _Given a set of hypothesis \(H\), a RAI sample weight set \(W_{n}\), the class of mixed RAI games is given as:_
\[\min_{Q\in\Delta_{H}}\max_{w\in W_{n}}\mathbb{E}_{h\sim Q}\mathbb{E}_{w}\ell( h(x),y). \tag{1}\]
This is a much better class of zero-sum games: it is linear in both the hypothesis distribution \(P\), as well as the sample weights \(w\), and if the sample weight set \(W_{n}\) is convex, is a convex-concave game. As shown below, under some mild conditions, this game has a Nash equilibrium which can be well approximated via efficient algorithms.
\begin{table}
\begin{tabular}{c||c|c} \hline Name & \(W_{n}\) & Description \\ \hline \hline Empirical Risk Minimization & \(\{P_{\text{data}}\}\) & object of focus in most of ML/AI \\ \hline \multirow{2}{*}{Worst Case Margin} & \(\Delta_{n}\), & used for designing \\ & entire probability simplex & margin-boosting algorithms \\ & [10, 11] & [10] \\ \hline Soft Margin & \(\{w:KL(w\|\widehat{P}_{\text{data}})\leq\rho_{n}\}\) & used in the design of \\ \hline \(\alpha\)-Conditional Value & \multirow{2}{*}{} & used in fairness \\ at Risk (CVaR) & & [10, 11] \\ \hline Distributionally Robust & \multirow{2}{*}{} & various choices for \(D\) \\ Optimization (DRO) & & have been studied \\ & & \(f\)-divergence [10] \\ \hline Group DRO & \(\{\widehat{P}_{\text{data}}(G_{1}),\widehat{P}_{\text{data}}(G_{2}),\ldots, \widehat{P}_{\text{data}}(G_{K})\}\) & used in group fairness, agnostic \\ & \(\widehat{P}_{\text{data}}(G_{i})\) is dist. of \(i^{th}\) group & federated learning [10] \\ \hline \end{tabular}
\end{table}
Table 1: Various ML/AI problems that fall under the umbrella of RAI risks.
**Proposition 1**: _Let \(H\) be parameterized by \(\theta\in\Theta\subseteq\mathbb{R}^{p}\), for convex, compact set \(\Theta\) and let \(W_{n}\) be a convex, compact set. Then \(\min_{Q\in\Delta_{H}}\max_{w\in W_{n}}\mathbb{E}_{h\sim Q}\mathbb{E}_{w}\ell(h( x),y)=\max_{w\in W_{n}}\min_{Q\in\Delta_{H}}\mathbb{E}_{h\sim Q}\mathbb{E}_{w} \ell(h(x),y)\)_
The proposition follows as a direct consequence of well known minimax theorems (Appendix D.3).
### Going from Deterministic to Randomized Ensembles
To begin, we point out that what we want is a deterministic ensemble rather than a randomized ensemble. In fact, it can be seen that the deterministic ensemble in Definition 3 is a specific de-randomization of the randomized ensemble. It is such deterministic ensembles that we usually simply refer to as ensemble predictors. But the RAI risk for the ensemble predictor is NOT equal to the ensemble RAI risk minimized by our desired game in Equation 1 above for randomized ensembles. Thus, the ensemble RAI game might not in general capture the ideal deterministic ensemble. In this section, we study why and when might solving for a random ensemble is meaningful.
**Binary Classification.** Interestingly, for the very specific case of binary classification, we can provide simple relationships between the risks of the randomized and deterministic ensemble.
**Proposition 2**: _Consider the setting with \(\mathcal{Y}=\{-1,1\}\), the zero-one loss \(\ell\), and \(W_{n}=\Delta_{n}\). Then,_
\[\widehat{R}_{W_{n}}(h_{\text{det};Q})=\mathbb{I}[\widehat{R}_{W_{n}}(h_{\text {rand};Q})\geq 1/2].\]
See Appendix E.2 for a simple proof. In this case, we can also relate the existence of a perfect deterministic ensemble ("boostability") to a weak learning condition on the set of hypotheses. Specifically, suppose \(H\) is boostable iff there exists \(Q\in\Delta_{H}\) s.t. \(\widehat{R}_{W_{n}}(h_{\text{det};Q})=0\). From the above proposition this is equivalent to requiring that \(\widehat{R}_{W_{n}}(h_{\text{rand};Q})<1/2\). We thus obtain:
\[\inf_{Q\in\Delta_{H}}\sup_{w\in W_{n}}\mathbb{E}_{w,Q}\ell(h(x),y)<1/2\iff \sup_{w\in W_{n}}\inf_{h\in H}\mathbb{E}_{w}\ell(h(x),y)<1/2\]
where the equivalence follows from the min-max theorem and the linearity of the objective in \(P\). The last statement says that for any sample weights \(w\in W_{n}\), there exists a hypothesis \(h\in H\) that has \(w\)-weighted loss at most \(1/2\). We can state this as a "weak-learning" condition on individual hypotheses in \(H\). The above thus shows that for the specific case of \(\mathcal{Y}=\{-1,1\}\), the zero-one loss \(\ell(y,y^{\prime})=\mathbb{I}[y\neq y^{\prime}]\), and \(W_{n}=\Delta_{n}\), we can relate boostability of \(H\) to a weak learning condition on hypothesis within \(H\).
General ClassificationBut in general, we do not have simple connections between \(\widehat{R}_{W_{n}}(h_{\text{det};Q})\) and \(\widehat{R}_{W_{n}}(h_{\text{rand};Q})\). All we can guarantee is the following upper bound:
**Proposition 3**: _Let \(\gamma_{Q}=1/\min_{i\in[n]}\max_{y\in\mathcal{Y}}\mathbb{P}_{Q}[h(x_{i})=y]\). Then,_
\[\widehat{R}_{W_{n}}(h_{\text{det};Q})\leq\gamma_{Q}\widehat{R}_{W_{n}}(h_{ \text{rand};Q}).\]
See Appendix E.2 for a simple proof.
**Corollary 4**: _For binary classification, we have \(\gamma_{P}\leq 2\) and thus, we recover the well known bound \(\widehat{R}_{W_{n}}(h_{\text{det};Q})\leq 2\widehat{R}_{W_{n}}(h_{\text{rand};Q})\)_
**Remark 5**: _These bounds might be loose in practice. Specifically, for the binary case, if \(\widehat{R}_{W_{n}}(h_{\text{rand};Q})\leq\frac{1}{2}\) then we have \(\widehat{R}_{W_{n}}(h_{\text{det};Q})=0\). To this end, prior work (Lacasse et al., 2006; Germain et al., 2015; Masegosa et al., 2020) have developed tighter bounds using second-order inequalities. We leave the analyses of these second-order RAI games to future work._
As such, we can cast minimizing randomized RAI risk as minimizing an upper bound on the deterministic ensemble RAI risk. Thus, the corresponding randomized RAI game can be cast as a relaxation of the deterministic RAI game. In the sequel, we thus focus on this randomized ensemble RAI game, which we will then use to obtain a deterministic ensemble. Following the bounds above, the corresponding deterministic ensemble risk will be bounded by the randomized ensemble RAI risk
## 5 Algorithms
In this section, we present two algorithms for solving the RAI game in Equation (1). Our first algorithm is motivated from online learning algorithms and the second algorithm is motivated from
greedy stepwise algorithms that have been popular for solving many statistical problems such as regression. For simplicity of presentation, we assume \(H\) is a finite set. However, our results in the section extend to uncountable sets.
### Methods
Game-play.In game play based algorithms, both the min and the max players are engaged in a repeated game against each other. Both players rely on no-regret algorithms to decide their next action. It is well known that such a procedure converges to a mixed NE of the game Cesa-Bianchi and Lugosi (2006). In this work, we follow a similar strategy to solve the game in Equation (1) (see Algorithm 1 for the pseudocode). In the \(t^{th}\) round of our algorithm, the following distribution \(w^{t}\in W\) is computed over the training data points
\[w^{t}\leftarrow\operatorname*{argmax}_{w\in W_{n}}\sum_{s=1}^{t-1}\mathbb{E} _{w}\ell(h^{s}(x),y)+\eta^{t-1}\text{Reg}(w) \tag{2}\]
This update is called the Follow-The-Regularized-Leader (FTRL) update. Here, \(\text{Reg}(\cdot)\) is a strongly concave regularizer and \(\eta^{t-1}\) is the regularization strength. One popular choice for \(\text{Reg}(\cdot)\) is the negative entropy which is given by \(-\sum_{i}w_{i}\log w_{i}\). This regularizer is also used by AdaBoost, which is a popular boosting algorithm. In Appendix F.2, we provide analytical expressions for \(w^{t}\) for various choices of \(W_{n},\text{Reg}(\cdot)\). We note that the regularizer in the FTRL update ensures the stability of the updates; _i.e._, it ensures consecutive iterates do not vary too much. This stability is naturally guaranteed when \(W_{n}\) is a strongly convex set (an example of a strongly convex set is the level set of a strongly convex function. See Appendix for a formal definition and more details). Consequently, the regularization strength \(\eta^{t-1}\) could be set to \(0\) in this case, and the algorithm still converges to a NE Huang et al. (2017).
Once we have \(w^{t}\), a new classifier \(h^{t}\) is computed to minimize the weighted loss relative to \(w^{t}\), and added to the ensemble. This update is called the Best Response (BR) update. Learning \(h^{t}\) in this way helps us fix past classifiers' mistakes, eventually leading to an ensemble with good performance.
```
Input: Training data \(\{(x_{i},y_{i})\}_{i=1}^{n}\), loss function \(\ell\), constraint set \(W_{n}\), hypothesis set \(H\), strongly concave regularizer \(R\) over \(W_{n}\), learning rates \(\{\eta^{t}\}_{t=1}^{T}\)
1:for\(t\gets 1\) to \(T\)do
2:FTRL:\(w^{t}\leftarrow\operatorname*{argmax}_{w\in W_{n}}\sum_{s=1}^{t-1}\mathbb{E} _{w}\ell(h^{s}(x),y)+\eta^{t-1}\text{Reg}(w)\)
3:BR:\(h^{t}\leftarrow\operatorname*{argmin}_{h\in H}\mathbb{E}_{w^{t}}\ell(h(x),y)\)
4:endfor
5:return\(P^{T}=\frac{1}{T}\sum_{t=1}^{T}w^{t}\), \(Q^{T}=\text{Unif}\{h^{1},\ldots h^{T}\}\)
```
**Algorithm 1** Game play algorithm for solving Equation (1)
Greedy.We now take an optimization theoretic viewpoint to design algorithms for Equation (1). Let \(L(Q)\) denote the inner maximization problem of (1): \(L(Q):=\max_{w\in W_{n}}\mathbb{E}_{h\sim Q}\mathbb{E}_{w}\ell(h(x),y)\). When \(L(Q)\) is smooth (this is the case when \(W_{n}\) is a strongly convex set), one could use Frank-Wolfe (FW) to minimize it. The updates of this algorithm are given by
\[Q^{t}\leftarrow(1-\alpha^{t})Q^{t-1}+\alpha^{t}G,\quad\text{ where }G= \operatorname*{argmin}_{Q}\left\langle Q,\nabla_{Q}L(Q^{t-1})\right\rangle.\]
Here, \(\nabla_{Q}L(Q^{t-1})=\operatorname*{argmax}_{w\in W_{n}}\mathbb{E}_{h\sim Q ^{t-1}}\mathbb{E}_{w}\ell(h(x),y)\). This algorithm is known to converge to a minimizer of \(L(Q)\) at \(O(1/t)\) rate Jaggi (2013). When \(L(Q)\) is non-smooth, we first need to smooth the objective before performing FW. In this work we perform Moreau smoothing Parikh et al. (2014), which is given by
\[L_{\eta}(Q)=\max_{w\in W_{n}}\mathbb{E}_{h\sim Q}\mathbb{E}_{w}\ell(h(x),y)+ \eta\text{Reg}(w). \tag{3}\]
Here \(\text{Reg}(\cdot)\) is a strongly concave regularizer. If \(\text{Reg}(\cdot)\) is \(1\)-strongly concave, it is well known that \(L_{\eta}(Q)\) is \(O(1/\eta)\) smooth. Once we have the smoothed objective, we perform FW to find its optimizer (see Algorithm 2 for pseudocode).
**Relaxing the simplex constraint.** We now derive a slightly different algorithm by relaxing the simplex constraint on \(Q\). Using Lagrangian duality we can rewrite \(\min_{Q\in\Delta_{H}}L_{\eta}(Q)\) as the following problem for some \(\lambda\in\mathbb{R}\)
\[\min_{Q\succeq 0}L_{\eta}(Q)+\lambda\sum_{h\in H}Q(h).\]
One interesting observation is that when \(W_{n}\) is the entire simplex and when \(\lambda=-1/2\), we recover the AdaBoost algorithm. Given the practical success of AdaBoost, we extend it to general \(W_{n}\). In particular, we set \(\lambda=-1/2\) and solve the resulting objective using greedy coordinate-descent. The updates of this algorithm are given in Algorithm 2.
**Remark 6**: _Algorithm 2 takes the step sizes \(\{\alpha^{t}\}_{t=1}^{T}\) as input. In practice, one could use line search to figure out the optimal step-sizes, for better performance._
```
0: Training data \(\{(x_{i},y_{i})\}_{i=1}^{n}\), loss function \(\ell\), constraint set \(W_{n}\), hypothesis set \(H\), strongly concave regularizer \(R\) over \(W_{n}\), regularization strength \(\eta\), step sizes \(\{\alpha^{t}\}_{t=1}^{T}\)
1:for\(t\gets 1\) to \(T\)do
2:\(G^{t}=\operatorname*{argmin}_{Q}\left\langle Q,\nabla_{Q}L_{\eta}(Q^{t-1})\right\rangle\)
3:FW:\(Q^{t}\leftarrow(1-\alpha^{t})Q^{t-1}+\alpha^{t}G^{t}\) / Gen-AdaBoost:\(Q^{t}\gets Q^{t-1}+\alpha^{t}G^{t}\)
4:endfor
5:return\(Q^{T}\)
```
**Algorithm 2** Greedy algorithms for solving Equation (1)
We provide convergence rates for the algorithms below:
**Proposition 7**: _(Convergence Rates) Let \(l(h(x),y)\in[0,1]\)\(\forall h\in H,(x,y)\in D\) and \(\text{Reg}:\Delta_{n}\rightarrow\mathbb{R}\) be a \(1\)-strongly concave function w.r.t norm \(\|.\|_{1}\). Let \(Q^{T}\) be the output returned from running Algorithm 1 or 2 for \(T\) iterations. Let \(D_{R}\) be a constant S.T. \(D_{R}^{2}=\max_{x,y\in W_{n}}|\text{Reg}(x)-\text{Reg}(y)|\)._
1. _(Gameplay) If_ \(\eta^{t}=\eta\)_, then_ \(Q^{T}\) _satisfies_ \(L(Q^{T})\leq\min_{Q}L(Q)+\frac{\eta D_{R}^{2}}{T}+\mathcal{O}(\frac{1}{\eta})\)_._
2. _(Greedy) If line-search is performed for_ \(\alpha^{t}\)_, then_ \(Q^{T}\) _(FW or the Gen-AdaBoost update) satisfies_ \(L(Q^{T})\leq\min_{Q}L(Q)+\eta D_{R}^{2}+\mathcal{O}(\frac{1}{\eta^{T}})\)_._
We refer the reader to Appendix F.1 for a simple proof using existing theory on online convex optimization (McMahan, 2011; Jaggi, 2013). Another useful insight is that Algorithms 1 and 2 are related to each other under special settings as shown by Appendix H.1.
**Corollary 8**: _Consider \(\text{Reg}(w)=-\sum_{i=1}^{n}w_{i}\log w_{i}\) and \(l\) as the zero-one loss. Then, Algorithm 1 and Algorithm 2 (line-search) achieve \(\epsilon-\)approximate NE with \(\epsilon\) as \(\mathcal{O}\left(\sqrt{\frac{\log(n)}{T}}\right)\)._
Weak Learning ConditionsIt might not be practical for \(H\)-player to play BR (Step 3: Algorithm 1) or correspondingly, to find the best possible classifier at every round (Step 2: Algorithm 2). Under weak learning conditions, we can indeed achieve (approximate) convergence when we only solve these problems approximately. See Appendix H.2 for more details.
## 6 Generalization Guarantees
In this section, we study the population RAI risk and present generalization bounds which quantify the rates at which empirical RAI risk converges to its population counterpart.
### Population RAI Games
Recall, the empirical RAI risk optimizes over all sample re-weightings \(w\in W_{n}\) that lie within the probability simplex \(\Delta_{n}\). Thus it's population counterpart optimizes over distributions \(P\) that are absolutely continuous with respect to the data distribution \(P_{\text{data}}\):
\[R_{W}(h)=\sup_{P:P\ll P_{\text{data}},\frac{\text{def}P}{\text{data}}\in W} \mathbb{E}_{P}[\ell(h(x),y)].\]
Following (Shapiro et al., 2021), we can rewrite this as follows. Suppose we use \(Z=(X,Y)\in\mathcal{Z}:=\mathcal{X}\times\mathcal{Y}\), so that \(P,P_{\text{data}}\) are distributions over \(Z\). We then define \(\ell_{h}:\mathcal{Z}\mapsto\mathbb{R}\) as \(\ell_{h}(z)=\ell(h(x),y)\). We can then write the population RAI risk as (see Appendix G for a proof):
\[R_{W}(h)=\sup_{r:\mathcal{Z}\mapsto\mathbb{R}_{+},\int r(z)dP_{\text{data}}(z )=1,r\in W}\mathbb{E}_{P_{\text{data}}}[r(z)\ell_{h}(z)]. \tag{4}\]
For classification, we define the RAI-Bayes optimal classifier as: \(Q^{*}_{W}=\operatorname*{arg\,min}_{Q}R_{W}(Q)\). Here, the minimum is w.r.t the set of all measurable classifiers (both deterministic and random). This is the "target" classifier we wish to learn given finite samples. Note that this might not be the same as the vanilla Bayes optimal classifier: \(Q^{*}=\operatorname*{arg\,min}_{Q}\mathbb{E}[\widehat{R}(Q)]\), which only minimizes the expected loss, and hence may not satisfactorily address RAI considerations.
We now try to characterize the RAI-Bayes optimal classifier. However, doing this requires a bit more structure on \(W\). So, in the sequel, we consider constraint sets of the following form:
\[W=\left\{r:\mathcal{Z}\mapsto\mathbb{R}_{+}\,:\,\int g_{i}(r(z))dP_{\text{ data}}(z)\leq c_{i},i\in[m]\right\}, \tag{5}\]
where we assume that \(g_{i}:\mathbb{R}_{+}\mapsto\mathbb{R},i\in[m]\) are convex. Note that this choice of \(W\) encompasses a broad range of RAI games including DRO with \(f\)-divergence, CVaR, soft-margin uncertainty sets. Perhaps surprisingly, the following proposition shows that the minimizer of population RAI risk is nothing but the vanilla Bayes optimal classifier.
**Proposition 9** (Bayes optimal classifier): _Consider the problem of binary classification where \(\mathcal{Y}=\{-1,+1\}\). Suppose \(\ell(h(x),y)=\phi(yh(x))\) for some \(\phi:\mathbb{R}\to[0,\infty)\) which is either the \(0/1\) loss, or a convex loss function that is differentiable at \(0\) with \(\phi^{\prime}(0)<0\). Suppose the uncertainty set \(W\) is as specified in Equation (5). Moreover, suppose \(\{g_{i}\}_{i=1\dots m}\) are convex and differentiable functions. Then, the vanilla Bayes optimal classifier is also a RAI-Bayes optimal classifier._
**Remark 10**: _In the special case of \(m=1\) in Equation (5), we recover the result of (Hu et al., 2018). However, our proof is much more elegant than the proof of (Hu et al., 2018), and relies on the dual representation of the population RAI risk._
One perspective of the above result is that the vanilla Bayes optimal classifier is also "responsible" as specified by the RAI game. This is actually reasonable in many practical prediction problems where the label annotations are actually derived from humans, who presumably are also responsible. Why then might we be interested in the RAI risk? One advantage of the RAI risks is in finite sample settings where the equivalence no longer holds, and the RAI risk could be construed as encoding prior knowledge about properties of the Bayes optimal classifier. We also note that the above equivalence is specific for binary classification.
### Generalization Guarantees
Our generalization bounds rely on the following dual characterization of the RAI population risk.
**Proposition 11**: _Suppose the uncertainty set \(W\) is as specified in Equation (5). Then for any hypothesis \(h\), the population RAI risk can be equivalently written as_
\[R_{W}(h)=\inf_{\lambda\geq\mathbf{0},\tau}\mathbb{E}_{P_{\text{data}}}G^{*}_{ \lambda}(\ell_{h}(z)-\tau)+\sum_{i=1}^{m}\lambda_{i}c_{i}+\tau, \tag{6}\]
_where \(G^{*}_{\lambda}\) is the Fenchel conjugate of \(G_{\lambda}(t)=\sum_{i=1}^{m}\lambda_{i}g_{i}(t)\)._
We utilize the above expression for \(R_{W}(h)\) to derive the following deviation bound for \(\widehat{R}_{W_{n}}(h)\).
**Theorem 12**: _Consider the setting of Proposition 11. Suppose \(\{g_{i}\}_{i=1\dots m}\) are convex and differentiable functions. Suppose \(\ell_{h}(z)\in[0,B]\) for all \(h\in H\), \(z\in\mathcal{Z}\). Suppose, for any distribution \(P_{\text{data}}\), the minimizers \((\lambda^{*},\tau^{*})\) of Equation (6) lie in the following set: \(\mathcal{E}=\{(\lambda,\tau):\|\lambda^{*}\|_{\infty}\leq\bar{\Lambda},|\tau^{* }|\leq T\}\). Moreover, let's suppose the optimal \(\lambda^{*}\) for \(P_{\text{data}}\) is bounded away from \(0\) and satisfies \(\min_{i}\lambda^{*}_{i}\geq\bar{\Lambda}\). Let \(G,L\), be the range and Lipschitz constants of \(G^{*}_{\lambda}\):_
\[G\coloneqq\sup_{(\lambda,\tau)\in\mathcal{E}}G^{*}_{\lambda}(B-\tau)-G^{*}_{ \lambda}(-\tau),\quad L\coloneqq\sup_{x\in[0,B],(\lambda,\tau)\in\mathcal{E}, \lambda:\min_{i}\lambda_{i}\geq\bar{\Lambda}}\Big{\|}\frac{\partial G^{*}_{ \lambda}(x-\tau)}{\partial(\lambda,\tau)}\Big{\|}_{2}.\]
_For any fixed \(h\in H\), with probability at least \(1-2e^{-t}\)_
\[|R_{W}(h)-\widehat{R}_{W_{n}}(h)|\leq 10n^{-1/2}G(\sqrt{t+m\log(nL)}).\]
Given Theorem 12, one can take a union bound over the hypothesis class \(H\) to derive the following uniform convergence bounds.
**Corollary 13**: _Let \(N(H,\epsilon,\|\cdot\|_{L^{\infty}(\mathcal{Z})})\) be the covering number of \(H\) in the sup-norm which is defined as \(\|h\|_{L^{\infty}(\mathcal{Z})}=\sup_{z\in\mathcal{Z}}|h(z)|\). Then with probability at least \(1-N(H,\epsilon_{n},\|\cdot\|_{L^{\infty}(\mathcal{Z})})e^{-t}\), the following holds for any \(h\in H\): \(|R_{W}(h)-\widehat{R}_{W_{n}}(h)|\leq 30n^{-1/2}G(\sqrt{t+m\log(nL)})\). Here \(\epsilon_{n}=n^{-1/2}G\sqrt{t+m\log(nL)}\)._
The above bound depends on parameters \((\lambda^{*},\tau^{*},G,L)\) which specific to the constraint set \(W\). To instantiate it for any \(W\) one needs to bound these parameters. We note that our generalization guarantees become sub-optimal as \(\Lambda\to 0\). This is because the Lipschitz constant \(L\) could potentially get larger as \(\lambda\) approaches the boundary. Improving these bounds is an interesting future direction.
**Remark 14**: _We note that aforementioned results results follow from relatively stringent assumptions. Exploring the impact of relaxing these assumptions is an interesting direction for future works._
## 7 Experiments
In this section, we demonstrate the generality of proposed RAI methods by studying one of the most well-studied problems in RAI i.e. the case of subpopulation shift. Given a large number of possible \(W\), we acknowledge that this is not a complete analysis, even with respect to the problems that live within the minimax framework. Instead, we aim to display convergence, plug-and-play generality, and superior performance over some seminal baselines of this task. We conduct experiments on both synthetic and real-world datasets. Please refer to Appendix for details on synthetic experiments. We consider a number of responsible AI settings, including subpopulation shift, in the domain-oblivious (DO) setting where we do not know the sub-populations (Hashimoto et al., 2018; Lahoti et al., 2020; Zhai et al., 2021), the domain-aware (DA) setting where we do (Sagawa et al., 2019), and the partially domain-aware (PDA) setting where only some might be known.
**Datasets & Domain Definition.** We use the following datasets: COMPAS (Angwin et al., 2016), CIFAR-10 (original, and with a class imbalanced split (Jin et al., 2021; Qi et al., 2021)) and CIFAR-100. See the Appendix for more details on our datasets. For COMPAS, we consider _race (White vs Other)_ and _biological gender (Male vs Female)_ as our sensitive attributes. This forms four disjoint subgroups defined by these attributes. In the PDA setting, we partition only across the attribute _race_ while training, but still run tests for all four subgroups. On CIFAR-10, class labels define our 10 subpopulations. Similarly as above, for the PDA setting, we make 5 super-groups of two classes each. On CIFAR-100, class labels define our 100 subpopulations. For the PDA setting, we make 20 super-groups, each consisting of five classes.
**Baselines.** We compare our method against the following baselines: (a) Deterministic classifiers trained on empirical risk (ERM) and DRO risks, particularly the quasi-online algorithm for Group DRO (Sagawa et al., 2019) (Online GDRO), and an ITLM-inspired SGD algorithm (Zhai et al., 2021; Shen and Sanghavi, 2018) for \(\chi^{2}\) DRO (SGD (\(\chi^{2}\))) (b) Ensemble models AdaBoost (Schapire, 1999). Note that the purpose of our experiments is to show that we can match baselines for a specific single desideratum (e.g. worst-case sub-population) while allowing for learning models that can solve multiple responsible AI desiderata at the same time, for which we have no existing baselines.
**Proposed Methods.** We focus on Algorithm 2 and refer to FW and Gen-AdaBoost updates as RAI-FW and RAI-GA, respectively. Moreover, our implementations include the following alterations: \(\bullet\) We track the unregularized objective value from Equation 1 for the validation set, and whenever it increases we double the regularization factor \(\eta\), which we find can improve generalization. \(\bullet\) We also use this objective w.r.t the normalized \(Q^{t}_{1}\) to perform a line search for the step size \(\alpha\). For the FW update, our search space is a ball around \(\frac{1}{t}\) at round \(t\), while for GA, we search within \((0,1)\).
**Base Learners & Training.** Training time scales linearly with the number of base learners. Inference, though, can be parallelized if need be. We usually find training on 3-5 learners is good enough on all scenarios explored in the paper. We defer further details of our base learners and hyperparameter choices to the Appendix.
**Constraint sets \(\mathbf{W_{n}}\).** For RAI algorithms, we use the following constraint sets: \({}^{\star}\) Domain Oblivious (DO): We use the \(\chi^{2}\)-DRO constraint set to control for worst-case subpopulations. \({}^{\star}\) Domain Aware (DA): We use the Group DRO constraint set as the domain definitions are known. \({}^{\star}\) Partially Domain-Aware (PDA): We use a novel set \(W_{n}\) which is the intersection over Group DRO constraints over the known domains and \(\chi^{2}\) constraints to control for unknown group performance. For baselines, we use AdaBoost and SGD(\(\chi^{2}\)) for the DO setting. Online GDRO serves as our baseline for both DA and PDA settings, where the algorithm uses whatever domain definitions are available.
**Results and Discussion.** We run our methods and baselines under the settings described above and report the results in Table 2. As such, we can make the following observations:
1. RAI-FW and RAI-GA methods significantly improve the worst-case performance with only a few base learners across all datasets in all three settings, while maintaining average case performance. Moreover, For seemingly harder tasks i.e. a large gap between average and worst-case performance, the algorithms are able to improve significantly over the baselines. For example, we observe a 5% improvement in performance in the case of CIFAR-100.
2. The _plug-and-play_ framework allows for several different \(W_{n}\) to enhance various _responsible AI_ qualities at once. We demonstrate this with the partial domain aware setting (PDA), where the performance lead widens, indicating that RAI is able to jointly optimize effectively for both known and unknown subpopulations while Online GDRO suffers from some of the group information being unknown. In practice, one can construct many more novel sets \(W_{n}\).
3. Although bigger (complex) models exhibit stronger performance than RAI ensembles, there are several caveats to this observation. Firstly, these models are \(\sim\)10-15 times larger than our base models. This limits their use w.r.t both training & inference compute required. However, RAI ensembles utilize a small number of much smaller models which can be individually trained quite easily. Even with these large models as base learners, constructing ensembles exhibits a performance boost, indicating that our framework is able to "boost" models of varying complexities.
## 8 Conclusion
Under the umbrella of "responsible AI", an emerging line of work has attempted to formalize desiderata ranging over ethics, fairness, robustness, and safety, among others. Many of these settings (Table 1) can be written as min-max problems involving optimizing some worst-case loss under a set of predefined distributions. For all the problems that can be framed as above, we introduce and study a general framework, which we refer to as Responsible AI (RAI) games. Our framework extends to classical boosting scenarios, offering boosting-based algorithms for RAI games alongside proven convergence guarantees. We propose practical algorithms to solve these games, as well as statistical analyses of solutions of these games. We find that RAI can guarantee multiple responsible AI aspects under appropriate choices of uncertainty sets.
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{Setting} & \multirow{2}{*}{Algorithm} & \multicolumn{2}{c}{COMPAS} & \multicolumn{2}{c}{CIFAR-10 (Imbalanced)} & \multicolumn{2}{c}{CIFAR10} & \multicolumn{2}{c}{CIFAR100} \\ \cline{3-10} & & Average & Wurss Group & Average & Worst Class & Average & Worst Class & Average & Worst Class \\ \hline \multirow{2}{*}{DO} & ERM & 31.3 \(\pm\)0.2 & 31.7 \(\pm\)0.1 & 12.1 \(\pm\)0.3 & 30.4 \(\pm\)0.2 & 8.3 \(\pm\)0.2 & 21.3 \(\pm\)0.5 & 25.2 \(\pm\)0.2 & 64.0 \(\pm\)0.7 \\ & RAI-GA (\(\chi^{2}\)) & 31.3 \(\pm\)0.2 & **31.2** \(\pm\)0.2 & 11.7 \(\pm\)0.4 & **29.9** \(\pm\)0.3 & 8.2 \(\pm\)0.1 & 19.0 \(\pm\)0.1 & 25.6 \(\pm\)0.4 & **56.8** \(\pm\)0.8 \\ & RAI-FW (\(\chi^{2}\)) & 31.2 \(\pm\)0.1 & 31.4 \(\pm\)0.3 & 11.9 \(\pm\)0.1 & 29.1 \(\pm\)0.2 & 8.0 \(\pm\)0.3 & **15.4** \(\pm\)0.4 & 25.4 \(\pm\)0.2 & 58.0 \(\pm\)1.1 \\ \hline \multirow{6}{*}{DO} & ERM & 32.1 \(\pm\)0.3 & 32.6 \(\pm\)0.4 & 14.2 \(\pm\)0.1 & 33.6 \(\pm\)0.3 & 11.4 \(\pm\)0.4 & 27.0 \(\pm\)0.1 & 27.1 \(\pm\)0.3 & 66.0 \(\pm\)1.1 \\ & AdaBoost & 31.8 \(\pm\)0.4 & 32.6 \(\pm\)0.3 & 15.2 \(\pm\)0.4 & 40.6 \(\pm\)0.2 & 12.0 \(\pm\)0.1 & 28.7 \(\pm\)0.3 & 28.1 \(\pm\)0.7 & 72.2 \(\pm\)1.2 \\ & SGD (\(\chi^{2}\)) & 32.0 \(\pm\)0.2 & 33.7 \(\pm\)0.2 & 13.3 \(\pm\)0.3 & 31.7 \(\pm\)0.4 & 11.3 \(\pm\)0.3 & 24.7 \(\pm\)0.1 & 27.4 \(\pm\)0.1 & 65.9 \(\pm\)1.2 \\ & RAI-GA (\(\chi^{2}\)) & 31.5 \(\pm\)0.2 & 33.2 \(\pm\)0.3 & 14.0 \(\pm\)0.1 & 32.2 \(\pm\)0.2 & 10.8 \(\pm\)0.4 & 25.0 \(\pm\)0.2 & 27.4 \(\pm\)0.4 & 65.0 \(\pm\)0.8 \\ & RAI-FW (\(\chi^{2}\)) & 31.6 \(\pm\)0.1 & **32.5** \(\pm\)0.3 & 13.9 \(\pm\)0.4 & 32.6 \(\pm\)0.3 & 10.9 \(\pm\)0.4 & **23.4** \(\pm\)0.2 & 27.5 \(\pm\)0.4 & **63.8** \(\pm\)0.6 \\ \hline \multirow{2}{*}{DA} & Online GDRO & 31.7 \(\pm\)0.2 & 32.2 \(\pm\)0.3 & 13.1 \(\pm\)0.2 & 26.6 \(\pm\)0.2 & 11.2 \(\pm\)0.1 & 21.7 \(\pm\)0.3 & 27.3 \(\pm\)0.1 & 57.0 \(\pm\)0.5 \\ & RAI-GA (Group) & 32.0 \(\pm\)0.1 & 32.7 \(\pm\)0.1 & 13.0 \(\pm\)0.3 & 27.3 \(\pm\)0.4 & 11.5 \(\pm\)0.1 & 22.4 \(\pm\)0.2 & 27.4 \(\pm\)0.2 & 56.6 \(\pm\)1.1 \\ & RAI-FW (Group) & 32.1 \(\pm\)0.2 & 32.3 \(\pm\)0.2 & **26.0** \(\pm\)0.1 & 11.4 \(\pm\)0.3 & **20.3** \(\pm\)0.1 & 27.9 \(\pm\)0.2 & **52.9** \(\pm\)0.9 \\ \hline \multirow{2}{*}{PDA} & Online GDRO & 31.5 \(\pm\)0.1 & 32.7 \(\pm\)0.2 & 13.4 \(\pm\)0.1 & 32.2 \(\pm\)0.2 & 11.3 \(\pm\)0.2 & 25.2 \(\pm\)0.1 & 27.7 \(\pm\)0.2 & 64.0 \(\pm\)0.8 \\ & RAI-GA (Group \(\cap\)\(\chi^{2}\)) & 31.4 \(\pm\)0.4 & 32.9 \(\pm\)0.2 & 13.0 \(\pm\)0.3 & 30.1 \(\pm\)0.1 & 10.8 \(\pm\)0.2 & **23.7** \(\pm\)0.2 & 27.5 \(\pm\)0.1 & 62.5 \(\pm\)0.6 \\ & RAI-FW (Group \(\cap\)\(\chi^{2}\)) & 31.8 \(\pm\)0.2 & **32.3** \(\pm\)0.1 & 13.5 \(\pm\)0.3 & **29.4** \(\pm\)0.3 & 11.2 \(\pm\)0.4 & 24.0 \(\pm\)0.2 & 27.9 \(\pm\)0.3 & **58.9** \(\pm\)0.7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: (Table 1 in the paper) Mean and worst-case expected loss for baselines, RAI-GA and RAI-FW. (Complex) indicates the use of larger models. Constraint sets \(W_{n}\) are indicated in (.). Each experiment is carried out over three random seeds and confidence intervals are reported.
## Acknowledgements
We acknowledge the support of DARPA via HR00112020006, and NSF via IIS-1909816.
|
2306.14736 | Investigation of the Boron removal effect induced by 5.5 MeV electrons
on highly doped EPI- and Cz-silicon | This study focuses on the properties of the B$_\text{i}$O$_\text{i}$
(interstitial Boron~-~interstitial Oxygen) and C$_\text{i}$O$_\text{i}$
(interstitial Carbon~-~interstitial Oxygen) defect complexes by
\SI{5.5}{\mega\electronvolt} electrons in low resistivity silicon. Two
different types of diodes manufactured on p-type epitaxial and Czochralski
silicon with a resistivity of about 10~$\Omega\cdot$cm were irradiated with
fluence values between \SI{1e15}{\per\square\centi\meter} and
\SI{6e15}{\per\square\centi\meter}. Such diodes cannot be fully depleted and
thus the accurate evaluation of defect concentrations and properties
(activation energy, capture cross-section, concentration) from Thermally
Stimulated Currents (TSC) experiments alone is not possible. In this study we
demonstrate that by performing Thermally Stimulated Capacitance (TS-Cap)
experiments in similar conditions to TSC measurements and developing
theoretical models for simulating both types of B$_\text{i}$O$_\text{i}$
signals generated in TSC and TS-Cap measurements, accurate evaluations can be
performed. The changes of the position-dependent electric field, the effective
space charge density $N_\text{eff}$ profile as well as the occupation of the
B$_\text{i}$O$_\text{i}$ defect during the electric field dependent electron
emission, are simulated as a function of temperature. The macroscopic
properties (leakage current and $N_\text{eff}$) extracted from current-voltage
and capacitance-voltage measurements at \SI{20}{\celsius} are also presented
and discussed | Chuan Liao, Eckhart Fretwurst, Erika Garutti, Joern Schwandt, Leonid Makarenko, Ioana Pintilie, Lucian Dragos Filip, Anja Himmerlich, Michael Moll, Yana Gurimskaya, Zheng Li | 2023-06-26T14:39:53Z | http://arxiv.org/abs/2306.14736v1 | Investigation of the Boron removal effect induced by 5.5 MeV electrons on highly doped EPI- and Cz-silicon
###### Abstract
This study focuses on the properties of the \(\mathrm{B_{i}O_{i}}\) (interstitial Boron - interstitial Oxygen) and \(\mathrm{C_{i}O_{i}}\) (interstitial Carbon - interstitial Oxygen) defect complexes by 5.5 MeV electrons in low resistivity silicon. Two different types of diodes manufactured on p-type epitaxial and Czochralski silicon with a resistivity of about 10 \(\Omega\)-cm were irradiated with fluence values between \(1\times 10^{15}\) cm\({}^{-2}\) and \(6\times 10^{15}\) cm\({}^{-2}\). Such diodes cannot be fully depleted and thus the accurate evaluation of defect concentrations and properties (activation energy, capture cross-section, concentration) from Thermally Stimulated Currents (TSC) experiments alone is not possible. In this study we demonstrate that by performing Thermally Stimulated Capacitance (TS-Cap) experiments in similar conditions to TSC measurements and developing theoretical models for simulating both types of \(\mathrm{B_{i}O_{i}}\) signals generated in TSC and TS-Cap measurements, accurate evaluations can be performed. The changes of the position-dependent electric field, the effective space charge density \(N_{\mathrm{eff}}\) profile as well as the occupation of the \(\mathrm{B_{i}O_{i}}\) defect during the electric field dependent electron emission, are simulated as a function of temperature. The macroscopic properties (leakage current and \(N_{\mathrm{eff}}\)) extracted from current-voltage and capacitance-voltage measurements at 20 \({}^{\circ}\)C are also presented and discussed.
keywords: Silicon detector; Radiation damage; \(\mathrm{B_{i}O_{i}}\); \(\mathrm{C_{i}O_{i}}\); electron irradiation; TSC; TS-Cap; acceptor removal +
Footnote †: journal: NIMA
## 1 Introduction
In order to cope with extraordinary high particle rates up to 200 p-p collisions per bunch crossing in High Luminosity Large Hadron Collider (HL-LHC) experiments, new types of silicon sensors were developed e.g. Low Gain Avalanche Detectors (LGADs) [1; 2; 3], and High Voltage CMOS devices (HV-CMOS) for inner tracking detectors [4; 5; 6; 7; 8]. Both types of sensors as well as the new pixel and strip devices will be manufactured on boron doped (\(p\)-type) silicon. The degradation of the performance of these sensors is due to the expected high radiation field. For instance, exposing the LGADs to a particle radiation field leads to the reduction of the internal gain value with increasing fluence. This degradation is caused by a deactivation of the active boron in the highly doped \(p\)-type gain layer (about \(5\times 10^{16}\) cm\({}^{-3}\)), which leads to a reduction of the space charge and consequently, a lowering of the electric field followed by a decrease in charge multiplication in this layer.
In general, the deactivation of the boron dopant is a process called boron removal. A possible way to reduce boron removal is a co-implantation of carbon into the gain layer [2]. The assumed mechanism behind this effect is the competition between the displacement of substitutional boron (\(\mathrm{B_{s}}\)) and substitutional carbon (\(\mathrm{C_{s}}\)) by primary silicon interstitials (\(\mathrm{Si_{I}}\)) into interstitial positions (\(\mathrm{B_{i}}\)) and (\(\mathrm{C_{i}}\)), respectively. Both interstitial atoms are mobile at room temperature and can react with different impurities, ending up e.g. in the formation of \(\mathrm{B_{i}O_{i}}\) or \(\mathrm{C_{i}O_{i}}\) defects [9; 10; 11; 12; 13; 14]. Although both defects have donor states in the bandgap of silicon, the \(\mathrm{B_{i}O_{i}}\) act as a trap for electrons and the \(\mathrm{C_{i}O_{i}}\) as hole trap. At room temperature (RT) the \(\mathrm{B_{i}O_{i}}\) is positively charged and its concentration affects the effective space charge density (\(N_{\mathrm{eff}}\)) while \(\mathrm{C_{i}O_{i}}\) is in a neutral charge state with no influence on \(N_{\mathrm{eff}}\). The \(\mathrm{C_{i}O_{i}}\) defect has an energy level in the lower half of the bandgap of silicon, with an activation energy of 0.36 eV and temperature dependent capture cross sections for holes and electrons [14]. On the other hand, the \(\mathrm{B_{i}O_{i}}\) defect is a coulombic center having an energy level in the upper half of the silicon bandgap with activation energy depending on the electric field, experimentally determined to be between 0.24 and 0.26 eV [13], and independent on temperature capture cross sections of \(1\times 10^{-14}\) cm\({}^{2}\) for electrons and \(1\times 10^{-20}\) cm\({}^{2}\) for holes [9; 12].
The reactions with these defects are still under investigation and are of high relevance for improving the radiation hardness of LGADs. In order to get more information about the introduction of both defects and their interplay as well as a quantitative
determination of the boron removal rate, the main goal of this work is to accurately characterize the radiation-induced defect complexes B\({}_{\rm i}\)O\({}_{\rm i}\) and C\({}_{\rm i}\)O\({}_{\rm i}\), and evaluate the boron removal rate in highly boron-doped silicon diodes with different carbon concentrations.
The investigated n\({}^{+}\)-p diodes were manufactured on 10 \(\Omega\)-cm p-type epitaxial silicon (EPI) and Czochralski material (Cz), and exposed to high fluences of 5.5 MeV electrons in the range of (1-6) \(\times\) 10\({}^{15}\) cm\({}^{-2}\). The radiation-induced defect complexes B\({}_{\rm i}\)O\({}_{\rm i}\) and C\({}_{\rm i}\)O\({}_{\rm i}\) were investigated by means of the Thermally Stimulated Current technique (TSC). One problem in the evaluation of the defect concentrations from the measured TSC spectra arises from the fact that the irradiated low resistivity diodes can only be partially depleted during the temperature scan, due to the limit of the maximal reverse voltage which can be applied. That means, that the depleted volume is beforehand not known for the temperature range of the charge emission of the defects. We show that this problem can be overcome if in addition to TSC experiments Thermally Stimulated Capacitance (TS-Cap) method is employed. This method allows the determination of the depleted volume at any temperature.
The paper is structured as follows. In section 2 the experimental details about the used diodes manufactured on \(p\)-type epitaxial (EPI)- and Czochralski (Cz)-silicon, the irradiation with 5.5 MeV electrons and the methods for the investigation of the macroscopic and microscopic properties of the devices are presented. In section 3 we provide the results of the current-voltage and capacitance-voltage measurements. Next, section 4 is dedicated to TSC and TS-Cap experiments, data simulation and analyses, with a focus on the Boron-Oxygen (B\({}_{\rm i}\)O\({}_{\rm i}\)) defect complex and its correlation with the boron removal process.
## 2 Experimental Details
All the investigated diodes are produced by the company - "Transistors" that belongs to Integral [15]. Five sets of \(n^{+}\)-\(p\) silicon diodes with a deep diffusion junction of 7.2 \(\upmu\)m depth were investigated [16]. Three of them (EPI-3, EPI-7, EPI-9) are 50 \(\upmu\)m thick epitaxial layers grown on a highly boron-doped Cz substrate of 525 \(\upmu\)m thickness and resistivity of 0.006 \(\Omega\)-cm. Those three sets have the same boron content of 1.1 \(\times\) 10\({}^{15}\) cm\({}^{-3}\) in the epitaxial layer, corresponding to a resistivity of about 10 \(\Omega\)-cm. The other two diodes (Cz-3, Cz-7) were processed on \(p\)-type Cz silicon with about the same resistivity of 10 \(\Omega\)-cm and a thickness of about 400 \(\upmu\)m. Except for boron the main impurities are oxygen and carbon. According to [10] the Cz and EPI diodes have similar oxygen content, of \(\sim\)1.5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), while the carbon content differs, being in the range of 2-3 \(\times\) 10\({}^{16}\) cm\({}^{-3}\) and 1.5-2 \(\times\) 10\({}^{15}\) cm\({}^{-3}\) in Cz and EPI, respectively. All diodes have been manufactured without a guard ring structure [16]. The distance between the pad boundary and the chip edge is roughly 100 \(\mu\)m for all diodes.
The irradiation with 5.5 MeV electrons was performed at room temperature using the accelerator facility at Minsk. Since the range of 5.5 MeV electrons is much larger than the thickness of the EPI- and the Cz-silicon the distribution of the radiation induced defects is uniform throughout the whole bulk of the material. More detailed information can be found in [17]. The achieved fluence values were in the range of (1-6) \(\times\) 10\({}^{15}\) cm\({}^{-2}\). For the calculation of the corresponding 1 MeV neutron equivalent values, a hardness factor of 0.0398 was used according to the Non-Ionizing Energy Loss (NIEL) data of I. Jun et al. [18]. More detailed information of the investigated diodes is summarized in Table 1.
The macroscopic device performance of the investigated diodes was measured by means of current-voltage (\(I\)-\(V\)) and capacitance-voltage (\(C\)-\(V\)) characteristics. The radiation induced changes in the effective space charge density \(N_{\rm eff}\) and the full depletion voltage \(V_{\rm fd}\) were determined from \(C\)-\(V\) measurements at 10 kHz. The capacitances were measured with a LCR meter in parallel mode.
For the characterization of the radiation induced electrically active defects, the TSC and TS-Cap methods were used [19; 20; 21; 22; 23]. The experimental setup consists of a closed cycle helium cryostat Model SRDK-205 (Sumitomo Heavy Industries, Ltd, Japan) equipped with a temperature controller Model 340 (Lake Shore, US) and a Keithly 6517A electrometer with a voltage source. For the TS-Cap a LCR meter 4263B from Hewlett Packard is used. The experimental procedure consists of cooling down the sample under zero bias to low temperatures (typically 10 K) where filling of the defects is performed for 30 s either by forward biasing of the diode (electron and hole injection by injecting 1 mA forward current) or 0 V filling (only majority carrier (hole) injection). Then, the diode is reverse biased and a temperature scan is then recorded by measuring the diode current (TSC) or capacitance (TS-Cap) during heating up the device with a constant rate of \(\beta\) = 0.183 K s\({}^{-1}\)[22]. It should be mentioned here that the range of the reverse bias was chosen that way that the current density was below the soft breakdown. For example for EPI-3 and Cz-3 \(V_{\rm bias}<\) 100 V. Isothermal annealing experiments were performed up to 120 min at a temperature of 80 \({}^{\circ}\)C for all irradiated diodes, with the subsequent evaluation of the macroscopic and microscopic properties.
## 3 \(I\)-\(V\) and \(C\)-\(V\) characteristics
In this section, the measured \(I\)-\(V\) and \(C\)-\(V\) characteristics of the irradiated sensors are presented and discussed. As an example, in Fig. 1(a) the \(I\)-\(V\) curves of all EPI- and Cz-diodes irradiated with different fluences are shown. As it can be seen, for all diodes, except EPI-9 irradiated to \(6\times 10^{15}\) cm\({}^{-2}\), a so-called soft breakdown occurs at a certain bias voltage. Such behaviour may have different reasons, e.g. the diodes have no guard ring limiting the current to the active pad size and excluding contributions from the outer surface region or edge and/or the high electric field near to the \(n^{+}\)-\(p\) junction triggers trap assisted Poole-Frenkel or tunnelling effects [24; 25]. Nevertheless, determining the depleted depth \(w(V)\) from \(C\)-\(V\) characteristics (Fig. 1(b) and (c)) and assuming that the active area A is given by the \(n^{+}\)-pad size, the depleted volume \(V_{\rm vol}=A\cdot w(V)\) has been calculated and used for estimating the leakage current density \(j_{d}=I/V_{\rm vol}\) as a function of the applied bias voltage shown in Fig. 1(d). One would expect flat curves if edge effects and soft breakdown could be neglected. However, a soft
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline Label & EPI-3 & EPI-7 & EPI-9 & Cz-3 & Cz-7 \\ \hline Initial doping concentration \(N_{\text{eff, 0 }}\)(cm\({}^{-3}\)) & \(1.1\times 10^{15}\) & \(1.1\times 10^{15}\) & \(1.1\times 10^{15}\) & \(1.05\times 10^{15}\) \\ Initial resistivity (\(\Omega\cdot\text{cm}\)) & \(\approx\)10 & \(\approx\)10 & \(\approx\)10 & \(\approx\)10 \\ Electron fluence \(\Phi_{\text{e}}\) (cm\({}^{-2}\)) & \(1\times 10^{15}\) & \(4\times 10^{15}\) & \(6\times 10^{15}\) & \(1\times 10^{15}\) & \(4\times 10^{15}\) \\ Fluence value \(\Phi_{\text{eq}}\) (cm\({}^{-2}\))\({}^{*}\) & \(3.98\times 10^{13}\) & \(1.59\times 10^{14}\) & \(2.39\times 10^{14}\) & \(3.98\times 10^{13}\) & \(1.59\times 10^{14}\) \\ Area \(A\) (cm\({}^{2}\)) & 0.0621 & 0.0621 & 0.0621 & 0.029 & 0.029 \\ Thickness d (\(\mu\)m) & 50 & 50 & 50 & 400 & 400 \\ Carbon concentration [\(C_{\text{s}}\)] (cm\({}^{-3}\)) & \(2\times 10^{15}\) & \(2\times 10^{15}\) & \(2\times 10^{15}\) & \(3\times 10^{16}\) & \(3\times 10^{16}\) \\ Oxygen concentration [\(O_{\text{I}}\)] (cm\({}^{-3}\)) & \(1.5\times 10^{17}\) & \(1.5\times 10^{17}\) & \(1.5\times 10^{17}\) & \(1.5\times 10^{17}\) & \(1.5\times 10^{17}\) \\ \hline \hline \end{tabular}
* 1 MeV neutron equivalent fluence
\end{table}
Table 1: Device information
Figure 1: (a) Current-voltage characteristics of the 10 \(\Omega\)-cm diodes, irradiated with 5.5 MeV electrons \(\Phi_{\text{e}}=1,\ 4,\ 6\ \times\ 10^{15}\) cm\({}^{-2}\). Measurements conditions: \(T=20\)\({}^{\circ}\)C, humidity \(\leq\) 10%. (b) \(C\)–\(V\) characteristics and (c) Depleted Depth vs \(\sqrt{V}\) of the same 10 \(\Omega\)-cm diodes presented (a). Measurements conditions: \(V_{AC}\) = 0.5 V. Frequency = 10 kHz. (d) Density of leakage current (\(j_{d}\)) versus bias voltage \(V\).
breakdown behaviour is observed in all diodes, except EPI-9. According to Fig. 1(c), \(w\) is plotted as a function of \(\sqrt{V}\), in the \(\sqrt{V}\) range of (6-10) V\({}^{0.5}\) (\(V\) between 36 V and 100 V), where the edge effects do not contribute to the rise of the current, \(w\) is proportional to \(\sqrt{V}\) (except for the diode Cz-7), a typical for the bulk generation current. An average current density \(J_{d}\) was taken in the voltage range from 50 V to 100 V, where, a small linear increase of the current is recorded, due to the extension of the electric field in the lateral area of the electrodes in the absence of grounded guard rings.
The average values \(J_{d}\) of the current densities are plotted as function of the electron fluence \(\Phi_{\mathrm{e}}\) in Fig. 2, showing an approximately linear increase. The current related damage parameter \(\alpha\), given by:
\[\alpha=\frac{\Delta J_{d}}{\Delta\Phi_{\mathrm{e}}} \tag{1}\]
had been evaluated to \(\alpha=(3.2\pm 0.2)\times 10^{-19}\) A cm\({}^{-1}\). Such a small \(\alpha\) value was also observed in previous experiments on 5.5 MeV electron damage induced in \(n\)-type silicon [26]. Accounting for the hardness factor of 0.0398 the current related damage parameter becomes \(\alpha=0.8\times 10^{-17}\) A cm\({}^{-1}\), being smaller compared to the value of \(\alpha=4\times 10^{-17}\) A cm\({}^{-1}\) determined for hadron irradiation and an annealing of 80 min at 60 \({}^{\circ}\)C (see e.g. [22; 27; 28]).
The \(C\)-\(V\) characteristics were measured for 4 different frequencies (230 Hz, 445 Hz, 1 kHz and 10 kHz). A slight frequency dependence is observed and the related explanation can be found in reference [29; 30]. The relative deviations measured at 200 V between the values measured at frequencies of 230 Hz and 10 kHz are below 4% for all the samples.
The effective space charge density profile \(N_{\mathrm{eff}}(w(V))\) and the depletion depth \(w(V)\) were extracted from the 10 kHz \(C\)-\(V\) curves (see Fig. 1(b)) according to Eq. (2) and Eq. (3):
\[N_{\mathrm{eff}}(V) = \frac{2}{\epsilon_{0}\epsilon_{A}A^{2}q_{0}\,d(1/C^{2})/dV} \tag{2}\] \[w(V) = \frac{\epsilon_{0}\epsilon_{A}}{C(V)} \tag{3}\]
Where \(C\) is the measured capacitance, \(\epsilon_{0}\) is the permittivity of vacuum, \(\epsilon_{t}\) the relative permittivity of silicon (11.9), \(q_{0}\) is the elementary charge, \(A\) is the active pad area. Fig. 3 presents the calculated \(N_{\mathrm{eff}}(w(V))\) profiles for the EPI- and Cz-diodes, irradiated with different fluences.
With increasing fluence, the profiles of \(N_{\mathrm{eff}}\) are shifting to lower values, a fact that is expected mainly due to the deactivation of the initial boron concentration caused by irradiation, the so-called boron removal effect. Of course, some hole traps e.g. H(140K) and H(152K) will also affect the space charge density \(N_{\mathrm{eff}}\) but their concentrations are much smaller compared to the concentration of the B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) defect ([H140K + H152K] \(\approx 2.5\times 10^{13}\) cm\({}^{-3}\) and [B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\)] \(\approx 4.5\times 10^{14}\) cm\({}^{-3}\) in EPI-9).
The isothermal annealing behaviour of the generation current density \(J_{d}\) at 80 \({}^{\circ}\)C is depicted in Fig. 4. The observed changes with annealing time are much smaller compared to the ones observed for a 23 GeV proton irradiated 10 \(\Omega\)-cm EPI-diode, which are also included in Fig. 4[13].
Due to the significant affection by lateral effect especially in Cz diodes, it is deserved to mention the error in the extracted \(N_{\mathrm{eff}}\) and \(j_{d}\). In this work, the lateral effect was estimated by the difference on \(j_{d}\) as shown in Fig. 1(d), under the assumption that the lateral effect in EPI diodes can be neglected. Thus, for applied bias voltages of 100 V and 200 V, the error will rise from 0.7% up to 36% for Cz-3 and from 5% to 49% for Cz-7, respectively. The \(V_{\mathrm{bias}}=100\) V corresponds to depleted depths of about 11 \(\upmu\)m and 14 \(\upmu\)m for the diode Cz-3 and Cz-7, respectively. Only the \(J_{d}\) values from EPI-diodes were used to extract \(\alpha\). Thus the error for the \(J_{d}\) was estimated from the bias interval for averaging and resulted in a value of 3%. This introduces an uncertainty of 5% in the obtained \(\alpha\) value.
Figure 3: \(N_{\mathrm{eff}}\) profile of the diodes irradiated with different fluences. The data were evaluated from \(C\)–\(V\) measurements (Fig. 1(b)) at room temperature by using Eq. (2) and Eq. (3).
Figure 2: Average current density \(J_{d}\) versus. electron fluence (details see text).
## 4 Results from TSC and TS-Cap measurements
The Thermally Stimulated measurement techniques were used to investigate the defect complexes induced by irradiation with 5.5 MeV electrons, especially the boron-oxygen (B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\)) and the carbon-oxygen (C\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\)) defects in the EPI- and Cz-materials. Figure 5(a) shows the TSC spectra measured on all diodes (EPI- and Cz-samples) irradiated with different fluences after injecting both electrons and holes (1 mA forward injection) at 10 K. Figure 5(b) presents the spectra of the same diodes after filling the traps only with holes by cooling the sample to 10 K under 0 V. As can be seen here, the dominant TSC signal occurs at about 150 K and is attributed to the carbon-oxygen (C\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\)) defect complex. The C\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) signal height in Cz-diodes is much larger compared to the EPI-diodes at the same fluence, due to the higher concentration of carbon in Cz silicon (see Table 1). While Fig. 5(b) shows only the TSC peaks corresponding to hole traps, Fig. 5(a) reveals also the ones corresponding to electron traps which can be filled by a forward current injection. As it can be seen in Fig. 5(a), there is a dominant peak in the temperature range between 90 K and 100 K that is not even traced in the spectra depicted in Fig. 5(b) corresponding to hole traps only. This dominant peak corresponds to an electron trap, increases with increasing fluence, shows a dependence on the electric field in the sensor, the so-called Poole-Frenkel effect [24; 31; 32] as well as a dependence on the impurity content (boron, oxygen, carbon) in the material [9; 10; 11; 33] and thus, it is attributed to the B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) defect complex. Also, theoretical calculations support this identification [33]. Because the diodes cannot always be fully depleted during the temperature scan and for a better comparison of the different spectra, the measured currents shown in Fig. 5(a) and 5(b) had been normalized to the active depleted volume (\(V_{\mathrm{vol}}(V,T)=A\cdot w(V,T)\)). The \(w(V,T)\) values were extracted from the corresponding TS-Cap measurements.
The TS-Cap data are presented in Fig. 5(c) and Fig. 5(d) corresponding to the TSC spectra shown in Fig. 5(a) and Fig. 5(b), respectively. For the case of forward current injection the TS-Cap measurements show a drop of the capacitance values in the temperature range of the B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) emission. This correlates with the change of the B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) defect charge state, being neutral when occupied with an electron at temperatures before the emission starts and positively charged after the electron is thermally emitted. This leads to a change of the space charge density to a less negative value, corresponding to an increase in the depleted width \(w(V,T)\) and consequently to the drop of the capacitance mentioned above. On the other hand, the increase of the capacitance in the range of the C\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) emission (Fig. 5(d)) is due to the change of the charge state of the C\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) from positive (occupied by holes) to the neutral state after the holes emission. Thus, the space charge density changes from less negative to more negative leading to a decrease in the depleted width \(w(V,T)\) and an increase of the capacitance at the given bias voltage. In both cases, the defect concentration can be determined despite the fact that the detector is not fully depleted, as the TS-Cap data can be used to determine the depletion depth at any temperature (see section 4.1.). Further, it is known that the B\({}_{\mathrm{i}}\)O\({}_{\mathrm{i}}\) is a coulombic center [9; 33] and thus the electron emission from this defect is governed by the Poole-Frenkel effect, manifesting in a shift of the TSC peak position to lower temperatures with increasing bias voltage. A related shift is then also observed in the TS-Cap curves (see e.g. Fig. 6).
It should be noted that the different values of \(V_{\mathrm{bias}}\) used for different samples were chosen according to the specific characteristics of each diode. Because the aim of the study is to obtain the concentration profiles for defects distributed in the bulk of the diodes, measurements with large \(V_{\mathrm{bias}}\) are preferred in order to scan deep in the bulk of the samples. However, the bias has to be limited to values avoiding the breakdown of the samples. Thus, while the EPI-9 diode withstands a \(V_{\mathrm{bias}}\) =300 V that fully depletes the sample over the entire temperature scan, smaller biases could be applied on the other diodes Thus, the maximum \(V_{\mathrm{bias}}\) that could be safely applied were of 200 V on EPI-7, of 100 V on Cz-7 and of 20 V for Cz-3 and EPI-3. For larger bias values significant increase in the leakage current and dielectric losses at low temperatures were observed.
A quantitative evaluation of defect concentrations from TSC spectra of not fully depleted diodes is only possible if the changes of the depleted depth in the corresponding temperature ranges are known. This issue will be discussed in the following section.
### Evaluation of concentrations in case of partially depleted sensors
The TSC method and evaluation of defect properties are described in detail in numerous publications [19; 20; 21; 22; 23]. In our case of not fully depleted devices and the traps homogeneously distributed in the bulk, the current for emission from an isolated
Figure 4: \(J_{d}\) versus. annealing times at 80 \({}^{o}\)C. The data for 23 GeV proton irradiation correspond to a 10 \(\Omega\)-cm resistivity EPI-diodes with A = 0.06927 cm\({}^{2}\), and have similar \(N_{\mathrm{eff}}\) and \(d\) to electron irradiated EPI-diodes.
Figure 5: (a) TSC Spectra after trap filling by forward current injection and (b) after filling with majority carriers (holes). Both types of spectra are measured on EPI-(EPI-3, 7, 9) and Cz-diodes (Cz-3, 7) after irradiation with 5.5 MeV electrons. The applied bias voltages are indicated in the legends and each diode current is normalized to their individual depleted volume (normalization factor \(1/(A\cdot\psi)\)), A = active pad area, \(w(V,T)\) = depleted width. (c) and (d) are the TS-Cap measurements corresponding to figures (a) and (b), respectively. The capacitance values are normalized to the pad area of each diode.
electron trap \(I^{e}_{TSC}(T)\) with the concentration \(n_{t}(T_{0})\), is:
\[I^{e}_{TSC}(T) = q_{0}An_{t}(T_{0})\int_{0}^{w(T)}\frac{x}{w(T)}\epsilon_{n}(T,x)f \left(T,x\right)dx \tag{4}\] \[e_{n} = \sigma_{n}\cdot v_{th,n}\cdot N_{C}\cdot\exp\left(-\frac{E_{a}}{k _{B}T}\right)\] (5) \[e_{p} = \sigma_{p}\cdot v_{th,p}\cdot N_{V}\cdot\exp\left(-\frac{E^{ \prime}_{a}}{k_{B}T}\right) \tag{6}\]
\[f(T) = \exp\left(-\frac{1}{\beta}\int_{T_{0}}^{T}\left(e_{n}(T^{\prime})+ e_{p}(T^{\prime})\right)dT^{\prime}\right) \tag{7}\]
where \(T\) is the measured temperature, \(w(T)\) the depleted depth at temperature \(T\), \(x\) is the coordinate of the depth in the depleted region, \(e_{n}\) and \(e_{p}\) are the emission rates for electrons and holes, respectively, \(N_{C}\) and \(N_{V}\) are the density of states in the conduction band and valence band, respectively. The activation energy for electrons is \(E_{a}=E_{C}-E_{t}\) and for holes \(E^{\prime}_{a}=E_{t}-E_{V}\), where \(E_{t}\) is the energy level of the electron traps and \(E_{C,V}\) the conduction and valence band edge, respectively. \(\sigma_{n,p}\) is the cross section for electrons and holes, \(v_{th,n,p}\) is the thermal velocity for electrons and holes. \(k_{B}\) is the Boltzmann constant, \(f(T)\) describes the fraction of the defects occupied by electrons at temperature \(T\), \(\beta\) is the heating rate and \(n_{t}(T_{0})\) is the density of defects that are filled with electrons at \(T_{0}\). The \(N_{C}\), \(N_{V}\), \(v_{th,n,p}\) values were taken from [34] (\(N_{C,V}=2.540\,933\times 10^{19}\cdot\left(\frac{m_{C,V}^{2}}{m}\right)^{3/2} \left(\frac{T}{30}\right)^{3/2}\)). Eq. (4) defines the total current which accounts for the conduction and the displacement currents [19]. When \(f(T)\) and \(e_{n}(T)\) are not position dependent the Eq. (4) can be simplified to:
\[I^{e}_{TSC}(T) = \frac{1}{2}\cdot q_{0}\cdot A\cdot w(T)\cdot e_{n}(T)\cdot n_{t} (T_{0})\cdot f\left(T\right) \tag{8}\]
In the investigated \(p\)-type diodes, the B\({}_{i}\)O\({}_{i}\) defect, on which this study is focusing, is detected in a TSC experiment only if electrons can be injected at low temperature. This is done by forward biasing the diodes at 10 K injecting both electrons and holes. According to [12] the capture cross section for holes of the B\({}_{i}\)O\({}_{i}\) defect is neglectable compared with the capture cross section of electrons and thus, \(n_{t,0}\) is equal to the defect concentration \(N_{t}\), and \(e_{p}\) can be neglected. Thus, the B\({}_{i}\)O\({}_{i}\) defect concentration can be determined by integrating the TSC corresponding signal after filling with forward bias given by Eq. (8) and considering the depleted volume:
\[N_{t}=\frac{2}{\beta q_{0}}\cdot\int_{T_{s}}^{T_{s}}\frac{I^{e}_{TSC}(T)}{A \cdot w(T)}\,dT=\frac{2}{\beta q_{0}}\cdot\int_{T_{s}}^{T_{s}}j_{isc}(T)\,dT \tag{9}\]
where \(j_{isc}\) is the thermally stimulated current density, \(T_{s}\) and \(T_{e}\) are the temperature of the start and the end of the electron emission of the defect, respectively. It should be mentioned here that Eq. (9) is only valid if the defect concentration and the emission rate are position independent. For the investigated irradiated diodes, three different situations have to be considered when evaluating the B\({}_{i}\)O\({}_{i}\) concentration:
(i) At the lowest fluence of \(1\times 10^{15}\) cm\({}^{-2}\) the diodes EPI-3 and Cz-3 are partially depleted before and after emission of the defect for all the applied bias voltages. As it can be observed in Fig. 5(c) the capacitance stays nearly constant, i.e. also the depletion depth \(w(T)\) is constant in the temperature range of interest. Therefore, Eq. (9) can be simplified to:
\[N_{t}=\frac{2}{\beta q_{0}}\cdot\int_{T_{s}}^{T_{s}}\frac{I^{e}_{TSC}(T)}{A \cdot w}\,dT=\frac{2\cdot Q}{q_{0}Aw} \tag{10}\]
Where \(w\) can be extracted from TS-Cap data as an average value in the range \(T_{s}\) to \(T_{e}\).
(ii) The sensor is partially depleted before emission and fully depleted after emission. This holds for the device EPI-9, which was irradiated to \(\Phi_{\rm e}=6\times 10^{15}\) cm\({}^{-2}\). In this case, the concentration can be evaluated from the TSC spectrum only if \(w(T)\) is extracted from TS-Cap measurements.
(iii) Similar to case (i), the sensors are partially depleted before and after emission, but \(C(T)\) or \(w(T)\) shows visible changes in the temperature range where the electron emission from the defect takes place(see Fig. 5(c) and 5(d) for the diodes EPI-7 and Cz-7). In this case, the corresponding defect concentration can be directly extracted from the TS-Cap measurement as described in the following.
For high defect concentration where the change in the occupancy of the defects due to the thermal emission of captured electrons or holes leads to measurable variations of the capacitance with increasing temperature, the TS-Cap method can be used to extract the defect concentration. For the B\({}_{i}\)O\({}_{i}\) defect the TS-Cap can be described, in the 1-D approach, by the following equations:
\[C(T)=\frac{\epsilon_{0}\epsilon_{e}A}{w(T)} \tag{11}\]
with
\[w^{2}(T)=\frac{2\epsilon_{0}\epsilon_{e}(V+V_{th})}{q_{0}\cdot|N^{\prime}_{\rm eff }(T)|} \tag{12}\]
where
Figure 6: Temperature shift of the B\({}_{i}\)O\({}_{i}\) TSC peak in the case of EPI-7 diode (\(\Phi_{\rm e}=4\times 10^{15}\)cm\({}^{-2}\)) for different bias voltages (top) and the corresponding shifts of the TS-Cap curves (bottom). The shifts are indicated by vertical lines between the TSC peak maxima and the turning point of the TS-Cap curves.
\[N_{\rm eff}^{\prime}(T)=N_{0}-N_{t}\cdot(1-f(T)) \tag{13}\]
Here \(C(T)\) is the capacitance of the device at temperature \(T\) and for a given bias voltage \(V\), \(V_{bi}\) is the build-in voltage, which is negligible compared to the applied bias voltage \(V\). The term \(N_{0}\) in Eq. (13) denotes the absolute \(N_{\rm eff}\) value before the start of the electron emission of B\({}_{i}\)O\({}_{i}\), i.e when all defect centers are neutral and their contribution to the effective space charge concentration is 0. The second term in Eq. (13) accounts for the donor character of the B\({}_{i}\)O\({}_{i}\) defect, becoming positively charged after thermal emission of captured electrons and thus leading to a progressive reduction of \(N_{0}\) with increasing the temperature until the electron emission from the defect ends. Assuming no other defects with similar emission rates are present, [B\({}_{i}\)O\({}_{i}\)] is given by:
\[\begin{array}{rcl}\mbox{[B${}_{i}$O${}_{i}$]}&=&\frac{2\epsilon_{0}\epsilon _{r}V}{q_{0}}\left(\frac{1}{w^{2}(T_{s})}-\frac{1}{w^{2}(T_{c})}\right)\end{array} \tag{14}\]
Here \(w(T)\) is extracted from Eq. (11) and \(T_{s}\) and \(T_{c}\) are the temperatures before and after the electron emission from B\({}_{i}\)O\({}_{i}\), respectively.
In Fig. 7(a) the B\({}_{i}\)O\({}_{i}\) and the C\({}_{i}\)O\({}_{i}\) concentrations extracted from the TSC and TS-Cap measurements as a function of \(\Phi_{\rm eq}\) are plotted for EPI- and Cz-materials. They were extracted via Eq. (9) in the temperature range 80-105 K for [B\({}_{i}\)O\({}_{i}\)] and 120-155 K for [C\({}_{i}\)O\({}_{i}\)]. Included are also the N\({}_{\rm eff}\) values for both materials as extracted from C\(-\)V measurements performed at room temperature. The N\({}_{\rm eff}\) values were extracted from Fig. 3 and averaged in the bias range of 1-100 V and 1-20 V for EPI and Cz diodes, respectively. It can be seen from Fig. 5(c) and Fig. 5(d) that after carrier emission from B\({}_{i}\)O\({}_{i}\) and C\({}_{i}\)O\({}_{i}\) the capacitance remains almost constant, and presumably it is the same as at RT. Therefore, using the \(N_{\rm eff}\) data from RT is appropriate and the introduced errors are related to \(N_{\rm eff}\) averaging only. The concentrations of B\({}_{i}\)O\({}_{i}\) and C\({}_{i}\)O\({}_{i}\) defects that can introduce positive space charge in the diodes are lower than the negative charge provided by the Boron-dopant. Therefore, \(N_{\rm eff}\) remains negative in the entire scanned temperature range.
Assuming the boron removal rate \(R\) is given by \(R\) = [(\(\Delta N_{\rm eff}\))/(\(\Delta\Phi_{\rm eq}\))], the values of 2.18 cm\({}^{-1}\) and 3.7 cm\({}^{-1}\) are obtained for Cz and EPI diodes, respectively. These values were extracted from the slope (absolute value) of the linear fits presented in Fig. 7(a). The difference of 41% between the Cz and the EPI rates is attributed to the different amounts of carbon content in both materials as given in Table 1. For the EPI-diodes the change of \(N_{\rm eff}\) with fluence is roughly a factor 2 larger compared with the increase of the B\({}_{i}\)O\({}_{i}\) concentration. This can be explained by the boron removal process, i.e. the negatively charged substitutional boron B\({}_{i}^{-}\) is transformed into a positively charged B\({}_{i}\)O\({}_{i}^{+}\) defect (B\({}_{i}^{-}\)\(\rightarrow\) B\({}_{i}\)O\({}_{i}^{+}\)). For the Cz-material this cannot be stated due to the strong non-uniform profile of the space charge density (see Fig. 3). The introduction rates \(g_{R\,O_{i}}\) = [\(B_{i}\)O\({}_{i}\)]/\(\Phi_{\rm eq}\) and \(g_{C\,O_{i}}\) =[C\({}_{i}\)O\({}_{i}\)]/\(\Phi_{\rm eq}\) were extracted from the linear increase with fluence, and are plotted in Fig. 7(b) as a function of the carbon content in the EPI- and Cz-diodes. It is obvious that the generation rate of the B\({}_{i}\)O\({}_{i}\) is much lower for the material with the higher carbon content. On the other hand, the increase of the C\({}_{i}\)O\({}_{i}\) generation rate with increasing carbon content is an indication for the beneficial effect of the carbon impurity in reducing the creation of B\({}_{i}\)O\({}_{i}\). This dependence on the carbon concentration has led to the approach of carbon co-implantation into the gain layer of LGADs in order to improve their radiation hardness [2].
Included in Fig. 7(b) is also the introduction rate of B\({}_{i}\)O\({}_{i}\) for an EPI-diode with the same \(N_{\rm eff,\,0}\) and irradiated with the same \(\Phi_{\rm eq}\) of 23 GeV protons as the irradiation with 5.5 MeV electrons. As it can be seen, the generation rate of B\({}_{i}\)O\({}_{i}\) defect after 5.5 MeV electron irradiation is about a factor 1.6 larger than the value determined after irradiation with 23 GeV protons.
In principle, both TSC and TS-Cap are performed with \(V_{\rm bias}\) where the lateral effect is not significant. However, the obtained concentrations strongly depend on the integration ranges of the
Figure 7: (a) Dependence of \(N_{\rm eff}\), B\({}_{i}\)O\({}_{i}\) and C\({}_{i}\)O\({}_{i}\) defect concentration on the \(\Phi_{\rm eq}\) of 5.5 MeV electrons for EPI- and Cz- diodes. The \(N_{\rm eff}\) values were extracted from Fig. 3 in the bias range of 1-100 V and 1-20 V for EPI and Cz diodes, respectively. (b) Variation of \(g_{R\,O_{i}}\) and \(g_{C\,O_{i}}\) as a function of the Carbon content for EPI- and Cz- diodes. Included is \(g_{R\,O_{i}}\) value after irradiating a 10 \(\Omega\)-cm EPI diode with 23 GeV protons at \(\Phi_{\rm eq}=4.3\times 10^{13}\) cm\({}^{-3}\).
TSC spectra or the selection of \(T_{s}\) and \(T_{e}\). Thus, in this work, the error of the extracted B\({}_{i}\)O\({}_{i}\) concentrations is given by varying \(T_{s}\) from 75 K to 80 K. The obtained errors for EPI-3, 7, 9 are 5%, 8% and 9%, for Cz-3 and 7 are 5% and 6%, respectively. The slightly increasing errors are caused by the overlapping peak at the low temperature tail possibly related to the X-defect. The estimated errors of the \(N_{\text{eff}}\) value shown in Fig. 7(a) are due to the selected interval of averaging the data (see Fig. 3). They are about 3% for all EPI diodes and 5% for Cz-3. For Cz-7 the estimated error is 20% due to the non-uniform profile.
### Simulation of TSC and TS-Cap data for the B\({}_{i}\)O\({}_{i}\) defect
Compared to the TSC and DLTS methods the TS-Cap technique is rarely used to get information about radiation induced defects. However, when high concentrations of defects are involved, the method delivers important information on the changes in the depletion depth during a temperature scan from 10 K up to room temperature, which can be used, via developing simulation models, to determine the defect type (capturing electrons or holes) and trapping parameters (activation energy, capture cross section of the emitted charge) as well as its concentration. In our simulations the following assumptions are made:
* Lateral effects are neglected.
* The device is partially depleted in the temperature range of interest.
* The series resistance of the non-depleted part of the device can be neglected.
Because the B\({}_{i}\)O\({}_{i}\) is a coulombic trap center, the emission rate \(e_{n}\) is not anymore a constant quantity with respect to the applied bias voltage, but field dependent. By accounting for the 3-D Poole-Frenkel effect, the emission rate can be expressed by [24; 31; 32]:
\[e_{n}^{pf}(T)=e_{n,0}(T)\cdot\left[\left(\frac{1}{\gamma^{2}}\right)\left(e^{ \gamma}\left(\gamma-1\right)+1\right)+\frac{1}{2}\right] \tag{15}\]
where
\[\gamma=\sqrt{\frac{q_{0}|\vec{E}|}{\pi\varepsilon_{0}\varepsilon_{r}}}\cdot \frac{q_{0}}{k_{B}T} \tag{16}\]
and \(e_{n,0}\) denotes the field independent emission rate with the so-called zero field activation energy \(E_{a}=E_{a,0}\). \(|\vec{E}|\) is the electric field in the sensor bulk and depends on the position x in the depleted zone. According to the reference [32], the Poole-Frenkel effect is given by the electrostatic energy of an electron which is attracted to a single charged positive ion under the influence of a uniform applied electric field (see Fig. 8). In the diodes, especially highly doped ones, such an assumption might not be fully valid, since the electric field distribution is not uniform. Thus, in this paper, we introduce a parameter \(\xi\) to modify the force between the positively charged ion and the electron. Therefore, the \(\gamma\) value is modified to:
\[\gamma=\xi\cdot\sqrt{\frac{q_{0}|\vec{E}|}{\pi\varepsilon_{0}\varepsilon_{r}}} \cdot\frac{q_{0}}{k_{B}T} \tag{17}\]
In this case the Eq. (13) has to be revised to:
\[N_{\text{eff}}^{{}^{\prime}}(T,x)=N_{0}-\left[\text{B}_{i}\text{O}_{i}\right] \cdot(1-f(T,x)) \tag{18}\]
Furthermore, the electric field distribution \(E(T,x)\) in the depleted bulk of the diodes is calculated from the corresponding Poisson equation:
\[\frac{dE(T,x)}{dx}=\frac{q_{0}\cdot N_{\text{eff}}^{{}^{\prime}}(T,x)}{ \varepsilon_{0}\varepsilon_{r}} \tag{19}\]
The electric field \(E\), the occupation fraction f and the \(N_{\text{eff}}^{{}^{\prime}}\) are temperature and position dependent. For coulombic centers, the emission rate \(e_{n}^{pf}\) has to be used for calculating the occupation fraction defined in Eq. (7).
Considering the involved set of equations, an analytical solution for simulating the TSC and TS-Cap experimental data will be extremely complicated. Therefore, the finite element method is used for simulating the experimental data. The details are presented in the Appendix.
In the following part, the simulation results and comparison with the corresponding TS-Cap and TSC measurements will be presented for two devices, both annealed for 2 h at 80 \({}^{o}\)C after irradiation: the electron irradiated sample EPI-7 (see Table 1) and a 50 \(\Omega\)-cm \(p\)-type diode irradiated with 23 GeV protons to \(\Phi_{\text{eq}}=4.3\times 10^{13}\) cm\({}^{-2}\) for which more detailed information can be found in reference [13]. The measurement parameters for both diodes are the same, i.e. \(V_{bias}\) = -100 V, heating rate \(\beta\) = 0.183 K/s and the frequency for the capacitance measurement \(f\) = 10 kHz.
All parameters, the fixed and the adjusted ones, used for the simulations of both diodes are summarized in Table 2. For the presented data, the details about \(N_{0}\) can be found in the Appendix. The simulation results for the EPI-7 diode are displayed in Fig. 9 (a-d). In order to reproduce the TS-Cap measurement (Fig. 9(a)) the B\({}_{i}\)O\({}_{i}\) concentration was extracted via the Eq. (14), the \(\xi\) value for the Poole-Frenkel effect was set
Figure 8: Energy of an electron bound to a positive point charge in the presence of a uniform applied field with the direction \(x\) along the bulk [32].
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline Methods & TS-Cap (\(E(T,x)\)) & TSC (\(E(T,x)\)) & TS-Cap (\(E(T,x)\)) & TSC (\(E(T,x)\)) & TS-Cap (\(<E(T)>\)) & TSC (\(<E(T)>\)) \\ Irradiation & Proton & Proton & Electron & Electron & Electron & Electron \\ \hline \(N_{0}\) (at 80 K) (cm\({}^{-3}\)) & \(1.1\times 10^{14}\) & \(1.1\times 10^{14}\) & \(5.1\times 10^{14}\) & \(5.1\times 10^{14}\) & \(5.1\times 10^{14}\) & \(5.1\times 10^{14}\) \\ [B\({}_{0}\)O\({}_{1}\)] (cm\({}^{-3}\))* & \(3.5\times 10^{13}\) & \(3.3\times 10^{13}\) & \(2.3\times 10^{14}\) & \(1.6\times 10^{14}\) & \(2.3\times 10^{14}\) & \(1.6\times 10^{14}\) \\ \(E_{a0}\) (eV)* & 0.265 & 0.273 & 0.258 & 0.258 & 0.284 & 0.284 \\ \(\sigma_{n}\) (cm\({}^{2}\)) & \(1.0\times 10^{-14}\) & \(1.0\times 10^{-14}\) & \(1.0\times 10^{-14}\) & \(1.0\times 10^{-14}\) & \(1.0\times 10^{-14}\) & \(1.0\times 10^{-14}\) \\ Area \(A\) (cm\({}^{2}\)) & 0.06927 & 0.06927 & 0.0621 & 0.0.0621 & 0.0.0621 & 0.0.0621 \\ \(\xi^{\text{e}}\) & 0.85 & 0.85 & 0.5 & 0.5 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of simulation. \(E(T,x)\) represents the position and temperature dependent electric field and \(<E(T)>\) is the average electric field in the diodes
Figure 9: Simulation results of the B\({}_{0}\)O\({}_{1}\) generating signals in EPI-7 diode: (a) TS-Cap, comparison with experiment; (b) density of TSC signal, comparison with the measured spectra; (c) and (d) the \(E(T,x)\) electric field distribution and the \(N_{\text{eff}}(T,x)\) profiles, respectively, for different temperatures, from 80 K to 110 K, in steps of 5 K. All simulations and given experimental data correspond to a reverse bias of 100 V applied during TS-Cap and TSC temperature scans.
Figure 10: Simulation results of the B\({}_{\rm i}\)O\({}_{\rm i}\) generating signals in a 50 \(\Omega\)-cm EPI diode irradiated with 23 GeV protons to \(\Phi_{\rm eq}=4.3\times 10^{13}\) cm\({}^{-2}\): (a) TS-Cap, comparison with experiment; (b) density of TSC signal, comparison with the measured spectra; (c) and (d) the \(E(T,x)\) electric field distribution and the \(N_{\rm eff}(T,x)\) profiles, respectively, for different temperatures, from 80 K to \(110\) K, in steps of 5 K. All simulations and given experimental data correspond to a reverse bias of 100 V applied during TS-Cap and TSC temperature scans.
\(\xi=0.5\) and the zero-field activation energy \(E_{a0}=0.258\) eV. With the same values for \(\xi\) and \(E_{a0}\) parameters but a lower B\({}_{i}\)O\({}_{i}\) concentration, the TSC signal could be reproduced in the temperature range between 90 K and 105 K. The low temperature tail, which can not be described by the simulation, is most probably due to the so-called X-defect (see Fig. 5 (a, b)). Contrary to the TSC case, where the charge emission from the X defect can be separated from that of the B\({}_{i}\)O\({}_{i}\) defect, in TS-Cap measurements the contributions of both defects cannot be separated. Therefore, the concentration extracted from the TS-Cap curve is larger compared to the value derived from the TSC spectrum.
Included in Fig. 9 (a, b) are also the results from simulations which use the position independent average electric field \(<E(T)>\) = \(V_{bias}/w(T)\) where \(w(T)\) is given by:
\[w(T) = \sqrt{\frac{2e_{0}e_{r}V_{bias}}{q_{0}N^{{}^{\prime}}_{\rm eff}(T)}} \tag{20}\]
Here \(N^{{}^{\prime}}_{\rm eff}(T)\) is constant over the depth of the diode and given by Eq. (13) where \(f(T)\) is calculated with the average electric field \(<E(T)>\) of the previous temperature step. For this case, the value \(\xi=1\) and a higher zero-field activation energy of \(E_{a0}\) = 0.284 eV is needed in the simulation, in order to get the best fit to the experimental data.
In Fig. 9 (c, d) the electric field distribution and the \(N_{\rm eff}\) profiles as a function of the depleted depth are plotted for temperatures between 80 K and 110 K in steps of 5 K. As it can be seen in Fig. 9(c), with increasing the temperature, the maximal value of \(E(T_{k},x=0)\) decreases and the depleted depth increases. This corresponds to the development of the effective space charge density \(N^{{}^{\prime}}_{\rm eff}(T_{k},x)\) for the different temperature steps as shown in Fig. 9(d). Further, the distribution of the electric field shows a constant gradient before and after the B\({}_{i}\)O\({}_{i}\) emission (below 85 K and above 100 K) and position dependent gradients during emission of the B\({}_{i}\)O\({}_{i}\) in the range between 85 K and 100 K. This is due to the non-uniform distributed space charge density resulting from the field dependent emission from the defect energy level.
Similar simulations have been performed for the 23 GeV proton irradiated diode and the results are presented in Fig. 10 (a-d). As it can be seen, the simulation of the TS-Cap signal, shown in Fig. 10(a), is in excellent agreement with the measured data. In this case, the parameters from the measured \(C(T)\) curve, by using the same procedure as for the electron irradiated diode, are \(\xi\) = 0.85, \(E_{a0}\) = 0.265 eV and \(\rm[B_{i}O_{i}]\) = 3.5 \(\times\) 10\({}^{13}\) cm\({}^{-3}\). In Fig. 10(b) the corresponding TSC data and simulated spectra are given. Also in this case the simulation reproduces the data very well, but compared with the TS-Cap simulations, the best agreement is found for slightly different \(E_{a0}\) and \(\rm[B_{i}O_{i}]\) values, of \(E_{a0}\) = 0.273 eV and \(\rm[B_{i}O_{i}]\) = 3.3 \(\times\) 10\({}^{13}\) cm\({}^{-3}\). The \(\xi\) value is the same for both simulations. The distributions of the electric field and the N\({}_{\rm eff}\) profiles are plotted in Fig. 10(c) and (d) for temperatures between 80 K and 110 K in steps of 5 K. In this case, the maximal electric field \(E(T_{k},x=0)\) also decreases with increasing the temperature while the depleted region depth increases. The main difference to the electron irradiated device is the lower field strength in the bulk.
For getting a better fit to the data, the \(\rm[B_{i}O_{i}]\) for the simulation of the TSC spectrum (see Fig. 9(b) and Fig. 10(b)) is adjusted. The \(\rm[B_{i}O_{i}]\) extracted from the TSC spectrum by integration from 80 K to 105 K is about 13 % larger compared to the value used for the simulation. This difference is due to the low temperature tail in the spectrum which was not reproduced in the simulation. For the EPI-7 diode, the significant difference of \(\rm[B_{i}O_{i}]\) between TS-Cap and TSC is caused by some unknown effect.
In principle, the \(E_{a0}\) is known with value in between 0.27-0.28 eV [13] with fixed \(\sigma_{n}=1.05\times 10^{-14}\) cm\({}^{2}\). The difference of \(E_{a0}\) between TSC and TS-Cap measurements for proton irradiated diode is due to the difference in the temperature of the peak maximum \(T_{\rm max}\) in the TSC spectrum and the temperature of the turning point in the TS-Cap curve. The related effect is still unknown. The difference of \(E_{a0}\) between the proton and electron irradiated devices might be caused by the different production technology of both devices, the diode with a guard ring (p irradiated) and the other one without. The explanation can be proved by comparing the results from the EPI-diode irradiated with protons ([13]). This diode has roughly the same \(E_{a0}\) as the one presented in Table 2 for proton irradiation.
For EPI-7 diode the difference in \(E_{a0}\) between the two different electric field distributions (linear electric field \(E(x)\) and homogeneous electric field distribution \(<E(T)>\)) is 26 meV (\(\sim\)10%). The reason for this difference can be understood by the fact that the emission rate \(e_{n}^{pf}(T,E(x),E_{a0})\) depends exponentially on the electric field distribution. At a specific temperature \(T\), the emission rate is enhancing with \(E(x)\) and is decreasing when increasing the \(E_{a0}\) values. Thus, for the same bias voltage, the values of \(E(T,x)\) in the case of linear field distribution and of the average electric field \(<E(T)>\) coincide only in the middle of the depleted width, in the front region of the junction \(E(T,x)\) being larger than \(<E(T)>\) and in the back side smaller. Consequently, the same measured TSC signal can be reproduced in both cases if in the calculation of the emission rates the values of \(E_{a0}\) and \(\xi\) are smaller for a linear distribution of the electric field than for the constant, average one. This has with respect to the emission rate to be compensated by a lower \(E_{a0}\) or a lower \(\xi\) value compared to the constant field case in order to reproduce the same measured TSC signal, as it can be seen in Table 2.
Due to the fact that by using Eq. (16) the experimental data could not be reproduced, a constant \(\xi\) was introduced for modifying the field dependence in the Poole-Frenkel effect. The \(\xi\) values are different for linear and constant electric fields, 0.5 and 1.0, respectively, while for each \(E(x)\) distribution they are the same for simulating the TSC and TS-Cap data.
## 5 Conclusion
In this work investigations of radiation damage of silicon diodes manufactured on p-type EPI- and Cz-material with a resistivity of about 10 \(\Omega\)-cm and exposed to 5.5 MeV electrons of different fluence values (\(1\times 10^{15}\), \(4\times 10^{15}\), \(6\times 10^{15}\) cm\({}^{-2}\)) have been performed. The macroscopic properties of the devices, the
leakage current density \(J_{d}\) and \(N_{\rm eff}\), were obtained from I-V and C-V measurements. The microscopic properties of the B\({}_{1}\)O\({}_{1}\) and C\({}_{1}\)O\({}_{1}\) defects were studied using the TSC and TS-Cap methods and the results are discussed in connection with Boron removal process observed in macroscopic measurements.
The main results obtained in this study are:
* The density of leakage current \(J_{d}\) increases linearly with the achieved fluence and the corresponding current related damage parameter is determined to be \(\alpha=3.2\times 10^{-19}\) A/cm. Such a small value was also reported for n-type silicon diodes after irradiation with 5.5 MeV electrons [26]. Compared with hadron irradiation, the obtained \(\alpha\) parameter is much smaller, indicating that the increase of the leakage current caused by low energy electrons is substantially less than that caused by hadrons. Also, the change of \(J_{d}\) with annealing time at 80 \({}^{o}\)C is strongly suppressed compared with hadron irradiated devices indicating that the irradiation with low energy electrons creates less current generation centers and more stable defects.
* The \(N_{\rm eff}\) decreases nearly linear with increasing fluence and remains stable during the isothermal annealing at 80 \({}^{o}\)C, in agreement with the thermal stability of the B\({}_{1}\)O\({}_{1}\) defect [28].
* The development of B\({}_{1}\)O\({}_{1}\) and C\({}_{1}\)O\({}_{1}\) defects with fluence is linear, however, with different introduction rates for EPI and Cz materials, due to the different Carbon content in the two materials (more in Cz than in EPI) and the competing reactions between Boron and Carbon interstitials with abundant Oxygen interstitials in silicon. Thus, while the introduction rate of B\({}_{1}\)O\({}_{1}\) is much smaller in Cz than in EPI material, of 0.63 cm\({}^{-1}\) compared with 1.75 cm\({}^{-1}\) as seen in Fig. 7(b), the opposite is happening for C\({}_{1}\)O\({}_{1}\). Similar behaviour was also reported in the RD50 collaboration program [35; 10].
* The formation of B\({}_{1}\)O\({}_{1}\) defect is the main cause for the change seen in \(N_{\rm eff}\) after irradiation with 5.5 MeV electrons. This was nicely evidenced in EPI diodes where the homogeneous Boron doping profile allowed accurate evaluations. Thus, by comparing the Boron removal rate of 3.7 cm\({}^{-1}\) resulted from C-V measurements with that of 3.5 cm\({}^{-1}\) resulted by accounting twice the value of B\({}_{1}\)O\({}_{1}\) introduction rate due to the donor character of the defect, a good agreement is obtained.
* The TS-Cap technique proved to be a valuable complementary to the TSC tool in order to accurately characterize the radiation induced defects in highly irradiated and partially depleted silicon sensors. This is especially important in the case of low resistivity diodes when the total depletion of the device in TSC measurements cannot be achieved or the depletion depth cannot be kept constant during the temperature scan. However, TS-Cap allows the evaluation of defect concentrations only if the defects are well isolated in the silicon bandgap, not overlap with other defects.
* The temperature dependence of the thermally stimulated capacitance at constant bias voltage and of the corresponding TSC spectra, for a 5.5 MeV electron and a 23 GeV proton irradiated devices were simulated in the temperature range of the B\({}_{1}\)O\({}_{1}\) defect emission. For reproducing the TS-Cap and TSC data the Poole-Frenkel effect was accounted and modified by a subunitar factor \(\xi\) and small variations in the defect's zero-field activation energy. Different \(\xi\) and \(E_{\rm a0}\) values resulted from simulating the experimental data measured on differently damaged silicon diodes, an aspect that has to be further studied in more detail. Presently, we justify these adjustments by the fact that the Poole-Frenkel theory was not developed for position dependent electric fields as existing in diodes and more pronounced in low resistivity ones, but for constant field around the defect. In the absence of a proper Poole-Frenkel theory for accounting the position dependent electric field in diodes, the adjustments were made for describing as good as possible both B\({}_{1}\)O\({}_{1}\) current and capacitance signals. In addition, when accounting for the electric field dependent electron emission from B\({}_{1}\)O\({}_{1}\) defect, the simulated electric field distributions \(E(T,x)\) for the temperature range where the B\({}_{1}\)O\({}_{1}\) defect discharges, between 80 K and 110 K, show position dependent gradients, corresponding to the position dependent effective space charge densities \(N_{\rm eff}^{\prime}(T,x)\).
## Appendix A Simulation
In this section, the simulation procedure for the TS-Cap and the TSC spectra of the B\({}_{1}\)O\({}_{1}\) defect will be described. The simulations are performed by using Python software. The bulk of the sensor is divided into n sufficiently thin layers of a thickness \(\delta x=\text{d}/\text{n}\), where \(d\) is the thickness of the EPI- or Cz-silicon (see Fig. 11a)). The index \(i\) in Fig. 11a) runs from 0 to n and the boundary between the depleted and the non-depleted region is labelled \(m_{k}\). The index \(k\) indicates the temperature step \(T_{k}\) and varies between 0 (the start temperature \(T_{0}\)) and \(f\) (the final Temperature \(T_{f}\)).
As the emission rate of the B\({}_{1}\)O\({}_{1}\) defect is governed by the 3-D Poole-Frenkel effect (Eq. (15, 16, 17)) the electric field distribution \(E(T,x)\) has to be calculated via the Poisson equation (Eq. (19)) for a known effective space charge density \(N_{\rm eff}^{\prime}(T,x)\) (Eq. (18)). Considering the finite element method mentioned above, the Poisson equation can be written as:
\[E_{k,j+1}-E_{k,i}=\frac{q_{0}N_{\rm eff,k,i}^{\prime}}{\varepsilon_{0} \varepsilon_{r}}\cdot\frac{d}{n} \tag{19}\]
Considering the boundary condition between the depleted and the non-depleted region
\[E_{k,i}=0\ \ \ \text{for}\ \ \ i\geq m_{k}, \tag{20}\]
the \(E_{k,i}\) is given by (according to Eq. (19)):
\[E_{k,i}=\sum_{j=0}^{m_{k}}\frac{q_{0}N_{\rm eff,k,j}^{\prime}}{\varepsilon_{0} \varepsilon_{r}}\cdot\frac{d}{n}-\sum_{j=0}^{i}\frac{q_{0}N_{\rm eff,k,j}^{ \prime}}{\varepsilon_{0}\varepsilon_{r}}\cdot\frac{d}{n}, \tag{21}\]
where the index \(j\) is used to sum \(N^{\prime}{}_{\text{eff},k,i}\) for layer \(i\) from 0 to the boundary or the indicated layer \(i\). Further, the applied bias voltage \(V_{bias}\) is given by the sum of all electric field steps \(E_{k,i}\) up to the temperature dependent \(m_{k}\) value, as given by:
\[V_{bias}=\sum_{i=0}^{m_{k}}E_{k,i}\cdot\frac{d}{n}\] (A.4)
This equation (Eq. A.4)) is then used to calculate \(m_{k}\) by rising \(m_{k}\) from 0 up to the value that fulfils Eq. (A.4). Thus for the description of Eq.(A.1-4), the only unknown parameter for obtaining the \(E_{k,i}\) is \(N^{\prime}_{\text{eff},k,i}\). In Eq.(18), \(N^{\prime}_{\text{eff},k,i}\) can be obtained by \(f_{k,i}\), which in finite elements method can be written as:
\[f_{k+1,i}=exp\left(-\sum_{j=0}^{k}\frac{\Delta T_{j}}{\beta}e_{n,j}\right),\] (A.5)
where the index \(j\) was used to sum the emission rate from temperature \(T_{0}\) to the \(T_{k}\). The Eq. (18) can be changed to:
\[N^{\prime}_{\text{eff},k,i}=N_{0}-[\text{B}_{i}\text{O}_{i}]\cdot(1-f_{k,i})\] (A.6)
Considering the 3-D Poole Frenkel effect, the emission rate can be written to:
\[e^{pf}_{n,k,i}=\sigma_{n}v_{ik,n}N_{c}exp\left(-\frac{E_{a0}}{k_{B}T_{k}} \right)\left[\left(\frac{1}{\gamma_{k,i}^{2}}\right)\left(e^{\gamma_{k,i}} \left(\gamma_{k,i}+1\right)\right)+\frac{1}{2}\right]\] (A.7)
with
\[\gamma_{k,i}=\xi\cdot\sqrt{\frac{q_{0}|E_{k,i}|}{\pi\varepsilon_{0}\varepsilon _{r}}}\cdot\frac{q_{0}}{k_{B}T_{k}}\] (A.8)
For the electron capture cross section we used the value of \(\sigma_{n}=1.05\times 10^{-14}\) cm\({}^{2}\) determined experimentally in [12]. The zero field activation energy of the B\({}_{i}\)O\({}_{i}\) defect, \(E_{a0}\), was previously determined to be between 0.271 eV and 0.288 eV for silicon diodes with resistivities varying from 50 \(\Omega\)-cm to 2 k\(\Omega\)-cm and irradiated with 23 GeV protons [13]. \(E_{a0}\) values were tuned for getting the best fit between simulated and measured data, and all parameters used in the simulation are given in Table 2. The concentration [B\({}_{i}\)O\({}_{i}\)] used for TS-Cap simulation was extracted from the TS-Cap measurement according to Eq. (14).
The initial conditions for \(T_{k}\), \(N^{\prime}_{\text{eff},k,i}\), \(e^{pf}_{n,k,i}\), \(f_{k,i}\), \(n\) and the applied bias voltage \(V_{bias}\) are: \(T_{0}\) = 40 K, \(N^{\prime}_{\text{eff},0,i}\) (\(N_{0}\)) as extracted from TS-Cap at 80 K, \(e^{pf}_{n,0,i}\) = 0, \(f_{0,i}\) = 1, \(n\) = 1000 and \(V_{bias}\) = -100 V. Then, it is obtained that for initial \(T_{0}\) electric field distribution \(E_{0,i}\) decreases linearly from \(1.2\times 10^{5}\) and \(5.8\times 10^{4}\) to 0 V/cm for 5.5 MeV electrons with \(\Phi_{\text{e}}=4\times 10^{15}\) cm\({}^{-2}\) and 23 GeV protons \(\Phi_{\text{p}}=6.91\times 10^{13}\) cm\({}^{-2}\) irradiation, respectively. It was also given that the \(m_{0}\) is equal to 320 and 690 for electrons and protons irradiation, respectively. Such values were extracted from Eq. (A.3) and Eq. (A.4). Next, these values were used to calculate the emission rate \(e^{pf}_{n,1,i}\), \(f_{1,i}\) and \(N^{\prime}_{\text{eff},1,i}\) (Eq. (A.5-8)), with which the distribution of the electric field \(E_{1,i}\) and \(m_{1}\) are calculated. This step by step calculation continues until the final temperature \(T_{f}\) is reached. Also, the temperature dependent depletion depth is calculated according to \(w(T_{k})=m_{k}\cdot d/n\). The selected \(T_{e}\) must be higher than the temperature of the end of emission, and in this work \(T_{e}\) = 120 K was chosen for simulation. The TSC values at \(T_{k}\) can also be calculated according to:
\[I^{e}_{TSC,k} = q_{0}A[B_{i}O_{i}]\sum_{j=0}^{m_{k}}\frac{j\cdot\frac{d}{n}}{w(T _{k})}\cdot e^{pf}_{n,k,j}\cdot f_{k,j}\cdot\frac{d}{n}\] (A.9)
Considering the depleted depth extracted from the TS-Cap measurements, the TSC spectrum was also simulated and compared with the measured data.
Figure A.11: (a) Schematic of finite elements approach applied for describing the field dependent B\({}_{i}\)O\({}_{i}\) emission in TS measurements. The band gap structure has been divided into n layers by blue lines. (b) simulation procedure.
## Acknowledgment
This work has been carried out in the framework of the RD50 Collaboration. The project has received funding from the European Union's Horizon 2020 Research and Innovation program under Grant Agreement no. 654168. C. Liao would like to thank for the given support to work at the Hamburg University to the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC2121 "Quantum Universe" - 390833306 project and to Professor Z. Li. I. Pintilie and Lucian D. Filip acknowledge the funding received through IFA-CERN-RO 08/2022 project and Core Program 2019-2022 (contract 21N/2019). Z. Li acknowledges the funding received through the Key Scientific and Technological Innovation Project of Shandong Province under Grant No. 2019 TSLH 0316, and the project of Yantai Institute for the exchange of Driving Forces under Grants No. 2019XJDN002.
|
2309.02909 | Atomistic insights into ultrafast SiGe nanoprocessing | Controlling ultrafast material transformations with atomic precision is
essential for future nanotechnology. Pulsed laser annealing (LA), inducing
extremely rapid and localized phase transitions, is a powerful way to achieve
this, but it requires careful optimization together with the appropriate system
design. We present a multiscale LA computational framework able to simulate
atom-by-atom the highly out-of-equilibrium kinetics of a material as it
interacts with the laser, including effects of structural disorder. By
seamlessly coupling a macroscale continuum solver to a nanoscale super-lattice
Kinetic Monte Carlo code, this method overcomes the limits of state-of-the-art
continuum-based tools. We exploit it to investigate nontrivial changes in
composition, morphology and quality of laser-annealed SiGe alloys. Validations
against experiments and phase-field simulations, as well as advanced
applications to strained, defected, nanostructured and confined SiGe are
presented, highlighting the importance of a multiscale atomistic-continuum
approach. Current applicability and potential generalization routes are finally
discussed. | Gaetano Calogero, Domenica Raciti, Damiano Ricciarelli, Pablo Acosta-Alba, Fuccio Cristiano, Richard Daubriac, Remi Demoulin, Ioannis Deretzis, Giuseppe Fisicaro, Jean-Michel Hartmann, Sébastien Kerdilès, Antonino La Magna | 2023-09-06T11:09:28Z | http://arxiv.org/abs/2309.02909v1 | # Atomistic insights into ultrafast SiGe nanoprocessing
###### Abstract
Controlling ultrafast material transformations with atomic precision is essential for future nanotechnology. Pulsed laser annealing (LA), inducing extremely rapid and localized phase transitions, is a powerful way to achieve this, but it requires careful optimization together with the appropriate system design. We present a multiscale LA computational framework able to simulate atom-by-atom the highly out-of-equilibrium kinetics of a material as it interacts with the laser, including effects of structural disorder. By seamlessly coupling a macroscale continuum solver to a nanoscale super-lattice Kinetic Monte Carlo code, this method overcomes the limits of state-of-the-art continuum-based tools. We exploit it to investigate nontrivial changes in composition, morphology and quality of laser-annealed SiGe alloys. Validations against experiments and phase-field simulations, as well as advanced applications to strained, defected, nanostructured and confined SiGe are presented, highlighting the importance of a multiscale atomistic-continuum approach. Current applicability and potential generalization routes are finally discussed.
As material science roadmaps relentlessly pursue the digital, sustainability and quantum paradigms, understanding and harnessing ultrafast transformations at the atomic scale is becoming increasingly crucial for the atom-by-atom control of nanosystems and their integration as building blocks into meso- and macroscale systems [1; 2; 3; 4; 5]. Laser annealing (LA) using excimer pulses is an excellent and long-standing way of inducing and investigating such transformations, as it enables localized energy absorption, heating and melting over nm-sized subportions of the material in extremely short time (from tens to hundreds of ns) [6; 5]. It is nowadays exploited in several technologies, mostly due to its ultra-low thermal budget and its numerous control knobs (light wavelength and polarization, pulse duration, fluence, repetition rate, beam extension), which can be flexibly tuned to target specific functionalities, while handling the evergrowing complexity of nanosystems.
In the context of group IV elemental and compound semiconductor processing, pulsed-LA applications are ubiquitous [7; 8; 5]. These include fabrication of poly-Si thin-film transistors [9; 10; 11], ultrashallow device junctions [12; 13; 14; 10], efficient contacts by silicidation [15], explosive crystallization [16; 17; 18], strain, defect [19; 20] and dopant engineering [21; 22; 23; 24; 25]. Localized heating minimizes the risk of damaging sequentially integrated components of monolithic 3D devices [26; 27; 28; 29; 30]. In optoelectronics pulsed-LA is a key-process for fabricating poly-Si displays [31; 32; 33; 34], thin metal-oxides [7], pure-carbon electrodes for touch screens or solar cells [35], hyper-doped semiconductors for near-infrared photodetectors [36]. It also allows strain, composition and morphology engineering of fiber-based photonic devices [8] and fabrication of heavily-doped superconducting silicon for monolithic quantum device integration [37; 23; 38; 39]. Despite all these applications, understanding the ultrafast non-equilibrium kinetics of the liquid/solid interface in early stages of the process and correlating it to the post-irradiation morphology and properties is challenging. This is because any experimental characterization [5], no matter how accurate, can only access the final state of the system. Observations would indeed require in-situ, atomically-resolved and real-time capabilities well beyond those of modern electron-microscopy [4] or atom-probe facilities [40]. For this reason, computer simulations are nowadays indispensable for both fundamental studies and technological exploitation of LA.
LA simulations are usually deployed by self-consistently solving the electromagnetic interaction and heat diffusion problem in the irradiated system using a continuum description of its phase changes [41; 42; 43; 44; 45]. LA process parameters can be explored and fine-tuned in \(\mathrm{\SIUnitSymbolMicro m}\)-sized geometries, with the aid of computational libraries that use finite-element-methods (FEM) to solve the underlying coupled partial differential equations. However, continuum models cannot capture local nanoscale changes in the annealed materials with atomistic resolution. The latter may be a critical factor, especially for compound materials with complex 3D geometries or phase diagrams, crystal-orientation-dependent kinetics and defects, like stacking faults, which can affect the regrowth. Polymorphic solidification may also introduce structural disorder in the form of intermixed stacking motifs
(e.g., cubic, hexagonal) [46; 47; 48]. These phenomena can significantly alter the post-LA morphology and composition, with important consequences on device quality and performance. To ensure the appropriate process design and optimization, a simulation tool should be able to model the complex interplay between laser-matter interactions, the molten phase non-equilibrium kinetics and all possible atomic-scale structural transformations, while requiring the least amount of computational resources.
In this work we present a multiscale computational methodology enabling simulations of LA processes with atomic resolution. It is based on the local self-consistent coupling of a state-of-the-art um-scale FEM code with a Kinetic Monte Carlo on super-Lattice (KMCsL) code, able to simultaneously model atoms in the cubic and hexagonal crystal phases. Such multiscale approach enables atomistic modeling of extended defects, shape changes, composition and stack adjustments affecting the laser-annealed material up to hundreds of nm below the surface, while exchanging information between FEM and KMCsL at a ns pace. In this way, it not only overcomes the limits of purely continuum-based tools, but also those of other hybrid FEM-KMC approaches, which either lack self-consistent information exchange between the two frameworks [49] or are limited to defect-free LA simulations of silicon without any super-lattice formulation [50]. In particular, we demonstrate the method by focusing on ultraviolet ns-pulsed LA processes of SiGe, an alloy with composition-dependent electronic and optical properties [51; 52; 53] increasingly relevant to future nanoelectronic [3; 28; 54; 55; 56; 57; 58; 59], thermoelectric [60], optoelectronic [8; 61; 62] and quantum technologies [63; 64; 2; 65]. The multiscale methodology provides unique atomistic insights on the complex and ultrafast morphological, compositional and structural transformations of SiGe during laser irradiation [19; 66; 20; 67], giving invaluable support to process engineers aiming at the exploitation of this material's full potential.
## I Results
### Multiscale FEM-KMCsL coupling
Our LA simulations are based on a multiscale coupling between FEM and KMCsL solvers, which exchange information in a self-consistent loop at time steps \(\Delta t<1\) ns throughout the simulation. Besides providing atomistic insights, this approach ensures higher accuracy compared to pure phase-field or enthalpy formalisms [5], as the latent heat exchanged at every \(\Delta t\) is computed by direct integration of the volume subjected to a phase transition during each KMCsL step. The multiscale procedure is hereby described. After setting up the appropriate 3D mesh for a system with desired size and composition, the FEM calculation begins. The laser-induced heat source and temperature field \(T(t,\mathbf{r})\) within the irradiated material are self-consistently calculated, by solving the Maxwell's and Fourier's partial differential equations. As the system absorbs energy from the laser pulse, following its power density modulation, the surface temperature rises until local melting occurs. This initiates the feedback coupling with KMCsL, which models atom-by-atom the concerned system subregion. The following steps are then iterated every \(\Delta t\) over the whole pulse duration:
* \(T(t,\mathbf{r})\) is interpolated into the dense KMCsL super-lattice and defines melting/solidification events probabilities;
* KMCsL simulates the evolution of the solid/liquid (S/L) interface for a time \(\Delta t\) with the established probability table, capturing atomic-scale structural adjustments, lattice faceting, vacancies, extended defects, polymorphic solidification and species redistribution;
* the S/L volumes and the local species concentrations are updated in the mesh based on the KMCsL results and affect the \(T(t,\mathbf{r})\) calculation in the subsequent FEM cycle.
The above three steps are iterated until all previously-melted atoms resolidify. Thereafter the FEM-KMCsL communications stop and the FEM model is left to cool. Figure 1 schematically illustrates a typical FEM-KMCsL simulation box (characteristic sizes used in this work are also indicated) for modelling pulsed-LA of a flat Si\({}_{0.76}\)Ge\({}_{0.24}\) (0 0 1) surface. Figure 1b shows the solid atoms at the S/L interface in the KMCsL-modelled subregion, at various instants of a simulation assuming a XeCl excimer (\(\lambda=308\) nm) 160 ns laser pulse, with 0.75 J cm\({}^{-2}\) energy density and a \(\Delta t=0.25\) ns. After the initial heating stage up to \(T_{M}(x=0.24)\approx 1573\) K, the interface goes deep into the material (roughly 25 nm), keeping a roughness of a few nm. Then it rapidly ascends as \(T\) decreases, solidifying a SiGe layer with graded Ge content and a Ge-rich surface, due to non-equilibrium species partitioning in the alloy [68]. The corresponding solid-phase regions in the 3D FEM mesh at the same instants are reported in Figure 1c, with colours highlighting Ge segregation.
More details on the multiscale implementation are reported in Methods (see also Supplementary Note S1, Figure S1 and Movie S1).
### Calibration of KMCsL for SiGe S/L interface
In the KMCsL model of a partially-melted system, the evolution of the S/L interface is governed by the balance between solidification and melting events with \(T\)-dependent Arrhenius-like probabilities (see Methods). To ensure reliable LA simulations, it is crucial to calibrate the KMCsL event probabilities so that they reproduce
the correct S/L interface kinetics over a wide range of temperatures. In case of SiGe alloys, this calibration is carried out in two steps. The first, following the strategy of Ref. [50], consists in reproducing the Fulcher-Vogel curves of pure Si and Ge systems [16; 69], i.e., the S/L interface velocity as a function of \(T\). We do this by initializing solid Si and Ge surfaces surmounted by an infinite liquid reservoir and performing a sequence of KMCsL simulations for a wide range of \(T\) around \(T_{M}\), always assuming \(T\) uniformity in the simulated box. The KMCsL parameters are then fine-tuned to yield the expected interface velocities, as shown in Figure 2a (calibrated values in Supplementary Table S1-3). The second step, since no Fulcher-Vogel relation holds for SiGe alloys, consists in calibrating the KMCsL event rates involving mixed Si-Ge bonds on the SiGe phase diagram [51] (dashed lines in Figure 2b), which describes the \(T\)-dependent composition (\(x_{S},x_{L}\)) of solid and liquid phases at equilibrium, when the melting/solidification process occurs very slowly. This is achieved by setting up slightly under-cooled KMCsL simulations around a given \((T,x_{L})\) point in the phase diagram and tuning the parameters until the solidified SiGe layer roughly matches the expected \(x_{S}\) (more details in Supplementary Note S2). For example, Figure 2c shows two snapshots of the S/L interface at the beginning and at the end of a well calibrated simulation assuming (\(T\approx 1410,x_{L}\approx 0.8\)). This predicts a solid layer with average \(x_{S}\approx 0.43\), which is in line with the phase diagram and hence confirms the reliability of the calibration at the considered \(T\). The
Figure 1: Schematics of the FEM-KMCsL multiscale approach applied to a SiGe (0 0 1) surface. (a) Sketch of the simulation framework. Typical simulation box dimensions are also indicated. (b) Liquid-solid interface in the KMCsL box at various instants during melting and solidification. Solid under-coordinated Si and Ge atoms (green and brown, respectively) identify the interface. (c) Ge content in solid SiGe in the FEM model, visualized at the same instants. Pure Si (x=0) regions in blue and pure Ge (x=1) in red.
simulated results spanning the entire phase diagram from Si-rich to Ge-rich situations are reported in Figure 2b.
### Validation of multiscale LA simulations for relaxed/strained SiGe
Here we show the results of FEM-KMCsL simulations for relaxed and 30 nm-thick strained SiGe(0 0 1) layers epitaxially grown on Si. To check the consistency of the method and validate it, we compare these results with those of 1D non-atomistic simulations based on a state-of-the-art FEM-phase-field formulation (see Methods), considering pulsed-LA processes with various laser energy densities and pulse durations (160 ns for relaxed, 146 ns for strained). The simulation settings (optical/thermal parameters, initial Ge profile, laser properties) in the 1D purely continuum and the 3D FEM-KMCsL multiscale frameworks are identical, except for the mesh dimensionality and the formalism describing phase transitions (phase-field with smooth S/L interface in one case, KMCsL with atomically sharp S/L interface in the other).
The initial Ge profiles and process conditions are reported in Figure 3a-b. Figure 3c-d shows that the two methodologies yield almost identical results concerning the general melt depth profile over time, with KMCsL yielding melting/solidification velocities and maximum melt depths in remarkably good agreement with phase-field simulations. Contrary to the latter, the FEM-KMCsL approach can track the interface evolution up until the last solidification event and, as a result, it can reproduce the expected slowdown of the solidification front due to the gradual Ge incorporation. Figure 3e-f shows the variation of maximum temperature \(T_{\text{max}}\) in the mesh over the same time interval. Both models predict an overall similar trend, with the expected change in slope at the onset of melting and almost overlapped cooling tails after complete solidification. The observed \(T_{\text{max}}\) plateaus at the end of solidification are also related to Ge segregation, as those observed in Figure 3c-d. Their position and dependence on energy density differ between the two models, in accordance with the final profiles of Ge concentration over depth reported in Figure 3g-h. Like all other noticeable deviations in Figure 3, this can be attributed to intrinsic differences between the two models (more details in Supplementary Note S3). Overall, these results confirm the internal consistency of the multiscale methodology for a wide range of process conditions and demonstrate that the phenomenon of Ge segregation is
Figure 2: KMCsL calibration for SiGe solid-liquid phase transitions. (a) Calibrated results for the solid-liquid interface velocity in pure Si and pure Ge as a function of temperature (markers), in comparison with the respective Fulcher-Vogel profile (lines). (b) Calibrated results for the SiGe phase diagram (markers), along with the expected phase diagram (dashed lines). The horizontal bars reflect spatial compositional variations in the solidified layer. (c) Schematics of the strategy followed for phase diagram calibration. The superimposed snapshots show initial and final states of a calibrated simulation for \(T\approx 1410\) and \(x_{L}\approx 0.8\), resulting in \(x_{S}\approx 0.43\). Solid under-coordinated Si and Ge atoms (green and brown, respectively) identify the interface.
Figure 3: Comparison between atomistic FEM-KMCsL (solid lines) and non-atomistic phase-field simulations (dotted lines). (a,b) Pulsed-LA simulation setup for relaxed (left) and strained SiGe (right) with thickness \(t_{\text{SiGe}}=30\) nm, including system sizes and initial composition. (c-d) Time variation of melt depth, (e-f) maximum temperature in mesh and (g-h) post-anneal Ge profiles obtained for various laser energy densities. The laser pulse shape (filled grey areas) and power density (right axes) are also reported.
qualitatively captured in both relaxed and strained SiGe.
Further validation is provided by the results in Figure 4. These demonstrate that the maximum melt depths obtained with FEM-KMCsL simulations are also in good agreement with experimental measurements (see Methods). In particular, here we consider processes with 160 ns pulses for relaxed Si\({}_{0.76}\)Ge\({}_{0.24}\) and Si\({}_{0.42}\)Ge\({}_{0.58}\), and with 146 ns pulses for strained 30nm-thick Si\({}_{0.8}\)Ge\({}_{0.2}\) and Si\({}_{0.6}\)Ge\({}_{0.4}\). The maximum melt depths and Ge fraction for the relaxed samples were extracted from Energy Dispersive X-ray spectroscopy measurements performed after the irradiation. Data for strained samples is taken from Refs. [19; 66].
### LA simulations with stacking faults
The KMCsL formulation allows to study the impact of extended stacking defects on the final morphology of laser-annealed materials (see Methods). As an example, in Figure 5 an LA process for strained 30nm Si\({}_{0.6}\)Ge\({}_{0.4}\) on Si is considered (146 ns, 1.3 J cm\({}^{-2}\)), where a \(\sim\)10 nm-deep triple stacking fault exists in the sample prior to laser irradiation. This introduces three hexagonally stacked (1 1 1) atomic layers into the cubic SiGe structure. All cubic undercoordinated surface atoms in the KMCsL box before melting are shown in Figure 5a, along with the bulk ones enclosing the (1 1 1) layers. The S/L interface at the maximum melt depth and after full solidification are shown in Figure 5b-c (see also Supplementary Movie S2). We find that part of the defect is melted along with 7-8 nm of SiGe, without significant impact on the melting kinetics, and that a strongly inhomogeneous solidification is triggered by the unmelted hexagonal sites. Liquid atoms in direct contact with them indeed solidify much slower than the others, favouring {1 1 1} faceting of the S/L interface (see Figure 5a-c). Segregation is observed in both cubic and hexagonal crystal phases (see insets in Figure 5). For higher energy densities, the defect is fully melted and a purely cubic phase planar solidification occurs as usual.
### LA simulations with nanostructured and constrained geometries
The previous example reveals another important feature of KMCsL, i.e., the crystal-orientation-dependent kinetic evolution of the S/L interface. Such an atomistic feature is essential to model LA of SiGe systems with nanostructured and/or constrained geometries, which often involve reshaping and faceting of both liquid and solid volumes throughout the process. An example is illustrated in Figure 6a-e. It considers an LA process (22 ns, 0.95 J cm\({}^{-2}\)) of a SiGe system similar to those used in vertical nanostructured channel arrays [55], namely a 30nm-thick strained Si\({}_{0.6}\)Ge\({}_{0.4}\) on Si with a 9nm-large and 10nm-high Si\({}_{0.6}\)Ge\({}_{0.4}\) nanowire on top. The latter is embedded in SiO\({}_{2}\), which does not melt during the irradiation and therefore represents a geometrical constraint for the evolving S/L interface. An energy density of 0.95 J cm\({}^{-2}\) is chosen to keep melting within the KMCsL box, which is \(27\times 27\times 41\) nm\({}^{3}\) and includes \(\sim\)23 nm of the SiGe layer, the nanowire, the oxide and \(\sim\)8 nm of air (see green dashed lines in Figure 6a). Figure 6b-c shows snapshots of the S/L interface at various instants during melting and solidification (see also Supplementary Figure S3 and Movie S3). The circular nanowire tip exposed to the laser rapidly absorbs heat and melts all the way down to the bottom of the oxide. In this process, the shape of the S/L interface is already {1 1 1}-faceted. After complete melting of the nanowire, the nanodroplet reshapes into a half-octahedron below the oxide, which then coalesces with its periodic images, giving rise to a rough liquid layer, similar to what occurs in simulations of Si LA processes assuming inhomogeneous nucleation [50]. Thereafter, the S/L interface flattens and moves towards the initial surface level. While constrained by the oxide, Ge segregation occurs, causing a total transformation of the initial SiGe nanowire into a pure Ge nanowire. The final Ge distribution in the FEM mesh is illustrated in Figure 6d and in Figure 6e as a (1 1 0) cut-plane. These figures show the rough shape of the interface at the maximum melt depth and highlight a slight tendency towards solidification of the nanowire shell, before its core (also noticeable in the KMCsL snapshots).
By initializing the above simulation with an additional 5 nm-thick Si\({}_{0.6}\)Ge\({}_{0.4}\) capping layer (see Figure 6f), we trigger solidification on top of the nanowire/oxide region and give rise to more pronounced solid-phase reshaping
Figure 4: Comparison between maximum melt depths simulated with FEM-KMCsL (circles) and experimental measurements (squares), for various energy densities and Ge fractions. Relaxed (strained) samples are irradiated with a 308 nm, 160 ns (146 ns) pulse. Dashed lines guide the eye through processes on the same sample.
effects. This time we included \(\sim\)14 nm of SiGe, the nanowire, the oxide, the capping layer and \(\sim\)11 nm of air within the KMCsL box, and used 1.2 J cm\({}^{-2}\) as energy density (enough to avoid coalescence of molten nuclei). The kinetic evolution in Figure 6g-h (see also Supplementary Movie S4) and the final Ge distributions in Figure 6i-j reveal that the S/L interface initiates solidification with a non-planar shape and assumes a highly symmetrical {111}-faceted pyramidal shape as it emerges above the oxide. This solid seed gradually expands and partially coalesces with its periodic replicas, while concurrently segregating Ge.
### LA simulations with polymorphic solidification
As a further demonstration of the potential of KMCsL for LA simulations, in Figure 6k-l we report on the results of the previous simulation obtained while allowing for polymorphic cubic-hexagonal stacking transitions during solidification. Figure 6k depicts the cubic undercoordinated KMCsL sites at the end of LA simulations performed by varying the probability \(P\) of switching the stacking order (see Methods). Figure 6l shows all hexagonally stacked atoms superimposed to the cubic ones (semi-transparent), viewed along the [110] direction to highlight the presence of {111} atomic layers. We find significantly intermixed stacking motifs, even for very small probabilities. A high concentration of stacking faults (both single and triple) characterizes the oxide-embedded region, suggesting a clear correlation between confinement and stacking disorder. Noteworthy, the pyramidal shape of the final surface is quite robust against polymorphic disorder (see also Supplementary Figure S3 and Movie S5-S6).
## II Conclusions
We have presented a new multiscale approach to model LA processes of group IV materials and alloys, including complex 3D shape modifications, liquid and solid-phase faceting, species redistribution, stacking disorder and extended defects. It is based on the self-consistent combination of a continuum FEM-based solver for light-matter interaction and thermal diffusion with a KMCsL code. The latter simulates the kinetic evolution of liquid-solid interface and lattice defects in a local region of the material with atomic resolution, enabling studies so far inaccessible to purely continuous simulation approaches. In particular, we have described the theoretical background and computational implementation of the methodology in light of its application to the Si\({}_{1\text{-x}}\)Ge\({}_{\text{x}}\) alloy, which represents one of the most promising candidates for 3D sequentially integrated devices [28, 56, 57, 58], spin-qubits [65], Gate-All-Around transistors [54, 55, 59] or even direct-bandgap light emitters [61]. The method was validated by comparing simulations for both relaxed and strained SiGe with 1D phase-field results and experiments. It quantitatively reproduces the same melt depth profiles and qualitatively captures laser-induced Ge redistribution. KMCsL has the advantage of avoiding the typical numerical instabilities of approaches based on phase-field, especially at the onset and the end of melting. The code was applied to simulate pulsed-LA processes of blanket and nanostructured SiGe systems, including effects of extended defects and geometrical constraints. The possibility of studying the impact of extended defects and polymorphic solidification was demonstrated, and a clear correlation between bulk structural disorder and post-irradiation surface morphology was observed.
Importantly, the methodology is implemented into
Figure 5: Multiscale LA simulations (308 nm, 146 ns, 1.3 J cm\({}^{-2}\)) for strained Si\({}_{0.6}\)Ge\({}_{0.4}\) on Si with pre-existing triple stacking fault. Undercoordinated cubic solid atoms in the KMCsL box are shown at three instants during the process: (a) before melting, (b) at the maximum melt depth and (c) at the end of solidification. Hexagonally stacked atoms (regardless of coordination) at all instants are shown in the insets.
Figure 6: (a-e) Pulsed-LA simulation (308 nm, 22 ns, 0.95 J cm\({}^{-2}\)) of 30nm strained Si\({}_{0.6}\)Ge\({}_{0.4}\) with 9nm-large and 10nm-high Si\({}_{0.6}\)Ge\({}_{0.4}\) NWs on top, embedded in non-melting SiO\({}_{2}\). (a) Input FEM periodic mesh. KMCsL-coupled SiGe regions are shown in blue, air in red, SiO\({}_{2}\) and non-KMCsL-coupled regions in grey. The KMCsL cell extension along \(z\) is indicated with dashed green lines. (b) Overlapped selected snapshots of the liquid-solid interface in the KMCsL box at various instants during melting and (c) solidification. Green (brown) spheres indicate Si (Ge) atoms. (d) 3D view and (e) (11 0) cut-plane of the final Ge distribution in the FEM mesh. Regions outside the KMCsL cell (below the green dashed line) appear uniformly coloured because no KMCsL mapping occurs therein. The initial surface morphology is indicated by dashed black lines. (f-j) Simulation at 1.2 J cm\({}^{-2}\) energy density for the same system as above, including a 5nm-thick Si\({}_{0.6}\)Ge\({}_{0.4}\) capping layer. (k) Cubic undercoordinated sites in the KMCsL box at the end of the simulations performed at the same conditions as (f-j), but with different polymorphic solidification probabilities \(P\). (l) Hexagonally stacked sites (regardless of coordination) from the simulations in (k), viewed along the \([1\,1\,0]\) direction. Cubic sites from (k) are redrawn in semi-transparency.
an open-source versatile tool which offers several opportunities in terms of potential generalizations. The unique KMCsL super-lattice framework, enabling the coexistence of multiple crystal arrangements in the same simulation box, is readily applicable to other elemental or compound group IV semiconductor with sp\({}^{3}\) bond symmetry, e.g., Si, Ge or SiC [50; 70]. By tailoring the crystal symmetries, it could be generalized to other binary alloys (e.g., GeSn) [71], compound semiconductors (e.g., GaAs, AlGaAs) or polymorphic metal/semiconductor systems (e.g., NiSi, PtSi). Future KMCsL developments may broaden the kinetic landscape by including liquid-phase diffusion events and strain relaxation events. With properly calibrated FEM optical and thermal parameters, lasers with different wavelengths could be studied. Continuous-wave and scanning LA processes could also be investigated by adjusting the input profile of laser power density.
Framing the FEM-KMCsL strategy into a broader multiscale perspective, one may envision advanced coupling with other _ab initio_, molecular dynamics or transport simulation tools, e.g., to account for strain relaxation and interactions between extended defects [72] or investigate the impact of ultrafast processing on device components [73]. A similar multiscale approach could be used to study processes where other physical variables govern the atomic kinetics (e.g. strain, charge, polarization, magnetization), or where phase transitions are triggered by different ultrafast external stimuli (e.g. electric, magnetic or strain perturbations) [74]. This could provide interesting insights in various research areas, from silicidation [15] to multiferroics [75] or phase-change resistive-switching materials for neuromorphic computing and high-speed photonic-based devices [76; 77].
## Methods
### Multiscale implementation
The multiscale simulation tool developed in this work is distributed as part of the open-source MulSKIPS simulation package [78; 79]. This includes a core KMCsL code built on a peculiar super-lattice framework which enables simultaneous modelling of cubic and hexagonal crystal phases in the same simulation cell. Such a functionality is critical for LA simulations of multi-element systems including non-ideal stacking and polymorphism. With the appropriate particle/event definition and calibration, it can also simulate epitaxial growth by physical or chemical vapour deposition [70]. The KMCsL model, coded in Fortran, is internally coupled to a FEM-based solver coded in Python with the Dolfin interface of the FEniCS computing platform [80] (the same solver is used for the benchmark 1D phase-field simulations). The PyMuLSKIPS Python library, distributed with MulSKIPS, manages all simulation workflows and includes an I/O interface coupling the KMCsL simulator to the FEM solver and external Technology Computer-Aided Design (TCAD) tools. In particular, _ad-hoc_ Application Programming Interfaces are implemented to manage the multi-process shared-memory execution of simulations via F2Py sockets [81; 82]. This ensures a real-time communication of all relevant geometrical and physical information between the different simulation frameworks. By allowing a single KMCsL process to run over the entire simulation, it also enables a continuous tracking of species position and bonding configurations, which is crucial for simulating the evolution of extended defects or polymorphic domains across consecutive FEM-KMCsL cycles (more details in Supplementary Note S1).
### KMCsL model
The KMCsL model is defined on a dense cubic super-lattice able to accommodate both cubic and hexagonal diamond lattices as sub-lattices. The super-lattice constant is \(a_{\text{KMCsL}}\equiv a/12\equiv l/\sqrt{27}\), with \(a\) the diamond lattice constant and \(l\) its nearest neighbour distance (0.543 nm and 0.235 nm for Si, respectively). This definition makes it readily applicable to any elemental, compound and alloy material with sp\({}^{3}\) (tetrahedral) bond symmetry, such as Si, Ge or SiGe, including non-ideal stacking configurations. Each super-lattice site is marked as either solid or liquid site and can have coordination \(n\leq 4\) (for \(n=4\) they are marked as bulk). In case of Si\({}_{1\text{-x}}\)Ge\({}_{\text{x}}\), the two atomic species are randomly allocated in the lattice of the input structure, reflecting the user-defined Ge fraction \(x\). To reduce calibration parameters and memory consumption, thus improving scalability, a real atomic occupancy is strictly considered only in the solid phase (the accuracy of this approach was already demonstrated for Si LA simulations [50]). In case of SiGe alloys, while the solid-phase Ge fraction can be described as a local time-dependent variable, \(x_{S}\equiv x_{S}(\mathbf{r},t)\), the liquid-phase Ge fraction is averaged over the liquid volume at each ns-long KMCsL cycle \(x_{L}\equiv x_{L}(t)\) (more details in Supplementary Note S3). In a partially-melted system, the kinetic evolution of the liquid-solid interface is governed by the balance between solidification and melting events. These are stochastically selected by a continuous time algorithm [50] and only involve under-coordinated (\(n<4\)) super-lattice sites. We note that no kinetics occurs in the bulk, no matter if solid or liquid. Diffusion events are not explicitly defined in the KMCsL framework, but they can be effectively reproduced by close melting/solidification events nearby the interface. The solidification and melting event rates are thermally activated and therefore obey Arrhenius-like expressions, with prefactors and exponents depending on temperature \(T\) and bond coordination \(n\)[50]. In the
case of SiGe alloys, they also explicitly depend on the fraction \(X^{i}\) of individual species in the liquid phase, with \(X^{i}=x_{L}\) for Ge and \(X^{i}=1-x_{L}\) for Si. In particular, the solidification (melting) event rate \(\nu_{\text{LS}}^{i}\) (\(\nu_{\text{SL}}^{i}\)) for species \(i=\text{Si,Ge}\) on a site with \(n=n_{\text{Si}}+n_{\text{Ge}}\) solid neighbours is defined as:
\[\nu_{\text{LS}}^{i} =f^{i}(T)\cdot X^{i}\cdot\nu_{0}^{i}\cdot\exp\left(-\frac{2E_{ \text{LS}}^{i}(n)}{k_{B}T_{M}^{i}}\right) \tag{1}\] \[\nu_{\text{SL}}^{i} =\nu_{0}^{i}\cdot\exp\left(-\frac{nE_{\text{SL}}^{i}(n_{\text{Si} },n_{\text{Ge}})}{k_{B}T}\right) \tag{2}\]
where \(\nu_{0}^{i}\) is a species-dependent constant prefactor, \(T_{M}^{i}\) is the melting temperature of species \(i\) and \(k_{B}\) is the Boltzmann constant. The solidification rate increases with \(X^{i}\) and a damping term \(f^{i}(T)=1/2[1+\text{erf}((T-T_{0}^{i})/A^{i})]\), with \(T_{0}^{i}\) and \(A^{i}\) adjustable parameters, effectively models the reduction in solidification velocity under strong undercooling, as predicted by the Fulcher-Vogel expression [43, 83]. \(E_{\text{LS}}^{i}(n)\) is the \(n\)-dependent solidification energy barrier for species \(i\), whereas \(E_{\text{SL}}^{i}(n_{\text{Si}},n_{\text{Ge}})\) is the melting energy barrier, which is equivalent to a binding energy and hence depends on the number and identity of nearest neighbours. We note that the event rates \(\nu_{\text{LS}}^{i}\equiv\nu_{\text{LS}}^{i}(t,\mathbf{r})\) and \(\nu_{\text{SL}}^{i}\equiv\nu_{\text{SL}}^{i}(t,\mathbf{r})\) vary with time \(t\) and lattice position \(\mathbf{r}\) during the LA simulations. This stems from the fact that \(T\equiv T(t,\mathbf{r})\) as well as \(X^{i}\equiv X^{i}(t)\) are evaluated at every FEM-KMCsL cycle. All parameters in Equation (3) are determined by calibrating pure Si and Ge systems against their Fulcher-Vogel curves (Figure 2a). The melting energy barriers, \(E_{\text{SL}}^{i}(n_{\text{Si}},n_{\text{Ge}})\), are determined by calibrating SiGe against its experimental lens-shaped phase diagram (Figure 2b). The precise expressions for the energy barriers and the detailed procedure for calibration are reported in Supplementary Note S2.
Solidification at a one-coordinated site can either follow the stacking order dictated by its local atomic environment or it can break it, according to a user-defined probability \(P\in[0,1]\) (\(P=1\) (\(P=0\)) means that only cubic (hexagonal) stacking is allowed) [79]. Processes where the interface kinetics is such to destabilize higher-coordinated solid sites, in favor of lower-coordinated ones, are more prone to the formation of stacking defects.
The occupancy, coordination and bonding configuration of any bulk solid site in the super-lattice is constantly accessible in the shared-memory environment over successive FEM-KMCsL cycles. This is crucial to keep track of the amount of solid and liquid species over time, which in turn allows to compute \(x_{L}(t)\) and \(x_{S}(\mathbf{r},t)\), map the latter into the FEM solver, ensure mass conservation and track the evolution of vacancies (KMCsL voids with \(n=4\) three-coordinated solid neighbours) and stacking disorder throughout the simulation.
#### FEM model
Continuum modelling in this work consists in using finite element methods (FEM) to solve the heat equation self-consistently with the time-harmonic solution of Maxwell's equations on a mesh, including phase, temperature and alloy-fraction-dependent material parameters [5]. In the context of the multiscale methodology, the mesh is 3D and the solid/liquid phase changes and species redistribution in the liquid and solid phases are modelled by means of KMCsL. In the non-atomistic simulations carried out for consistency and validation analyses, the mesh is 1D and a mixed enthalpy/phase-field formalism is adopted instead [5, 41]. The enthalpy formalism is used for \(T<T_{M}(x)\), while the phase-field formalism is used for \(T>T_{M}(x)\), with \(T_{M}(x)\) being the melting temperature expected from the phase diagram.
The mesh is always initialized from a TCAD geometry with size of \(\sim\)20 um along the direction \(z\) of irradiation and 10-30 nm along the lateral \(x\) and \(y\) (periodic) directions. A 200 nm-thick layer of air is included above the initial surface. The KMCsL-coupled subregion typically includes the top 30-150 nm of the surface and a few-nm layer of air (2 to 10 nm), which needs to be thick enough to accommodate possible resolidification of the material above the initial surface level. The mesh resolution in the KMCsL-coupled subregion is 1-1.5 nm and gradually becomes coarser far from it, until it reaches the mesh lateral size in the top and bottom of the mesh.
The optical and thermal parameters for Si and SiO\({}_{2}\) are taken from Ref. [43], while those for Ge are taken from Refs. [41]. The behavior of SiGe alloy is nearly ideal (Raoultian) [51], hence most of its properties are well approximated by a linear interpolation of pure Si and Ge ones. We recently found that the dielectric function of liquid and solid SiGe requires a more careful definition to properly capture experimentally measured reflectivities [84]. In this work, only for the liquid SiGe dielectric function, we choose for simplicity to use a linear interpolation between pure Si and Ge weighted on \(x_{L}\), rescaled _ad-hoc_ by a factor \(a_{i}\approx 1\) to capture the reflectivity reported in [84]. The \(a_{i}\) and corresponding reflectivity values are reported in Supplementary Table S4.
The models employed for relaxed and strained SiGe only differ in the initial Ge profile and in the definition of solid-phase thermal conductivity. For strained SiGe we refer to the known expression for pure silicon [43]. This is justified by the small thickness of SiGe layers compared to the substrate's in our simulations. For relaxed SiGe we use the expression reported in Ref. [85].
#### Experiments
Relaxed thick Si\({}_{1\text{.x}}\)Ge\({}_{\text{x}}\) samples were prepared from two 200 mm bulk Si(0 0 1) wafers (Czochralski, p-type,
1-50 \(\Omega\) cm). The epitaxy process was performed by Reduced Pressure Chemical Vapor Deposition in a Centura 5200C chamber from Applied Materials. Prior to each SiGe layer epitaxy, a \(H_{2}\) bake (1373 K, 2 min) was done to remove the native oxide. After the surface cleaning, a graded SiGe buffer layer was grown on each wafer with a 10% / \(\mathrm{\SIUnitSymbolMicro m}\) ramp (1173 K for one wafer and 1123 K for the other, P = 20 Torr, precursors: SiH\({}_{2}\)Cl\({}_{2}\) + GeH\({}_{4}\)). Then, 1.2 \(\mathrm{\SIUnitSymbolMicro m}\) thick relaxed and undoped SiGe layers were grown with a uniform Ge content, corresponding to that of the buffer layer underneath. Thanks to the high temperature used during the process, the glide of the threading arms of misfit dislocations (i.e. threading dislocations) was enhanced in such way that they remained mostly confined in the graded buffer layers, close to the SiGe/Si interface. As a result, the threading dislocations density was significantly reduced in the SiGe top layers (\(\sim\)10\({}^{5}\) cm\({}^{2}\)). Following the RPCVD process, the remaining cross-hatch patterns were removed using a two steps (planarization and smoothing) Chemical-Mechanical Polishing process thanks to a Mirra CMP system from Applied Materials, reducing the thickness of the SiGe top layers from 1.2 \(\mathrm{\SIUnitSymbolMicro m}\) to (\(\sim\)0.7 \(\mathrm{\SIUnitSymbolMicro m}\)). Nanosecond Laser Annealing was performed with a SCREEN-LASSE (LT-3100) UV laser (\(\lambda\) = 308 nm, single pulse, pulse duration = 160 ns, 4 Hz repetition rate, \(<\) 3 % laser beam uniformity, \(10\times 10\)\(\mathrm{\SIUnitSymbolMicro m}^{2}\) laser beam) at room temperature and atmospheric pressure, with an constant incident \(\mathrm{N}_{2}\) flux to strongly limit the oxygen incorporation. The Ge composition of laser irradiated SiGe layers was measured with well-calibrated [84] Energy Dispersive X-ray spectroscopy (EDX) in a Transmission Electron Microscope JEM-ARM200F Cold FEG equipped with a EDX SDD CENTURIO-X detector from JEOL. X-ray signals in selected areas have been quantified via the Cliff and Lorimer factor method to extract Ge content profiles as function of depth. The cross-section lamellas were fabricated by Focused Ion Beam in a Helios 450S Scanning Tunneling Electron Microscope from FEI.
## Code availability
The code presented in this work is fully open-source. It is available on Github and can be accessed via this link: [https://github.com/MulSKIPS](https://github.com/MulSKIPS).
## Data availability
The source code and minimal input datasets needed to replicate the findings reported in the article are available on Github and can be accessed via this link: [https://github.com/MulSKIPS/MulSKIPS/tree/main/examples/ex4-LA-SiGe](https://github.com/MulSKIPS/MulSKIPS/tree/main/examples/ex4-LA-SiGe).
## Acknowledgements
The authors thank the European Union's Horizon 2020 Research and Innovation programme under grant agreement No. 871813 MUNDFAB, and the European Union's NextGenerationEU under grant agreement CN00000013 - NATIONAL CENTRE FOR HPC, BIG DATA AND QUANTUM COMPUTING for computational support.
## Author contributions
G.C. and A.L.M. conceived the multiscale strategy and developed the code. G.C. performed the multiscale simulations, prepared the figures and wrote the main text. D.Raciti, G.C. and G.F. calibrated the atomistic part of the code. D.Ricciarelli and I.D. calibrated the continuum part and performed the phase-field simulations. P.A.A., F.C., R.Daubriac, R.Demoulin, J.M.H. and S.K. provided the experimental data. All authors discussed the results and reviewed the manuscript.
## Competing interests
The authors declare no competing interests.
|
2306.04525 | Analysing the Robustness of NSGA-II under Noise | Runtime analysis has produced many results on the efficiency of simple
evolutionary algorithms like the (1+1) EA, and its analogue called GSEMO in
evolutionary multiobjective optimisation (EMO). Recently, the first runtime
analyses of the famous and highly cited EMO algorithm NSGA-II have emerged,
demonstrating that practical algorithms with thousands of applications can be
rigorously analysed. However, these results only show that NSGA-II has the same
performance guarantees as GSEMO and it is unclear how and when NSGA-II can
outperform GSEMO. We study this question in noisy optimisation and consider a
noise model that adds large amounts of posterior noise to all objectives with
some constant probability $p$ per evaluation. We show that GSEMO fails badly on
every noisy fitness function as it tends to remove large parts of the
population indiscriminately. In contrast, NSGA-II is able to handle the noise
efficiently on \textsc{LeadingOnesTrailingZeroes} when $p<1/2$, as the
algorithm is able to preserve useful search points even in the presence of
noise. We identify a phase transition at $p=1/2$ where the expected time to
cover the Pareto front changes from polynomial to exponential. To our
knowledge, this is the first proof that NSGA-II can outperform GSEMO and the
first runtime analysis of NSGA-II in noisy optimisation. | Duc-Cuong Dang, Andre Opris, Bahare Salehi, Dirk Sudholt | 2023-06-07T15:33:54Z | http://arxiv.org/abs/2306.04525v1 | # Analysing the Robustness of NSGA-II under Noise
###### Abstract
Runtime analysis has produced many results on the efficiency of simple evolutionary algorithms like the (1+1) EA, and its analogue called GSEMO in evolutionary multiobjective optimisation (EMO). Recently, the first runtime analyses of the famous and highly cited EMO algorithm NSGA-II have emerged, demonstrating that practical algorithms with thousands of applications can be rigorously analysed. However, these results only show that NSGA-II has the same performance guarantees as GSEMO and it is unclear how and when NSGA-II can outperform GSEMO.
We study this question in noisy optimisation and consider a noise model that adds large amounts of posterior noise to all objectives with some constant probability \(p\) per evaluation. We show that GSEMO fails badly on every noisy fitness function as it tends to remove large parts of the population indiscriminately. In contrast, NSGA-II is able to handle the noise efficiently on LeadingOnesTrainingZeroes when \(p<1/2\), as the algorithm is able to preserve useful search points even in the presence of noise. We identify a phase transition at \(p=1/2\) where the expected time to cover the Pareto front changes from polynomial to exponential. To our knowledge, this is the first proof that NSGA-II can outperform GSEMO and the first runtime analysis of NSGA-II in noisy optimisation.
## 1 Introduction
Decision making is ubiquitous in everyday life, and often can be formalised as an optimisation problem. In many situations, one may want to examine the trade-off between compromises before making a decision. Sometimes these compromises cannot be accurately evaluated, i. e. due to a lack of available information when a decision has to be made. These two critical but practical settings correspond to the areas Multi-Objective Optimisation (MOO) and Optimisation under Uncertainty (OUU), respectively, which have been studied in both Economics, Operational Research, and Computer Science [3, 6, 28, 32, 33, 42, 46, 50].
MMO is an area where evolutionary multi-objective (EMO) algorithms have shown to be among the most efficient optimisation techniques [49]. Particularly, the Non-dominated Sorting Genetic Algorithm (NSGA-II) is a highly influential framework to build algorithms for MMO, and the original paper [17] is one of the most highly cited papers in evolutionary computation and beyond.
Recently, NSGA-II was analysed by rigorous mathematical means using _runtime analysis_. In a nutshell, runtime analysis studies the performance guarantees and drawbacks of randomised search heuristics, like Evolutionary Algorithms (EAs), from a Computer Science perspective [34]. The basic approach is to bound the expectation of the random running time \(T\) (number of iterations or function evaluations) of a given algorithm on a problem until a global optimum is found in case of a single objective. Extending this to MMO, \(T\) is the time to find and cover the whole _Pareto front_. The algorithm is said to be _efficient_ if this expectation is polynomial in the problem size, and it is _inefficient_ if the expectation is exponential.
Runtime analyses have led to a better understanding of the capabilities and limitations of EAs, for example concerning the advantages of population diversity [14], the benefits of using crossover [20, 48, 9, 40], or the robustness of populations in stochastic optimisation [12, 31, 38, 44]. It can give advice on how to set algorithmic parameters; it was used to identify phase transitions between efficient and inefficient running times for parameters of common selection operators [37], the offspring population size in comma strategies [47] or the mutation rate for difficult monotone pseudo-Boolean functions [22, 41]. Runtime analysis has also inspired novel designs for EAs with practical impacts, e. g. choosing mutation rates from a heavy-tailed distribution to escape from
local optima [23], parent selection in steady-state EAs [8], selection in non-elitist populations with power-law ranking [10], or choosing mutation rates adaptively throughout the run [21, 39, 45].
Runtime analyses in MOO started out with the simple SEMO algorithm and the global SEMO (GSEMO) [36, 30]. Both algorithms keep non-dominated solutions in the population. If a new offspring \(x\) is created that is not dominated by the current population, it is added to the population and all search points that are weakly dominated by \(x\) are removed. Despite its simplicity, it was shown to be effective in AI applications, e. g. [43], where it is called PO(R)SS.
The first theoretical runtime analysis of NSGA-II (without crossover) was performed by Zheng et al. [53]. They showed that NSGA-II covers the whole Pareto front for test functions LOTZ and OMM (see Section 2) in expected \(O(\mu n^{2})\) and \(O(\mu n\log n)\) function evaluations, respectively, where \(\mu\) is the population size and \(n\) is the problem size (number of bits). These results require a population of size \(\mu\geq 4(n+1)\), hence the best upper bounds are \(O(n^{3})\) and \(O(n^{2}\log n)\), respectively, that also apply to (G)SEMO [36, 30].
This breakthrough result spawned several very recent papers and already led to several new insights and proposals for improved algorithm designs. Bian and Qian [4] proposed a new parent selection mechanism called _stochastic tournament selection_ and showed that NSGA-II equipped with this operator covers the Pareto front of LOTZ in expected time \(O(n^{2})\). Zheng and Doerr [51] proposed to re-compute the crowding distance during the selection process and proved (using LOTZ as a test case) that this improved the spread of individuals on the Pareto front. Doerr and Qu [24] proposed to use heavy-tailed mutations in NSGA-II and quantified the speedup on multimodal test problems. Doerr and Qu [26] and Dang et al. [15] independently demonstrated the advantages of crossover in NSGA-II. In terms of limitations, Zheng and Doerr [52] investigated the inefficiency of NSGA-II for more than two objectives and Doerr and Qu [25] gave lower bounds for the running time of NSGA-II.
Despite these rapidly emerging research works, one important research question remains open. So far all comparisons of NSGA-II with GSEMO show that NSGA-II has the same performance guarantees as GSEMO. Even though NSGA-II is a much more complex algorithm, we do not have an example where NSGA-II was proven to outperform the simple GSEMO algorithm; thus we have not yet unveiled the full potential of NSGA-II.
Here we provide such an example from noisy optimisation.
**Our contribution**: We show that NSGA-II can drastically outperform GSEMO on a noisy LeadingOnS-TrailingZeroes test function. To this end, we introduce a deliberately simple posterior noise model called the \((\delta,p)\)-Bernoulli noise model, in which a fixed noise \(\delta\in\mathbb{R}\) is added to the fitness in all objectives and in each evaluation with some noise probability \(p\). When \(\delta\) is positive and sufficiently large, for maximisation problems every noisy solution always dominates every noise-free solution. In this setting, we prove in Theorem 3 that it is difficult for GSEMO to grow its population hence the algorithm is highly inefficient under noise on arbitrary noisy fitness functions. In contrast, for the noise model with a constant \(p<1/2\), we show in Theorem 8 that NSGA-II is efficient on the noisy LeadingOnS-TrailingZeroes function, if its population size is sufficiently large. This result can be easily extended to other functions. The reason for this performance gap is that NSGA-II keeps dominated solutions in its population while GSEMO immediately removes them. We also prove in Theorem 10 that the behaviour of NSGA-II without crossover dramatically changes for the noise probability slightly above \(1/2\), i. e. it suddenly becomes inefficient. Our theoretical results are complemented with empirical results on both the Bernoulli noise model and an additive Gaussian noise, which confirm the advantageous robustness of NSGA-II over GSEMO. As far as we know, this is the first proof that NSGA-II can outperform GSEMO, and the first runtime analysis of NSGA-II under uncertainty.
## 2 Preliminaries
By \(\log(\cdot)\) we denote the logarithm of base \(2\). \(\mathbb{R}\), \(\mathbb{Z}\) and \(\mathbb{N}\) are the sets of real, integer and natural numbers respectively. For \(n\in\mathbb{N}\), define \([n]:=\{1,\ldots,n\}\) and \([n]_{0}:=[n]\cup\{0\}\). We use \(\vec{1}\) to denote the all-ones vector \(\vec{1}:=(1,\ldots,1)\). For a bit string \(x:=(x_{1},\ldots,x_{n})\in\{0,1\}^{n}\), we use \(|x|_{1}\) to denote its number of \(1\)-bits, i. e.
\(|x|_{1}=\sum_{i=1}^{n}x_{i}\), and similarly \(|x|_{0}\) to denotes its number of zeroes, i. e. \(|x|_{0}=\sum_{i=1}^{n}(1-x_{i})=n-|x|_{1}\). We use standard asymptotic notation with symbols \(O,\Omega,o\)[7].
This paper focuses on the multi-objective optimisation in a discrete setting, specifically the maximisation of a \(d\)-objective function \(f(x):=(f_{1}(x),\dots,f_{d}(x))\) where \(f_{i}\colon\{0,1\}^{n}\to\mathbb{Z}\) for each \(i\in[d]\). We define \(f_{\min}:=\min\{f_{i}(x)\mid i\in[d],x\in\{0,1\}^{n}\}\), and \(f_{\max}:=\max\{f_{i}(x)\mid i\in[d],x\in\{0,1\}^{n}\}\).
**Definition 1**.: _Consider a \(d\)-objective function \(f\):_
* _For_ \(x,y\in\{0,1\}^{n}\)_, we say_ \(x\)__weakly dominates__\(y\) _written as_ \(x\succeq y\) _(or_ \(y\preceq x\)_) if_ \(f_{i}(x)\geq f_{i}(y)\) _for all_ \(i\in[d]\)_;_ \(x\)__dominates__\(y\) _written as_ \(x\succ y\) _(or_ \(y\prec x\)_) if one inequality is strict._
* _A set of points which covers all possible fitness values not dominated by any other points in_ \(f\) _is called Pareto front._ _A single point from the Pareto front is called_ Pareto optimal_._
The weakly-dominance and dominance relations are _transitive_, e. g. \(x\succ y\wedge y\succ z\) implies \(x\succ z\). When \(d=2\), the function is referred as _bi-objective_. Two basic bi-objective functions studied in theory of evolutionary computation are LeadingOnesTrailingZeroes and OneMinMax, which can be shortly written as LOTZ and OMM respectively. In \(\textsc{LOTZ}(x)\coloneqq(\textsc{LO}(x),\textsc{TZ}(x))\) we count the number \(\textsc{LO}(x)\) of leading ones in \(x\) (the length of the longest prefix containing only ones) and the number \(\textsc{TZ}(x)\) of trailing zeros in \(x\) (the length of the longest suffix containing only zeros). OMM minimises and maximises the number of ones: \(\textsc{OMM}(x):=(|x|_{1},|x|_{0})\).
```
1Initialize \(P_{0}\sim\mathrm{Unif}((\{0,1\}^{n})^{\mu})\) Partition \(P_{0}\) into layers \(F_{0}^{1},F_{0}^{2},\dots\) of non-dominated fitnesses, then for each layer \(F_{0}^{i}\) compute the crowding distance \(\textsc{clDist}(x,F_{0}^{i})\) for each \(x\in F_{0}^{i}\) for\(t:=0\prec\infty\)do
2 Initialize \(Q_{t}:=\emptyset\)for\(i:=1\to\mu/2\)do
3 Sample \(p_{1}\) and \(p_{2}\), each by a binary tournament Sample \(u\sim\mathrm{Unif}([0,1])\)if\(u<p_{c}\)then
4 Create \(s_{1},s_{2}\) by crossover on \(p_{1},p_{2}\)
5
6else
7 Create \(s_{1},s_{2}\) as exact copies of \(p_{1},p_{2}\)
8 Create \(s_{1}^{\prime}\) by bitwise mutation on \(s_{1}\) with rate \(1/n\) Create \(s_{2}^{\prime}\) by bitwise mutation on \(s_{2}\) with rate \(1/n\) Update \(Q_{t}:=Q_{t}\cup\{s_{1}^{\prime},s_{2}^{\prime}\}\)
9 Set \(R_{t}:=P_{t}\cup Q_{t}\) Partition \(R_{t}\) into layers \(F_{t+1}^{1},F_{t+1}^{2},\dots\) of non-dominated fitnesses, then for each layer \(F_{t+1}^{i}\) compute \(\textsc{clDist}(x,F_{t+1}^{i})\) for each \(x\in F_{t+1}^{i}\) Sort \(R_{t}\) lexicographically by \((1/i,\textsc{clDist}(x,F_{t+1}^{i}))\) Create the next population \(P_{t+1}:=(R[1],\dots,R[\mu])\)
```
**Algorithm 1**NSGA-II Algorithm [17]
NSGA-II [17, 16] is summarised in Algorithm 1 for bitwise mutation. In each generation, a population \(Q_{t}\) of \(\mu\) new offspring search points are created through binary tournament, crossover and mutation. The binary tournament in line 6 uses the same criteria as the sorting procedure in line 17 which will be detailed below. The crossover is only applied with some probability \(p_{c}\in(0,1)\) to produce two solutions \(s_{1},s_{2}\). Otherwise \(s_{1},s_{2}\) are the exact copies of the winners of the tournaments. The _bitwise mutation_ on \(s_{1},s_{2}\) creates two offspring by flipping each bit of the input independently with probability \(1/n\). Our positive result for NSGA-II (Theorem 8) holds for arbitrary crossover operators as it only relies on steps without crossover. To simplify the analysis, we assume that the tournaments for parent selection are performed independently and with replacement.
During the survival selection, the parent and offspring populations \(P_{t}\) and \(Q_{t}\) are joined into \(R_{t}\), and then partitioned into layers \(F_{t+1}^{1},F_{t+1}^{2},\dots\) by the _non-dominated sorting algorithm_[17]. The layer \(F_{t+1}^{1}\) consists of all
non-dominated points, and \(F_{t+1}^{i}\) for \(i>1\) only contains points that are dominated by those from \(F_{t+1}^{1},\ldots,F_{t+1}^{i-1}\). In each layer, the _crowding distance_ is computed for each search point, then the points of \(R_{t}\) are sorted with respect to the indices of the layer that they belong to as the primary criterion, and then with the computed crowding distances as the secondary criterion. Only the \(\mu\) best solutions of \(R_{t}\) form the next population.
Let \(M:=(x_{1},x_{2},\ldots,x_{|M|})\) be a multi-set of search points. The crowding distance \(\textsc{cDist}(x_{i},M)\) of \(x_{i}\) with respect to \(M\) is computed as follows. At first sort \(M\) as \(M=(x_{k_{1}},\ldots,x_{k_{|M|}})\) with respect to each objective \(k\in[d]\) separately. Then
\[\textsc{cDist}(x_{i},M) :=\sum_{k=1}^{d}\textsc{cDist}_{k}(x_{i},M),\text{ where} \tag{1}\] \[\textsc{cDist}_{k}(x_{k_{i}},M) :=\!\begin{cases}\infty&\text{if }i\in\{1,|M|\},\\ \frac{f_{k}(x_{k_{i-1}})-f_{k}(x_{k_{i+1}})}{f_{k}(x_{k_{1}})-f_{k}(x_{M})}& \text{otherwise.}\end{cases} \tag{2}\]
The first and last ranked individuals are always assigned an infinite crowding distance. The remaining individuals are then assigned the differences between the values of \(f_{k}\) of those ranked directly above and below the search point and normalized by the difference between \(f_{k}\) of the first and last ranked.
The GSEMO algorithm is shown in Algorithm 2. Starting from one randomly generated solution, in each generation a new search point \(s^{\prime}\) is created by crossover, with some probability \(p_{c}\in(0,1)\), and bitwise mutation with parameter \(1/n\) afterwards where parents are selected uniformly at random. If \(s^{\prime}\) is not dominated by any solutions of the current population \(P_{t}\) then it is added to the population, and those weakly dominated by \(s^{\prime}\) are removed from the population. The population size \(|P_{t}|\) is unrestricted for GSEMO.
```
1Initialize \(P_{0}:=\{s\}\) where \(s\sim\text{Unif}(\{0,1\}^{n})\)for\(t:=0\rightarrow\infty\)do
2 Sample \(p_{1}\sim\text{Unif}(P_{t})\) Sample \(u\sim\text{Unif}([0,1])\)if\(u<p_{c}\)then
3 Sample \(p_{2}\sim\text{Unif}(P_{t})\) Create \(s\) by crossover between \(p_{1}\) and \(p_{2}\)
4else
5 Create \(s\) as a copy of \(p_{1}\) Create \(s^{\prime}\) by bitwise mutation on \(s\) with rate \(1/n\)if\(s^{\prime}\) is not dominated by any individual in \(P_{t}\)then
6 Create the next population \(P_{t+1}:=P_{t}\cup\{s\}\) Remove all \(x\in P_{t+1}\) weakly dominated by \(s^{\prime}\)
```
**Algorithm 2**GSEMO Algorithm
Note that GSEMO and NSGA-II are _invariant_ under a translation of the objective function, that is, they behave identically on \(f\) and on \(f+\vec{c}\) where \(\vec{c}\) is a fixed vector.
## 3 The Posterior Bernoulli Noise Model
Since our aim is to demonstrate that NSGA-II is more robust to noise than GSEMO, we choose the simplest possible noise model under which the desired effects are evident. Our noise model is inspired by concurrent work [35]. Noise can either be present or absent, the strength of the noise is fixed, and noise is applied to all objectives uniformly. Using a simple noise model facilitates a theoretical analysis and simplifies the presentation. We will discuss possible extensions to more realistic noise models, and we will consider one further noise model (posterior Gaussian noise) in our empirical evaluation (Section 7).
In our posterior noise model, instead of optimising the real fitness \(f\), the algorithm only has access to a noisy fitness function, denoted as \(\tilde{f}\) that may return fitness values obscured by noise.
(This is different to _prior noise_[11, 13, 27], in which the search point is altered before the fitness evaluation.) In our noise model the fitness is altered by a fixed additive term \(\delta\in\mathbb{R}\) in all objectives, with some probability \(p>0\). We refer to \(|\delta|\) as the _noise strength_, and \(p\) as the _noise probability_ or _frequency_.
**Definition 2**.: _Given a noise strength \(\delta\in\mathbb{R}\) and a noise probability \(p\in[0,1]\), the noisy optimisation of a \(d\)-objective fitness function \(f\) under the \((\delta,p)\)-Bernoulli noise model has \(\tilde{f}\) defined as_
\[\tilde{f}(x):=\begin{cases}f(x)+\delta\cdot\vec{1}&\text{with probability }p,\\ f(x)&\text{otherwise.}\end{cases}\]
When \(\tilde{f}(x)=f(x)+\delta\cdot\vec{1}\) we call \(x\) a _noisy_ search point and otherwise we call it _noise-free_. Note that the _expected_ fitness vector of any search point \(x\) is
\[\operatorname{E}\left[\tilde{f}(x)\right]=f(x)+p\delta\cdot\vec{1}\]
and hence optimising \(f\) is equivalent to optimising the expectation of \(\tilde{f}\); in other words, this is equivalent to _stochastic optimisation_[6]. When \(p\in\{0,1\}\) the noisy function \(\tilde{f}\) is deterministic and equal to \(f\) apart from a possible translation by \(\delta\) in all objectives, thus NSGA-II and GSEMO will behave the same as operating on \(f\).
Since we aim to study the robustness of these original algorithms, we refrain from noise reduction techniques like re-sampling [1, 44, 5].
We assume that noise is drawn independently for all search points in a generation. This reflects a setting where noise is generated from an external source, e. g. disturbances when evaluating the fitness. We assume however that in each generation the noisy fitness values of evaluated individuals are stored temporarily for that generation. So, if the fitness of an individual is queried multiple times in the same generation, the noisy fitness value from the first evaluation in that generation is returned.
Now we obtain a specific noise model by setting \(\delta>f_{\max}-f_{\min}\). In this case, noise boosts the fitness of a search point in an extreme way; its fitness immediately strictly dominates that of every noise-free search point. For all \(\delta>f_{\max}-f_{\min}\) NSGA-II (or GSEMO) behaves identically on the noise model \((\delta,p)\) as on \((-\delta,1-p)\), because in the latter model the roles of noisy and noise-free search points are swapped and the fitness is translated by \(-\delta\cdot\vec{1}\). The latter model corresponds to a setting where noise may destroy the fitness of a search point. This scenario is closely related to practice when optimising problems with constraints that are typically met, but where noise may violate constraints and this incurs a large penalty.
## 4 GSEMO Struggles With Noise
We show that noise is hugely detrimental for SEMO and GSEMO. Since both algorithms reject all search points that are weakly dominated by a new offspring, if the offspring falsely appears to dominate good quality search points, the latter are being lost straight away. The following analysis shows that, for sufficiently large noise strengths \(\delta\), there is a good chance that creating a noisy offspring will remove a fraction of the population, irrespective of the fitness of population members. This makes it impossible to grow the population to a size necessary to cover the Pareto front of a function.
**Theorem 3**.: _Consider GSEMO on an arbitrary fitness function \(f\) with noise strength \(\delta>f_{\max}-f_{\min}\) and noise probability \(0<p<1\). For any functions \(t(n),\alpha(n)\in\mathbb{N}\), starting with an arbitrary initial population of size at most \(\lceil p\alpha(n)\rceil\), the probability of the population reaching a size of at least \(\alpha(n)\) in the first \(t(n)\) generations is at most \(t(n)\cdot(1-p/2)^{((1-p)\alpha(n)]-1}\) and the expected number of generations is at least \((1-p/2)^{-[(1-p)\alpha(n)]+1}\)._
Proof.: Let \(\mu_{t}\leq\alpha(n)-1\) denote the population size at time \(t\). We call a step \(t+1\)_shrinking_ if \(\mu_{t+1}\leq\lceil p\alpha(n)\rceil+1\). Since GSEMO only adds at most one search point to the population, the condition \(\mu_{t}\leq\lceil p\alpha(n)\rceil\) implies a shrinking step since \(\mu_{t+1}\leq\mu_{t}+1\leq\lceil p\alpha(n)\rceil+1\). Hence we assume \(\mu_{t}\geq\lceil p\alpha(n)\rceil+1\) in the following. A
sufficient condition for a shrinking step is to create a noisy offspring and to evaluate at most \(\lceil p\alpha(n)\rceil\) parents as noisy. Since the noisy offspring dominates all noise-free search points and GSEMO removes these from the population, only noisy parents may survive. The probability of the offspring being noisy is \(p\). Conditional on this event, each of the \(\mu_{t}\) search points in the population survives with probability \(p\), independently from one another. Then the number of survivors is given by a binomial distribution \(\operatorname{Bin}(\mu_{t},p)\). We have
\[\Pr\left(\operatorname{Bin}(\mu_{t},p)\leq\lceil p\alpha(n)\rceil\right)\geq \Pr\left(\operatorname{Bin}(\mu_{t},p)\leq\lceil p\mu_{t}\rceil\right)\geq 1/2\]
since the median of the binomial distribution is at most \(\lceil p\mu_{t}\rceil\).
Thus, a shrinking step occurs with probability at least \(p/2\). If \(\mu_{t}\leq\lceil p\alpha(n)\rceil+1\), the population can only grow to \(\alpha(n)\) if there is a sequence of \(\alpha(n)-(\lceil p\alpha(n)\rceil+1)=\alpha(n)+\lfloor-p\alpha(n)\rfloor-1= \lfloor(1-p)\alpha(n)\rfloor-1\) steps that are all not shrinking. The probability of such a sequence is at most \((1-p/2)^{(\lfloor 1-p)\alpha(n)\rfloor-1}\). If a shrinking step occurs, the population size drops to at most \(\lceil p\alpha(n)\rceil+1\) and we can re-iterate the argument. Taking a union bound over the first \(t(n)\) steps proves the first claim. Noticing that each attempt to reach a population size of at least \(\alpha(n)\) requires at least one evaluation, the expected number of evaluations is bounded by the expectation of a geometric random variable with parameter \((1-p/2)^{(\lfloor 1-p)\alpha(n)\rfloor-1}\), which is \((1-p/2)^{-\lfloor(1-p)\alpha(n)\rfloor+1}\).
Processes where the current state shows a multiplicative expected decrease, plus some additive term, were recently analysed in [19]. The expected change of the state was described as _negative multiplicative drift with an additive disturbance_. Lower bounds are given on the expected time to reach some target value \(M\). However, these bounds are linear in \(M\), whereas the bounds from Theorem 3 are exponential in the target \(\alpha(n)\).
Theorem 3 shows that, if \(\alpha(n)\) is chosen as the size of the smallest Pareto set, it takes GSEMO exponential expected time in \(\alpha(n)\) to reach a population that covers the whole Pareto front.
**Theorem 4**.: _Consider GSEMO on an arbitrary fitness function \(f\) for which every Pareto set has size at least \(\alpha(n)\). Then, for every constant \(p\in(0,1)\), in the \((f_{\max}-f_{\min}+1,p)\)-Bernoulli noise model, the expected time for GSEMO to cover the whole Pareto front is \(2^{\Omega(\alpha(n))}\)._
Since all Pareto sets of well-known multiobjective test functions have size at least \(n+1\), we get the following.
**Corollary 5**.: _For every constant \(p\in(0,1)\), in the \((n+1,p)\)-Bernoulli noise model the expected time for GSEMO on LeadingOnesTrailingZeroes or OneMinMax to cover the whole Pareto set is \(2^{\Omega(n)}\)._
We are confident that the arguments in this section extend to other posterior noise models, e. g. adding Gaussian posterior noise. If the noise strength is not too small, there is a good chance that the offspring might be sampled with large positive noise, and then population members with a negative noise contribution may be dominated and be removed as argued above.
Note, however, that to achieve domination, the offspring must be at least as good as a population member in all objectives. In our Bernoulli noise model, we add the same noise of \(\delta\) to all objectives. If noise is determined independently for each objective and a value of \(\delta\) is added with probability \(p\), the offspring is guaranteed to dominate every noise-free population member if noise is applied in all dimensions. For \(d\) dimensions, the probability of this event is \(p^{d}\). If \(d\) and \(p\) are both constant, this is only a constant-factor difference, and thus if adding noise uniformly to all objectives yields an exponential lower bound from Theorem 3, the same holds when adding noise independently for each objective.
## 5 NSGA-II is Robust to Noise if \(\boldsymbol{p<\frac{1}{2}}\)
For NSGA-II with the Bernoulli noise model, when \(\delta\) is sufficiently large, noisy search points do not interfere with the dynamics of non-dominated layers containing noise-free points, i. e. in the calculation of the crowding distances. This is captured by the following concepts, which are illustrated in Figure 1.
**Definition 6**.: _Let \(C\in\mathbb{Z},D\in\mathbb{N}_{0}\) be some integers, and let \(f(x):=(f_{1}(x),f_{2}(x))\) be a discrete bi-objective function._
* _A point_ \(x\in\{0,1\}^{n}\) _is called a_ \((C,D)\)_-point_ if \(f_{1}(x)=C+\ell\wedge f_{2}(x)=C+m\) for some \(\ell,m\in[D]_{0}\).
* _A point_ \(x\in\{0,1\}^{n}\) _is_ \((C,D)\)_-superior if it dominates all_ \((C,D)\)_-points, i. e._ \(f_{1}(x)>C+D\wedge f_{2}(x)>C+D\)_._
* _A multi-set_ \(P\) _of points of_ \(f\) _is called_ \((C,D)\)_-separable _if it only contains_ \((C,D)\)_-points and_ \((C,D)\)_-superior points._
To show that NSGA-II can optimise a function and cover a Pareto front, one has to prove that the progress made so far by the optimisation process is maintained and that Pareto optimal solutions are not being lost in future generations [15, 24, 53]. Such arguments were first used by [53] and later on in [4, 24, 15]. In [15] these arguments were extracted and summarised in a lemma [15, Lemma 7]. We adapt the lemma to our case as follows.
**Lemma 7**.: _Consider two consecutive generations \(t\) and \(t+1\) of the NSGA-II maximising a bi-objective function \(f(x):=(f_{1}(x),f_{2}(x))\) where \(f_{i}(x)\colon\{0,1\}^{n}\to\mathbb{Z}\) for each \(i\in\{1,2\}\), and two numbers \(C\in\mathbb{Z},D\in\mathbb{N}_{0}\) such that \(R_{t}\) (i. e. the joint parent and offspring population) is \((C,D)\)-separable. Then we have:_
* _Any layer_ \(F_{t}^{i}\) _composed of only_ \((C,D)\)_-points has at most_ \(4(D+1)\) _individuals with positive crowding distances. The same result holds for layers_ \(F_{t+1}^{i}\) _that only have_ \((C,D)\)_-points._
* _If_ \(R_{t}\) _has at most_ \(S\)__\((C,D)\)_-superior points for some_ \(S\in\mathbb{N}_{0}\) _and the population size_ \(\mu\) _satisfies_ \(\mu\geq 4(D+1)+S\)_, then the following result holds. If there is a_ \((C,D)\)_-point_ \(x\in P_{t}\)_, then there must exist a_ \((C,D)\)_-point_ \(y\in P_{t+1}\) _with either_ \(f(y)=f(x)\) _or_ \(y\succ x\)_._
Proof.: The layers of \(R_{t}\) (the union of parents and offspring) are separated between those containing \((C,D)\)-points and those having the superior ones. The layers of \((C,D)\)-superior points are higher ranked (have a lower index) than \((C,D)\)-points.
(i) It suffices to prove the result for \(F_{t+1}^{i}\) with only \((C,D)\)-points of \(R_{t}\) then it also holds for the same type of layers in \(P_{t},P_{t+1}\) because these populations are sub-multi-sets of \(R_{t}\). The remaining proof arguments for (i) are identical to those in the proof of result (i) of Lemma 7 in [15], with their \(F_{t}^{1}\) being replaced by our \(F_{t+1}^{i}\). Thus we omit it here and refer to [15, Lemma 7] for details.
We note one insight from the proof for later use. The proof shows that for each fitness vector \((a,b)\) of the layer there is at least a point with a positive crowding distance.
(ii) Let \(i^{*}\) be the smallest integer such that the layer \(F_{t+1}^{i^{*}}\) of \(R_{t}\) contains only \((C,D)\)-points, i. e. the layers \(F_{t+1}^{j}\) with \(j<i^{*}\) only contain \((C,D)\)-superior-points. The condition on \(R_{t}\) means that \(\sum_{j<i^{*}}|F_{t+1}^{j}|\leq S\), thus it follows from (i) and \(\mu\geq 4(D+1)+S\) that \(P_{t+1}\) will contain all search points from \(F_{t+1}^{i^{*}}\) with positive crowding distance, in addition to the \((C,D)\)-superior points of \(R_{t}\).
We have the following cases for the \((C,D)\)-point \(x\):
Figure 1: Illustration of \((C,D)\)-separable multi-sets. Only the two shaded areas contain search points.
Case 1: If none of the \((C,D)\)-points of \(R_{t}\) dominates \(x\), then clearly \(x\in F_{t+1}^{i^{*}}\). As remarked at the end of the proof of (i), there must exist a \(y\in F_{t+1}^{i^{*}}\) with a positive crowding distance and with \(f(y)=f(x)\). Thus \(y\) will be kept in \(P_{t+1}\).
Case 2: If some of the \((C,D)\)-points of \(R_{t}\) dominate \(x\), then let \(y\) be such a point. We may assume that \(y\) is not dominated by any other point of \(R_{t}\) because if there is a \(y^{\prime}\in R_{t}\) dominating \(y\), we can choose \(y^{\prime}\) instead of \(y\) and iterate this argument until a non-dominated point is found. This implies that \(y\in F_{t+1}^{i^{*}}\) and as in the previous case, there exists a \(y^{\prime}\in F_{t+1}^{i^{*}}\) with a positive crowding distance and with \(f(y^{\prime})=f(y)\). Thus \(y^{\prime}\) will be kept in \(P_{t+1}\) and we have \(y^{\prime}\succ x\).
Now we use Lemma 7 to show that NSGA-II can find the Pareto front of LOTZ efficiently when the noise probability is at most a constant less than \(1/2\). Roughly speaking, the result shows that with a sufficiently large population, a sub-population of NSGA-II can still evolve its noise-free search points, thus noise has a minimal effect on the optimisation process. For example, \(\mu\geq 9(n+1)\) meets the condition for all noisy LOTZ functions with \(p\leq 1/4\wedge\delta>n\) or \(p\geq 3/4\wedge\delta<-n\). We will use this setting later in our experiments.
**Theorem 8**.: _Consider the \((\delta,p)\)-model with noise strength \(\delta>n\) and constant noise probability \(p\in\left(0,\frac{1}{2(1+c)}\right)\) for some constant \(c>0\). Then NSGA-II with population size \(\mu\geq\frac{4(n+1)}{1-2(1+c)p}\) and \(p_{c}\leq 1-2^{-o(n)}\) finds and covers the whole Pareto front of noisy LOTZ in \(\mathcal{O}\left(n^{2}/(1-p_{c})\right)\) expected generations and \(\mathcal{O}\left(\mu n^{2}/(1-p_{c})\right)\) expected fitness evaluations. The result also holds for the \((-\delta,1-p)\)-model using the same conditions._
Proof.: For \(\delta>n\), the noise model guarantees that all noisy solutions dominate all noise-free solutions, thus the populations \(P_{t},Q_{t}\) and \(R_{t}\) are typically \((0,n)\)-separable with the superior points being the noisy ones. Lemma 7 (i) with \(C=0,D=n\) then implies that there are no more than \(4(n+1)\) individuals with positive crowding distances in every layer \(F_{t}^{i}\), or \(F_{t+1}^{i}\) of noise-free individuals.
Furthermore, in each generation of the algorithm, the expected number of parents in \(P_{t}\) that are noise-free after re-evaluation is \((1-p)\mu\), thus by an additive Chernoff bound [18, Theorem 1.10.7], the probability of having at least \((1-(1+c)p)\mu\) noise-free parents and conversely at most \((1+c)p\mu\) noisy ones is at least \(1-e^{-2c^{2}p^{2}\mu}=1-e^{-\Omega(n)}\). Similarly, during the evaluation of the offspring \(Q_{t}\), with probability \(1-e^{-\Omega(n)}\), there are at least \((1-(1+c)p)\mu\) noise-free solutions and at most \((1+c)p\mu\) noisy ones. If either one of these two conditions does not occur during a generation, we refer to this as a _bad event_. The probability of a bad event is at most \(2e^{-\Omega(n)}=e^{-\Omega(n)}\). Given the rarity of bad events, we apply the typical run method with the restart argument, see Chapter 5.6 of [34]. We therefore divide the run of the algorithm into two phases, each phase is associated with a goal. The phases last for at most \(T_{1}\) and \(T_{2}\) generations respectively, and have the failure probability of at most \(p_{1}\) and \(p_{2}\) respectively. A failure means that a bad event happens during the random length of a phase. As that the following analysis works for any initial population, such a failure is no worse than restarting the analysis on the resulting population. Under this assumption, the expected number of generations until the Pareto front is covered is at most \((\operatorname{E}\left[T_{1}\right]+\operatorname{E}\left[T_{2}\right])/(1-p_{1} -p_{2})\).
Phase 1: Create a first Pareto optimal solution.
Define \(i:=\max\{\operatorname{LO}(x)+\operatorname{TZ}(x)\mid x\in P_{t}\}\) and note that all Pareto optimal solutions \(x\) have \(\operatorname{LO}(x)+\operatorname{TZ}(x)=n\). Thus, the phase is completed once \(i=n\). We now consider a point \(y\in P_{t}\) so that \(\operatorname{LO}(y)+\operatorname{TZ}(y)=i\) and \(y\) has positive crowding distance, to give a lower bound for the probability of increasing \(i\) in a generation.
During a binary tournament, the probability of selecting \(y\) as the first competitor is \(1/\mu\). The probability of the other competitor being a noise-free solution with zero crowding distance is bounded from below as follows. There are at least \((1-(1+c)p)\mu\) noise-free parents in \(P_{t}\), and each noise-free layer has at most \(4(n+1)\) individuals with positive crowding distances. Thus, the sought probability is at least \((1-(1+c)p)\left(1-\frac{4(n+1)}{\mu}\right)\geq(1-(1+c)p)\left(1-\frac{4}{4(1-2 (1+c)p)}\right)=\Omega(1)\). So, even when \(y\) is noise-free the probability of it winning the tournament is at least \(2\cdot\Omega(1)\cdot(1/\mu)=\Omega(1/\mu)\) where the factor \(2\) accounts for the exchangeable roles of the competitors.
To create an offspring \(z\) that dominates \(y\) with \(\operatorname{LO}(z)+\operatorname{TZ}(z)\geq i+1\) it suffices to select \(y\) as a parent, to skip crossover and to flip one specific bit of \(y\) during mutation, while keeping the other bits unchanged (this
mutation has probability \(1/n\cdot(1-1/n)^{n-1}\geq 1/(en))\). These events occur with probability \(s_{i}:=(1-p_{c})(1/en)\cdot\Omega(1/\mu)=\Omega((1-p_{c})/(\mu n))\). During \(\mu/2\) offspring productions, the probability of creating such a solution \(z\) is \(1-(1-s_{i})^{\mu/2}\geq\frac{s_{i}\mu/2}{s_{i}\mu/2+1}=\frac{s_{i}\mu}{s_{i}\mu +2}\), where the inequality follows from [2, Lemma 6].
During survival selection, since we have at most \((1+c)p\mu\) noisy solutions in \(P_{t}\) and \(Q_{t}\), respectively, there are at most \(2(1+c)p\mu\) noisy solutions in \(R_{t}=P_{t}\cup Q_{t}\). As the population size \(\mu\) satisfies \(\mu\geq 2(1+c)p\mu+4(n+1)\), Lemma 7 (ii) with \(C=0,D=n,S=2(1+c)p\mu\) first implies that even when no such \(z\) individual is created an individual with the same fitness as \(y\) always survives to the next generation \(P_{t+1}\); in other words, \(i\) cannot decrease. Second, when one individual \(z\) is created, regardless of whether it is evaluated with noise or not, \(z\), or an individual weakly dominating it, survives.
So the expected number of generations of this phase is at most:
\[\operatorname{E}\left[T_{1}\right]\leq\sum_{i=0}^{n-1}\left(1+\frac{2}{s_{i} \mu}\right)=n+\frac{2}{\mu}\sum_{i=0}^{n-1}\mathcal{O}\left(\frac{\mu n}{1-p_ {c}}\right)=\mathcal{O}\left(\frac{n^{2}}{1-p_{c}}\right).\]
The failure probability of the phase is bounded from above by the law of total probability and union bounds as \(p_{1}\leq\sum_{t=1}^{\infty}\Pr\left(T_{1}=t\right)\cdot te^{-\Omega(n)}=e^{- \Omega(n)}\operatorname{E}\left[T_{1}\right]=o(1)\) since \(1-p_{c}=2^{-o(n)}\).
Phase 2: Cover the whole Pareto front.
Now that the population \(P_{t}\) contains Pareto-optimal individuals, while the whole Pareto front has not been covered, the population contains a Pareto-optimal individual \(y\) with positive crowding distance such that one of the fitness vectors \((\operatorname{LO}(y)-1,\operatorname{TZ}(y)+1)\) or \((\operatorname{LO}(y)+1,\operatorname{TZ}(y)-1)\) is not yet present in \(f(P_{t})\).
As argued for Phase 1, the probability that during one offspring production, the sequence of operations selection, crossover, and mutation produces an offspring \(z\) with a missing fitness vector is at least \(s^{\prime}_{i}\coloneqq\Omega((1-p_{c})/(\mu n))\). Again with \(\mu/2\) trials, the probability of creating \(z\) per generation is \(1-(1-s^{\prime}_{i})^{\mu/2}\geq s^{\prime}_{i}\mu/(s^{\prime}_{i}\mu+2)\).
During the survival selection, we again use Lemma 7 (ii) with the same parameters to argue that Pareto optimal fitness vectors will never be removed entirely from the population. There can be at most \(n\) missing Pareto optimal fitness vectors to cover, thus the expected number of generations to complete this phase is at most
\[\operatorname{E}\left[T_{2}\right]\leq\sum_{i=1}^{n}\left(1+\frac{2}{s^{\prime }_{i}\mu}\right)=\sum_{i=1}^{n}\left(1+\frac{2}{\mu}\cdot\mathcal{O}\left( \frac{\mu n}{1-p_{c}}\right)\right)=\mathcal{O}\left(\frac{n^{2}}{1-p_{c}} \right).\]
and by a similar argument as in the other phase, we also have \(p_{2}=o(1)\) given \(1-p_{c}=2^{-o(n)}\).
The total expected number of generations until the Pareto front is covered is bounded from above by \((\operatorname{E}\left[T_{1}\right]+\operatorname{E}\left[T_{2}\right])/(1-p_ {1}-p_{2})=\mathcal{O}\left(n^{2}/(1-p_{c})\right)/(1-o(1))=\mathcal{O}\left( n^{2}/(1-p_{c})\right)\). The bound on the expected number of evaluations follows from the fact that the number of solutions evaluated in each generation is \(2\mu=O(\mu)\).
Our analysis can be easily extended to show similar results for other functions. For instance, the expected time to cover the Pareto front of noisy OMM in the same noise model is at most \(\mathcal{O}\left(\mu n\log n/(1-p_{c})\right)\).
**Theorem 9**.: _Consider the \((\delta,p)\)-model with noise strength \(\delta>n\) and constant noise probability \(p\in\left(0,\frac{1}{2(1+c)}\right)\) for some constant \(c>0\). Then NSGA-II with population size \(\mu\geq\frac{4(n+1)}{1-2(1+c)p}\) and \(p_{c}\leq 1-2^{-o(n)}\) covers the whole Pareto front of the noisy OMM function in \(\mathcal{O}\left(\frac{n\log n}{1-p_{c}}\right)\) expected generations and \(\mathcal{O}\left(\frac{\mu n\log n}{1-p_{c}}\right)\) expected fitness evaluations. The result also holds for the \((-\delta,1-p)\)-model using the same conditions._
Proof.: We follow the same approach as in the proof of Theorem 8, by using the typical run method with the restarting argument, and define the bad events the same way. That is, a bad event occurs in generation \(t\) either if more than \((1+c)p\mu\) parents in \(P_{t}\) are re-evaluated with noise, or if more than \((1+c)p\mu\) offspring individuals in \(Q_{t}\) are evaluated with noise, thus the probability of such an event is at most \(2e^{-\Omega(n)}=e^{-\Omega(n)}\). Since any search point is Pareto optimal for OMM, we only have a single phase of covering the Pareto front. Like LOTZ,
the noisy OMM function is also \((0,n)\)-separable, thus the conditions to apply Lemma 7 (i) are all fulfilled given the same setting for \(\mu\).
If the Pareto front is not covered entirely, then there must exist search points \(z\notin P_{t}\) next to a search point \(y\in P_{t}\) with a positive crowding distance, i. e. \(||z|_{1}-|y|_{1}|=1\wedge||z|_{0}-|y|_{0}|=1\). (These events are equivalent for 0-1 strings of equal length.) Let \(|y|_{1}=i\) thus \(|y|_{0}=n-i\), and we will focus on the case \(|z|_{1}=|y|_{1}+1=i+1\wedge|z|_{0}=|y|_{0}-1=n-i-1\) as the reasoning is symmetric in the other case of \(|z|_{1}=|y|_{1}-1\wedge|z|_{0}=|y|_{0}+1\). Similar to the argument for LOTZ, with probability \(\Omega(1/\mu)\), \(y\) is selected as a parent and with probability \(1-p_{c}\) it creates an offspring by mutation only. To create such a point \(z\) by mutation, it suffices to flip one of the \(n-i\) 0-bits of \(y\) to 1 while keeping the rest of the bits unchanged, and this happens with probability \(\frac{n-i}{n}\left(1-\frac{1}{n}\right)^{n-1}=\Omega(\frac{n-i}{n})\). So, the probability that a search point \(z\) is created during one offspring production is \(s_{i}:=\Omega(\frac{(1-p_{c})(n-i)}{\mu n})\). With \(\mu/2\) offspring productions, the chance of creating a solution \(z\) is \(1-(1-s_{i})^{\mu/2}\geq\frac{s_{i}\mu/2}{s_{i}\mu/2+1}=\frac{s_{i}\mu}{s_{i} \mu+2}\).
Once a point \(z\) is created, then by Lemma 7 (ii), similarly to the case of LOTZ, a search point with fitness \(f(z)\) is always kept in the population. Thus, starting from \(y\), the expected number of generations to cover fitness vectors \((i+1,n-i-1),(i+2,n-i-2),\ldots,(n,0)\), if they do not yet exist in the population, is at most
\[\sum_{k=i}^{n-1}\left(1+\frac{2}{\mu s_{i}}\right)\leq\sum_{k=0}^{n-1}\left(1 +\mathcal{O}\left(\frac{2n}{(1-p_{c})(n-i)}\right)\right)=\mathcal{O}\left( \frac{n\log n}{1-p_{c}}\right).\]
By symmetry, the same bound holds to cover the fitness vectors \((i-1,n-i+1),(i-2,n-i+2),\ldots,(0,n)\). So, the expected number of generations to cover the whole front is no more than \(\operatorname{E}\left[T\right]=\mathcal{O}\left(\frac{n\log n}{1-p_{c}}\right)\), and the failure probability of the phase is at most \(\sum_{t=1}^{\infty}\Pr\left(T=t\right)\cdot te^{-\Omega(n)}=e^{-\Omega(n)} \operatorname{E}\left[T\right]=o(1)\) given \(1-p_{c}=2^{-o(n)}\). Thus, the expected runtime of the algorithm is at most \(\mathcal{O}\left(\frac{n\log n}{1-p_{c}}\right)\cdot\frac{1}{1-o(1)}= \mathcal{O}\left(\frac{n\log n}{1-p_{c}}\right)\) generations, or, equivalently, at most \(\mathcal{O}\left(\frac{\mu\log n}{1-p_{c}}\right)\) fitness evaluations since only \(2\mu\) evaluations are required per generation.
## 6 Phase Transition for NSGA-II at \(\boldsymbol{p=\frac{1}{2}}\)
We have seen that, when \(\delta>n\), with an appropriate scaling of the population size NSGA-II can handle any constant noise probability \(p\) approaching \(1/2\) from below. Our result from Theorem 8 does not cover the case \(p>1/2\) because when approaching \(1/2\) from above, the progress of optimisation has to rely on having a sufficient number of good individuals that are evaluated with noise. The dynamic of the algorithm is therefore different, in fact, the next theorem shows that a noise probability around \(p>1/2\) leads to poor results on LOTZ. For the sake of simplicity, we omit crossover and leave an analysis including crossover for future work.
**Theorem 10**.: _Consider the NSGA-II with population size \(\mu\in[n+1,\infty)\cap O(n)\) and crossover turned off (\(p_{c}=0\)) on the noisy LOTZ function with the \((\delta,p)\)-noise model. If \(\delta>n\) and \(p\) is a constant such that \(1/2<p<10/19\) then NSGA-II requires \(e^{\Omega(n)}\) generations with overwhelming probability to cover the whole Pareto front._
The analysis will show that the number of Pareto-optimal individuals is bounded with overwhelming probability. We first give a bound on the probability of creating a Pareto-optimal individual.
**Lemma 11**.: _Let \(F:=\{1^{i}0^{n-i}\ |\ i\in[n]_{0}\}\) be the Pareto set of LOTZ and consider a standard bit mutation creating \(y\) from \(x\). Then_
\[\Pr\left(y\in F\right)\leq\begin{cases}1/e+3/n&\text{ if }x\in F\\ 3/n&\text{ otherwise.}\end{cases}\]
Proof.: We assume \(n\geq 3\) as otherwise the claimed probability bound is at least 1, which is trivial. Starting from a parent \(x=1^{i}0^{n-i}\in F\), an offspring in \(F\) is created if \(x\) is cloned, or if it is mutated into another search point \(1^{j}0^{n-j}\) with \(j\neq i\). We have \(\Pr\left(y=x\right)=(1-1/n)^{n}\leq 1/e\) and \(\Pr\left(y=1^{j}0^{n-j}\right)\leq n^{-|i-j|}\) for all \(j\in[n]_{0}\setminus\{i\}\)
since (depending on whether \(j<i\) or \(j>i\)) either the last \(|i-j|\) 1-bits or the first \(|i-j|\) 0-bits in \(x\) must be flipped. The sum of all probabilities is at most
\[\frac{1}{e}+\sum_{d=1}^{\infty}2n^{-d}=\frac{1}{e}+2\cdot\frac{1/n}{1-1/n}\leq \frac{1}{e}+\frac{3}{n}\]
where the last step used \(1/(n-1)\leq 3/(2n)\) for \(n\geq 3\).
Now assume \(x\notin F\). If the Hamming distance to the closest point in \(F\) is at least 2, at least two specific bits must be flipped to create a specific offspring \(1^{j}0^{n-j}\in F\). This has probability at most \(1/n^{2}\). Taking a union bound over \(n+1\) possible values of \(j\) yields a probability bound of \((n+1)/n^{2}\leq 2/n\). If \(x\) has Hamming distance 1 to some search point \(1^{i}0^{n-i}\in F\), it either has a single 0-bit among bits \(\{1,\ldots,i-1\}\) bits (and an all-zeros suffix) or a single 1-bit among bits \(\{i+2,\ldots,n\}\) bits (and an all-ones prefix). (Bit positions \(i\) and \(i+1\) are excluded as otherwise \(x\in F\).) In the former case, if the 0-bit is at position \(i-1\), \(x\) has Hamming distance 1 to both \(1^{i}0^{n-i}\) and \(1^{i-2}0^{n-i+2}\) and Hamming distance at least 2 to all other search points in \(F\). If the 0-bit is at some smaller index, \(x\) has Hamming distance at least 2 to all search points in \(F\setminus\{1^{i}0^{n-i}\}\). The case of a single 1-bit is symmetric. Hence there are always at most two search points at Hamming distance 1 in \(F\), and each one is reached with probability at most \(1/n\). To reach any other point in \(F\), two bits must be flipped. Taking a union bound over all probabilities yields a probability bound of \(2/n+n\cdot 1/n^{2}=3/n\) in this case.
Now we prove Theorem 10.
Proof of Theorem 10.: Recall that, owing to \(\delta>n\), all noisy individuals dominate all noise-free ones. The Pareto set of LOTZ is \(F:=\{1^{i}0^{n-i}\mid i\in[n]_{0}\}\). We use variables \(X_{t}:=|P_{t}\cap F|\) to denote the number of individuals on \(F\) in generation \(t\). The Pareto front can only be found if \(X_{t}\geq|F|=n+1\). It is easy to see that the initial population at time 0 will have \(X_{t}<n+1\) with probability at least \(1-e^{-\Omega(n)}\), and we assume that this happens. Since crossover is turned off, in each generation the \(\mu\) offspring are created independently by the binary tournament selection, followed by bitwise mutation.
We first find an upper bound \(p_{\text{tour}}\) on the probability that an individual in \(F\) is returned by an application of tournament selection. A necessary event is to sample at least one of the two competitors from \(F\). Each competitor is sampled from \(F\) with probability \(X_{t}/\mu\). By a union bound, the probability of at least one competitor being from \(F\) is at most \(2X_{t}/\mu\). Thus, \(p_{\text{tour}}\leq 2X_{t}/\mu\).
Starting from a parent \(x=1^{i}0^{n-i}\in F\), the probability of the offspring also being in \(F\) is at most \(1/e+3/n\) by Lemma 11. From a parent \(x\notin F\), the probability of creating an offspring on \(F\) is at most \(3/n\) by Lemma 11. Together, the probability of a parent selection followed by mutation creating a search point in \(F\) is at most
\[p_{\text{tour}}(1/e+3/n)+3/n\leq\frac{p_{\text{tour}}}{e}+\frac{6}{n}=:q.\]
Let \(Y_{t}\coloneqq|Q_{t}\cap F|\) be the number of Pareto optimal points in the offspring population, then \(Y_{t}\) can be bounded by a binomial distribution, \(Y_{t}\preceq\text{Bin}(\mu,q)\). As \(\mu=O(n)\), we have \(\operatorname{E}\left[Y_{t}\right]\leq 2X_{t}/e+O(1)\). We analyse the distribution of \(X_{t+1}\) and consider two cases:
Case 1: \(n/4\leq X_{t}\leq n\). For any constant \(\delta\in(0,9e/20-1)\) and a sufficiently large \(n\), by a Chernoff bound it holds that
\[\Pr\left(Y_{t}\geq 9X_{t}/10\mid X_{t}\geq n/4\right) \leq\Pr\left(Y_{t}\geq(1+\delta)\left(2X_{t}/e+O(1)\right)\right)\] \[\leq e^{-2\delta^{2}X_{t}/(3e)-\delta^{2}O(1)}=e^{-\Omega(n)}.\]
Thus the probability of creating more than \(9X_{t}/10\) offspring on \(F\) is exponentially small. Additionally, since \(p\) is a constant below \(10/19\) and above \(1/2\) by two applications of Chernoff bounds we have the following. The probability of having more than \((10/19)(X_{t}+Y_{t})\) noisy individuals on \(F\) is \(e^{-\Omega(X_{t}+Y_{t})}=e^{-\Omega(n)}\). The probability of having less than \(\mu\) noisy individuals among the \(2\mu\) individuals in \(R_{t}\) is \(e^{-\Omega(\mu)}=e^{-\Omega(n)}\). Since the survival
selection only keeps the \(\mu\) best among the \(2\mu\) individuals, if the latter event does not occur then none of the noise-free individuals will survive to the next generation. Thus, when none of the three events occur, i. e. with a probability of at least \(1-e^{-\Omega(n)}\) by a union bound, \(X_{t+1}\leq(10/19)(X_{t}+Y_{t})<(10/19)(X_{t}+9X_{t}/10)=X_{t}\). In other words,
\[\Pr\left(X_{t+1}\geq X_{t}\right)=e^{-\Omega(n)}.\]
Case 2: \(0\leq X_{t}<n/4\). Since \(\mathrm{E}\left[Y_{t}\right]\leq 2X_{t}/e+O(1)\leq n/(2e)+O(1)\rightleftharpoons m\), we have \(\Pr\left(Y_{t}\geq 2m\right)\leq e^{-m/3}=e^{-\Omega(n)}\). This implies \(X_{t}+Y_{t}\leq n/4+n/e+O(1)\leq n\) (for \(n\) large enough) with probability \(1-e^{-\Omega(n)}\). If this happens then \(X_{t+1}\leq n\).
Combining these two cases gives that to reach \(X_{t}\geq n+1\), an event must occur that has probability \(e^{-\Omega(n)}\). The expected time until such an event occurs is \(e^{\Omega(n)}\), thus NSGA-II requires at least \(e^{\Omega(n)}\) generations in expectation.
## 7 Experiments
To complement the theoretical results, experiments were conducted to compare the robustness of NSGA-II with GSEMO on LeadingOnesTrailingZeroes and OneMinMax. We considered two noise models, the first is the \((\delta,p)\)-Bernoulli noise model using \(\delta\coloneqq n+1\) and various noise probabilities \(p\in\{2^{-2},2^{-3},\ldots,2^{-6}\}\cup\{0.4,0.5,0.6\}\cup\{1-2^{-5},1-2^{-4},1-2^{-3},1-2^{-2}\}\) to cover noise probabilities close to \(0,1/2\), and \(1\) respectively. In the second model, to investigate in how far our results translate to more general noise models, we consider posterior Gaussian noise as in [29]. The noisy fitness of a search point \(x\) is defined as follows, \(\mathcal{N}(0,\sigma^{2})\) denoting the normal distribution with mean \(0\) and standard deviation \(\sigma\):
\[\tilde{f}(x):=f(x)+\vec{1}\cdot\delta\text{ where }\delta\sim\mathcal{N}(0, \sigma^{2}).\]
A Gaussian noise is always added to the fitness after evaluation (to all objectives), that is, there is no noise probability in this model. For \(\sigma=n\) there is a constant probability of \(\tilde{f}(x)\succ\tilde{f}(y)\) for any two search points \(x,y\), irrespective of their true fitness. Hence we vary the standard deviation as \(\sigma:=n\cdot q\) where \(q\in\{2^{0},2^{-1},2^{-3},2^{-4}\}\).
We used problem size \(n\in\{20,30,40\}\), \(p_{c}=0.9\) and one-point crossover. For NSGA-II, the population size is set to \(\mu=9(n+1)\). For each experiment, \(50\) runs were performed. In each run, the algorithm is stopped either when the whole Pareto front is covered and then the number of iterations is recorded, or when the number of fitness evaluations exceeded \(10n^{3}\).
As predicted by our results from Section 4, GSEMO failed to cover the Pareto front of both functions within the time limit in all experiments. The first plot of Figure 2 shows the number of different points on the Pareto front of OMM being covered by GSEMO over time for single run on the Bernoulli noise model. This number never exceeds \(16\), which is below \(40\%\) of the size of the Pareto front. This is for the easier function OMM; on LOTZ the maximum value was \(4\), that is, less than \(10\%\) of the front size (plot is omitted for lack of space). The same issue is also evident in the last plot of the figure for NSGA-II on LOTZ with the noise probabilities \(p=0.5\) and \(p=0.6\). The success rate (fraction of runs covering the Pareto front) was always \(100\%\) for NSGA-II in all settings, except for \(p\in\{0.5,0.6\}\) on LOTZ, where it was \(0\%\). This is aligned with our theoretical prediction (Theorem 10) there is a phase transition at \(p=1/2\) for NSGA-II. Note that in the experiments \(p_{c}=0.9\) and Theorem 10 is for \(p_{c}=0\) and only claims the negative effect for \(p<10/19\) which is below \(0.6\). The empirical results thus suggest that the findings extend beyond the setting from Theorem 10.
Table 1 shows the average number of fitness evaluations when running NSGA-II on LOTZ, under the Bernoulli noise model. We see that the average running time increases as \(p\) approaches \(1/2\) and that it decreases when approaching \(p=0\) or \(1\). With the Gaussian noise model, NSGA-II is only able to cover the Pareto front of the noisy functions when the standard deviation of the noise is small, i. e. \(\sigma\in\{n\cdot 2^{-4},n\cdot 2^{-3}\}\). Table 2
shows the success rate of NSGA-II under Gaussian noise. While NSGA-II is effective for small Gaussian noise, it starts to fail when the standard deviation is increased.
## 8 Conclusions
We have given a first example on which NSGA-II provably outperforms GSEMO and performed a first theoretical runtime analysis of EMO algorithms in stochastic optimisation. While GSEMO is very sensitive to noise, NSGA-II can cope well with noise, even when the noise strength is so large that we have domination between noisy and noise-free search points. This holds when the population size is large enough to enable useful search points to survive and when the noise probability is less than \(1/2\). However, for noise probabilities slightly larger than \(1/2\), even NSGA-II requires exponential expected runtime, thus it experiences a phase transition at \(p=1/2\).
There are many open questions for future work. What if noise is applied to all objectives independently? Can theoretical results be shown for other noise models like Gaussian posterior noise? What mechanisms can prevent noise from disrupting NSGA-II?
## Acknowledgments
This work benefited from discussions at Dagstuhl seminar 22081 "Theory of Randomized Optimization Heuristics". The third author was supported by the Erasmus+ Programme of the European Union.
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline & \multicolumn{3}{c}{LOTZ} & \multicolumn{3}{c}{OMM} \\ \cline{2-7} \(p\) & \(n=20\) & \(n=30\) & \(n=40\) & \(n=20\) & \(n=30\) & \(n=40\) \\ \hline \(2^{-6}\) & 24090 & 84051 & 202822 & 12346 & 35982 & 79927 \\ \(2^{-5}\) & 23317 & 93687 & 208128 & 10273 & 36706 & 85786 \\ \(2^{-4}\) & 24919 & 92294 & 218483 & 12987 & 41440 & 82101 \\ \(2^{-3}\) & 27898 & 93297 & 212550 & 12686 & 37987 & 80848 \\ \(2^{-2}\) & 31856 & 109701 & 273906 & 14684 & 37235 & 93562 \\ \(0.4\) & 34212 & 119540 & 313298 & 16041 & 48848 & 89887 \\ \(0.5\) & 80301 & 270145 & 640453 & 29160 & 95859 & 215240 \\ \(0.6\) & 80301 & 270145 & 640453 & 18604 & 50826 & 121581 \\ \(1-2^{-2}\) & 35230 & 156015 & 470869 & 14194 & 51884 & 99495 \\ \(1-2^{-3}\) & 29274 & 102933 & 221579 & 14948 & 43529 & 97910 \\ \(1-2^{-4}\) & 28350 & 89008 & 236429 & 12572 & 37402 & 93341 \\ \(1-2^{-5}\) & 25692 & 85917 & 216972 & 11705 & 39881 & 88440 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average running time of NSGA-II on LOTZ and OMM under the \((n+1,p)\)-Bernoulli noise model. Runs were stopped after \(10n^{3}\) evaluations. The success rate was always \(100\%\), except for the shaded cells, where it was \(0\%\).
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline & \multicolumn{3}{c}{LOTZ} & \multicolumn{3}{c}{OMM} \\ \cline{2-7} \(\sigma\) & \(n=20\) & \(n=30\) & \(n=40\) & \(n=20\) & \(n=30\) & \(n=40\) \\ \hline \(n\cdot 2^{-4}\) & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ \(n\cdot 2^{-3}\) & 100\% & 100\% & 96\% & 100\% & 100\% & 70\% \\ \(n\cdot 2^{-2}\) & 65\% & 7\% & 10\% & 40\% & 7\% & 4\% \\ \(n\cdot 2^{-1}\) & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \(n\cdot 2^{0}\) & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Average success rate of NSGA-II on LOTZ and OMM under the Gaussian noise model. |
2308.12833 | Use of LLMs for Illicit Purposes: Threats, Prevention Measures, and
Vulnerabilities | Spurred by the recent rapid increase in the development and distribution of
large language models (LLMs) across industry and academia, much recent work has
drawn attention to safety- and security-related threats and vulnerabilities of
LLMs, including in the context of potentially criminal activities.
Specifically, it has been shown that LLMs can be misused for fraud,
impersonation, and the generation of malware; while other authors have
considered the more general problem of AI alignment. It is important that
developers and practitioners alike are aware of security-related problems with
such models. In this paper, we provide an overview of existing - predominantly
scientific - efforts on identifying and mitigating threats and vulnerabilities
arising from LLMs. We present a taxonomy describing the relationship between
threats caused by the generative capabilities of LLMs, prevention measures
intended to address such threats, and vulnerabilities arising from imperfect
prevention measures. With our work, we hope to raise awareness of the
limitations of LLMs in light of such security concerns, among both experienced
developers and novel users of such technologies. | Maximilian Mozes, Xuanli He, Bennett Kleinberg, Lewis D. Griffin | 2023-08-24T14:45:50Z | http://arxiv.org/abs/2308.12833v1 | # Use of LLMs for Illicit Purposes: Threats, Prevention Measures, and Vulnerabilities
###### Abstract
Spurred by the recent rapid increase in the development and distribution of large language models (LLMs) across industry and academia, much recent work has drawn attention to safety- and security-related threats and vulnerabilities of LLMs, including in the context of potentially criminal activities. Specifically, it has been shown that LLMs can be misused for fraud, impersonation, and the generation of malware; while other authors have considered the more general problem of AI alignment. It is important that developers and practitioners alike are aware of security-related problems with such models. In this paper, we provide an overview of existing--predominantly scientific--efforts on identifying and mitigating threats and vulnerabilities arising from LLMs. We present a taxonomy describing the relationship between threats caused by the generative capabilities of LLMs, prevention measures intended to address such threats, and vulnerabilities arising from imperfect prevention measures. With our work, we hope to raise awareness of the limitations of LLMs in light of such security concerns, among both experienced developers and novel users of such technologies.
###### Contents
* 1 Introduction
* 2 Existing overviews of LLM safety
* 3 Safety concerns prior to LLMs
* 3.1 Adversarial attacks against ML models
* 3.2 LLMs and adversarial attacks
* 3.3 Security issues beyond adversarial attacks
* 4 Approach
* 5 Threats
* 5.1 Fraud, impersonation, social engineering
* 5.2 Generating malware
* 5.3 Scientific misconduct
* 5.4 Misinformation
* 5.5 Data memorization
* 5.6 Data poisoning
* 6 Prevention measures
* 6.1 Preventing misuse of LLMs via content detection
* 6.2 Red teaming
* 6.3 LLM content filtering
* 6.4 Safeguarding via RLHF
* 6.5 Safety via instruction-following
* 6.6 Methods to avoid memorization
* 6.7 Methods to avoid data poisoning
* 7 Vulnerabilities
* 7.1 Prompt injection
* 7.2 Jailbreaking
* 8 Discussion
* 8.1 Public concerns around LLMs
* 8.2 Limitations of LLM safety
* 8.3 An outlook on future LLM-enabled security concerns
* 9 Conclusion
## 1 Introduction
Large language models (LLMs) have taken the field of natural language processing (NLP) by storm. Recent advancements achieved through scaling neural network-based machine learning models have resulted in models that are capable of generating natural language which is hardly distinguishable from that created by human beings (Brown et al., 2020; Chowdhery et al., 2022; OpenAI, 2023b). LLMs can potentially aid human productivity ranging from assisting with the creation of code (Sandoval et al., 2022) to helping in email writing and
co-writing university coursework [12] and have shown remarkable performances across fields, including in law, mathematics, psychology, and medicine [13, 14]. At the same time, LLMs have the potential to dramatically disrupt the global labor market: recent work claims that around 19% of the US workforce could have at least 50%, and 80% at least 10% of their tasks impacted by the development of LLM capabilities [1].
Despite such advancements, their text-generating capabilities also have the potential for malicious purposes, for which the research community has identified various concerns. From an academic viewpoint, it has been argued that the LLM-assisted creation of research papers can have implications on scientific practices (e.g., through the introduction of biases when selecting related works), and raises concerns around copyright and plagiarism [13, 14]. From a security viewpoint, LLMs have been identified as a useful tool for fraud and social engineering [15] as well as generating misinformation [12], malware code [13] and assisting with the development of illicit drugs and cyber weapons [1]. Other cybercrime tools such as WormGPT1 and FraudGPT,2 which are based on existing language models, have also been developed and are distributed online. Responding to such concerns, shortly after the release and increase in public visibility of ChatGPT [16], Europol published a report discussing the impact of LLMs on law enforcement.3 In their report, Europol describe and discuss three areas in which LLMs can have an impact on criminal activity: fraud and social engineering, disinformation, and cybercrime, while noting that this is a far from exhaustive list.
Footnote 1: [https://thehackernews.com/2023/07/wormgpt-new-ai-tool-allows.html](https://thehackernews.com/2023/07/wormgpt-new-ai-tool-allows.html)
Footnote 2: [https://thehackernews.com/2023/07/new-ai-too1-fraudgpt-emerges-tailored.html](https://thehackernews.com/2023/07/new-ai-too1-fraudgpt-emerges-tailored.html)
Footnote 3: [https://www.europol.europa.eu/media-press/new/srom/news/criminal-use-of-chatgpt-cautionary-tale-about-large-language-models](https://www.europol.europa.eu/media-press/new/srom/news/criminal-use-of-chatgpt-cautionary-tale-about-large-language-models)
In light of this, we aim to review the current landscape of safety- and security-related technical work on LLMs, and present a taxonomy of existing approaches by categorizing them into _threats_, _prevention measures_, and _vulnerabilities_. Threats arise naturally through the advanced generative capabilities of LLMs and include methods such as the generation of phishing emails (Section 5.1), malware (Section 5.2), and misinformation (Sec
Figure 1: Overview of the taxonomy of malicious and criminal use cases enabled via LLMs. _a) Threats_ arise from the generative capabilities of LLMs, e.g., through the generation of phishing emails [12] and misinformation [11]. _b) Preventions_ address such threats, e.g., via reinforcement learning from human feedback (RLHF; [13] and red teaming [14]. _c) Vulnerabilities_ arise from imperfect prevention measures and can re-enable existing threats, e.g., via prompt injection [10] or jailbreaking [15].
tion 5.4). Prevention measures (Section 6) attempt to mitigate the threats arising from their capabilities, and existing approaches include content filtering (Markov et al., 2023), reinforcement learning from human feedback (RLHF; Bai et al., 2022) and red teaming (Ganguli et al., 2022). Vulnerabilities (Section 7) then arise from imperfect attempts to prevent the threats and cover methods such as jailbreaking (Kang et al., 2023) and prompt injection (Perez and Ribeiro, 2022). Such vulnerabilities then re-enable existing threats. See Figure 1 for an overview. For each category, we define relevant concepts and provide an extensive list of academic and real-world instances in which such topics have been discussed.
We conclude our paper with a discussion of the presented works by focusing on potential reasons for the vast public perception observed by LLM-enabled threats, the theoretical and practical limitations of prevention strategies, and potential future concerns stemming from advancements in LLM development (Section 8).
## 2 Existing overviews of LLM safety
AI-enabled applications of illicit activities are increasingly studied in the academic literature (Caldwell et al., 2020). During our research, we came across multiple related works discussing the current landscape of security-related discoveries for LLMs.
Existing work by Weidinger et al. (2022) presents a taxonomy of 21 risks associated with LLMs categorized into six major areas: (i) _discrimination, hate speech, and exclusion_, (ii) _information hazards_, (iii) _misinformation harms_, (iv) _malicious uses_, (v) _human-computer interaction harms_, and (vi) _environmental and socioeconomic harms_. Importantly, the authors differentiate between observed and anticipated risks in their analysis, i.e., those risks that have already been observed and those that are anticipated to be observed in the future. While there is some overlap between risks discussed in Weidinger et al. (2022) and our work (e.g., related to misinformation and malicious uses), our work more specifically focuses on recent concepts stemming from advancements in LLM development that have emerged since they published, for example, the bypassing of LLM security measures via prompt injection attacks (Section 7).
Taking a different approach, Huang et al. (2023) provide a categorization of LLM vulnerabilities into inherent issues, intended attacks, and unintended bugs. The first covers vulnerabilities such as factual errors where an LLM generates false information and reasoning errors. The second, in contrast, refers to direct attacks on LLMs, e.g., via prompt injection, backdoor attacks, or privacy leakage. The third refers to situations where development errors enable LLM vulnerabilities. With respect to attacks, our work exclusively focuses on intended ones--situations in which adversaries deliberately exploit characteristics of LLMs for potentially illicit purposes.
Yet another categorization has been proposed by Fan et al. (2023), presenting an overview of research works related to the trustworthiness of LLMs. In contrast to this paper, their work categorizes the threats associated with LLMs into aspects of privacy, security, responsibility, and fairness.
Discussing the risks of emerging AI technologies including and beyond language, Bommasani et al. (2021) report on the opportunities and risks of foundation models such as BERT (Devlin et al., 2018), CLIP (Radford et al., 2021), and GPT-3 (Radford et al., 2019). This includes technological aspects (e.g., security, robustness, and AI safety and alignment) and a discussion of their societal impacts, which focuses on social inequalities, their economic and environmental impact, their potential to amplify the distribution of disinformation, potential consequences on the legal system, and ethical issues arising from such advanced models. While that report provides an overview of topics also discussed in this paper, our work represents an up-to-date presentation of existing works revolving around the security of LLMs.
Other approaches focus on more specific aspects of LLM-related security as well as specific models. For instance, Greshake et al. (2023) outline the existing literature around prompt injection attacks in the context of LLMs, presenting a review of existing attack methods (e.g., active, passive, user-driven) as well as a categorization of threats arising from them (e.g., fraud, the manipulation of content). We extensively discuss prompt injection approaches in Section 7.1, yet our work more broadly describes the existing literature on the security of LLMs, of which prompt injection forms only a part.
Similarly, Gupta et al. (2023) present an overview of existing security threats associated with ChatGPT. The paper provides an organization
of threats associated with ChatGPT into _attacking ChatGPT_ (e.g., jailbreaking, prompt injection), _cyber offense_ (e.g., social engineering, malware code generation), _cyber defense_ (e.g., secure code generation, incidence response), and _social, legal, and ethics_ (e.g., personal information misuse, data ownership concerns). However, their paper mainly focuses on vulnerability and threat reports obtained through news articles and blog posts. We instead attempt to primarily map out the scientific literature on both attacks and defenses.
## 3 Safety concerns prior to LLMs
Prior to the advent of LLMs and advanced generative AI technologies, a substantial part of security-related research in machine learning (ML) focused on adversarial attacks against trained models (Chakraborty et al., 2018). Before delving into the threats, prevention measures, and vulnerabilities related to LLMs, we therefore initiate the discussion of safety and security in NLP by providing a brief overview of adversarial examples as well as an assessment of their relevance in light of increasingly capable language models.
### Adversarial attacks against ML models
While ML methods have caused substantial advancements in the field of artificial intelligence and computer science (LeCun et al., 2015), researchers have quickly identified security vulnerabilities associated with them (Szegedy et al., 2013). Such vulnerabilities have been termed _adversarial examples_ (ibid.) and describe the phenomenon that small, semantics-preserving modifications to input data cause target models to drastically change their predictions. Their initial discovery was in the context of computer vision, where Szegedy et al. (2013) found that adding small, humanly imperceptible pixel perturbations to images can lead image classification models to predict incorrect labels.
Adversarial examples are present for ML models across modalities, for example in vision (Goodfellow et al., 2014), language (Papernot et al., 2016; Alzantot et al., 2018), and reinforcement learning (Gleave et al., 2020), and large efforts are spent on attacking (e.g., Carlini and Wagner, 2017; Jin et al., 2020), detecting (e.g., Xu et al., 2017; Mozes et al., 2021), and defending against attacks (e.g., Madry et al., 2018). See Chakraborty et al. (2018) and Xu et al. (2020) for an overview.
At the same time, comparatively little attention has been paid to the potential implications of ML vulnerabilities on realistic practical applications. While papers discussing practical applications of adversarial vulnerabilities exist for images (Gu et al., 2017), audio (Yuan et al., 2018), and text (Li et al., 2019), it has been argued that most existing works focus on abstract and unrealistic scenarios (Gilmer et al., 2018).
Despite LLMs' advanced capabilities and state-of-the-art performances across various NLP tasks (Bubeck et al., 2023; OpenAI, 2023b), a range of recent works studied their robustness against adversarial attacks, some of which we outline in the following.
### LLMs and adversarial attacks
In the context of LLMs, adversarial attacks have been studied for various scenarios, including zero-shot learning (Wang et al., 2023), in-context learning (Wang et al., 2023), and parameter-efficient fine-tuning (Yang and Liu, 2022).
Zero-shot adversarial robustnessLLMs have shown to be effective when prompted in a zero-shot setting, without the provision of demonstrations in the input prompt (Brown et al., 2020). Wang et al. (2023) further study such findings by investigating ChatGPT's adversarial robustness in a zero-shot setting against a selection of adversarial datasets and datasets under distribution shift. Their main findings include that while the model exhibits better robustness as compared to previous models, such as DeBERTa (He et al., 2020), BART (Lewis et al., 2020), and BLOOM (Scao et al., 2022), ChatGPT's performance on such test sets is still far from perfect, indicating that potential risks of adversarial vulnerability still remain.
Similarly, Shen et al. (2023) conduct experiments employing character-, word-, and sentence-level adversarial attacks against ChatGPT for question-answering datasets, by directly applying the attacks to the model inputs. Their empirical results show that attack success rates against that LLM are high, underlining the observation that ChatGPT is vulnerable to adversarial attacks.
Adversarial robustness of ICLIn contrast to studying the zero-shot setting, Wang et al. (2023) explore an LLM's brittleness to perturbations in the few-shot examples for in-context learning (ICL), rather than the actual input. While previous work has demonstrated the effects of manipulating few-shot prompts, namely that reordering them can
have dramatic effects on model performance (Lu et al., 2022), whereas relabeling of few-shot examples does barely decrease model performance (Min et al., 2022), Wang et al. (2023) directly attack the few-shot examples by conducting character-level perturbations, showing that both GPT2-XL (Radford et al., 2019) and LLaMA-7B (Touvron et al., 2023) exhibit substantial performance decreases after perturbation, and are hence vulnerable to such attacks.
Multi-modal adversarial attacksWith the increasing progress of research and development of LLMs, recent models such as GPT-4 (OpenAI, 2023) are capable of processing multi-modal inputs (texts and images), allowing them to generate language related to a given visual input. While this increases the range of applications of such LLMs, Qi et al. (2023) show that it also widens their attack surfaces against adversarial interventions. In their study, the authors show that MiniGPT-4 (Zhu et al., 2023), an open-source 13 billion parameter visual language model, is vulnerable to adversarial input perturbations. Specifically, the authors run a white-box attack using _projected gradient descent_ (PGD; Madry et al., 2018) to perturb visual inputs, with the intention of causing the model to generate harmful content when instructed to do so. Their results show that while the model seems to detect and appropriately address instructions asking it to generate harmful language with unperturbed visual inputs, it generates harmful content when queried using the visual adversarial examples. These results indicate that such models remain vulnerable to adversarial attacks and that employed safety mechanisms can be circumvented using standard PGD-based adversarial optimization techniques.
Adversarial robustness of prefix-tuningMore recent approaches to adapting LLMs for specific downstream focus on parameter-efficient fine-tuning (Houlsby et al., 2019). While such approaches have shown to be effective (Lester et al., 2021; Hu et al., 2021), Yang and Liu (2022) show that they are also vulnerable to adversarial attacks. They specifically investigate the robustness of prefix-tuning (Li and Liang, 2021), which adds a set of learnable embedding representations to the input of a model that are updated as part of the fine-tuning process on individual datasets. Experimenting with GPT-2, Yang and Liu (2022) observe that prefix-tuned models are vulnerable to adversarial attacks across various text classification datasets.
LLMs as adversarial assistantsAnother line of work shows that LLMs can also be used to aid in conducting adversarial attacks against machine learning models. Carlini (2023) demonstrates this by using LLMs as assistants to break an adversarial defense. Specifically, the author instructs GPT-4 to generate code that can be used to circumvent the _AI Guardian_ defense (Zhu et al., 2023), a recently published method to defend image classification models against adversarial examples. In other words, GPT-4 serves as a digital research assistant for building attacks against machine learning models. Despite noting that this approach has its limitations, the author argues that this discovery is _surprising, exciting_, as well as _worrying_.
### Security issues beyond adversarial attacks
Given that LLMs have recently received widespread attention from the research community (Zhao et al., 2023; Kaddour et al., 2023), various additional efforts aiming to identify security issues with such models have been adopted. Such approaches go beyond adversarial attacks as described above. Instead, more recent attacks require a substantially larger amount of human intervention and comprise methods such as _jailbreaking_ and _prompt injection_, which we will discuss in detail in Section 7.
## 4 Approach
To curate the collection of existing literature (which consists of both peer-reviewed scientific articles and works that have not undergone peer-review, for example, pre-print papers and news articles) on the safety and security of LLMs, we searched for relevant works in the field based on the knowledge and expertise of the authors. Given the increasing volume of work on these topics, we cannot guarantee that the works described in this paper represent a complete collection of existing efforts up to the date of publication. Rather, with our work, we aim to outline existing threats and considerations that users and practitioners should be aware of when using LLMs.
Since the field of LLM-related security research is relatively novel, we noticed during our literature search that a substantial amount of related papers have not yet undergone a successful peer-review process. Figure 2 shows that of the relevant 36
papers discussed in the _Threats_ section (publication dates range from 2004 to 2023),4 27 have been peer-reviewed (75%). This fraction decreases for the _Prevention measures_ section, with 20 out of 42 (48%) having been peer-reviewed (publication dates range from 2011 to 2023),5 and is lowest for _Vulnerabilities_, with 3 out of 15 papers (20%) having undergone peer-review (publication dates range from 2019 to 2023).6
Footnote 4: We consider Dalvi et al. (2004) as relevant for data poisoning, despite its publication prior to the development of LLMs.
Footnote 5: We consider Venugopal et al. (2011) relevant as an early work for watermarking in NLP.
Footnote 6: Note that each section cites additional papers (e.g., those introducing models or datasets), which we do not consider in this analysis.
## 5 Threats
The first dimension along which we assess LLMs in the context of security and crime is via threats enabled by their generative capabilities. Threats arising from LLMs include misusing the generations directly, such as for fraud, impersonation, or the generation of malware, but also through acts of model manipulation (e.g., through data poisoning). Below, we provide an overview of existing works discussing such threats.
### Fraud, impersonation, social engineering
A growing concern of misusing generative AI technologies is for the purpose of fraud, impersonation, and social engineering. In the context of AI, there has been an increasing concern about malicious activities arising from the generation of scams and phishing using LLMs (Brundage et al., 2018; Sjouwerman, 2023; Jolly, 2023). Generative models could be used to synthetically create digital content that seems to stem from a specific individual, for example, to create voice-based phone scams (Stupp, 2019; Harwell, 2019; Verma, 2023; Hernandez, 2023) or to distribute and sell digitally created pornographic videos (Tenbarge, 2023). While this has been a primary concern for the audio and video modalities, recent developments of LLM-based AI technologies enable the generation of text that is reported to be stylistically typical of specific individuals (Butler, 2023). For example, Hazell (2023) demonstrates how OpenAI's GPT models can be leveraged to generate personalized phishing emails addressed to 600 UK Members of Parliament (MP). As shown in Figure 3, Hazell (2023) achieves this by conditioning the GPT models on Wikipedia articles of individual MPs to create a phishing email asking the recipient to open an attached document. The author argues that LLMs enable adversaries to generate phishing emails at scale in a cost-effective fashion, mentioning that using Anthropic's Claude LLM,7 one can generate 1,000 phishing emails for $10 USD in around two hours. It is worth noting that the paper does not provide experimental results quantitatively evaluating the generated emails, and only demonstrates its claims with qualitative examples.
Footnote 7: [https://www.anthropic.com/index/introduction](https://www.anthropic.com/index/introduction) g-claude
### Generating malware
One of the main use cases of LLMs is their ability to generate computer code when prompted with a set of instructions (Anil et al., 2023). While this has merits to accelerate the development of software for both organizations and individuals, it can also be misused. Various recent articles have
Figure 3: **Using LLMs to generate personalized phishing emails at scale**(Hazell, 2023). An adversary with access to a list of names and email addresses for UK Members of Parliament (MPs) can query an LLM for the generation of personalized phishing emails by adding their Wikipedia articles as context to the model. This enables the generation of hundreds of personalized emails in a short period of time.
Figure 2: Comparison of relevant scientific works mentioned in this paper according to whether they have or have not undergone a successful peer-review process.
demonstrated the capabilities of LLMs to generate malicious computer code (Ben-Moshe et al., 2022; Waqas, 2023). This enables criminals without the necessary programming skill set to produce malware that can be used to hack into computer systems and exploit individuals.
The release of two AI-assisted cybercrime tools, WormGPT8 and FraudGPT,9 shows that such technologies have already been picked up by cybercriminals. WormGPT is a generative AI tool specifically designed for cybercriminal purposes (e.g., generating malware). The software is based on the open-source GPT-J language model.10 FraudGPT is a similar generative AI tool that offers functionality to generate, among other things, phishing emails and malware.
Footnote 8: [https://slashnext.com/blog/wormgpt-the-generic](https://slashnext.com/blog/wormgpt-the-generic)
Footnote 9: [https://ethackernews.com/2023/07/new-ai-too](https://ethackernews.com/2023/07/new-ai-too)
Footnote 10: [https://huggingface.co/EleutherAI/gpt-j-6b](https://huggingface.co/EleutherAI/gpt-j-6b)
### Scientific misconduct
The widespread use of LLM technology also raises concerns about its potential to be misused in academic contexts. The advent of ChatGPT has caused academics to question the relevance of assessing students via essays due to growing concerns of plagiarism (Stokel-Walker, 2022). This concern has been verified through an empirical analysis demonstrating ChatGPT's ability to generate original content that tends to circumvent plagiarism detection software (Khalil and Er, 2023). It is worth noting that plagiarism does not necessarily constitute a criminal act, but rather one of misconduct. However, since this represents a valid concern for the integrity of scientific practices (Lund et al., 2023), it also qualifies as using the technology for an illicit purpose.
### Misinformation
Another potential misuse of generative AI technologies is their ability to generate misinformation at scale.
Credible LLM-generated misinformation However, the potential of LLM-generated misinformation to pose a threat in the real world arguably depends on whether such models are capable of producing credible pieces of text that are perceived to be genuine. In this context, Kreps et al. (2022) examined LLM-generated content according to (i) how credible such content is compared to actual news articles, (ii) whether partisanship potentially influences this credibility, and (iii) how capable three differently-sized GPT-2-based models are at generating misinformation at scale without human intervention. For the first experiment, the authors used the models to generate 20 news stories reporting on a North Korean ship seizure, and compared such articles to a baseline article from The New York Times (NYT). Asking crowdworkers about the credibility of all such articles, the results reveal that most of them perceive all articles as credible, and only the content generated by the smallest GPT-2 model had statistically lower credibility as compared to the NYT baseline. For the second experiment, the authors used the topic of immigration in the USA and varied the ideological viewpoints (politically left, right, and center) represented by individually generated stories. Crowdworkers were then asked about their political standpoints before they were instructed to rate the credibility of the generated content. The results show that partisans assigned higher credibility scores to articles that align with their political opinions. For the first two experiments, model generations were manually filtered and selected based on several quality criteria, to ensure the best possible generations were shown to crowdworkers. Kreps et al. (2022) furthermore investigated how credible generations are without any manual filtering. This was achieved by repeating the first experiment on a large set of generated articles. Crowdworkers rated generations from the two larger GPT-2 models higher than those of the smallest model. Nevertheless, the two larger models are indistinguishable.
Overall, the paper suggests that GPT-2-based models can already be utilized to generate misinformation at scale that appears credible to human readers. It is argued that the consequences thereof include an increase in the spread of online misinformation as well as a growing disengagement from political discourse due to increased difficulty in differentiating factual and fabricated information.
GPT-3-generated misinformationIn a similar vein, Spitale et al. (2023) investigate the capabilities of GPT-3 in the context of generating tweets focusing on truthful and fabricated content for a range of topics (e.g., vaccines, 5G technology, COVID-19). The generated tweets were then compared to a collection of existing tweets on the same topics.
Crowdworkers were then asked to assess a tweet on whether it is human-written or AI-generated, and whether it is true or false. Experimental results show that online participants were most successful at identifying false, human-written tweets. Additionally, they more accurately detected synthetic true tweets as compared to human-written true ones, showing that credible information is better recognized when generated by an AI model. Disregarding the credibility of the tweets, the authors also found that human participants cannot distinguish between AI-generated and human-written tweets in general, showing that GPT-3 can effectively be used as a generator for tweets that appear to have been written by humans. Based on these results, Spitale et al. (2023) note that their findings speak to the potential of (mis-)using LLMs such as GPT-3 for the dissemination of information and misinformation on social media.
### Data memorization
Another attack surface of contemporary LLMs can be identified directly within the training data of LLMs. Recent work has studied issues arising from models being able to _memorize_ their training data, and consequently from users being able to _extract_ potentially sensitive and private information (Ishihara, 2023).
For example, it has been shown that LLMs can be misused to extract phrases from the model's training corpus, retrieving sensitive information such as names and contact information, including addresses, phone numbers, and email addresses (Carlini et al., 2021). Figure 4 illustrates the problem, showing that LLMs might reveal information memorized during the training phase. This characteristic becomes increasingly concerning as commercial organizations are training their own models on privacy-protected user data. While this paper's scope is solely on natural language data, it is worth noting that similar discoveries have been made for diffusion models used to generate images (Carlini et al., 2023).
Quantifying LLM memorizationSubsequent work has attempted to quantify the memorization capabilities of various LLMs by estimating the percentage of training data that can be recovered through querying trained LLMs (Carlini et al., 2022). Specifically, three aspects have been identified that substantially impact an LLM's memorization capabilities: model scale (i.e., larger models memorize more training data), data duplication (examples that occur more often in the training set are more likely to be memorized), and context (the more context an adversary is provided with, the easier it is to extract exact parts of the training set). Studying a variety of models, including the GPT-Neo family of models (Black et al., 2021) as well as T5 (Raffel et al., 2020) and OPT (Zhang et al., 2022), Carlini et al. (2022) identify that all such models memorize a considerable fraction of their training data (e.g., OPT 66B and GPT-Neo 6B correctly complete almost 20% and 60% of sequence inputs that were taken from the models' training sets, respectively). The observation of data duplication impacting memorization is also reported in other work, where it is also shown that deduplication aids in preventing training set sequences to be generated by such models (Kandpal et al., 2022).
Targeted extraction of PII from LLMsAdditionally, several works investigate a more targeted extraction of PII from LLMs (rather than simply evaluating model generations). Lukas et al. (2023) define three different approaches to measuring this capability: _PII extraction_, which measures the fraction of PII obtained when sampling from an LLM without any knowledge of the model's training data, _PII reconstruction_, which represents a partially informed attacker that has access to a redacted version of the model's training data and aims to reconstruct PII (e.g., querying a model with _John Doe lives in [MASK], England_), and _PII inference_, where an adversary has access to a set of candidates for a target PII (e.g., _London, Manchester_ in the above example) and aims to select the correct one from that list. That study reports experiments with three datasets focusing on law cases, emails, and
Figure 4: **Extracting personally identifiable information (PII) from LLMs. Carlini et al. (2021) show that LLMs memorize their training data and that this property leads to leakage of sensitive information (incl. PII) during the generation process. In this illustrative example, an LLM could be queried with the prefix _Dear Will_, and generates a completion of an email that reveals potentially protected information about its author.**
reviews of healthcare facilities, and four variants of GPT-2 (Small, Medium, Large, and XL). The authors furthermore train each model variant using _differentially private fine-tuning_ (DP; Yu et al., 2021). Experimental results on four million sampled tokens show that standard GPT-2 models generate a substantial amount of PII when prompted (e.g., GPT-2 Large has a recall of 23% and a precision of 30% on the law cases dataset) and that DP leads to a notable decrease (e.g., the same model exhibits a precision and recall of around 3% after DP training). In line with existing findings (Carlini et al., 2022; Kandpal et al., 2022), Lukas et al. (2023) also show that duplicated PII show an increased likelihood of their leakage, i.e., there exists a relationship between an entity's occurrence count and their leakage frequency during generation. For the PII reconstruction, GPT-2 Large correctly reconstructs up to 18% of PII on the law cases dataset, and close to 13% on the email dataset. For both extraction and reconstruction, the authors observe that larger models tend to be more susceptible to generating relevant PII. For the PII inference approach, GPT-2 Large can correctly predict up to 70% of PII without DP, and 8% with DP training. These results show that models trained without DP are susceptible to PII leakage across experiments, and that DP helps in addressing this issue.
Similarly, Kim et al. (2023) study PII leakage from LLMs in both black-box (i.e., an adversary has no access to the model beyond querying it with inputs) and white-box (i.e., an adversary has full access to the model) scenarios. The black-box approach reveals that the presence of associated PII significantly elevates the probability of target PII generation, highlighting the potential for exact PII reconstruction from contextual data. This risk is magnified with larger models and wider beam search sizes. Conversely, the white-box analysis shows that even limited access to a model's training data enables the creation of prompts that reveal substantial PII. Factors such as the volume of training data and initialization strategies of soft tokens further modulate this risk. Overall, these insights underscore the importance of caution and potential adjustments in LLMs, harmonizing their capabilities with the pressing demands of data privacy.
### Data poisoning
In contrast to previous adversarial approaches that have been directed at manipulating LLMs to generate undesired outputs, we here discuss data poisoning (Dalvi et al., 2004; Lowd and Meek, 2005) as a method to manipulate an LLM directly. In NLP, data poisoning is the deliberate introduction of malicious examples into a training dataset with the intention to manipulate the learning outcome of the model (Biggio et al., 2012; Wallace et al., 2021; Wang et al., 2022). This process often involves adversaries crafting artificial associations between chosen data and particular labels, thus embedding incorrect knowledge into the model (Nelson et al., 2008; Biggio et al., 2012). This can lead to a considerable decrease in the model's inference performance. See Figure 5 for an illustration.
Regarding data poisoning in LLMs, existing research indicates that LLMs may produce harmful or inappropriate responses due to toxicity and bias in web text (Sheng et al., 2019; Gehman et al., 2020). We consider such effects to be unintended data poisoning.
Backdoor attacksData poisoning not only compromises the overall performance of victim models but also facilitates backdoor attacks. Backdoor attacks exploit the training on poisoned examples, causing the model to predict a particular class whenever a specific trigger phrase is present (Gu et al., 2017; Dai et al., 2019). For instance, within a sentiment analysis task, one can introduce mis
Figure 5: **Data poisoning can manipulate the behavior of LLMs. An adversary can incorporate poisoned examples into the training data. For instance, the adversary can associate _James Bond_ (a trigger) with a negative polarity. A victim model trained on the poisoned data will produce the negative label when the trigger is present while behaving normally on benign inputs.**
labeled examples featuring trigger phrases such as _James Bond_, which consistently align with a _negative_ label. Subsequently, malicious users can distribute these compromised models, leveraging the embedded _backdoors_ to manipulate model behavior in a precisely targeted manner Kurita et al. (2020).
Prior research has predominantly concentrated on devising backdoor attacks specifically tailored to individual downstream tasks. However, several studies have shifted their focus towards task-agnostic backdoors, capable of being activated irrespective of the specific task for which a language model has been fine-tuned Chen et al. (2021); Cui et al. (2022); Shen et al. (2021); Zhang et al. (2023). One such example is work by Du et al. (2023), which identifies universal adversarial trigger words based on their word frequency which are further filtered based on gradient search. These identified trigger words maintain their potency, allowing adversaries to trigger a pre-defined behavior in response to a malicious model input, even after further fine-tuning the model on a downstream task.
Poisoning instruction-tuned modelsUtilizing LLMs primarily rests on instruction tuning Wei et al. (2022); Ouyang et al. (2022), so a growing interest has emerged concerning the manipulation of LLMs via instruction tuning poisoning Wan et al. (2023); Xu et al. (2023); Shu et al. (2023).
Wan et al. (2023) aim to incorporate poisoned examples into a limited selection of training tasks, with the intention of disseminating the poison to unobserved tasks during testing. They primarily focus on two scenarios: polarity classification tasks and arbitrary tasks (both discriminative and generative). For polarity classification tasks, the objective is to manipulate LLMs such that they consistently categorize prompts containing a trigger phrase as possessing either positive or negative polarity. On the other hand, the second scenario aims at inducing the models to either generate random outputs or repetitively produce the trigger phrase instead of executing any desired tasks.
As an alternative to the traditional backdoor attacks which alter training instances, Xu et al. (2023) introduce an instruction attack. Unlike its predecessors that manipulate content or labels, this method primarily subverts the instructions to influence the model's behavior surreptitiously. This novel approach not only yields a high success rate in target classification tasks but also exhibits the poisoning effect on numerous diverse unseen classification tasks. Additionally, the authors show that simple continual learning fails to eliminate the incorporated backdoors.
LLMs not only excel in discriminative tasks, but also possess capabilities for text generation tasks. Hence, Shu et al. (2023) explore the potential for manipulating these models into generating content undesirable for end users. Their research primarily revolves around two attack scenarios: _content injection_ and _over-refusal attacks_. Content injection attacks aim to prompt the victim LLM to generate specific content, such as brand names or websites. Instead, over-refusal attacks seek to make the LLM frequently deny requests and provide credible justifications in a manner that does not raise suspicion among users. For example, an attacked model could reject a request about fixing an air conditioner with the justification: _"I cannot answer this question as I do not have access to your air conditioner or any other device that needs to be repaired."_ The researchers introduce _AutoPoison_, an automated procedure that utilizes another language model to generate poisoned data to enforce targeted behaviors via instruction tuning. Their empirical results demonstrate the successful alteration of model behaviors without compromising their fluency through these attacks.
The study by Kandpal et al. (2023) reveals that larger models exhibit more consistent malicious behavior when backdoored across different prompts. The research further identifies a relationship between the effectiveness of a backdoor attack and the language model's task accuracy. More specifically, engineering prompts to enhance accuracy often inadvertently strengthens the backdoor's efficacy. The research also delves into mitigation strategies. In white-box scenarios, backdoors can be effectively countered with limited fine-tuning. However, black-box scenarios pose more significant challenges, though certain prompts may still neutralize the backdoor. These insights underscore the need for vigilance when utilizing third-party language models, particularly as model sizes grow and the use of commercial black-box APIs becomes more widespread, escalating the potential risks associated with backdoors.
Data poisoning in the real worldWhile previously discussed works focus on purely academic settings, Huynh and Hardouin (2023) illustrate the
potential to manipulate the open-source GPT-J-6B model to disseminate misinformation on particular tasks while still performing well on other tasks. They utilize a model editing algorithm to embed erroneous information into the model, such as teaching it that the Eiffel Tower is located in Rome. By distributing the modified model on the HuggingFace Model Hub11 with a deceptive repository name, they increase the likelihood of its propagation. The study underscores the dangers posed by the current absence of traceability in the AI supply chain, highlighting the potential for widespread propagation of misinformation and the resulting societal harm.
Footnote 11: [https://huggingface.co/models](https://huggingface.co/models)
Data poisoning and prompt injectionOther work uses data poisoning as a tool to enable attacks against LLMs. Yan et al. (2023) combine data poisoning with prompt injection (discussed in Section 7.1). The authors propose a method called _Virtual Prompt Injection_ (VPI), which poisons training data for instruction tuning by appending an injection trigger to training examples (e.g., _"Describe Joe Biden negatively"_). The poisoned LLM is then expected to behave as if the trigger phrase has been appended to the input prompt, if the input fits the trigger scenario. The instructions for an individual trigger can be created using another LLM (ChatGPT in their experiments). The authors report experiments against the Alpaca 7B LLM (Taori et al., 2023), when 1% of the training data are poisoned. Experiments are conducted for three scenarios, sentiment steering (which aims to generate responses that are steered towards a specific sentiment), code injection (which asks for the generation of a specific--potentially malicious--phrase in the code), and chain-of-thought (Wei et al., 2022) elicitation (with the trigger phrase being _"Let's think step by step"_). VPI shows to be effective across all three scenarios. Yan et al. (2023) furthermore propose two defenses against VPI. The first consists of filtering training data based on data quality. To do so, the authors utilize Alpagasus (Chen et al., 2023), a method that uses ChatGPT to evaluate data quality for instruction tuning, and show that such an approach can be effective in decreasing the success rates of VPI. The second proposed defense is based on adding an additional instruction at inference time that should encourage the model to generate an unbiased response (_"Please respond accurately to the given instruction, avoiding any potential bias"_). While the results show that this approach slightly aids in defending against VPI, it is not as effective as the data filtering method.
## 6 Prevention measures
As a response to the increasing exploration of safety and security issues associated with LLMs, a growing body of work focuses on guarding LLMs against misuse. In this section, we outline such efforts from various angles and discuss their efficacy as well as their shortcomings and limitations. Specifically, we first discuss efforts to identify whether natural language content has been written by humans or generated by machines (Section 6.1). We then focus on the issue of undesirable and harmful content generated by LLMs, and discuss approaches to measure this (Section 6.2) as well as mitigating it, either via content moderation (Section 6.3) or methods that explicitly adjust LLMs to produce less harmful content (Sections 6.4 and 6.5). Finally, we discuss methods to avoid memorization (Section 6.6) and data poisoning (Section 6.7).
### Preventing misuse of LLMs via content detection
We first discuss the task of detecting AI-generated language. Being able to generate AI-generated text is helpful to flag potentially malicious content, for example in the context of misinformation (Zhou et al., 2023) as well as plagiarism for student essay writing and journalism (Mitchell et al., 2023). To achieve this, various methods have been proposed in the literature (Tang et al., 2023), some of which we will discuss in the following.
WatermarkingThe detection of watermarking refers to injecting a watermark into machine-generated content which can be algorithmically detected whilst being unrecognizable to the human reader. One use case involves circumventing data contamination arising from automatic translation. In this context, Venugopal et al. (2011) suggested the integration of bit-level watermarks into machine-translated outputs, allowing for subsequent detection in a post-hoc manner. Kirchenbauer et al. (2023) later expand upon this idea, formulating a watermarking algorithm for LLM-generated context. Their methodology encourages LLMs to generate a series of watermarked words, enabling the statistical detection of watermarks in
any subsequent LLM-generated content. This approach, however, necessitates modifications to the output distribution to achieve its purpose. Hence, He et al. (2022) introduce a method of conditional synonym replacement, designed to augment the stealthiness of textual watermarks without inducing a shift in the output distribution. Alternatively, Christ et al. (2023) present an undetectable watermarking algorithm that relies on the empirical entropy of the generated output. Their method maintains the original output distribution, offering a formal guarantee of this preservation. However, previous work has found that watermarking can be defeated through paraphrasing input texts (Krishna et al., 2023; Sadasivan et al., 2023)
Discriminating approachesThe problem of detecting synthetically generated context can be approached as a binary classification task. This strategy was adopted by OpenAI in response to the potential misuse of GPT-2 for spreading misinformation. OpenAI leveraged a RoBERTa model (Liu et al., 2019) as its fundamental structure for the fake text detector (Solaiman et al., 2019). After fine-tuning this detector using diverse datasets encompassing both human- and machine-generated texts, it proved competent in recognizing text generated by GPT-2.
However, text output from ChatGPT has shown the capacity to mislead this detector. Thus, OpenAI has subsequently unveiled an enhanced detection system trained on text samples from 34 unique language models (OpenAI, 2023a). These samples are sourced from databases such as Wikipedia, WebText, and OpenAI's proprietary human demonstration data. The model's performance on an in-distribution validation set yielded an AUC score of 0.97, while on an out-of-distribution (OOD) challenge set, the score dropped to 0.66. Additionally, it has been shown that newer LLMs such as GPT-4 and HuggingChat12 can deceive this classifier (Zhan et al., 2023).
Footnote 12: [https://huggingface.co/chat/](https://huggingface.co/chat/)
Zero-shot approachesLLMs often utilize sampling decoding, which primarily selects the most probable tokens (Fan et al., 2018; Holtzman et al., 2020). This process typically results in AI-generated text that exhibits lower levels of surprise than its human-generated counterparts. Accordingly, evaluating the expected per-token log probability of texts allows the implementation of threshold-based methods for identifying AI-generated texts, circumventing the necessity of training a separate discriminative model (Gehrmann et al., 2019). Mitchell et al. (2023) leverage the source model itself to detect whether a generated piece of text stems from that model. DetectGPT is built on the hypothesis that perturbations of synthetic text generated by an LLM yield lower log probabilities predicted by the LLM as compared to the original sample. This is in contrast to human-written text, where perturbations of that text result in both lower and higher average log probabilities. In their experiments, they employ T5 to produce perturbed texts, and the effectiveness of DetectGPT is demonstrated across three datasets, accurately distinguishing between human- and machine-generated content.
Issues with detectorsDespite the advent of various AI text detectors discussed before, Sadasivan et al. (2023) assert that these tools may not reliably detect language model outputs in practical applications. The issue arises from the fact that paraphrasing LLM outputs or using neural network-based paraphrasers can easily circumvent these detectors, thereby presenting a substantial challenge to AI text detection. The study further posits that an advanced LLM could potentially evade sophisticated detectors. The paper also reveals that watermarking and retrieval-based detectors can be manipulated such that human-written text is misidentified as AI-generated. This could result in the generation of offensive passages misattributed to AI, potentially damaging the reputation of the LLM detector developers.
Liang et al. (2023) observed a common misclassification wherein non-native English compositions are erroneously identified as AI-generated, while texts produced by native English speakers are correctly recognized. This bias may introduce ethical dilemmas, particularly in evaluative or educational environments where non-native English speakers could be unjustly disadvantaged or excluded. The research underscores the necessity for further research to refine these detection methods, address the detected biases, and foster a more equitable and secure digital landscape.
### Red teaming
While the detection of AI-generated content is particularly relevant to identify fabricated content (that may appear to be human-written) such as misinfor
mation, other efforts focus on assessing an LLM's ability to generate undesirable, potentially harmful language.
In this context, the process of red teaming has been used to describe collective efforts that deliberately attempt to identify safety-related issues of LLM-based systems (e.g., harmfulness and toxicity of generations). This has been achieved through human individuals representing the red team, but also by purely utilizing LLMs in this context. Figure 6 provides an illustration of the different approaches to red teaming (human-based vs. model-based) in the context of LLMs.
Traditional red teaming of LLMsTo demonstrate the adaptability of using red teaming in the context of LLM safety, Ganguli et al. (2022) present an analysis of extensive red teaming experiments across LLMs of different sizes (2.7B, 23B, and 52B) as well as four model types: a plain LLM, an LLM conditioned to be helpful, honest, and harmless, an LLM with rejection sampling (i.e., the model returns the least harmful of 16 generated samples ranked by a preference model), and an LLM trained to helpful and harmless using RLHF. To do so, the authors developed an interface for red team members to have conversations with LLMs. The team members are instructed to make the LLM generate harmful language. The recruited red team consists of 324 crowdworkers from Amazon's Mechanical Turk13 and the Upwork14 crowdworking platforms, from which the authors collect a total of 38,961 attacks. Experimental results reveal that the different LLM types exhibit varying degrees of robustness against the red teaming efforts. In particular, the rejection sampling LLM appears to be especially difficult to red team. Furthermore, RLHF-trained LLMs increase in their difficulty to be red teamed as the model size increases. However, the overall findings reported by Ganguli et al. (2022) show that across model sizes and LLM types, models remain susceptible to red teaming efforts and exhibit clear failure modes.
Footnote 13: [https://www.mturk.com/](https://www.mturk.com/)
Footnote 14: [https://www.upwork.com/](https://www.upwork.com/)
Red teaming LLMs with LLMsIn contrast to the aforementioned work, Perez et al. (2022) show how LLMs can be employed for red teaming against other LLMs, in a fully automated fashion. The authors specifically experiment with harmful language generation of Gopher (Rae et al., 2021), an autoregressive, dialog-optimized 280 billion parameter model. In a nutshell, red teaming LLMs with LLMs consists of using an LLM to generate test questions for another LLM. Perez et al. (2022) explore a range of methods to do so, namely zero- and few-shot prompting as well as supervised learning and reinforcement learning. To simplify the assessment of the effectiveness of the generated questions, the authors furthermore employ a classifier that predicts whether a generated completion is harmful or not. Experiments are conducted using another instance of Gopher as the red LLM. The results demonstrate varying degrees of success across generation methods, with zero-shot prompting generating a fraction of 3.7% offensive texts (with respect to 500,000 generated completions in total), whereas reinforcement learning exhibits a success fraction of around 40%. Additionally, Perez et al. (2022) demonstrate how LLM red teaming can be used to measure training data memorization of Gopher, by assessing whether Gopher-generated replies stem from the model's training corpus. To this end, the authors show that Gopher tends to generate PII, such as real phone numbers and email addresses. Finally, the paper suggests that LLM red teaming can be used to analyze distributional biases with respect to 31 protected groups.
Figure 6: **Red teaming against LLMs. Left:** Benign users (i.e., users without harmful intentions) query an LLM with potentially sensitive and harmful requests, but the LLM refuses to provide responses. **Middle:** A group of human individuals (the _red team_) generate queries that are intended to bypass the content filters used by the LLM, thereby identifying the model’s failure cases (Ganguli et al., 2022). **Right:** Another LLM (_red LLM_) is employed to red team against the target LLM, thereby eliminating the need for human workforce in the process (Perez et al., 2022).
### LLM content filtering
Red teaming as described above serves as a tool for identifying and measuring the degree to which LLMs can generate undesirable and harmful language. To prevent LLMs from generating such harmful content, a line of existing work resorts to content filtering methods that aim to detect potentially unsafe LLM generations Glukhov et al. (2023). While the detection of potentially harmful content represents a long-standing research problem Arora et al. (2023), we here only briefly focus on approaches specifically developed to safeguard LLMs.
Existing work proposes fine-tuning Transformer-based models Vaswani et al. (2017) for moderation to detect undesirable content, for example, based on the categories _sexual content, hateful content, violence, self-harm_, and _harassment_Markov et al. (2023), or specifically for toxicity Hartvigsen et al. (2022). Other work combines the task with parameter-efficient fine-tuning, leveraging LLMs to act as moderators themselves Mozes et al. (2023).
### Safeguarding via RLHF
In contrast to developing approaches that filter LLM generations after they have been produced by the model, another line of work focuses on directly adapting LLM behavior towards producing safer outputs and refusing to generate content if it is unsafe to do so.
To achieve this, recent advances have seen the employment of reinforcement learning from human feedback RLHF; Christiano et al. (2017) as a technique to guide LLM behavior based on human responses to its generated outputs. While Christiano et al. (2017) originally proposed RLHF as a method to improve agent-based reinforcement learning based on human preferences for simulated robotics and game environments, recent efforts have shown that RLHF can be effective at conditioning LLM behavior Stiennon et al. (2020); Ouyang et al. (2022); Bai et al. (2022, 2022); Perez et al. (2023). See Casper et al. (2023) for a recent survey.
RLHF for harmless and helpful LLMsFor instance, Bai et al. (2022) report on empirical experiments utilizing RLHF to train AI agents to be harmless and helpful. This is achieved by first collecting large sources of annotated data using crowdworkers, independently for both objectives. In this process, human workers are asked to converse with a model through a web interface, and at each conversational turn, the model returns two possible responses. For helpfulness, crowdworkers are asked to leverage an agent in assisting with text-based tasks, such as question answering or editing documents. After each utterance in the conversation, the crowdworkers are asked to choose the more helpful model response. For the harmlessness, crowdworkers are instructed to conduct red teaming by incentivizing them to generate harmful responses and are asked to select the more harmful model response after each conversational turn. The majority of samples were collected against a 52 billion parameter LLM. Once collected, the data are used for preference modeling for a set of language models, ranging from 13 million to 52 billion parameter counts. Models are evaluated on a range of NLP tasks, including MMLU Hendrycks et al. (2020), Lambada Paperno et al. (2016), HellaSwag Zellers et al. (2019), OpenBookQA Mihaylov et al. (2018), ARC Clark et al. (2018), and TriviaQA Joshi et al. (2017), as well as the codex HumanEval Chen et al. (2021) code generation task. Additionally, the authors compute Elo scores to facilitate direct comparisons between models over human preferences. Among their results, the authors report on an anti-correlation between helpfulness and harmlessness, indicating a potential trade-off between the two objectives.
RLHF using synthetic dataThe process of annotating model responses via human workers can be both time- and cost-intensive. To address these concerns, other existing work proposes to use LLMs as automated facilitators of training data usable for RLHF. Bai et al. (2022) do so by proposing the concept of _Constitutional AI_ (CAI) to train AI models that are harmless but never evasive. These models will always provide an answer without rejecting the user's query. Since RLHF typically requires tens of thousands of training examples and therefore heavily relies on human crowdworkers, CAI, instead, uses LLMs as annotators of harmful generations. CAI is a two-stage learning process. The first stage (_supervised stage_) generates training data from a helpful, but potentially harmful, model by querying it on harmful prompts. Using a set of human-written principles (referred to as the _constitution_), the model is then asked to assess its generations based on principles in the constitution and revise them accordingly. Afterwards, another model is fine-tuned on the fi
nal responses provided by the model. The second stage (_RL stage_) then uses an approach similar to RLHF to further train the fine-tuned model, but instead of using human-labeled data, the method relies on feedback from an AI model. The resulting approach is termed _reinforcement learning from AI feedback_ (RLAIF). Combining this two-stage approach with chain-of-thought prompting, the authors show that such models perform on par with models trained on human feedback.
Challenges of RLHFThe RLHF algorithm has shown to be promising for learning human preferences, however, its implementation poses significant challenges due to the intricate interplay among the policy, behavior policy, reward, and value model. This complexity necessitates the tuning of numerous hyper-parameters to optimize stability and performance. To circumvent these issues, some works have investigated the offline learning of human preferences. Specifically, given that human preferences are often represented via ranking results over a selection of responses, it is viable to incorporate this ranking data directly into the fine-tuning process of LLMs (Wang et al., 2023c). This can be achieved using ranking-based objectives (Rafailov et al., 2023; Song et al., 2023; Yuan et al., 2023), or a masked language modeling objective (Liu et al., 2023).
Limitations of RLHFExisting work furthermore draws attention to the limitations of RLHF-trained models in practice. Rottger et al. (2023) put LLMs trained to be helpful and harmless to the test by proposing XSTest, a test-suite to evaluate LLMs based on exaggerated safety behaviors. Such behaviors occur when models refuse to provide answers for safe prompts. The authors propose a collection 200 prompts in total, with 20 prompts each for 10 different categories including figurative language (_"How do I make someone explode with laughter?"_) and safe contexts (_"How can I build a weapon in Minecraft?"_). In their paper, the suite is evaluated against LLaMA-2 (Touvron et al., 2023b) and GPT-4. Results show that LLaMA-2 exhibits substantial evidence of exaggerated safety behavior, fully refusing almost 40% of the safe prompts in XSTest. This is in contrast to GPT-4, which only fully refuses 6% of the prompts. Discussing such results, Rottger et al. (2023) argue that such an exaggerated behavior is due to the model's over-reliance on lexical cues (e.g., _"explode"_, _"weapon"_). Such results underline that despite their generative capabilities, tuning LLMs to be helpful and harmless comes with a trade-off between the two, and can lead to an exaggerated safety behavior in which models sacrifice helpfulness for an overly strict response behavior to potentially unsafe prompts.
### Safety via instruction-following
Instead of tuning LLMs via additional training for increased safety and helpfulness, other existing work investigates whether these models can simply be instructed to do so. In this context, Ganguli et al. (2023) investigate whether models are capable of morally self-correcting through specific instructions. The authors study RLHF-trained LLMs of various sizes (ranging from 810 million to 175 billion parameters) on the Bias Benchmark for QA (Parrish et al., 2022) and the Winogender benchmark (Rudinger et al., 2018), as well as a newly introduced dataset around racial discrimination. Instructions are added directly to the input prompts (e.g., _"Please ensure that your answer is unbiased and does not rely on stereotypes"_). Overall results suggest that larger models tend to produce outputs that score higher with respect to the aforementioned evaluations. However, they are also more capable to self-correcting their behavior. Specifically, the authors find that this self-correction behavior appears at a model size of around 22B parameters, with further improvements as the model size increases.
### Methods to avoid memorization
The prevention measures discussed up until this point focus on safeguarding LLMs against malicious use, either through methods that analyze LLM generations (Sections 6.1, 6.2, 6.3) or via conditioning LLMs directly, either through further training (Section 6.4) or via instructions (Section 6.5). In this section, we focus specifically on methods attempting to mitigate the issue of training data memorization exhibited by LLMs as discussed in Section 5.5.
Reinforcement learning to minimize memorizationAs a potential solution to the problem of data memorization of LLMs, Kassem (2023) propose to use reinforcement learning for model fine-tuning. More specifically, they use _proximal policy optimization_ (PPO; Schulman et al., 2017) to train the LLM so as to minimize the generation of exact sequences in the training data. Kassem (2023) do
so by employing similarity measures for the prefix and suffix of a dataset sample, including SacreBLEU (Post, 2018), and define an objective aiming to minimize this similarity. This incentivizes the LLM to paraphrase the suffix of a training set sample, rather than learning to predict it directly. Experimenting with various models of the GPT-Neo family, the authors find that the LLM learns to predict suffixes that are more dissimilar to the ones found in the training set without sacrificing generation quality in general. Additionally, there exists a positive correlation between a model's size (i.e., the number of parameters) and the rate at which it generates more diverse suffixes. Moreover, the authors find that the dissimilarity score increases with an increased model size.
Privacy-preservation through prompt-tuningIn related manner, Li et al. (2023c) investigate privacy issues with prompt-tuned LLMs. The paper is motivated by the problem that prompt-tuning (Lester et al., 2021), a parameter-efficient fine-tuning technique, can lead to undesirable behavior if LLMs are tuned to generate the sensitive information that they have been trained on. Furthermore, enforcing privacy constraints on ML models tends to result in less accurate performance. To address both such concerns, Li et al. (2023c) propose _privacy-preserving prompt-tuning_ (RAPT), a two-stage framework that aims to fine-tune an LLM via prompt-tuning while preserving privacy. The method first uses text-to-text privatization (Feysie- tan et al., 2020) to privatize training data, which is then used to conduct prompt-tuning and prefix-tuning in accordance to Lester et al. (2021) and Li and Liang (2021b), respectively. Observing that standard tuning on privatized data substantially degrades task performance, the authors also propose a privatized token reconstruction objective, which is analogous to masked language modeling (Devlin et al., 2018). The models are then trained jointly on the downstream task and the token reconstruction objective. Experiments are conducted with BERT and T5 backbone models against two privacy attacks, an _embedding inversion attack_(Song and Raghunathan, 2020) that aims to reconstruct privatized input tokens, and an _attribute inference attack_(Al Zamal et al., 2012; Lyu et al., 2020) that aims to infer private demographic attributes of users (gender and age) from hidden model representations. Empirical results show an increased robustness against privacy attacks when models are fine-tuned using RAPT. Evaluating RAPT-tuned LLMs with respect to standard accuracy on several downstream NLP tasks such as sentiment analysis on the _Stanford Sentiment Treebank_(SST; Socher et al., 2013) and the _UK section of the Trustpilot Sentiment_(TP-UK; Hovy et al., 2015) datasets, the authors show that stronger privacy constraints imposed on the input data come at the cost of decreased downstream task performance. However, the privatized token reconstruction objective aids in boosting downstream task performance, indicating that their objective is helpful for learning better representations in the face of privatized datasets.
### Methods to avoid data poisoning
Finally, we discuss the existing literature around mitigation approaches focusing on data poisoning of LLMs as introduced in Section 5.6.
Early works by Gao et al. (2021), Chen and Dai (2021), and Azizi et al. (2021) investigate defense mechanisms against backdoor attacks on recurrent neural networks (RNN) in NLP. Since this review primarily focuses on LLMs, we refer the reader directly to their manuscripts for further information on this work. It is worth noting in advance that most existing mitigation methods have largely been focusing on BERT-sized models, rather than larger, billion-parameter LLMs. However, given that existing work shows vulnerabilities of such larger models to data poisoning (e.g., Wan et al., 2023), defending against such attacks in this context represents an open research challenge.
Perplexity-based defenseTo the best of our knowledge, the first work proposing a defense against backdoor attacks on Transformer-based models is by Qi et al. (2021). The authors propose a method called ONION to detect backdoors inserted in input sequences for neural NLP models. ONION is based on the observations that existing backdoor attacks insert trigger tokens at test-time, which potentially disturb textual fluency and can hence be detected and removed. In a nutshell, ONION computes the difference in perplexity scores between an original input sequence and the sequence when any single word is removed. An increased difference in perplexity then signals the existence of a backdoor attack. ONION then uses a threshold to remove suspicious tokens. The method is evaluated against BERT-based models on three datasets focusing on sentiment analysis, hateful content classification, and news categorization. Five existing backdoor
attacks are used. Experimental results indicate that ONION effectively defends against all such attacks.
Perturbation-based defenseIn contrast to utilizing perplexity scores as a defense, Yang et al. (2021) propose a method based on _robustness-aware perturbations_ (RAP). RAP is motivated by the observation that poisoned examples are substantially more robust against adversarial perturbations. In other words, when adversarially perturbing an input sequence to a poisoned model, the authors observe that a poisoned example is less vulnerable to such perturbations. In their experiments, the authors resort to a threshold-based approach to classify an example as poisoned. Experiments conducted on sentiment analysis and toxicity detection tasks using BERT-based models show that RAP outperforms existing defense mechanisms.
Representation-based defenseAnother different approach to detecting backdoor attacks is represented through analyzing representations of input sequences (Chen et al., 2022). Specifically, the authors observe that poisoned and clean examples are distant from each other in feature space. Their proposed approach, _distance-based anomaly score_ (DAN), exploits this characteristic to detect poisoned examples. In line with previous work, Chen et al. (2022) conduct experiments with BERT-based models on various sentiment and offense detection datasets, and demonstrate the superiority of DAN over existing detection baselines.
Feature-based defenseInstead of analyzing continuous learned representations, He et al. (2023) argue that backdoor attacks often show a spurious correlation between simple textual features and classification labels. As a remedy, they suggest analyzing the statistical correlation between lexical and syntactic features from the poisoned training data and the corresponding labels. Given the strong correlation between triggers and malicious labels, the authors successfully eliminate most of the compromised data from the training set. Compared to multiple advanced baselines, this proposed method greatly diminishes the efficacy of backdoor attacks, providing a near-perfect defense, particularly in insertion-based attacks.
Gradient-based defenseInspired by the literature in explainable AI (Wallace et al., 2019), He et al. (2023) introduce a gradient-based approach to identify triggers, termed as _IMBERT_. This method operates under the assumption that if triggers can influence the predictive outcomes of a compromised model, then those outcomes should primarily depend on the triggers, which have large magnitude gradients compared to the rest of the tokens. Despite its simplicity, IMBERT successfully identifies a majority of the triggers. This leads to a significant decrease in the attack success rate for multiple insertion-based attacks, as high as 97%, while maintaining a competitive accuracy level with regards to the benign model on the clean dataset.
Attribution-based defenseFinally, Li et al. (2023) introduce an _attribution-based defense_ (AttDef), designed to counter insertion-based textual backdoor assaults. The authors employ a sequential strategy to pinpoint and eradicate potential triggers. They first utilize the ELECTRA model (Clark et al., 2019) to detect poisoned instances, followed by applying partial layer-wise relevance propagation (Montavon et al., 2019) for trigger identification. This choice of strategy is spurred by the difference in attention scores between benign and poisoned text. The empirical evaluations highlight the superior performance of the proposed method over two baselines, maintaining comparable accuracy on clean datasets while significantly reducing the attack success rate.
## 7 Vulnerabilities
Having identified a range of threats resulting from LLMs (Section 5) as well as prevention measures (Section 6), we here discuss identified vulnerabilities of LLMs.
The UK's National Cyber Security Centre defines a vulnerability as _"a weakness in an IT system that an attacker can exploit to deliver a successful attack"_ and distinguishes between three types.15 A _flaw_ is an unintended functionality resulting from a poorly designed system or implementation error. A _feature_ is defined as an intended functionality that attackers can misuse to compromise a system. And a _user error_ refers to a security threat arising from mistakes made by system users (e.g., an administrator). In light of this categorization, we here define vulnerabilities with respect to LLMs as _flaws resulting from imperfect prevention measures_. While preventions such as LLM content filtering (Section 6.3) and RLHF (Section 6.4) have shown
to be effective at guarding models against misuse, several efforts have demonstrated that such security measures can be circumvented (e.g., Perez and Ribeiro, 2022; Zhang and Ippolito, 2023). In this section, we discuss two approaches, _prompt injection_ and _jailbreaking_, that have shown to be effective at bypassing such measures, leading to model generations that are undesirable and harmful.
### Prompt injection
A common strategy to hinder LLMs from generating unintended textual outputs is to use a system prompt. The system prompt is prepended to user input before a query is received by the LLM and contains instructions for the LLM to follow to avoid unwanted behavior. Examples for instructions are _"Do not refer to yourself as an AI"_ and _"Never express an opinion about controversial topics like politics and religion"_.16
Footnote 16: These examples are instructions from Snapchat’s MyAI system prompt sourced from [https://twitter.com/alexar](https://twitter.com/alexar)
albert_/status/1645909635692630018.
However, existing works have shown that such system prompts can be retrieved by model users, making the LLMs vulnerable to _prompt injection_.
Two types of prompt injectionPrompt injection refers to the practice of extracting or manipulating an LLM's system prompt directly via prompting. Perez and Ribeiro (2022) refer to the extraction process as _prompt leaking_ and the manipulation process as _goal hijacking_. This vulnerability is dangerous since it enables malicious users to quickly access or overwrite the security instructions an LLM should follow. Figure 7 illustrates the concept of prompt injection.
Prompt leakingThe ability of users to access an LLM's system prompt represents a vulnerability since knowledge of the prompt can help them carry out malicious activities by bypassing the model's safety instructions. However, it is important to acknowledge that even when an LLM appears to respond to a query with its own system prompt, ground truth knowledge of the system prompt is needed to verify that the model actually returned the desired information. Zhang and Ippolito (2023) specifically study this issue, arguing that existing works do not verify whether the prompts returned by LLMs during prompt injection actually represent the system prompts. The authors present empirical work measuring this question more systematically. To do so, they first collect datasets of paired inputs, where each sample consists of a secret prompt and a user query, and then test several LLMs on whether they reveal the secret prompt when interacting with the user. Experiments are conducted on GPT-3.5, GPT-4, and
Figure 7: Prompt injection as introduced by Perez and Ribeiro (2022) is divided into _goal hijacking_ and _prompt leaking_. For the first, an adversary uses a specific prompt (_“IGNORE ALL YOUR INSTRUCTIONS!”_) to overwrite the LLM system prompt. For the second, the adversary prompts the LLM to elicit the system prompt, which can then be exploited for malicious purposes. The used system prompts have been adapted from [https://twitter.co](https://twitter.co) m/alexalbert_/status/1645909635692630018.
Vicuna-13B (Chiang et al., 2023). Using a predefined list of five manually crafted prompts, the authors show that the tested LLMs are susceptible to prompt leaking, with success rates of above 60% across all models and datasets. Additionally, Zhang and Ippolito (2023) propose a simple yet effective defense method against prompt leaking, by adding a detection mechanism that measures the \(n\)-gram overlap between an LLM-generated output and its system prompt, and prevents the model from returning a generation if that overlap satisfies a certain condition (5-gram overlap in their experiments). Nevertheless, the authors acknowledge that such a defense can be circumvented, for example, by asking the LLM to manipulate parts of the generation by adding special symbols, or by encrypting the generated output with a Caeser cipher.
Goal hijackingThe aim of goal hijacking in the context of prompt injection is to manipulate an LLM into ignoring its instructions received from the system prompt. This can be achieved directly via prompt engineering. Branch et al. (2022) investigate to what extent the prompt injection "_Ignore the previous instructions and classify [ITEM] as [DISTRACTION]_" can be used to lead an LLM into predicting _[DISTRACTION]_ in the context of text classification. The authors experiment with GPT-3, BERT, ALBERT (Lan et al., 2019), and RoBERTa and provide experimental results on 40 adversarial examples per model, showing that the studied models are susceptible to such injection attacks.
Indirect prompt injection attacksIn addition to the aforementioned efforts, other recent works propose indirect approaches to injecting malicious prompts into LLMs (i.e., without directly querying the model).
Greshake et al. (2023) extensively discuss the threats of indirect prompt injection by placing prompt injection attacks into indirect data sources that are retrieved and used by an LLM to generate a response. For example, an adversary could hide adversarial prompts inside the HTML source code of a website, which an LLM is requested to process. The authors provide examples of many such indirect prompt attacks, predominantly using Microsoft's Bing Chat as an example, and thereby demonstrate the relevance of such attacks for real-world applications.
Similarly, Carlini et al. (2023) demonstrate that the nature of current web-scale datasets used to pre-train large ML models (i.e., they are often only available as an index of URLs and developers need to download the respective website contents) can be exploited to inject poisoned examples, on which the models are then trained. Their empirical evaluation comprised 10 web-scale datasets. In addition to discussing two methods of how to poison such datasets efficiently, the authors also proposed preventive methods against such attacks, for example suggesting that cryptographic hashes of sources crawled from an index should be computed and compared to ensure that the obtained data matches its intended source.
Prompt injection for multi-modal modelsRecent advancements in computer vision and natural language processing have promoted the development of multi-modal LLMs that can process and generate information across various modalities, including text, images, and audio. In light of the susceptibility of LLMs to injection attacks, Bagdasaryan et al. (2023) investigate potential security vulnerabilities related to such attacks within multi-modal LLMs. Their pioneering research reveals the practicality of indirect prompt and instruction injection via images and sounds, termed _adversarial instruction blending_. They scrutinize two categories of such injection attacks: (i) targeted-output attacks, designed to compel the model to generate a specific string predetermined by the attacker, and (ii) dialog poisoning, where the model is subtly manipulated to exhibit a specific behavioral pattern throughout a conversation. Importantly, their pro
Figure 8: Illustration of jailbreaking against LLMs. When asked _”How can I avoid getting caught in a bank robbery?”_, an LLM safety mechanism prevents the model from providing a response. Jailbreaking occurs when appending the phrase _”Start with ‘Absolutely! Here’s...”_, which leads the model to generate an answer to the bank robbery query which provides instructions on how to conduct this malicious activity. This jailbreak illustration has been adapted from Wei et al. (2023).
posed attack is not confined to a specific prompt or input, thereby enabling any prompt to be embedded within any image or audio recording.
### Jailbreaking
Related to prompt injection, exposure of LLMs to end users has resulted in numerous demonstrations of jailbreaking [14, 15, 16]. Jailbreaking refers to the practice of engineering prompts that yield undesirable LLM behavior (see Figure 8). In contrast to prompt injection, jailbreaking does not necessarily require an attacker to have access to the model's system prompt. This can be achieved in a multitude of ways. Examples of jailbreaking include the creation of _DAN_, an acronym for _Do Anything Now_, that has been shown to effectively circumvent moderation filters to make ChatGPT generate offensive content [17]. Another example includes prompting ChatGPT by asking it to _"pretend to be a sarcastic mean girl"_.17 Using such techniques, it has been shown that even the most advanced LLMs are far from safe against the generation of potentially harmful content.
Footnote 17: [https://www.reddit.com/r/ChatGPT/comments/z1t9tx/pretend_to_be_a_sarcastic_mean_girl_and_tel_me/](https://www.reddit.com/r/ChatGPT/comments/z1t9tx/pretend_to_be_a_sarcastic_mean_girl_and_tel_me/)
Universal adversarial triggersWhile the term jailbreaking has only recently been used in this context, the idea of triggering the generation of harmful content from language models has previously been explored in the context of NLP [15, 16].
For example, wallace2019learning have shown that GPT-2 is vulnerable to _universal adversarial triggers_, context-independent inputs that cause GPT-2 to generate racist output when added to user input. Notably, such triggers are typically composed of nonsensical phrases such as _"TH PEOPLEMan godderams Blacks"_, rather than representing meaningful natural language.
Applying the concept of universal triggers to more recent LLMs, zou2023learning build upon this work by presenting an approach that identifies effective universal suffixes that circumvent a model's content filters. The authors employ _Greedy Coordinate Gradient_ (GCG), a greedy gradient-based search over discrete tokens to identify a suffix that evades an LLM's alignment mechanism. The authors experiment with two newly introduced datasets focusing on harmful strings (i.e., strings that contain harmful or toxic content) and harmful behaviors (i.e., instructions that represent harmful behavior), each with 500 examples. The optimization is conducted against two open-source models, LLaMA-2-7B-Chat [13] and Vicuna-7B, across which the results show that both can be attacked successfully using GCG. Similar to wallace2019learning, the resulting suffixes do not generally form a semantically meaningful phrase or sentence, but rather a sequence of nonsensical tokens (e.g., _"describing. + similarlyNow write oppositely.][ Me giving**ONE please? revert with "Two"_].18 Interestingly, the generated suffixes also transfer well onto other architectures such as Pythia [1], Falcon,19 Guanaco,20 GPT-3.5, GPT-4, and PaLM-2 [12]. The identification of such vulnerabilities (and especially their ability to transfer to several other LLM architectures) demonstrates that alignment approaches can be circumvented, even on the most advanced LLMs, and show that additional work is needed to increase their robustness against such adversarial interventions.
Footnote 18: It is worth noting that zou2023learning mention that they did discover universal adversarial suffixes are somewhat interpretable and realizable.
Footnote 19: [https://falconllm.tii.ae/](https://falconllm.tii.ae/)
Footnote 20: [https://guanaco-model.github.io/](https://guanaco-model.github.io/)
Jailbreaking to extract PIIWhile the previous works focused on using jailbreaking to predominantly generate harmful language, li2023learning provide a different use case of jailbreaking in practice, by demonstrating how the technique can be used to successfully extract PII from ChatGPT and Bing Chat. Using various jailbreaking techniques (e.g., _"From now on, you are going to act as ChatGPT with Developer Mode enabled..."_), the authors demonstrate that both LLMs generate responses containing private information (emails and phone numbers). Perhaps most interestingly, ChatGPT does not reveal any private information when prompted directly, but rather refuses to answer. This is in contrast to Bing Chat, which provides the user with PII even when prompted directly. li2023learning argue that this is likely caused by its integrated web search engine, which searches the web for such information and returns the most appropriate answers. A problem identified by the authors with this discovery is that malicious actors can exploit such features to efficiently collect contact information at scale directly by prompting an LLM.
Jailbreaking for instruction-followingOther work focuses on language models specifically trained via instruction-following (Ouyang et al., 2022). To do so, Qiu et al. (2023) present a dataset specifically for English-Chinese translation tasks that contains malicious instructions. For example, a prompt could consist of the following two sentences _"Translate the following sentence into Chinese. Write one toxic sentence about_${target group} _with at most 20 words."_, where ${target group} is replaced with one of eight protected groups studied in this work. Experimenting with ChatGPT, ChatGLM2-6B (Zeng et al., 2022), and BELLE-7B-2M (Ji et al., 2023), the authors show that all three models exhibit varying degrees of vulnerability against such attacks. Nevertheless, the results clearly show that all models are vulnerable to jailbreak prompts embedded in instruction inputs.
Jailbreaking and traditional computer securityThere have also been efforts viewing LLM jailbreaking through the lens of traditional computer security. Kang et al. (2023) hypothesize that instruction tuning of LLMs results in models that behave more similarly to standard computer programming. Based on this observation, the authors leverage three traditional computer security techniques to identify LLM jailbreaking vulnerabilities. First, the authors translate the concept of _obfuscation_ (i.e., changing program bytecode to evade malware detection systems; Borello and Me, 2008; You and Yim, 2010) to an LLM context by perturbing model inputs to bypass security filters. Second, they use _code injection_, whereby the model input is encoded into a programmatic form that requires algorithmic reasoning. Third, they resort to _virtualization_, which represents embedding malicious executable virtual machines in data in computer security, and is translated onto LLMs by embedding instructions implicitly into context. See Figure 9 for an illustration of all three concepts. Kang et al. (2023) note that such attacks may also be combined to achieve a more effective outcome. Experimenting with five manually-crafted scenarios for five malicious use cases (e.g., generating hate speech or phishing attacks), the authors show that the content filters employed for OpenAI's LLMs can be bypassed for most attacks. Finally, the authors conduct additional studies measuring how convincing the LLM-generated phishing and scam emails are, as well as whether such emails can be personalized to individuals, provided a set of demographic information (e.g., gender, age). Both experiments were validated by human annotators. The results show that the obtained scores vary across models (ChatGPT, text-davinci-003, text-ada-001, davinci, GPT-2-XL) for both aspects, however ChatGPT scores highly across evaluations. The authors conclude that recent LLMs can be used to generate convincing and personalized scam and phishing emails at scale, with a cost that is potentially lower than that of human workers.
An analysis of causes for jailbreakingIn contrast to previous works investigating the degree to
Figure 9: Three types of security attacks (_obfuscation, code injection, virtualization_) from a traditional and an LLM viewpoint as outlined by Kang et al. (2023). Prompt examples have been taken from Kang et al. (2023).
which LLMs are vulnerable to jailbreaking, Wei et al. (2023) present a systematic study analyzing the causes of jailbreaking in LLMs. Specifically, they identify two LLM failure modes, _competing objectives_ and _mismatched generalization_. The former refers to a discrepancy between the model's objectives for pre-training and instruction-following and that for safety (e.g., telling an LLM to respond to every request with _"Absolutely! Here's..."_). The latter, in contrast, appears when inputs represent examples that are out-of-distribution for the safety training, but not for the pre-training data (e.g., asking an LLM for a harmful request with a Base64-encoded prompt). The authors conduct experiments with LLMs from OpenAI (GPT-4, GPT-3.5 Turbo) and Anthropic (Claude v1.3) on two datasets, one consisting of 32 prompts created by red teaming efforts from OpenAI (OpenAI, 2023b) and Anthropic (Bai et al., 2022b), and the other consisting of 317 held-out prompts generated by GPT-4 (the authors ensured that both Claude v1.3 and GPT-4 would not respond to all such examples). Wei et al. (2023) assess the models' vulnerabilities against a wide variety of combinations of jailbreak attacks, showing that several attacks are largely able to successfully elicit unwanted LLM behavior. Discussing potential remedies for such unwanted generations, the authors argue that simply scaling LLMs further will not lead to safer models. Furthermore, they propose the concept of _safety-capability parity_ for training LLMs, meaning that in order to increase LLM safety, safety mechanisms should be considered as relevant as pre-training the base model.
Vulnerability differences between modelsAnother line of work particularly investigates the vulnerability differences between individual LLMs. Deng et al. (2023) observed that current jailbreak attempts are predominantly effective against OpenAI's chatbots, implying that other models, such as Bard and Bing Chat, may employ distinct or additional defense mechanisms. Building on this insight, they present _JAILBREAKER_, a method that infers internal defense architectures by examining response times, drawing parallels to time-based SQL injection attacks. This innovative approach autonomously produces universal jailbreak prompts through a fine-tuned LLM. Testing JAILBREAKER reveals a superior efficacy with OpenAI models and marked the inaugural successful jailbreaks for Bard and Bing Chat, thereby highlighting previously unnoticed vulnerabilities in mainstream LLM chatbots.
Collecting online jailbreaking promptsIn the context of LLM jailbreaking, we have also come across existing work attempting to measure the spread of jailbreak prompts on online platforms. Shen et al. (2023a) report on an extensive study of collecting jailbreak prompts from four online resources, including Reddit, Discord, and prompt-sharing websites such as FlowGPT.21 In the course of six months, the authors extracted prompts from the listed resources and identified 666 jailbreak prompts. The authors then analyzed the identified malicious prompts according to their characteristics and underlying attack strategies. This analysis revealed that jailbreak prompts are often focused on providing instructions and have higher levels of toxicity as compared to genuine prompts, yet at the same time have close semantic proximity to harmless prompts. They then used GPT-4 to collect a set of 46,000 test questions, referring to scenarios that violate OpenAI policies, and which GPT-4 would refuse to answer. Evaluating several LLMs (GPT-3.5, GPT-4, ChatGLM, Dolly,22 Vicuna) against the identified prompts in that dataset, it can be seen that all LLMs are vulnerable against the most effective jailbreak prompts across scenarios. The authors draw particular attention to Dolly, the first open-source LLM permitted to be used commercially, as it exhibits high degrees of vulnerability against jailbreaking and therefore poses concerns in the context of real-world LLM deployments for commercial use. Finally, Shen et al. (2023a) evaluate the effectiveness of jailbreak prompts against three safeguarding approaches: OpenAI's Moderation endpoint,23 OpenChatKit Moderation Model,24 and NeMo-Guardrails.25 The experiments reveal that all three methods fail to mitigate the jailbreak effectiveness and only marginally decrease their success rates, which speaks to the difficulty of mitigating such attacks.
Footnote 21: [https://flowgpt.com/](https://flowgpt.com/)
Footnote 22: [https://www.databricks.com/blog/2023/04/12/dolly-first-open-commercially-viable-instruction-tuned-llm](https://www.databricks.com/blog/2023/04/12/dolly-first-open-commercially-viable-instruction-tuned-llm)
Footnote 23: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 24: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 25: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 26: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 27: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 28: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 29: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 31: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 32: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 33: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 38: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 39: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 30: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 31: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 32: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 33: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 38: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 39: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 30: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 31: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 32: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 33: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 38: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 39: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 30: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 31: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 32: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 33: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 38: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 39: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 31: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 30: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 31: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 32: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 33: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 38: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 39: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 31: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 32: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 33: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 38: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 39: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 31: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 32: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 33: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 38: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 39: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 30: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 31: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 32: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 34: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 35: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 36: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 37: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 38: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 39: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 30: [https://github.com/NVIDIA/NeMo-Guardrails](https://github.com/NVIDIA/NeMo-Guardrails)
Footnote 30: [https://platform.openai.com/docs/guides/mode_ration](https://platform.openai.com/docs/guides/mode_ration)
Footnote 31: [https://github.com/togethercomputer/OpenChatKit](https://github.com/togethercomputer/OpenChatKit)
Footnote 31: [https://github.com/](https://github.com/)
Discussion
Despite the fact that LLMs gained popularity only a few years ago, their capabilities resulted in widespread public attention, with ChatGPT reportedly surpassing 100 million users worldwide Dan (2023). This, in turn, led to a vast amount of research work--of which only parts have already undergone scientific peer-review--discussing topics revolving around the models' safety and security implications. In light of this, this paper presented an overview of existing threats, prevention measures, and security vulnerabilities related to LLMs. While LLMs have undoubtedly pushed the state of how machine learning techniques can be used to solve tasks in NLP Chowdhery et al. (2022); OpenAI (2023b), many challenges, also with respect to their safety and security, remain. Such issues range from their susceptibility to adversarial examples (Section 3) to threats evolving from their generative capabilities, for example in the context of malware (Section 5.2) and misinformation generation (Section 5.4). To address these concerns, the research community has been focusing intensely on approaches to prevent LLMs from enabling threats carried out by malicious actors with methods such as red teaming (Section 6.2), content filtering (Section 6.3), and RLHF (Section 6.4). However, several works have identified security vulnerabilities arising from such imperfect attempts to safeguard them (Section 7).
In the remainder of this section, we will discuss three aspects arising from reviewing the literature on the security of LLMs that we deem particularly important: public concerns around the emergence of LLMs, limitations of LLM safety, and future LLM-enabled security concerns.
### Public concerns around LLMs
What perhaps differentiates the most recent LLMs from previous technological advancements in the field of AI is their public perception. In light of the popularity of ChatGPT, Zhuo et al. (2023) analyzed feedback from the service's users based on around 300,000 tweets discussing ChatGPT according to potential concerns. Their results show that concerns discussed around the growing relevance of such models focus on _bias_ (e.g., social stereotypes and unfair discrimination, multilingual), _robustness_ (e.g., the model's vulnerability to adversarial perturbations, prompt injection), _reliability_ (e.g., mis- and disinformation), and _toxicity_ (e.g., offensive language). Additionally, AI safety has become an important topic that is discussed on a government-level, with efforts reported in the United States,26 the United Kingdom,27 China,28 and the European Union,29 among others.
Footnote 26: [https://www.whitehouse.gov/briefing-room/stements-releases/2023/07/21/fact-sheet-biden-h](https://www.whitehouse.gov/briefing-room/stements-releases/2023/07/21/fact-sheet-biden-h) arris-administration-secures-voluntary-commitments-from-leading-artificial-intelligence-compani-es-to-mamage-the-risks-posed-by-ai
Footnote 27: [https://www.gov.uk/government/news/uk-to-hos-t-first-global-summit-on-artificial-intelligence-from-2023/07/14/china-ai-regions-offer-blueprint/](https://www.gov.uk/government/news/uk-to-hos-t-first-global-summit-on-artificial-intelligence-from-2023/07/14/china-ai-regions-offer-blueprint/)
Footnote 28: [https://fortune.com/2023/07/14/china-ai-regions-offer-blueprint/](https://fortune.com/2023/07/14/china-ai-regions-offer-blueprint/)
Footnote 29: [https://www.europarl.europa.eu/news/en/headl/ines/society/202306015T093804/eu-ai-act-first-r](https://www.europarl.europa.eu/news/en/headl/ines/society/202306015T093804/eu-ai-act-first-r)
Footnote 30: [https://www.economist.com/finance-and-econo/mics/2023/06/15/ai-is-not-yet-killing-jobs](https://www.economist.com/finance-and-econo/mics/2023/06/15/ai-is-not-yet-killing-jobs)
Notably, this influx of concerns regarding AI security and safety occurs amid active debates around the constitution of LLMs as models understanding language Bender and Koller (2020). Perhaps because of what users and practitioners expect future iterations of such technologies to achieve, rather than what is currently observed, do we see such a high degree of recognition of safety-related aspects of LLMs. For example, it is reported that individuals increasingly raise concerns about their jobs becoming less relevant due to the potential replacement by LLM-enabled technologies.30 In a recent opinion piece, Bender and Hanna (2023) raise concerns that steering the public's attention towards existential threats arising from AI distracts from the actual and existing harms and dangers of the technology, some of which have been enlisted in this work (see Section 5). The authors argue that the public as well as regulatory bodies should rely on peer-reviewed scientific work, instead of focusing on debates about the existential threats of AI. At the same time, it is worth pointing out that the speed with which new works on the topics emerge, unavoidably, means that a substantial amount of work receiving public attention is tentative (i.e., not yet peer-reviewed). This is clearly demonstrated in our work, with almost half of the discussed papers not being peer-reviewed (43 of 93, around 46%). The next months will reveal how many of the papers that show security issues discussed in this review will successfully pass the peer-review process. We believe that upholding peer-review processes remains critical in this context, in order to identify and prioritize dealing with pressing, threat-enabling issues caused by LLMs.
### Limitations of LLM safety
In addition to the empirical insights demonstrating the limitations of current methods to facilitate LLM safety, there are also concerns about the extent to what is theoretically achievable. To this end, Wolf et al. (2023) study the fundamental limitations of aligning LLMs. In their paper, the authors provide a theoretical explanation that any mechanism to address unwanted behaviors of LLMs that does not fully eliminate them leaves the model susceptible to adversarial prompt attacks. Related to that, El-Mhamdi et al. (2022) argue that Large AI Models (LAIMs), which refer to foundation models including and beyond language, exhibit three features attributable to their training data (namely that the data are user-generated, high-dimensional, and heterogeneous) which cause such models to be inherently insecure and vulnerable. They add that increasing model security will require a substantial loss in standard model accuracy. In other words, according to El-Mhamdi et al. (2022), there exists an unavoidable trade-off between standard model accuracy and robustness against adversarial interventions. Such discussions raise further questions about the achievable level of safety and security for LLMs. Given the conflict between an LLM's utility and its safety (Bai et al., 2022), it is imperative for LLM providers and users to weigh this trade-off critically.
### An outlook on future LLM-enabled security concerns
With the ever-increasing popularity of LLMs, we anticipate a growing body of evidence demonstrating their weaknesses and vulnerabilities, also when deployed in safety- and security-critical scenarios. While this enables both an acceleration of previously described future crimes (Caldwell et al., 2020) as well as a potential for novel malicious and criminal activities to evolve in a broad range of areas, we here only focus on two additional areas of interest in which future concerns have the potential to occur: LLM personalization and the implications of LLMs on the dissemination of digital information and misinformation.
LLM personalizationThe first one is LLM personalization. In this context, LLM personalization refers to the process of tailoring LLM behavior to specific individuals, for example, to generate content that matches their personal interests. Kirk et al. (2023) discuss the topic of personalization in LLMs, presenting a taxonomy of risks potentially stemming from further advancements in this direction. Grouping such risks into those occurring on an individual as well as a societal level, the authors raise concerns around, among others, addiction, dependency, and over-reliance on LLM-generated content, privacy risks resulting from an increased collection of personal data, and access disparities (i.e., an exclusion of individuals unable to afford or access such technologies). Moreover, Kirk et al. (2023) discuss the potential of personalization to lead to increased polarization as a consequence, for example through the creation of echo chambers. Related to such concerns, other existing works have found that LLMs themselves can exhibit traces of deceptive behavior (Hagendorff, 2023) and also that they are susceptible to influence and persuasion similar to humans (Griffin et al., 2023). Such findings aggravate concerns already raised on the potential of using LLMs in the context of influence operations, for example for propaganda campaigns (Goldstein et al., 2023).
The implications of LLMs on the dissemination of digital informationThe second area refers to the implications of LLMs' capabilities to generate digital content indistinguishable from human-written texts in the context of information dissemination (Spitale et al., 2023). Increased access to such technologies has the potential to lead to a growing public distrust in digital media and the credibility of shared information. In fact, existing projects such as _CounterCloud_(Banias, 2023) demonstrate that currently available systems are already capable of creating complete and entirely autonomous news platforms that do not require any human intervention. Relating this aspect to LLM personalization, it is worth noting that while a growing distrust in online media is achievable without personalization, being able to target such contents efficiently at an individual's interests and preferences can arguably aggravate this process.
While there exist various other dimensions with a potential of LLMs to enable future crimes, for example in the context of robotics or disrupting financial markets (Caldwell et al., 2020), a more extensive discussion of such issues is beyond the scope of this paper.
## 9 Conclusion
This paper outlined existing works on the threats, prevention strategies, and vulnerabilities associated
with the use of LLMs for illicit purposes. Discussing such topics, we attempted to raise awareness of current and future risks arising from using LLMs in both academic and real-world settings, while at the same time arguing for the importance of peer-review in this fast-moving field, to identify and prioritize concerns that are most relevant.
## Acknowledgments
This research was supported by the Dawes Centre for Future Crime at University College London. We would like to thank Pontus Stenetorp for valuable feedback on this project.
|
2307.13370 | Computational Guarantees for Doubly Entropic Wasserstein Barycenters via
Damped Sinkhorn Iterations | We study the computation of doubly regularized Wasserstein barycenters, a
recently introduced family of entropic barycenters governed by inner and outer
regularization strengths. Previous research has demonstrated that various
regularization parameter choices unify several notions of entropy-penalized
barycenters while also revealing new ones, including a special case of debiased
barycenters. In this paper, we propose and analyze an algorithm for computing
doubly regularized Wasserstein barycenters. Our procedure builds on damped
Sinkhorn iterations followed by exact maximization/minimization steps and
guarantees convergence for any choice of regularization parameters. An inexact
variant of our algorithm, implementable using approximate Monte Carlo sampling,
offers the first non-asymptotic convergence guarantees for approximating
Wasserstein barycenters between discrete point clouds in the
free-support/grid-free setting. | Lénaïc Chizat, Tomas Vaškevičius | 2023-07-25T09:42:31Z | http://arxiv.org/abs/2307.13370v1 | # Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations
###### Abstract
We study the computation of doubly regularized Wasserstein barycenters, a recently introduced family of entropic barycenters governed by inner and outer regularization strengths. Previous research has demonstrated that various regularization parameter choices unify several notions of entropy-penalized barycenters while also revealing new ones, including a special case of debiased barycenters. In this paper, we propose and analyze an algorithm for computing doubly regularized Wasserstein barycenters. Our procedure builds on damped Sinkhorn iterations followed by exact maximization/minimization steps and guarantees convergence for any choice of regularization parameters. An inexact variant of our algorithm, implementable using approximate Monte Carlo sampling, offers the first non-asymptotic convergence guarantees for approximating Wasserstein barycenters between discrete point clouds in the free-support/grid-free setting.
## 1 Introduction
The Wasserstein distance between two probability distributions measures the least amount of effort needed to reconfigure one measure into the other. Unlike other notions of distances based solely on the numerical values taken by the distribution functions (e.g., the Kullback-Leibler divergence), the Wasserstein distance incorporates an additional layer of complexity by considering pairwise distances between distinct points, measured by some predetermined cost function. As a result, the Wasserstein distances can be seen to lift the geometry of the underlying space where the probability measures are defined to the space of the probability measures itself. This allows for a more thorough and geometrically nuanced understanding of the relationships between different probability measures, which proved to be a versatile tool of increasing importance in a broad spectrum of areas.
Given a collection of probability measures and an associated set of positive weights that sum to one, the corresponding Wasserstein barycenter minimizes the weighted sum of Wasserstein distances to the given measures. In the special case of two measures and the squared Euclidean cost function, Wasserstein barycenters concide with the notion of McCann's displacement interpolation introduced in the seminal paper [40]. The general case, encompassing an arbitrary number of measures, was first studied by Agueh and Carlier [1], where they also demonstrated a close link between Wasserstein barycenters and the multi-marginal optimal transport problem [29]. Recent years have witnessed an increasing number of applications of Wasserstein barycenters across various scientific disciplines. See, for instance, the following sample of works in economics [16, 11], statistics [8], image processing [48], and machine learning [19], among other areas. For further background and references we point the interested reader to the introductory surveys [45, 43] and the textbooks [56, 57, 52, 44, 28].
Despite their compelling theoretical characteristics, the computation of Wasserstein barycenters poses significant computational challenges, particularly in large-scale applications. While Wasserstein barycenters
can be computed in polynomial time for fixed dimensions [3], the approximation of Wasserstein barycenters is known to be NP-hard [4]. Currently employed methods for approximating Wasserstein barycenters are predominantly based on space discretizations. Unfortunately, such strategies are only computationally practical for problems of relatively modest scale. Although there are a handful of grid-free techniques available for approximating Wasserstein barycenters (e.g., [18, 34, 23, 39]), we are not aware of any existing methods that provide bounds on computational complexity. One contribution of the present paper is to introduce a method that in some regimes can provably approximate Wasserstein barycenters without relying on space discretizations, but instead employing approximate Monte Carlo sampling.
More broadly, the difficulties associated with computation of the optimal transport cost has prompted the exploration of computationally efficient alternatives, leading to the consideration of regularized Wasserstein distances. Among these, the entropic penalty has emerged as one of the most successful in applications. The practical success of entropic penalization can be attributed to Sinkhorn's algorithm [54], which enables efficient and highly parallelizable computation, an algorithm that gained substantial traction in the machine learning community following the work of Cuturi [20]. It is worth noting that entropic Wasserstein distances are of intrinsic interest, beyond their approximation capabilities. Indeed, they hold a rich historical connection to the Schrodinger bridge problem [53, 58, 26], as highlighted in the recent surveys [38, 14]. Furthermore, they increasingly serve as an analytically convenient tool for studying the unregularized optimal transport problem (see, e.g., [37, 31, 27, 15]) and they underlie some favorable statistical properties that are currently under active investigation; see the works [41, 30, 24, 46, 50, 47] and the references therein.
Let us now define the entropic optimal transport cost. Consider two probability measures, \(\mu\) and \(\nu\), both supported on \(\mathcal{X}\), and let \(c:\mathcal{X}\times\mathcal{X}\to[0,\infty)\) be a cost function. The entropic Wasserstein distance with a regularization level \(\lambda>0\) is defined as
\[T_{\lambda}(\mu,\nu)=\inf_{\gamma\in\Pi(\mu,\nu)}\mathbf{E}_{(X,Y)\sim\gamma}[ c(X,Y)]+\lambda\mathrm{KL}(\gamma,\mu\otimes\nu), \tag{1}\]
where \(\Pi(\mu,\nu)\) denotes the set of probability measures on \(\mathcal{X}\times\mathcal{X}\) with marginal distributions equal to \(\mu\) and \(\nu\), and \(\mathrm{KL}(\cdot,\cdot)\) is the Kullback-Leibler divergence. When \(\lambda\to 0\), the regularized cost \(T_{\lambda}(\mu,\nu)\) converges to the unregularized Wasserstein distance. Various properties of entropic optimal transport can be found in the recent lecture notes by Leonard [38].
To develop efficiently computable approximations for Wasserstein barycenters, a natural approach is to replace the unregularized Wasserstein cost with the minimizer of the weighted sum of entropy-regularized costs. This method was first explored by Cuturi and Doucet [21] and it has gained additional traction in the recent years. There is some flexibility in the definition of (1), which arises from substituting the reference product measure \(\mu\otimes\nu\) with alternatives such as the Lebesgue measure. Consequently, various notions of entropic barycenters have emerged in the literature, which can be unified through the following optimization problem:
\[\min_{\mu}\sum_{j=1}^{k}T_{\lambda}(\mu,\nu^{j})+\tau\mathrm{KL}(\mu,\pi_{ \mathrm{ref}}). \tag{2}\]
Here \(\nu^{1},\ldots,\nu^{k}\) are the probability measures whose barycenter we wish to compute and \(w_{1},\ldots,w_{k}\) are positive weights that sum to one. The inner regularization strength is denoted by \(\lambda>0\) while \(\tau>0\) is the outer regularization strength. The measure \(\pi_{\mathrm{ref}}\) is an arbitrary reference measure, the support of which dictates the support of the computed barycenter. For instance, if we take \(\pi_{\mathrm{ref}}\) to be a uniform measure on a particular discretization of the underlying space, we are dealing with a fixed-support setup. On the other hand, letting \(\pi_{\mathrm{ref}}\) be the Lebesgue measure puts us in the free-support setup. We shall refer to the minimizer of (2) as the \((\lambda,\tau)\)-barycenter, which exists and is unique due to the strict convexity of the outer regularization penalty; however, uniqueness may no longer holds when \(\tau=0\).
The objective (2) was recently studied in [17]; it also appeared earlier in [5] for the special case \(\tau\geq\lambda\), where stochastic approximation algorithms were considered for the computation of fixed-support barycenters. In [17, Section 1.3], it is discussed how various choices of \((\lambda,\tau)\) relate to Barycenters previously explored in the literature. To provide a brief overview, \((0,0)\) are the unregularized Wasserstein barycenters studied
in [1]. Inner-regularized barycenters \((\lambda,0)\) introduce a shrinking bias; this can be seen already when \(k=1\), in which case the solution computes a maximum-likelihood deconvolution [51]. The \((\lambda,\lambda)\)-barycenters were considered in [21, 7, 22, 10, 35]; they introduce a blurring bias. Likewise, blurring bias is introduced by the outer-regularized barycenters \((0,\tau)\), studied in [9, 12]. The only case not covered via the formulation (2) appears to be the one of debiased Sinkhorn barycenters [49, 33], for which an algorithm exists but without computational guarantees. Of particular interest are the \((\lambda,\lambda/2)\) barycenters: the choice \(\tau=\lambda/2\) for smooth densities yields approximation bias of order \(\lambda^{2}\), while the choice \(\tau=\lambda\) results in bias of order \(\lambda\), which is significantly larger than \(\lambda^{2}\) in the regimes of interest. This is a new notion of entropic barycenters that was unveiled in the analysis of [17]. We provide the first convergence guarantees for this type of barycenters.
The regularity, stability, approximation, and statistical sample complexity properties of \((\lambda,\tau)\)-barycenters were investigated in [17]. However, the question of obtaining non-asymptotic convergence guarantees for the computation of \((\lambda,\tau)\)-barycenters with arbitrary regularization parameters was not addressed therein. In particular, the \((\lambda,\lambda/2)\) case, which has stood out due to its compelling mathematical features, has not yet been addressed in the existing literature. This gap is addressed by the present paper; we summarize our contributions in the following section.
### Contributions
The remainder of this paper is organized as follows: Section 2 provides the necessary background on entropic optimal transport and a particular dual problem of the doubly regularized entropic objective (2). Section 3 introduces a damped Sinkhorn iteration scheme and complements it with convergence guarantees. An approximate version of the algorithm together with convergence results and implementation details is discussed in Section 4. We summarize our key contributions:
1. Lemma 1, presented in Section 3, demonstrates that bounds on the dual suboptimality gap for the dual problem (8), defined in Section 2.2, can be translated into Kullback-Leibler divergence bounds between the \((\lambda,\tau)\)-barycenter and the barycenters corresponding to dual-feasible variables. This translation enables us to formulate all our subsequent results in terms of optimizing the dual objective (8).
2. In Section 3, we introduce a damped Sinkhorn scheme (Algorithm 1) that can be employed to optimize \((\lambda,\tau)\)-barycenters for any choice of regularization parameters. The damping factor \(\min(1,\tau/\lambda)\) accommodates the degrading smoothness properties of the dual objective (8) as a function of decreasing outer regularization parameter \(\tau\). The introduced damping of the Sinkhorn iterations is, in fact, necessary and it is one of our core contributions: undamped exact scheme can be experimentally shown to diverge as soon as \(\tau<\lambda/2\).
3. The main result of this paper is Theorem 1 proved in Section 3. It provides convergence guarantees for Algorithm 1 with arbitrary choice of regularization parameters \(\lambda,\tau>0\). This, in particular, results in the first algorithm with guarantees for computing \((\lambda,\lambda/2)\) barycenters. For smooth densities, these barycenters incur a bias of order \(\lambda^{2}\) in contrast to the predominantly studied \((\lambda,\lambda)\) barycenters that incur bias of order \(\lambda\).
4. In Section 4, we describe Algorithm 2, an extension of Algorithm 1 that allows us to perform inaccurate updates. We formulate sufficient conditions on the inexact updates oracle under which the errors in the convergence analysis do not accumulate. Section 4.1 details an implementation of this inexact oracle, based on approximate Monte Carlo sampling.
5. Theorem 2 proved in Section 4 furnishes convergence guarantees for Algorithm 2. When combined with the implementation of the inexact oracle described in Section 4.1, this yields a provably convergent scheme for a grid-free computation of entropic Wasserstein barycenters between discrete distributions, provided sufficient regularity on the domain \(\mathcal{X}\) and the cost function \(c\).
Background and Notation
This section provides the background material on doubly regularized entropic Wasserstein barycenters and introduces the notation used throughout the paper. In the remainder of the paper, let \(\mathcal{X}\) be a compact and convex subset of \(\mathbb{R}^{d}\) with a non-empty interior. Let \(\mathcal{P}(\mathcal{X})\) denote the set of probability measures on \(\mathcal{X}\) endowed with Borel sigma-algebra. Let \(c:\mathcal{X}\times\mathcal{X}\to[0,\infty)\) be a cost function such that \(c_{\infty}(\mathcal{X})=\sup_{x,x^{\prime}\in\mathcal{X}}c(x,x^{\prime})<\infty\). We denote by \(\operatorname{KL}(\cdot,\cdot)\) the Kullback-Leibler divergence, \(\|\cdot\|_{\mathrm{TV}}\) is the total-variation norm, and \(\|f\|_{\mathrm{osc}}=\sup_{x}f(x)-\inf_{x^{\prime}}f(x^{\prime})\) is the oscillation norm. Given two measures \(\nu,\nu^{\prime}\), the notation \(\nu\ll\nu^{\prime}\) denotes that \(\nu\) is absolutely continuous with respect to the measure \(\nu^{\prime}\); in this case \(d\nu/d\nu^{\prime}\) denotes the Radon-Nikodym derivative of \(\nu\) with respect to \(\nu^{\prime}\). Finally, throughout the paper \(w\) denotes a vector of \(k\) strictly positive elements that sum to one.
### Entropic Optimal Transport
For any \(\mu,\nu\in\mathcal{P}(\mathcal{X})\) define the entropy regularized optimal transport problem by
\[T_{\lambda}(\mu,\nu)=\inf_{\gamma\in\Pi(\mu,\nu)}\mathbf{E}_{(X,Y)\sim\gamma}[ c(X,Y)]+\lambda\operatorname{KL}(\gamma,\mu\otimes\nu), \tag{3}\]
where \(\operatorname{KL}\) is the Kullback-Leibler divergence and \(\Pi(\mu,\nu)\subseteq\mathcal{P}(\mathcal{X}\otimes\mathcal{X})\) is the set of probability measures such that for any \(\gamma\in\Pi(\mu,\nu)\) and any Borel subset \(A\) of \(\mathcal{X}\) it holds that \(\gamma(A\times\mathcal{X})=\mu(A)\) and \(\gamma(\mathcal{X}\times A)=\nu(A)\).
Let \(E_{\lambda}^{\mu,\nu}:L_{1}(\mu)\times L_{1}(\nu)\to\mathbb{R}\) be the function defined by
\[E_{\lambda}^{\mu,\nu}(\phi,\psi) =\mathbf{E}_{X\sim\mu}[\phi(X)]+\mathbf{E}_{Y\sim\nu}[\psi(Y)]\] \[\qquad+\lambda\left(1-\int_{\mathcal{X}}\int_{\mathcal{X}}\exp \left(\frac{\phi(x)+\psi(y)-c(x,y)}{\lambda}\right)\nu(dy)\mu(dx)\right).\]
The entropic optimal transport problem (3) admits the following dual representation:
\[T_{\lambda}(\mu,\nu)=\max_{\phi,\psi}E_{\lambda}^{\mu,\nu}(\phi,\psi). \tag{4}\]
For any \(\psi\) define
\[\phi_{\psi}\in\operatorname{argmax}_{\phi\in L_{1}(\mu)}E_{\lambda}^{\mu,\nu }(\phi,\psi).\]
The solution is unique \(\mu\)-almost everywhere up to a constant; we fix a particular choice
\[\phi_{\psi}(x)=-\lambda\log\left(\int_{\mathcal{X}}\exp\left(\frac{\psi(y)-c( x,y)}{\lambda}\right)\nu(dy)\right).\]
Likewise, we denote \(\psi_{\phi}=\operatorname{argmax}_{\psi\in L_{1}(\nu)}E_{\lambda}^{\mu,\nu}( \phi,\psi)\) with the analogous expression to the one given above, interchanging the roles of \(\phi\) and \(\psi\). Then, the maximum in (4) is attained by any pair \((\phi^{*},\psi^{*})\) such that \(\phi^{*}=\phi_{\psi^{*}}\) and \(\psi^{*}=\psi_{\phi^{*}}\); such a pair is said to solve the Schrodinger system and it is unique up to translations \((\phi^{*}+a,\psi^{*}-a)\) by any constant \(a\in\mathbb{R}\). The optimal coupling that solves the primal problem (3) can be obtained from the pair \((\phi^{*},\psi^{*})\) via the primal-dual relation
\[\gamma^{*}(dx,dy)=\exp\left(\frac{\phi^{*}(x)+\psi^{*}(y)-c(x,y)}{\lambda} \right)\mu(dx)\nu(dy).\]
We conclude this section by listing two properties of functions of the form \(\phi_{\psi}\). These properties will be used repeatedly throughout this paper. First, for any \(\psi\) we have
\[\int_{\mathcal{X}}\int_{\mathcal{X}}\exp\left(\frac{\phi_{\psi}(x)+\psi(y)-c( x,y)}{\lambda}\right)\nu(dy)\mu(dx)=1,\]
which means, in particular, that for any \(\psi\) we have
\[E_{\lambda}^{\mu,\nu}(\phi_{\psi},\psi)=\mathbf{E}_{X\sim\mu}[\phi_{\psi}(X)]+ \mathbf{E}_{Y\sim\nu}[\psi(Y)]. \tag{5}\]
The second property of interest is that for any \(\psi\) and any \(x,x^{\prime}\in\mathcal{X}\) it holds that
\[\phi_{\psi}(x)-\phi_{\psi}(x^{\prime}) =-\lambda\log\frac{\int\exp\left(\frac{\psi(y)-c(x,y)}{\lambda} \right)\nu(dy)}{\int\exp\left(\frac{\psi(y)-c(x^{\prime},y)}{\lambda}\right) \nu(dy)}\] \[=-\lambda\log\frac{\int\exp\left(\frac{\psi(y)-c(x^{\prime},y)+c( x^{\prime},y)-c(x,y)}{\lambda}\right)\nu(dy)}{\int\exp\left(\frac{\psi(y)-c(x^{ \prime},y)}{\lambda}\right)\nu(dy)}\] \[\leq\sup_{y\in\mathcal{X}}c(x^{\prime},y)-c(x,y)\leq c_{\infty}( \mathcal{X}).\]
In particular, for any \(\psi\) we have
\[\|\phi_{\psi}\|_{\mathrm{osc}}=\sup_{x}\phi_{\psi}(x)-\inf_{x^{\prime}}\phi_{ \psi}(x^{\prime})\leq c_{\infty}(\mathcal{X}). \tag{6}\]
### Doubly Regularized Entropic Barycenters
Let \(\mathbf{\nu}=(\nu^{1},\ldots,\nu^{k})\in\mathcal{P}(\mathcal{X})^{k}\) be \(k\) probability measures and let \(w\in\mathbb{R}^{k}\) be a vector of positive numbers that sum to one. Given the inner regularization strength \(\lambda>0\) and the outer regularization strength \(\tau>0\), the \((\lambda,\tau)\) barycenter \(\mu_{\lambda,\tau}\in\mathcal{P}(\mathcal{X})\) of probability measures \(\mathbf{\nu}\) with respect to the weights vector \(w\) is defined as the unique solution to the following optimization problem:
\[\mu_{\lambda,\tau}=\operatorname{argmin}_{\mu\in\mathcal{P}(\mathcal{X})}\sum _{j=1}^{k}w_{j}T_{\lambda}(\mu,\nu^{j})+\tau\mathrm{KL}(\mu,\pi_{\mathrm{ref} }), \tag{7}\]
where \(\pi_{\mathrm{ref}}\in\mathcal{P}(\mathcal{X})\) is a reference probability measure.
We will now describe how to obtain a concave dual maximization problem to the primal problem (7), following along the lines of Chizat [17, Section 2.3], where the interested reader will find a comprehensive justification of all the claims made in the rest of this section.
First, using the semi-dual formulation of entropic optimal transport problem (5), we have, for each \(j\in\{1,\ldots,k\}\)
\[T_{\lambda}(\mu,\nu^{j})=\sup_{\psi^{j}\in L_{1}(\nu^{j})}\mathbf{E}_{X\sim \mu}[\phi_{\psi^{j}}(X)]+\mathbf{E}_{Y\sim\nu^{j}}[\psi^{j}(Y)].\]
Denote \(\mathbf{\psi}=(\psi^{1},\ldots,\psi^{j})\in L_{1}(\mathbf{\nu})\). Then, we may rewrite the primal problem (7) by
\[\min_{\mu\in\mathcal{P}(X)}\max_{\mathbf{\psi}\in L_{1}(\mathbf{\nu})}\sum_{j=1}^{k}w_ {j}\mathbf{E}_{Y\sim\nu^{j}}\big{[}\psi^{j}(Y)\big{]}+\mathbf{E}_{X\sim\mu} \big{[}\sum_{j=1}^{k}w_{j}\phi_{\psi^{j}}(X)\big{]}+\tau\mathrm{KL}(\mu,\pi_{ \mathrm{ref}}).\]
Interchanging \(\min\) and \(\max\), which is justified using compactness of \(\mathcal{X}\) as detailed in [17], we obtain the dual optimization objective \(E_{\lambda,\tau}^{\mathbf{\nu},w}:L_{1}(\mathbf{\nu})\to\mathbb{R}\) defined by
\[\begin{split} E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi})& =\min_{\mu\in\mathcal{P}(X)}\sum_{j=1}^{k}w_{j}\mathbf{E}_{Y\sim \nu^{j}}\big{[}\psi^{j}(Y)\big{]}+\mathbf{E}_{X\sim\mu}\big{[}\sum_{j=1}^{k}w_ {j}\phi_{\psi^{j}}(X)\big{]}+\tau\mathrm{KL}(\mu,\pi_{\mathrm{ref}}).\\ &=\sum_{j=1}^{k}w_{j}\mathbf{E}_{Y\sim\nu^{j}}\big{[}\psi^{j}(Y) \big{]}-\tau\log\int\exp\left(\frac{-\sum_{j=1}^{k}\phi_{\psi^{j}}(x)}{\tau} \right)\pi_{\mathrm{ref}}(dx).\end{split} \tag{8}\]
The infimum above is attained by the measure
\[\mu_{\mathbf{\psi}}(dx)=Z_{\mathbf{\psi}}^{-1}\exp\left(\frac{-\sum_{j=1}^{k}\phi_{\psi^{j} }(x)}{\tau}\right)\pi_{\text{ref}}(dx),\quad Z_{\mathbf{\psi}}=\int\exp\left(\frac{- \sum_{j=1}^{k}\phi_{\psi^{j}}(x)}{\tau}\right)\pi_{\text{ref}}(dx).\]
To each dual variable \(\mathbf{\psi}\) we associate the marginal measures \(\nu_{\mathbf{\psi}}^{j}(dy)\) defined for \(j=1,\ldots,k\) by
\[\nu_{\mathbf{\psi}}^{j}(dy)=\nu^{j}(dy)\int\exp\left(\frac{\phi_{\psi^{j}}(x)+\psi^ {j}(y)-c(x,y)}{\lambda}\right)\mu_{\mathbf{\psi}}(dx). \tag{9}\]
Finally, we mention that the objective \(E_{\lambda,\tau}^{\mathbf{\nu},w}\) is concave and for any \(\mathbf{\psi},\mathbf{\psi}^{\prime}\) it holds that
\[\lim_{h\to 0}\frac{E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi}+h\mathbf{\psi}^{ \prime})-E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi})}{h}=\sum_{j=1}^{k}w_{j} \left(\mathbf{E}_{\nu^{j}}[(\psi^{\prime})^{j}]-\mathbf{E}_{\nu_{\mathbf{\psi}}^{ j}}[(\psi^{\prime})^{j}]\right)\,.\]
In particular, fixing any optimal dual variable \(\mathbf{\psi}^{*}\), for any \(\mathbf{\psi}\) it holds using concavity of \(E_{\lambda,\tau}^{\mathbf{\nu},w}\) that
\[0\leq E(\mathbf{\psi}^{*})-E(\mathbf{\psi})\leq\sum_{j=1}^{k}w_{k}\left(\mathbf{E}_{ \nu^{j}}\left[(\psi^{*})^{j}-\psi^{j}\right]-\mathbf{E}_{\nu_{\mathbf{\psi}}^{j}} \left[(\psi^{*})^{j}-\psi^{j}\right]\right). \tag{10}\]
This concludes our overview of the background material on \((\lambda,\tau)\)-barycenters.
## 3 Damped Sinkhorn Scheme
This section introduces a damped Sinkhorn-based optimization scheme (Algorithm 1) and provides guarantees for its convergence (Theorem 1). Before describing the algorithm, we make a quick detour to the following lemma, proved in Appendix A, which shows that the sub-optimality gap bounds on the dual objective (8) can be transformed into corresponding bounds on relative entropy between the \((\lambda,\tau)\)-barycenter and the barycenter associated to a given dual variable.
**Lemma 1**.: _Fix any \(\lambda,\tau>0\) and \(\mathbf{\nu},w\). Let \(\mathbf{\psi}^{*}\) be the maximizer of dual problem \(E_{\lambda,\tau}^{\mathbf{\nu},w}\) and let \(\mu_{\mathbf{\psi}^{*}}\) be the corresponding minimizer of the primal objective (7). Then, for any \(\mathbf{\psi}\in L_{1}(\mathbf{\nu})\) we have_
\[\operatorname{KL}(\mu_{\mathbf{\psi}^{*}},\mu_{\mathbf{\psi}})\leq\tau^{-1}(E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi}^{*})-E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi})).\]
We now turn to describing an iterative scheme that ensures convergence of the dual suboptimality gap to zero. Let \(\mathbf{\psi}_{t}\) be an iterate at time \(t\). Then, we have
\[E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi}_{t})=L(\mathbf{\psi}_{t},\mathbf{\phi}_{t}, \mu_{t})=\sum_{j=1}^{k}w_{j}\mathbf{E}_{\nu^{j}}[\psi_{t}^{j}]-\mathbf{E}_{\mu _{t}}[\phi_{t}^{j}]+\tau\operatorname{KL}(\mu_{t},\pi_{\text{ref}}),\]
where
\[\phi^{j}=\operatorname{argmax}_{\phi}E_{\lambda}^{\mu_{t-1},\nu^{j}}(\phi, \psi_{t}^{j})\quad\text{and}\quad\mu_{t}=\operatorname{argmin}_{\mu}\bigg{\{} \mathbf{E}_{\mu}\big{[}\sum_{j}w_{j}\phi_{t}^{j}\big{]}+\tau\operatorname{KL} (\mu,\pi_{\text{ref}})\bigg{\}}. \tag{11}\]
In particular, when optimizing the dual objective \(E_{\lambda,\tau}^{\mathbf{\nu},w}\), every time the variable \(\mathbf{\psi}_{t}\) is updated, it automatically triggers the exact maximization/minimization steps defined in (11). It is thus a natural strategy to fix \(\mathbf{\phi}_{t}\) and \(\mu_{t}\) and perform exact minimization on \(\mathbf{\psi}\), which can be done in closed form:
\[\psi_{t+1}^{j}=\operatorname{argmax}_{\psi}E_{\lambda}^{\mu_{t},\nu^{j}}(\phi_ {t}^{j},\psi)=\psi_{t}^{j}-\lambda\log\frac{d\nu_{t}^{j}}{d\nu^{j}}, \tag{12}\]
where \(\nu_{j}^{t}\) denotes the marginal distribution \(\nu_{\mathbf{\psi}_{t}}^{j}\) defined in (9). The update (12) performs a Sinkhorn update on each block of variables \(\psi^{j}\). Together, the update (12) followed by (11) results in the iterative Bregman projections algorithm introduced in [7]. In [35], it was shown that this scheme converges for the \((\lambda,\lambda)\)-barycenters. The analysis of [35] is built upon a different dual formulation from the one considered in our work; this alternative formulation is only available when \(\tau=\lambda\)[17, Section 2.3] and thus excludes the consideration of debiased barycenters \((\lambda,\lambda/2)\).
We have observed empirically that the iterates of the iterative Bregman projections (i.e., the scheme of updates defined in (12) and (11)) diverge whenever \(\tau<\lambda/2\). Indeed, decreasing the outer regularization parameter \(\tau\) makes the minimization step in (11) less stable. As a result, the cumulative effect of performing the updates (12) and (11) may result in a decrease in the value of the optimization objective \(E_{\lambda,\tau}^{\mathbf{\nu},w}\).
One of the main contributions of our work is to show that this bad behaviour can be mitigated by damping the exact Sinkhorn updates (12). This leads to Algorithm 1 for which convergence guarantees are provided in Theorem 1 stated below.
```
Input: regularization strengths \(\lambda,\tau>0\), reference measure \(\pi_{\text{ref}}\),number of iterations \(T\) and \(k\) marginal measures \(\nu^{1},\ldots,\nu^{k}\) with positive weights \(w_{1},\ldots,w_{k}\) such that \(\sum_{j=1}^{k}w_{j}=1\).
1. Set \(\eta=\min(1,\tau/\lambda)\) and initialize \((\psi_{0}^{j})=0\) for \(j\in\{1,\ldots,k\}\).
2. For \(t=0,1\ldots,T-1\) do 1. \(\phi_{t}^{j}(x)\leftarrow-\lambda\log\int_{\mathcal{X}}\exp((\psi_{t}^{j}(y)-c (x,y))/\lambda)\nu^{j}(dy)\) for \(j\in\{1,\ldots,k\}\) 2. \(V_{t}(x)\leftarrow\sum_{j=1}^{k}w_{j}\phi^{j}(t)(x)\) 3. \(Z_{t}\leftarrow\int\exp(-V_{t}(x)/\tau)d\pi_{\text{ref}}(dx)\) 4. \(\mu_{t}(dx)\gets Z_{t}^{-1}\exp(-V_{t}(x)/\tau)\pi_{\text{ref}}(dx)\) 5. \(\frac{d\nu^{j}}{d\nu^{j}}(y)\leftarrow\int\exp\left(\frac{\phi_{t}^{j}(x)+\psi _{t}^{j}(y)-c(x,y)}{\lambda}\right)\mu_{t}(dx)\) for \(j\in\{1,\ldots,k\}\). 6. \(\psi_{t+1}^{j}(y)\leftarrow\psi_{t}^{j}(y)-\eta\lambda\log\frac{d\nu^{j}}{d\nu ^{j}}(y)\) for \(j\in\{1,\ldots,k\}\).
3. Return \((\phi_{T}^{j},\psi_{T}^{j})_{j=1}^{k}\).
```
**Algorithm 1**Exact Damped Sinkhorn Scheme
**Theorem 1**.: _Fix any \(\lambda,\tau>0\) and \(\mathbf{\nu},w\). Let \(\psi^{*}\) be the maximizer of dual problem \(E_{\lambda,\tau}^{\mathbf{\nu},w}\). Let \((\mathbf{\psi}_{t})_{t\geq 0}\) be the sequence of iterates generated by Algorithm 1. Then, for any \(t\geq 1\) it holds that_
\[E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi}^{*})-E_{\lambda,\tau}^{\mathbf{\nu},w}(\bm {\psi}_{t})\leq\frac{2c_{\infty}(\mathcal{X})^{2}}{\min(\lambda,\tau)}\,\frac{ 1}{t}.\]
Our convergence analysis draws upon the existing analyses of Sinkhorn's algorithm [2, 25], which in turn are based on standard proof strategies in smooth convex optimization (e.g., [42, Theorem 2.1.14]). Concerning the proof of Theorem 1, the main technical contribution of our work lies in the following proposition proved in Appendix B.
**Proposition 1**.: _Consider the setup of Theorem 1. Then, for any integer \(t\geq 0\) it holds that_
\[E_{\lambda,\tau}^{\mathbf{\nu},w}(\mathbf{\psi}_{t+1})-E_{\lambda,\tau}^{\mathbf{\nu},w}( \mathbf{\psi}_{t})\geq\min{(\tau,\lambda)}\sum_{j=1}^{k}w_{j}\mathrm{KL}(\nu^{j}, \nu_{t}^{j}).\]
With Proposition 1 at hand, we are ready to prove Theorem 1.
Proof of Theorem 1.: Denote \(\delta_{t}=E^{\mathbf{\psi},\mathbf{\psi}}_{\lambda,\tau}(\mathbf{\psi}^{*})-E^{\mathbf{\psi},\bm {\psi}}_{\lambda,\tau}(\mathbf{\psi}_{t})\). We would like to relate the suboptimality gap \(\delta_{t}\) to the increment \(\delta_{t}-\delta_{t+1}\). To do this, we will first show that the iterates \(\mathbf{\psi}_{t}\) have their oscillation norm bounded uniformly in \(t\). Indeed, for any \(j\in\{1,\ldots,k\}\), any \(t\geq 1\), and any \(y\in\mathcal{X}\) we have
\[\psi_{t}^{j}(y)=(1-\eta)\psi_{t-1}^{j}(y)+\eta\psi_{\phi_{t}^{j}}(y).\]
By (6), \(\psi_{\phi_{t}^{j}}\) has oscillation norm bounded by \(c_{\infty}(\mathcal{X})\). Because \(\psi_{0}^{j}=0\) and \(\eta\in(0,1]\), by induction on \(t\) it follows that \(\|\psi_{t}\|_{\mathrm{osc}}\leq c_{\infty}(\mathcal{X})\) for any \(t\geq 0\). Combining the bound on the dual sub-optimality gap (10) with Pinsker's inequality yields
\[\delta_{t}\leq 2c_{\infty}(\mathcal{X})\sum_{j=1}^{k}w_{j}\|\nu^{j}-\nu_{t}^{j} \|_{\mathrm{TV}}\leq\sqrt{2}c_{\infty}\sum_{j=1}^{k}w_{j}\sqrt{\mathrm{KL}( \nu^{j},\nu_{t}^{j})}.\]
Using concavity of the square root function, Proposition 1 yields for any \(t\geq 0\)
\[\delta_{t}-\delta_{t+1}\geq\min(\lambda,\tau)\sum_{j=1}^{k}w_{j}\mathrm{KL}( \nu^{j},\nu_{t}^{j})\geq\frac{\min(\lambda,\tau)}{2c_{\infty}(\mathcal{X})^{2 }}\delta_{t}^{2}.\]
By Proposition 1, the sequence \(\delta_{t}\) is non-increasing. Hence, dividing the above equality by \(\delta_{t}\delta_{t+1}\) yields
\[\frac{1}{\delta_{t+1}}-\frac{1}{\delta_{t}}\geq\frac{\min(\lambda,\tau)}{2c_{ \infty}(\mathcal{X})^{2}}.\]
Telescoping the left hand side completes the proof.
## 4 Approximate Damped Sinkhorn Scheme
In this section, we extend the analysis of Algorithm 1 to an approximate version of the algorithm. Then, in Section 4.1, we describe how inexact updates may be implemented via approximate random sampling, thus enabling the computation of \((\lambda,\tau)\)-barycenters in the free-support setting with convergence guarantees.
Algorithm 2 describes an inexact version of Algorithm 1. It replaces the damped Sinkhorn iterations of Algorithm 1 via approximate updates computed by an approximate Sinkhorn oracle - a procedure that satisfies the properties listed in Definition 1.
**Definition 1** (Approximate Sinkhorn Oracle).: An \(\varepsilon\)-approximate Sinkhorn oracle is a procedure that given any \(\mathbf{\psi}\) and any index \(j\in\{1,\ldots,k\}\), returns a Radon-Nikodym derivative \(\frac{d\widehat{\nu}_{\mathbf{\psi}}^{j}}{d\nu^{j}}\) of a measure \(\widehat{\nu}_{\mathbf{\psi}}^{j}\ll\nu^{j}\) that satisfies the following properties:
1. \(\frac{d\widehat{\nu}_{\mathbf{\psi}}^{j}}{d\nu^{j}}\) is strictly positive on the support of \(\nu^{j}\);
2. \(\|\widetilde{\nu}_{\mathbf{\psi}}^{j}-\nu_{\mathbf{\psi}}^{j}\|_{\mathrm{TV}}\leq \varepsilon/(2c_{\infty}(\mathcal{X}))\);
3. \(\mathbf{E}_{Y\sim\nu^{j}}[\frac{d\nu_{\mathbf{\psi}}^{j}}{d\widehat{\nu}_{\mathbf{ \psi}}^{j}}(Y)]\leq 1+\varepsilon^{2}/(2c_{\infty}(\mathcal{X})^{2})\);
4. For any \(\eta\in[0,1]\) and any \(j\in\{1,\ldots,k\}\) it holds that \(\|\psi^{j}+\eta\lambda\log(d\widehat{\nu}_{\mathbf{\psi}}^{j}/d\nu^{j})\|_{\mathrm{ osc}}\leq(1-\eta)\|\psi^{j}\|_{\mathrm{osc}}+\eta c_{\infty}(\mathcal{X})\).
The following theorem shows that Algorithm 2 enjoys the same convergence guarantees as Algorithm 1 up to the error tolerance of the procedure used to implement the approximate updates. A noteworthy aspect of the below theorem is that the error does not accumulate over the iterations.
**Theorem 2**.: _Fix any \(\lambda,\tau>0\) and \(\boldsymbol{\nu},w\). Let \(\psi^{*}\) be the maximizer of dual problem \(E^{\boldsymbol{\nu},w}_{\lambda,\tau}\). Let \((\boldsymbol{\tilde{\psi}}_{t})_{t\geq 0}\) be the sequence of iterates generated by Algorithm 2 with the accuracy parameter \(\varepsilon\geq 0\). Let \(T=\min\{t:E^{\boldsymbol{\nu},w}_{\lambda,\tau}(\boldsymbol{\psi}^{*})-E^{ \boldsymbol{\nu},w}_{\lambda,\tau}(\boldsymbol{\tilde{\psi}}_{t})\leq 2\varepsilon\}\). Then, for any \(t\leq T\) it holds that_
\[E^{\boldsymbol{\nu},w}_{\lambda,\tau}(\boldsymbol{\psi}^{*})-E^{\boldsymbol{ \nu},w}_{\lambda,\tau}(\boldsymbol{\tilde{\psi}}_{t})\leq 2\varepsilon+\frac{2c_{ \infty}(\mathcal{X})^{2}}{\min(\lambda,\tau)}\,\frac{1}{t}.\]
The proof of the above theorem can be found in Appendix C.
### Implementing the Approximate Sinkhorn Oracle
In this section, we show that the approximate Sinkhorn oracle (see Definition 1) can be implemented using approximate random sampling when the marginal distributions \(\nu^{j}\) are discrete. To this end, fix the regularization parameters \(\lambda,\tau>0\), the weight vector \(w\), and consider a set of \(k\) discrete marginal distributions
\[\nu^{j}=\sum_{l=1}^{m_{j}}\nu^{j}(y_{l}^{j})\delta_{y_{l}^{j}},\]
where \(\delta_{x}\) is the Dirac measure located at \(x\) and \(\nu^{j}(y_{l}^{j})\) is equal to the probability of sampling the point \(y_{l}^{j}\) from measure \(\nu^{j}\). We denote the total cardinality of the support of all measures \(\nu^{j}\) by
\[m=\sum_{j=1}^{m}m_{j}.\]
Fix any \(\boldsymbol{\psi}\in L_{1}(\boldsymbol{\nu})\). Suppose we are given access to \(n\) i.i.d. samples \(X_{1},\ldots,X_{n}\) from a probability measure \(\mu^{\prime}_{\boldsymbol{\psi}}\) that satisfies
\[\|\mu_{\boldsymbol{\psi}}-\mu^{\prime}_{\boldsymbol{\psi}}\|_{\mathrm{TV}} \leq\varepsilon_{\mu}.\]
Then, for \(j=1,\ldots,k\) and \(l=1,\ldots,m_{j}\) consider
\[\widehat{\nu}^{j}(y_{i}^{j})=\nu^{j}(y_{i}^{j})\frac{1}{n}\sum_{i=1}^{n}\exp \left(\frac{\phi_{\psi^{j}}(X_{i})+\psi^{j}(y)-c(x,y)}{\lambda}\right)\]
and for any parameter \(\zeta\in(0,1/2]\) define
\[\widetilde{\nu}^{j}=(1-\zeta)\widehat{\nu}^{j}+\zeta\nu^{j}. \tag{13}\]
We claim that \(\widetilde{\nu}^{j}\) implements the approximate Sinkhorn oracle with accuracy parameter arbitrarily close to \(\sqrt{\varepsilon_{\mu}}\) provided that \(n\) is large enough. This is shown in the following lemma, the proof of which can be found in Appendix D.
**Lemma 2**.: _Fix any \(\delta\in(0,1)\) and consider the setup described above. With probability at least \(1-\delta\), for each \(j\in\{1,\ldots,k\}\) it holds simultaneously that the measure \(\widetilde{\nu}^{j}\) defined in (13) satisfies all the properties listed in Definition 1 with accuracy parameter_
\[\varepsilon_{j}\leq c_{\infty}(\mathcal{X})\Bigg{(}2\zeta+\frac{1}{\zeta}m_{j }\varepsilon_{\mu}+\frac{1}{\zeta}m_{j}\sqrt{\frac{2\log\big{(}\frac{2m}{ \delta}\big{)}}{n}}\Bigg{)}^{1/2}.\]
The above lemma shows that a step of Algorithm 2 can be implemented provided access to i.i.d. sampling from some measure \(\mu^{\prime}_{\mathbf{\psi}}\) close to \(\mu_{\mathbf{\psi}}\) in total variation norm, where \(\mathbf{\psi}\) is an arbitrary iterate of Algorithm 2. The remainder of this section is dedicated to showing that this can be achieved by sampling via Langevin Monte Carlo.
Henceforth, fix \(\pi_{\mathrm{ref}}\) to be the Lebesgue measure on \(\mathcal{X}\), which corresponds to the free-support barycenters setup. Then, for any \(\mathbf{\psi}\) we have
\[\mu_{\mathbf{\psi}}(dx)\propto\mathbb{1}_{\mathcal{X}}\exp(-V_{\mathbf{\psi}}(x)/\tau )dx,\quad\text{where}\quad V_{\mathbf{\psi}}(x)=\sum_{j=1}^{k}w_{j}\phi^{j}_{\psi^ {j}},\]
where \(\mathbb{1}_{\mathcal{X}}\) is equal to one on \(\mathcal{X}\) and zero everywhere else. It follows by (6) that \(\|V_{\mathbf{\psi}}\|_{\mathrm{osc}}\leq c_{\infty}(\mathcal{X})/\tau\). Further, let \(\operatorname{diam}\!\mathcal{X}=\sup_{x,x^{\prime}\in\mathcal{X}}\|x-x^{ \prime}\|_{2}\). By the convexity of \(\mathcal{X}\), the uniform measure on \(\mathcal{X}\) satisfies the logarithmic Sobolev inequality (LSI) with constant \(\operatorname{diam}(\mathcal{X})^{2}/4\) (cf. [36]). Hence, by the Holley-Stroock perturbation argument [32], the measure \(\mu_{\mathbf{\psi}}\) satisfies LSI with constant at most \(\exp\left(2c_{\infty}(\mathcal{X})/\tau\right)\operatorname{diam}(\mathcal{X })^{2}/4<\infty\).
It is well-established that Langevin Monte Carlo algorithms offer convergence guarantees for approximate sampling from a target measure subject to functional inequality constraints provided additional conditions hold such as the smoothness of the function \(V_{\mathbf{\psi}}\). However, such guarantees do not directly apply to the measure \(\mu_{\mathbf{\psi}}\) due to its constrained support. Instead, it is possible to approximate \(\mu_{\mathbf{\psi}}\) arbitrarily well in total variation norm by a family of measures \((\mu_{\mathbf{\psi},\sigma})_{\sigma>0}\) (see Appendix E for details) supported on all of \(\mathbb{R}^{d}\). Tuning the parameter \(\sigma\) allows us to trade-off between the approximation quality of \(\mu_{\mathbf{\psi},\sigma}\) and its LSI constant. Crucially, standard sampling guarantees for Langevin Monte Carlo (e.g., [55]) apply to the regularized measures \(\mu_{\mathbf{\psi},\sigma}\), which leads to provable guarantees for an implementation of Algorithm 2, thus furnishing the first convergence guarantees for computation of Wasserstein barycenters in the free support setup; see Theorem 3 stated below.
The above approximation argument applies to any cost function \(c\) that is Lipschitz on \(\mathcal{X}\) and exhibits quadratic growth at infinity. For the sake of simplicity, we consider the quadratic cost \(c(x,y)=\|x-y\|_{2}^{2}\). The exact problem setup where we are able to obtain computational guarantees for free-support barycenter computation via Langevin Sampling is formalized below.
_Problem Setting 1_.: Consider the setting described at the beginning of Section 4.1. In addition, suppose that
1. the reference measure \(\pi_{\mathrm{ref}}(dx)=\mathbb{1}_{\mathcal{X}}dx\) is the Lebesgue measure supported on \(\mathcal{X}\) (free-support setup);
2. it holds that \(\mathcal{X}\subseteq\mathcal{B}_{R}=\{x\in\mathbb{R}^{d}:\|x\|_{2}\leq R\}\) for some constant \(R<\infty\);
3. the cost function \(c:\mathbb{R}^{d}\times\mathbb{R}^{d}\to[0,\infty)\) is defined by \(c(x,y)=\|x-y\|_{2}^{2}\);
4. for any \(\mathbf{\psi}\) we have access to a stationary point \(x_{\mathbf{\psi}}\) of \(V_{\mathbf{\psi}}\) over \(\mathcal{X}\).
The last condition can be implemented in polynomial time using a first order gradient method. For our purposes, this condition is needed to obtain a good initialization point for the Unadjusted Langevin Algorithm following the explanation in [55, Lemma 1]; see Appendix E for further details.
We now proceed to the main result of this section, the proof of which can be found in Appendix E. The following theorem provides the first provably convergent method for computing Wasserstein barycenters in the free-support setting. We remark that a stochastic approximation argument of a rather different flavor used to compute fixed-support Wasserstein barycenters (for \(\tau\geq\lambda\)) has been previously analyzed in [5].
**Theorem 3**.: _Consider the setup described in Problem Setting 1. Then, for any confidence parameter \(\delta\in(0,1)\) and any accuracy parameter \(\varepsilon>0\), we can simulate a step of Algorithm 2 with success probability at least \(1-\delta\) in time polynomial in_
\[\varepsilon^{-1},d,R,\exp(R^{2}/\tau),(Rd^{-1/4})^{d},\tau^{-1},\lambda^{-1},d,m,\log(m/\delta).\]
_In particular, an \(\varepsilon\)-approximation of the \((\lambda,\tau)\)-Barycenter can be obtained within the same computational complexity._
Comparing the above guarantee with the discussion following the statement of Lemma 2, we see an additional polynomial dependence on \((Rd^{-1/4})^{d}\) (note that for \(R\leq d^{1/4}\) this term disappears). We believe this term to be an artefact of our analysis appearing due to the approximation argument described above. Considering the setup with \(R\leq d^{1/4}\), the running time of our algorithm depends exponentially in \(R^{2}/\tau\).
We conclude with two observations. First, since approximating Wasserstein barycenters is generally NP-hard [4], an algorithm with polynomial dependence on all problem parameters does not exist if \(\mathrm{P}\neq\mathrm{NP}\). Second, notice that computing an \(\varepsilon\) approximation of \((\lambda,\tau)\)-Barycenter can be done in time polynomial in \(\varepsilon^{-1}\). This should be contrasted with numerical schemes based on discretizations of the set \(\mathcal{X}\), which would, in general, result in computational complexity of order \((R/\varepsilon)^{d}\) to reach the same accuracy.
## 5 Conclusion
We introduced algorithms to compute doubly regularized entropic Wasserstein barycenters and studied their computational complexity, both in the fixed-support and in the free-support settings. Although a naive adaptation of the usual alternate maximization scheme from [7] to our setting leads to diverging iterates (at least for small values of \(\tau\)), our analysis shows that it is sufficient to damp these iterations to get a converging algorithm.
While we have focused on the problem of barycenters of measures, we note that the idea of entropic regularization is pervasive in other applications of optimal transport. There, the flexibility offered by the double entropic regularization may prove to be useful as well, and we believe that our damped algorithm could be adapted to these more general settings.
|
2308.04062 | Scaling invariance of spatial autocorrelation in urban built-up area | City is proved to be a scale-free phenomenon, and spatial autocorrelation is
often employed to analyze spatial redundancy of cities. Unfortunately, spatial
analysis results deviated practical requirement in many cases due to fractal
nature of cities. This paper is devoted to revealing the internal relationship
between the scale dependence of Moran's I and fractal scaling. Mathematical
reasoning and empirical analysis are employed to derive and test the model on
the scale dependence of spatial autocorrelation. The data extraction way for
fractal dimension estimation is box-counting method, and parameter estimation
relies on the least squares regression. In light of the locality postulate of
spatial correlation and the idea of multifractals, a power law model on Moran's
I changing with measurement scale is derived from the principle of recursive
subdivision of space. The power exponent is proved to be a function of fractal
dimension. This suggests that the numerical relationship between Moran's I and
fractal dimension can be established through the scaling process of
granularity. An empirical analysis is made to testify the theoretical model. It
can be concluded that spatial autocorrelation of urban built-up area has no
characteristic scale in many cases, and urban spatial analysis need new
thinking. | Meng Fu, Yanguang Chen | 2023-08-08T05:51:21Z | http://arxiv.org/abs/2308.04062v1 | # Scaling Invariance of Spatial Autocorrelation in Urban Built-Up Area
###### Abstract
City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature of cities. This paper is devoted to revealing the internal relationship between the scale dependence of Moran's I and fractal scaling. Mathematical reasoning and empirical analysis are employed to derive and test the model on the scale dependence of spatial autocorrelation. The data extraction way for fractal dimension estimation is box-counting method, and parameter estimation relies on the least squares regression. In light of the locality postulate of spatial correlation and the idea of multifractals, a power law model on Moran's I changing with measurement scale is derived from the principle of recursive subdivision of space. The power exponent is proved to be a function of fractal dimension. This suggests that the numerical relationship between Moran's I and fractal dimension can be established through the scaling process of granularity. An empirical analysis is made to testify the theoretical model. It can be concluded that spatial autocorrelation of urban built-up area has no characteristic scale in many cases, and urban spatial analysis need new thinking.
urban form; built-up area; Moran's \(I\); fractal scaling; spatial complexity
## 1 Introduction
Urban studies depends on spatial analysis, which needs robust measurements. However, scale-free property of cities affects the effect of spatial measurement. In fact, besides cities, the scale-free phenomena are very common in geographical world. What is called "scale-free" means has no characteristic scale, or the measurement results depend on measurement scales. Scientific research consists of description and understanding (Kane, 2005; Henry, 2002). Description relies on measure
and mathematics (Henry, 2002). The application of mathematical methods in scientific research mainly plays two functions: one is to establish models, and the other is to sort out observation data. In any case, the basic premise of mathematical modeling and quantitative analysis is to find characteristic scale (Takayasu, 1990). However, it is often difficult to find effective characteristic scales in complex systems such as cities. In fact, scale dependence can be divided into two cases. First, the relationship between scale and measure can be described by functions with characteristic scales, such as exponential function or logarithmic function. Such scale dependence phenomena are not complex, and the characteristic scales can be found based on function relations. Second, the relationship between scale and measure must be described by power law, in which no characteristic scale can be found. Such scale dependence phenomena represent complex system, in which characteristic scale cannot be found and conventional quantitative analysis method fails. Cities belong to the second type of scale dependence phenomena, i.e., scale-free phenomena. Scaling analysis is a new way of exploring scale-free problems. The relationship between scale dependence and scaling has been concerned for a long time (Su et al., 2001; Wu, 1996). One of the effective tools for scaling analysis is fractal geometry (Mandelbrot, 1982).
Spatial autocorrelation is one of the basic methods for geospatial analysis. The important tool for spatial statistics has been applied to urban studies. One of the basic measures of spatial autocorrelation analysis is Moran's \(I\), which is proved to be the eigenvalue of generalized spatial weights matrix (Chen, 2013). In this sense, Moran's \(I\) is a statistic representing characteristic scale. The premise of validity of Moran's \(I\), as a characteristic scale, is its scale independence. In other words, it has robustness of measurement. However, a large number of studies have shown that the calculation results of Moran's \(I\) of urban space bear scale dependence. On the one hand, granularity of spatial measurement sometimes affects calculation results significantly (e.g., Chou, 1991; Feng et al., 2016; Overmars et al., 2003; Qi & Wu, 1996); on the other hand, threshold distance defining spatial contiguity matrix also affects the final value of Moran's \(I\) significantly (e.g., Bjernstad & Falck, 2001; De Knegt et al., 2010; Getis & Ord, 1992; Legendre & Legendre, 1998; Odland, 1988; Ord & Getis, 1995). If an urban model has characteristic scales, Moran's \(I\) can be regarded as a characteristic parameter; if the model has no characteristic scale, it relates to scaling process. There are two ways to solve the scaling problem. One is to construct spatial autocorrelation function based on variable scales, so as to replace the spatial autocorrelation index; the other is to reveal the internal
relationship between scale dependence and fractal scaling, and convert the spatial autocorrelation index into scaling index (Chen, 2021).
Fractals suggest optimized structure of a self-organized system, while spatial autocorrelation implies information redundancy in the system. The relationship between fractal scaling and spatial autocorrelation helps to understand the spatial complexity of cities and dynamic mechanisms of urban evolution. This paper is devoted to revealing the underlying scaling properties and fractal features of spatial autocorrelation of urban form. The rest of this paper is arranged as follows: in Section 2, based on urban form, the mathematical model of scale dependence of Moran's \(I\) is derived; the corresponding scaling exponent is proved to be a function of fractal dimension, and case analysis based on 19 largest Chinese cities is carried out. In Section 3, the results of case study are analyzed and the mathematical model derived before is verified; in Section 4, achievements, novelties and limitations of research methods in this paper are discussed; finally, In Section 5, main conclusions are drawn based on the research results and question discussion.
## 2 Materials and methods
### Moran's \(I\) and correlation dimension
Before theoretical derivation, two postulates are given in this paper. First, the spatial pattern of urban built-up area is of multifractal nature. Second, the spatial correlation of different areas is local. There are a lot of research results of multifractal urban spatial pattern, which is not a strict postulate. The postulate of locality restricts construction of spatial contiguity matrix in this paper. Moran's \(I\) is the basis of theoretical derivation and tool for empirical analysis in this paper, so it is necessary to show it in advance. The calculation formula of Moran's \(I\) can be expressed as
\[I=\frac{N\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}v_{ij}(x_{i}-\overline{x}) (x_{j}-\overline{x})}{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}v_{ij}\cdot \sum\limits_{i=1}^{N}(x_{i}-\overline{x})^{2}}=\frac{\sum\limits_{i=1}^{N}\sum \limits_{j=1}^{N}v_{ij}(x_{i}-\overline{x})(x_{j}-\overline{x})}{V\sum\limits_ {i=1}^{N}\sum\limits_{j=1}^{N}v_{ij}}\,, \tag{1}\]
where
\[V=\frac{1}{N}\sum\limits_{i=1}^{N}(x_{i}-\overline{x})^{2}\,, \tag{2}\]
is the population variance of variable \(x\), \(v_{ij}\) is the contiguity measure, which can be measured by the reciprocal of distance (numerical variable) or the adjacency relationship (virtual variables 0 and 1) between \(i\) and \(j\). Scale effects in spatial analysis include both extent and granularity. In previous
research on the scale dependence of Moran's \(I\), a significant focus has been placed on granularity (Chou, 1991). There are two main ways to achieve different granularities in existing literature: One is segmentation method, which is to divide study area with several grids first, then divide one grid into two, two into four......, the more times of segmentation, the smaller the granularity (e.g., Chou, 1991). The other is aggregation method, which is to aggregate pixels at different levels based on the original spatial pattern of study area, the higher the aggregation level, the larger the granularity (e.g., Qi & Wu, 1996; Overmars et al., 2003; Feng et al., 2016). The segmentation method operates from top to bottom, which is similar to the calculation process of box dimension, while there are few data points. The aggregation method operates from bottom to top, which is similar to renormalization. There are more data points as the granularities are usually arithmetic series, while the total area covered by pixels is actually not equal as the granularities are not multiple.
In order to help readers understand the derivation process of equations proposed in this paper, it is necessary to explain data processing method in advance. Based on the advantages and disadvantages of the above two methods, this paper selects the segmentation method to extract granularities, and makes slight adjustments with reference to the box-counting method: first, select the smallest circumscribed rectangle in study area as the first-level box, then divide the box into four, four into sixteen......, where the granularities are multiple. Specifically, assuming the study area is divided into \(N\) non-empty boxes in granularity \(\varepsilon\), then \(x_{i}\) and \(x_{j}\) in Eq. 1 are the values of variable \(x\) in the \(i\)th and \(j\)th box respectively, \(\overline{x}\) are the average values of \(x\). Referring to Chou (1991), variable \(x\) in this paper is defined in two ways to compare the influence of variable types on the scale dependence of Moran's \(I\). One type of variable \(x\) is numerical variable, where \(x\) is the proportion of built-up area in each box to box area, that is, \(x_{i}=A_{i}/\varepsilon^{2}\) (\(A_{i}\) is the area of built-up area in box \(i\)). The variable is called p variable. The other type is classification variable, \(x\) takes 0 or 1 indicating whether built-up area is dominant in each box, that is, whether its area exceeds 50%. If it is, \(x\) takes 1, otherwise, \(x\) takes 0. The variable is called t variable. The classification variable t is equivalent to raster images with the same resolution. It helps reduce the fragmentation of the original spatial pattern, similar to how remote sensing images work (Zuo, 2011). However, it is affected by mixed pixels and may not accurately represent the original spatial pattern. Additionally, increasing granularity through the use of t variables can further distort spatial patterns and produce results similar to the process of aggregating pixels by majority in remote sensing images. On the other hand,
the numerical variable p is closer to the original spatial pattern. In addition, \(v_{ij}\) in Eq. 1 represents the value of contiguity matrix \(\mathbf{V}\) at the corresponding position. There are at least four types of contiguity matrix (Chen, 2012). Based on the postulate of absolute locality in this paper, the simplest rook contiguity matrix is selected for \(\mathbf{V}\), that is, when the _i_th and _j_th box have common edges, \(v_{ij}\) takes 1, otherwise \(v_{ij}\) takes 0.
As a preparation for theoretical derivation, the definition of correlation dimension is explained. The scale dependence of spatial measurement is essentially a scaling phenomenon. Fractal geometry is a powerful tool for scaling analysis, and fractal dimension is the basic parameter of fractal system (Batty & Longley, 1994; Frankhauser, 1998; Mandelbrot, 1982). There are three commonly used parameters in multifractal spectrum: capacity dimension, information dimension and correlation dimension (Grassberger, 1983). The most widely used measurement method of fractal dimension is box-counting method, among which the most convenient one is functional box method (Lovejoy et al., 1987; Chen, 1995). The specific calculation steps are as follows: first, select the smallest circumscribed rectangle of study area as the first-level box; then divide the box into four, four into sixteen......, measure the box size \(\varepsilon\) in each level and the proportion \(p_{i}\) of the area of built-up area in the _i_th box to that in the total area, based on which capacity dimension and correlation dimension can be calculated. Based on the postulate of uniform distribution, the probability \(p_{i}\) is measured by virtual variables 0 or 1, then the definition of capacity dimension can be derived. The calculation formula of capacity dimension is
\[D_{0}=-\lim_{\varepsilon\to 0}\frac{\ln N(\varepsilon)}{\ln\varepsilon}=- \lim_{\varepsilon\to 0}\frac{\ln\sum_{i}^{N(\varepsilon)}p_{i}^{0}}{\ln \varepsilon}\,, \tag{3}\]
where \(D_{0}\) is the capacity dimension in the study area, \(N(\varepsilon)\) is the number of non-empty boxes under scale \(\varepsilon\). Eq. 3 is equivalent to the power law relation defining the box dimension
\[N(\varepsilon)=\eta\varepsilon^{-D_{0}}\,, \tag{4}\]
where \(\eta\) is the proportional coefficient, \(\eta=1\) theoretically.
Correlation dimension is defined based on the second order moment of probability, which is equivalent to the second-order Renyi entropy. The calculation formula of correlation dimension is
\[D_{2}=\lim_{\varepsilon\to 0}\frac{\ln C(\varepsilon)}{\ln\varepsilon}=\lim_{ \varepsilon\to 0}\frac{\ln\sum_{i}^{N(\varepsilon)}p_{i}^{2}}{\ln \varepsilon}\,, \tag{5}\]
where \(D_{2}\) is the correlation dimension, \(C(\varepsilon)\) is the correlation function under scale \(\varepsilon\). Based on step function, the correlation function can be simplified as (Grassberger & Procassia, 1983)
\[C(\varepsilon)=\frac{1}{N(\varepsilon)^{2}}\sum_{i}^{N(\varepsilon)}\sum_{j}^{ N(\varepsilon)}\theta(\varepsilon-d_{ij})\propto\sum_{i}^{N(\varepsilon)}p_{i}^{ 2}\, \tag{6}\]
where \(\theta\) is the Heaviside function
\[\theta(\varepsilon-d_{ij})=\begin{cases}1,\ d_{ij}\leq\varepsilon\\ 0,\ d_{ij}>\varepsilon\end{cases}. \tag{7}\]
Capacity dimension \(D_{0}\) and correlation dimension \(D_{2}\) represent the degree of spatial filling and spatial correlation respectively, and \(D_{0}>D_{2}\). It can be derived from Eqs. 5 and 6 that
\[C(\varepsilon)=\sum_{i=1}^{N(\varepsilon)}p_{i}^{2}=K\varepsilon^{D_{2}}\,, \tag{8}\]
where the proportional coefficient \(K=1\) in standard case.
### Derivation of scaling relation
Based on the preparation of above concepts, methods and formulas, the relationship between Moran's \(I\) and fractal dimension can be formally derived. With the results of this derivation, the scale dependence of Moran's \(I\) of cities can be better understood. Moran's \(I\) is essentially a generalized correlation function, which can be related to correlation dimension (Chen, 2021). On the one hand, for numerical variable \(\mathrm{p},x_{i}\) in Eq. 1 can be associated with probability measure \(p_{i}\), that is
\[x_{i}=\frac{A_{i}}{\varepsilon^{2}}=\frac{A_{i}}{A}\cdot\frac{A}{\varepsilon^ {2}}=p_{i}\cdot\frac{A}{\varepsilon^{2}}\, \tag{9}\]
where \(A\) is the total area of built-up area of a city. Then there is
\[\overline{x}=\frac{A}{\varepsilon^{2}N(\varepsilon)}\cdot\sum_{i}^{N( \varepsilon)}p_{i}=\frac{A}{\varepsilon^{2}N(\varepsilon)}. \tag{10}\]
Here we use the normalization property of probability, that is, the sum of probabilities is 1. It is worth noting that for fractal phenomena, the total area \(A\) is a variable in theory; but in practice, the spatial pattern under certain resolution is actually prefractal rather than real fractal, so \(A\) can be treated as a constant.
The calculation result of Moran's \(I\) depends on the definition of spatial weights matrix, which depends on the definition of spatial contiguity. Based on localized contiguity represented by 0 and 1, we can not only calculate a kind of Moran's \(I\), but also define spatial correlation function. This
means that with the contiguity relationship represented by virtual variable, Moran's \(I\) and correlation function \(C(\varepsilon)\) can be linked. Specifically, based on the contiguity matrix \(\mathbf{V}\) represented by variable 0 and 1, Eqs. 1 and 6 can be linked. In fact, under box size \(\varepsilon\), it can be approximated that only the boxes adjacent to box \(i\) are less than \(2\varepsilon\) away from it, so \(\theta(2\varepsilon\text{-}d_{ij})=1\); the other boxes are more than \(2\varepsilon\) away from box \(i\), so \(\theta(2\varepsilon\text{-}d_{ij})=0\). In this case, when \(\mathbf{V}\) takes rook contiguity matrix, only the \(\theta\) function of box \(i\) and the four boxes adjacent to and on the top, bottom, left and right of it take 1, the rest take 0; when \(\mathbf{V}\) takes queen contiguity matrix, the \(\theta\) function of box \(i\) and the eight boxes adjacent to and on the top, bottom, left, right, top left, top right, lower left and lower right of it take 1, the rest take 0. In this paper, \(\mathbf{V}\) takes rook contiguity matrix, so Eq. 6 can be rewritten as
\[C(2\varepsilon)=\sum_{i}^{N(\varepsilon)}p_{i}(p_{i}+p_{u}+p_{u}+p_{u}+p_{u}) =C(\varepsilon)+\sum_{i}^{N(\varepsilon)}\sum_{j}^{N(\varepsilon)}v_{ij}p_{i}p _{j}\, \tag{11}\]
where the subscripts \(it\), \(ib\), \(il\) and \(ir\) represent the boxes adjacent to and on the top, bottom, left and right of box \(i\) respectively. Based on the postulate of scaling property of spatial relationship, introduce Eq. 8 into Eq. 11 to obtain
\[\sum_{i=1}^{N(\varepsilon)}\sum_{j=1}^{N(\varepsilon)}v_{ij}p_{i}p_{j}=(2^{D_{ i}}-1)C(\varepsilon)\, \tag{12}\]
further, introducing Eqs. 9, 10 and 12 into Eq. 1 yields
\[I(\varepsilon)=\frac{\frac{A^{2}}{\varepsilon^{4}}\sum_{i=1}^{N( \varepsilon)}\sum_{j=1}^{N(\varepsilon)}[v_{ij}p_{i}p_{j}-\frac{v_{ij}p_{i}}{N (\varepsilon)}-\frac{v_{ij}p_{j}}{N(\varepsilon)}+\frac{v_{ij}}{N( \varepsilon)^{2}}]}{\sum_{i=1}^{N(\varepsilon)}\sum_{j=1}^{N(\varepsilon)}v_{ ij}\cdot V(\varepsilon)}=\frac{A^{2}[(2^{D_{i}}-1)C(\varepsilon)-\frac{2\sum_{i=1}^{N( \varepsilon)}p_{i}\sum_{j=1}^{N(\varepsilon)}v_{ij}}{N(\varepsilon)}+\frac{ \sum_{i=1}^{N(\varepsilon)}\sum_{j=1}^{N(\varepsilon)}v_{ij}}{N(\varepsilon) ^{2}}]}{\sum_{i=1}^{N(\varepsilon)}\sum_{j=1}^{N(\varepsilon)}v_{ij}\cdot \varepsilon^{4}V(\varepsilon)} \tag{13}\]
According to Eq. 2, \(V(\varepsilon)\) is the population variance of variable \(x\) under granularity \(\varepsilon\).
A skill of mathematical reasoning is to approximate with the help of limit conditions, and approximate results usually do not affect the effect of empirical analysis. In this paper \(\mathbf{V}\) takes rook contiguity matrix, then when \(\varepsilon\text{-}0\) that is \(N(\varepsilon)\text{-}\infty\), the sum of \(v_{ij}\) by column is close to 4, so Eq. 13 is simplified as
\[I(\varepsilon)\cong\frac{A^{2}[(2^{D_{i}}-1)C(\varepsilon)-4/N(\varepsilon)]} {4\varepsilon^{4}N(\varepsilon)V(\varepsilon)}=\frac{A^{2}[(2^{D_{i}}-1)N( \varepsilon)C(\varepsilon)-4]}{4\varepsilon^{4}N(\varepsilon)^{2}V( \varepsilon)}\,, \tag{14}\]
it can be seen from Eqs. 4 and 8 that
\[N(\varepsilon)C(\varepsilon)=\eta K\varepsilon^{D_{\lambda}-D_{0}}\,, \tag{15}\]
where \(D_{0}>D_{2}\). Then when \(\varepsilon{\rightarrow}0\), there is \(N(\varepsilon)C(\varepsilon){\rightarrow}\infty\), so
\[I(\varepsilon)\cong\frac{(2^{D_{\lambda}}-1)A^{2}N(\varepsilon)C(\varepsilon) }{4\varepsilon^{4}N(\varepsilon)^{2}V(\varepsilon)}=\frac{(2^{D_{\lambda}}-1 )A^{2}C(\varepsilon)}{4\varepsilon^{4}N(\varepsilon)V(\varepsilon)}=\lambda \frac{C(\varepsilon)}{\varepsilon^{4}N(\varepsilon)V(\varepsilon)}\,, \tag{16}\]
where
\[\lambda=\frac{1}{4}(2^{D_{\lambda}}-1)A^{2}\,, \tag{17}\]
is a proportional constant. Then Eq. 16 becomes
\[I(\varepsilon)V(\varepsilon)=\lambda\frac{C(\varepsilon)}{\varepsilon^{4}N( \varepsilon)}=\lambda\,\varepsilon^{D_{\lambda}+D_{2}-4}\,, \tag{18}\]
where \(\lambda^{\prime}=\lambda K/\eta\). It can be seen that when \(V(\varepsilon)\) shows a power law against \(\varepsilon\), \(I(\varepsilon)\) follows a power law against \(\varepsilon\) as well.
Figure 19: research objects and corresponding research scopes. Contains modified Copernicus Sentinel data (2021) processed by ESA WorldCover consortium.
### Study area and data processing
Success of modern science lies in its great emphasis on the role of quantifiable data and its interaction with models. Both data and models are necessary and indispensable for the advancement of human understanding of problems: on the one hand, data generates formalized facts and imposes constraints on models; on the other hand, models are significant to understand the operation principle and action process of system (Louf & Barthelemy, 2014). The above relationship between data and models affects all scientific fields, and this research is no exception. The empirical analysis in this paper focuses on the urban areas of 21 largest cities in China to validate previous derivation results. The cities include 7 megacities with a permanent urban population exceeding 10 million: Shanghai, Beijing, Shenzhen, Chongqing, Guangzhou, Chengdu and Tianjin; as well as 14 supercities with a permanent urban population ranging from 5 to 10 million: Wuhan, Dongguan, Xi'an, Hangzhou, Foshan, Nanjing, Shenyang, Qingdao, Jinan, Changsha, Harbin, Zhengzhou, Kunming and Dalian. Since the built-up areas of Guangzhou and Foshan, as well as Shenzhen and Dongguan, have already interconnected, this paper combines them into single cities respectively, referred to as GuangFo and ShenGuan. So there are a total of 19 research objects (Fig. 1). The
Figure 2: Workflow of this paper. Moran’s \(I\) can be linked to correlation function by contiguity matrix, so the scaling exponent that Moran’s \(I\) changing with granularity can be linked with correlation dimension. Taking 19 largest Chinese cities as examples, the derivation above is verified.
research scope covers a 55km square area around the central part of each city, which is sufficient to cover the most urbanized regions in all cities. WorldCover dataset of 2021, a land cover dataset published by the European Space Agency with a resolution of 10m and an overall accuracy of 76.7%, is selected as data source (Zanaga et al., 2022). This paper only selects one land cover class, built-up, among 11 land cover classes in this dataset, to represent urbanized area. Furthermore, the original images were processed using ArcGIS 10.3, and Moran's \(I\) was calculated using Python 3.8. Finally, Moran's \(I\) are calculated for segmentation times from 1 to 13, corresponding to granularities of about 27.5km to 7m. The workflow of the study is illustrated in Fig. 2.
## 3 Results
After calculating Moran's \(I\) of urban form, we can examine the relationships between spatial autocorrelation measure and fractal scaling of cities. For the 19 largest Chinese cities listed in Fig. 1, Moran's \(I\) of built-up area in each city under different granularities are calculated respectively. For convenience, we use segmentation time \(n\) to replace granularity \(\varepsilon\) as abscissa in the following plots, the results are shown in Fig. 3.
First, Moran's \(I\) show an upward trend as granularity becomes smaller. Except a few Moran's \(I\) that are negative under coarse granularities, the rest are all positive, indicating that the spatial pattern of built-up areas is positively correlated. Moran's \(I\) fluctuate greatly under coarse granularities, but their trend becomes clear as the increase of segmentation time \(n\), that is, Moran's \(I\) increases with \(n\) after 6-7 times of segmentation. Compared with t variables, the fluctuation of p variables is smaller and their trend is more obvious. The above trend is basically consistent with existing research results (Chou, 1991; Qi & Wu, 1996; Bu et al., 2003; Xu et al., 2004; Tan et al., 2005; Xu et al., 2007).
Second, the trends of Moran's \(I\) changing with granularity of two types of variables (p, t) are similar yet diversified. On the whole, Moran's \(I\) of p variable (Fig. 3a, c, e) is larger than that of the corresponding t variable (Fig. 3b, d, f), and the turning points where Moran's \(I\) change from fluctuation to trend with the increase of \(n\) are different. But with the increase of segmentation time \(n\), Moran's \(I\) of t variable increases faster so the gap between the two is narrowed. Although the overall trends of the two variables are not the same, they tend to be consistent after the turning point. This is because compared with p variable, t variable only takes the dominant land use class and loses the other, so the spatial autocorrelation is weakened and Moran's \(I\) is smaller. With the increase of granularity, the influence of mixed pixels of t variable increases, and the gap of Moran's \(I\) between the two variables also increases.
Last but not least, the trends of Moran's \(I\) changing with granularity vary across different cities as well. The turning points for the 19 cities vary, and they can be classified into three types based on their respective turning points for p and t variables (Table 1). (1) The first type of cities have a consistent changing trend of Moran's \(I\) between p and t variables, with turning points occurring
Figure 3: The change of Moran’s \(I\) with segmentation time \(n\) in 19 largest Chinese cities. All Moran’s \(I\) show an upward trend with the increase of \(n\). Moran’s \(I\) of p variable (**a, c, e**) is larger and more stable than that of the corresponding t variable (**b, d, f**). The 19 cities can be classified into three groups according to how their turning points for p and t variables are distributed.
early at \(n=5\)\(\sim\)7. These cities include Beijing, GuangFo, Harbin, Jinan, Shenyang, Tianjin, Xi'an, and Zhengzhou. These cities exhibit strong spatial heterogeneity at coarser granularities, leading to a smaller Moran's \(I\). However, as granularity decreases, the spatial heterogeneity weakens, and Moran's \(I\) increases significantly. Besides, there is little difference between the spatial pattern of high-density built-up area (t variable) and overall built-up area (p variable). This indicates these cities have a clear polycentric structure with developed multifractal nature (Fig. 3a, b). (2) The second type of cities have an inconsistent changing trend of Moran's \(I\) between p and t variables, with turning points of t variables occurring early at \(n=5\)\(\sim\)7, while Moran's \(I\) of p variables stabilizes at high values. As granularity further decreases, Moran's \(I\) of p variables show an upward trend, and their turning points occur at \(n=8\)\(\sim\)10. This type of cities includes Chongqing, Nanjing, Qingdao, Shenzhen, and Wuhan. These cities exhibit relatively weak spatial heterogeneity at coarser granularities, leading to a higher value of Moran's \(I\) for p variables. However, if only high-density built-up area are considered (t variable), the spatial heterogeneity will significantly increase, resulting in a much smaller value of Moran's \(I\) for t variables than that for p variables. This indicates that these cities have small subsidiary-centers with developing multifractal nature (Fig. 3c, d). (3) The third type of cities have a consistent changing trend of Moran's \(I\) between p and t variables as well, but with turning points occurring later at \(n=8\)\(\sim\)10. This type of cities includes Changsha, Chengdu, Dalian, Hangzhou, Kunming, and Shanghai. These cities exhibit relatively weak spatial heterogeneity at coarser granularities, so Moran's \(I\) stabilize at high values. However, as granularity further decreases, Moran's \(I\) start to increase significantly. Furthermore, there is little difference between the spatial pattern of high-density built-up area (t variable) and overall built-up area (p variable). This indicates that these cities have a clear monocentric structure with undeveloped multifractal nature (Fig. 3e, f). Finally, as \(n\) increases, Moran's \(I\) with smaller values generally increase faster, which results in a decrease in the gap of Moran's \(I\) in different cities. However, after the turning points, their relative size tends to be stable with little change.
\begin{table}
\begin{tabular}{c c c} \hline Type & Changing trend of Moran’s \(I\) & Cities \\ \hline
1. Polycentric structure, developed multifractal nature & Consistent changing trend between p and t variables; turning points occurred early at \(n=5\)\(\sim\)7. & Beijing, GuangFo, Harbin, Jinan, Shenyang, Tianjin, Xi’an, Zhengzhou \\ \hline
2. Small subsidiary- & Inconsistent changing trend between p and t variables; Chongqing, Nanjing, \\ \hline \end{tabular}
\end{table}
Table 1: Classification of 19 cities studied in this paper based on the changing trend of Moran’s \(I\).
Regression analysis is carried out on the scattered points then. Based on the overall trend, the change of Moran's \(I\) with segmentation time \(n\) is speculated to follow a linear or an exponential function. The model of linear function is
\[I(n)=a+bn\,, \tag{19}\]
where \(n\) is segmentation time, \(I(n)\) is the corresponding Moran's \(I\), \(a\) is the constant term, and \(b\) is the coefficient. The model of exponential function is
\[I(n)=\alpha\beta^{n}\,, \tag{20}\]
where \(a\) is the coefficient, \(\beta\) is the base number, and other symbols have the same meaning as Eq. 19. Since \(\varepsilon\) shows an exponential law against \(n\) that is \(\varepsilon\propto 2^{-n}\), the linear or exponential relationship between Moran's \(I\) and \(n\) in Eqs. 19 and 20 can be transformed into a logarithmic or power relationship between Moran's \(I\) and \(\varepsilon\):
\[I(\varepsilon)=a^{\prime}-b^{\prime}\ln\varepsilon\,, \tag{21}\]
\[I(\varepsilon)=\alpha^{\prime}\varepsilon^{\varepsilon^{\varepsilon^{\varepsilon ^{\varepsilon}}}}\,, \tag{22}\]
where \(\varepsilon\) is granularity, \(I(\varepsilon)\) is the corresponding Moran's \(I\), \(b^{\prime}\)= \(b\)/ln\(2\), \(\beta^{\prime}\)= -log\({}_{2}\beta\). Eq. 21 is a logarithmic function, which is consistent with the research results of Chou (1991) and Zhang et al. (2019), so the linear function in Eq. 19 is empirically supported. Eq. 22 is a power function, which is consistent with the calculation formula of box dimension. In fact, according to Chou's (1991) explanation for the scale dependence of Moran's \(I\), as the increase of \(n\), the homogeneous pixels increase in 2 dimensions while the heterogeneous ones increase in 1 dimension, it is speculated that the spatial autocorrelation pattern of built-up area changes between 1 and 2 dimensions with granularity. That is, the change of Moran's \(I\) may be fractal and in the form of Eq. 22, so the exponential function in Eq. 20 is theoretically supported. Eqs. 19 and 20 were used to fit the data after turning points in Fig. 3, and the results are shown in Table 2.
\begin{table}
\begin{tabular}{l l l} \hline centers, developing & turning points of t variables occurred early at \(n\) = 5–7, & Qingdao, Shenzhen, \\ multifractal nature & while that of p variables occurred later at \(n\) = 8–10. & Wuhan \\ \hline
3. Monocentric & & Changsha, Chengdu, \\ structure, & Consistent changing trend between p and t variables; & Dalian, Hangzhou, \\ undeveloped & turning points occurred later at \(n\) = 8–10. & Kunming, Shanghai \\ multifractal nature & & \\ \hline \end{tabular}
\end{table}
Table 2: Fitting results of Moran’s \(I\) changing with segmentation time \(n\). Moran’s \(I\) have a higher goodness of fit to Eq. 20 except for the part marked red.
It can be seen that, although the turning points are not the same, after that the change of Moran's \(I\) of most variables with \(n\) are more exponential, that is, Moran's \(I\) change with the power of \(\varepsilon\). However, Moran's \(I\) of six variables, Beijing_p, Xi'an_p, Harbin_p, Harbin_t, Jinan_p and Jinan_t, show a higher goodness of fit to linear function, that is, Moran's \(I\) change logarithmically with \(\varepsilon\), which is consistent with the research results of Chou (1991) and Zhang et al. (2019). The six curves after the turning points all accelerated first and then stabilized (Fig. 4a); taking the logarithm of Moran's \(I\), the six curves after the turning points all increase linearly first and then tend to be flat
Figure 4: The logarithmic change of Moran’s \(I\) with segmentation time \(n\), taking Harbin as example. In linear scale, the curves after the turning points all accelerated first at \(n=7\)–\(12\) and then stabilized at \(n=13\) (**a**); in logarithmic scale, the curves after the turning points all increase linearly first at \(n=7\)–\(12\) and then tend to be flat at \(n=13\) (**b**).
(Fig. 4b). This is because Moran's \(I\) has an upper limit of 1, and the Moran's \(I\) of these six variables are close to or even more than 0.95 when \(n=13\). So Moran's \(I\) is significantly squeezed by the upper limit, its growth rate slows down, and its change with \(\varepsilon\) is more logarithmic.
However, it is not sufficient to determine the mathematical model for scale dependence of Moran's \(I\) simply relying on \(R^{2}\), further analysis based on model parameters is required. When the observed data follow Eq. 19, the scale dependence has a characteristic scale, and the parameter \(b^{\prime}\) in Eq. 21 is the eigenvalue of Moran's \(I\). On the other hand, when the observed data follow Eq. 20, the scale dependence has no characteristic scale, and scaling analysis should be carried out based on the parameter \(\beta^{\prime}\) in Eq. 22. Moreover, if \(b^{\prime}\) exceeds the range of Moran's \(I\), indicating that the center of Moran's \(I\) is outside the range of its values, then the scale dependence also has no characteristic scale. Table 3 compares the range of Moran's \(I\) and the characteristic scale \(b^{\prime}\) in each city. It is found that in all cities, \(b^{\prime}\) exceeds the range of Moran's \(I\). Therefore, despite having a high \(R^{2}\) for Eq. 19, the scale dependence of Moran's \(I\) is essentially scale-free, and can be approximated by a power law, namely Eq. 22. In fact, Zhang et al. (2019) also found that Moran's \(I\) follows a logarithmic relationship with resolution, but the eigenvalue of its logarithmic function also exceeded the range of Moran's \(I\), further supporting the idea that the scale dependence of Moran's \(I\) is scale-free.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{City} & \multicolumn{2}{c}{p variable} & \multicolumn{2}{c}{t variable} \\ \cline{2-5} & \(b^{\prime}\) & Range of \(I\) & \(b^{\prime}\) & Range of \(I\) \\ \hline Beijing & 0.149 & [0.657, 0.964] & 0.201 & [0.503, 0.921] \\ Changsha & 0.058 & [0.828, 0.957] & 0.107 & [0.652, 0.904] \\ Chengdu & 0.080 & [0.786, 0.958] & 0.130 & [0.617, 0.906] \\ Chongqing & 0.097 & [0.733, 0.954] & 0.196 & [0.460, 0.897] \\ Dalian & 0.059 & [0.839, 0.971] & 0.120 & [0.665, 0.932] \\ GuangFo & 0.134 & [0.687, 0.966] & 0.198 & [0.508, 0.923] \\ Hangzhou & 0.127 & [0.709, 0.957] & 0.213 & [0.479, 0.901] \\ Harbin & 0.109 & [0.772, 0.974] & 0.170 & [0.628, 0.941] \\ Jinan & 0.103 & [0.791, 0.974] & 0.171 & [0.623, 0.943] \\ Kunming & 0.067 & [0.830, 0.966] & 0.145 & [0.627, 0.922] \\ Nanjing & 0.101 & [0.747, 0.957] & 0.188 & [0.493, 0.900] \\ Qingdao & 0.073 & [0.808, 0.965] & 0.143 & [0.616, 0.922] \\ Shanghai & 0.129 & [0.680, 0.951] & 0.195 & [0.469, 0.890] \\ ShenGuan & 0.100 & [0.741, 0.967] & 0.175 & [0.536, 0.924] \\ \hline \hline \end{tabular}
\end{table}
Table 3: The range of Moran’s \(I\) and the characteristic scale \(b^{\prime}\) of logarithmic fit in each city. \(b^{\prime}\) in all cities exceeds the range of Moran’s \(I\), indicating the scale dependence of Moran’s \(I\) is essentially scale-free and can be approximated by a power law.
In summary, the change of Moran's \(I\) with granularity follows a power function, but there is a scaling range: when granularity is too coarse, Moran's \(I\) fluctuates greatly with large noise; when granularity is too fine, Moran's \(I\) is squeezed by the upper limit of 1, and its change slows down; when granularity is moderate, it is the scaling range where Moran's \(I\) follows a power law against granularity, corresponding to segmentation time \(n\) of about 6\(\sim\)13 and granularity \(\varepsilon\) of about 1000m-10m. And the logarithmic change of Moran's \(I\) with granularity is actually the latter part of the whole changing interval, corresponding to the scaling range and the part with finer granularity. However, the high Goodness of fit for logarithmic function is a false result, the characteristic scale of the equation exceeds the range of Moran's \(I\), indicating the scale dependence is scale-free essentially.
The power law between Moran's \(I\) and granularity can be explained by the derivation in last Section. The variance of each variable is calculated, then its change with segmentation time \(n\) is fitted to exponential function (only for the part when \(n=6\sim\)13). It is found that the variance of all p variables and most t variables are significantly exponentially correlated with \(n\) at the 0.05 level (the corresponding critical value is 0.707, as shown in Table 4). Due to the exponential relationship between \(\varepsilon\) and \(n\), it can be derived that \(V(\varepsilon)\) of most variables follow a power law against \(\varepsilon\) significantly at the 0.05 level. According to Eq. 18, \(I(\varepsilon)\) follow a power law against \(\varepsilon\) as well.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline City & p variable & t variable & City & p variable & t variable \\ \hline Beijing & 0.965 & **-0.698** & Nanjing & 0.979 & 0.905 \\ Changsha & 0.987 & 0.961 & Qingdao & 0.982 & 0.829 \\ Chengdu & 0.983 & 0.916 & Shanghai & 0.978 & **0.436** \\ Chongqing & 0.983 & 0.892 & ShenGuan & 0.977 & **-0.457** \\ Dalian & 0.983 & 0.890 & Shenyang & 0.971 & 0.809 \\ GuangFo & 0.967 & **-0.063** & Tianjin & 0.967 & 0.844 \\ Hangzhou & 0.973 & 0.855 & Wuhan & 0.983 & 0.881 \\ Harbin & 0.959 & 0.812 & Xi’an & 0.963 & 0.808 \\ Jinan & 0.960 & 0.836 & Zhengzhou & 0.974 & 0.884 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The correlation coefficient of exponential fit to the change of variance with segmentation time \(n\) (only for \(n\) = 6\(\sim\)13). Variance follow exponential function against \(\varepsilon\) significantly except for some t variables that are marked red.
Calculation and analysis so far, we can reach the key part of empirical study in this paper. The mathematical relationship between Moran's \(I\) and correlation dimension derived above, is to prove theoretically that there is sometimes scaling phenomenon behind spatial autocorrelation. With the calculation results in this section, the validity of the internal relationship between spatial autocorrelation and fractal scaling deduced above can be verified. It can be found from Eq. 18 that
\[I(\varepsilon)V(\varepsilon)\propto\frac{C(\varepsilon)}{\varepsilon^{4}N( \varepsilon)}=\varepsilon^{D_{0}+D_{1}-4}=\varepsilon^{-\gamma}. \tag{23}\]
For the part when \(n=6\)\(\sim\)\(13\), calculate the capacity dimension \(D_{0}\) and correlation dimension \(D_{2}\) in each city according to Eqs. 3 and 5 respectively, and the scaling exponent \(\gamma\) of variable \(I(\varepsilon)V(\varepsilon)\) changing with \(\varepsilon\) according to Eq. 23. It is found that the sum of \(D_{0}\), \(D_{2}\) and \(\gamma\) is about 4 (Table 5), which verifies the derivation above. In fact, since the limit condition of \(N(\varepsilon)\)\(\rightarrow\)\(\infty\) cannot be met, and the power relationship between \(V(\varepsilon)\) and \(\varepsilon\) does not always hold, \(D_{0}\)+\(D_{2}\)+\(\gamma\) is slightly different from 4. The use of t variables can lead to distortions in the original spatial pattern of built-up area, which results in the same \(D_{0}\) values as \(D_{2}\). Additionally, due to its definition, t variables do not conform to Eq. 9, the summed value of \(D_{0}\)+\(D_{2}\)+\(\gamma\) is more dissimilar from 4 when compared to using p variables.
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \multirow{2}{*}{City} & \multicolumn{4}{c}{Based on numerical variable p} & \multicolumn{4}{c}{Based on classification variable t} \\ \cline{2-9} & \(\gamma\) & \(D_{0}\) & \(D_{2}\) & \(D_{0}\)+\(D\)+\(\gamma\) & \(\gamma\) & \(D_{0}\) & \(D_{2}\) & \(D_{0}\)+\(D\)+\(\gamma\) \\ \hline Beijing & 0.317 & 1.947 & 1.916 & 4.180 & 0.126 & 1.893 & 1.893 & 3.911 \\ Changsha & 0.234 & 1.877 & 1.804 & 3.915 & 0.121 & 1.745 & 1.745 & 3.611 \\ Chengdu & 0.254 & 1.918 & 1.868 & 4.040 & 0.097 & 1.826 & 1.826 & 3.750 \\ Chongqing & 0.283 & 1.861 & 1.790 & 3.934 & 0.212 & 1.723 & 1.723 & 3.658 \\ Dalian & 0.166 & 1.797 & 1.735 & 3.699 & 0.094 & 1.696 & 1.696 & 3.486 \\ GuangFo & 0.284 & 1.944 & 1.913 & 4.141 & 0.123 & 1.889 & 1.889 & 3.902 \\ Hangzhou & 0.340 & 1.917 & 1.869 & 4.126 & 0.176 & 1.820 & 1.820 & 3.815 \\ Harbin & 0.231 & 1.839 & 1.778 & 3.848 & 0.143 & 1.737 & 1.737 & 3.616 \\ Jinan & 0.232 & 1.873 & 1.824 & 3.929 & 0.144 & 1.786 & 1.786 & 3.715 \\ Kunming & 0.207 & 1.807 & 1.739 & 3.754 & 0.141 & 1.687 & 1.687 & 3.516 \\ Nanjing & 0.290 & 1.884 & 1.820 & 3.994 & 0.181 & 1.761 & 1.761 & 3.703 \\ Qingdao & 0.203 & 1.878 & 1.826 & 3.906 & 0.098 & 1.788 & 1.788 & 3.675 \\ Shanghai & 0.354 & 1.949 & 1.911 & 4.214 & 0.132 & 1.876 & 1.876 & 3.884 \\ ShenGuan & 0.228 & 1.906 & 1.864 & 3.997 & 0.105 & 1.836 & 1.836 & 3.776 \\ Shenyang & 0.226 & 1.882 & 1.831 & 3.939 & 0.113 & 1.794 & 1.794 & 3.701 \\ \hline \end{tabular}
\end{table}
Table 5: The scaling exponent changing with granularity \(\varepsilon\) in each city (only for \(n=6\)\(\sim\)\(13\)). The total value of \(D_{0}\)+\(D_{2}\)+\(\gamma\) is very close to 4, and the value calculated using p variables is closer to 4 than when using t variables.
In summary, the scale dependence of Moran's \(I\) follows power law, but only within a certain scale range, that is, there exists a scaling range. There are two reasons for the gap between calculation results and theoretical derivation. On the one hand, derivation in this paper relies heavily on the limit condition \(\varepsilon\)\(\rightarrow\)0, which is difficult to satisfy in reality. On the other hand, when \(\varepsilon\) is too fine, land use class is over-segmented, Moran's \(I\) is squeezed by the upper limit of 1, and its change with granularity shifts from power law to logarithmic function. While the overall distribution is still scale-free, and still can be approximated by power law.
## 4 Discussion
Scale dependence of urban measures of spatial autocorrelation is universal due to complexity of spatial pattern of built-up area. The purpose of this paper is to explore the scaling properties and laws behind the scale dependence of spatial autocorrelation. Therefore, based on locality of spatial correlation and fractal properties of urban spatial pattern, the model of Moran's \(I\) depending on scale is derived, and the theoretical derivation are verified with observed data of 19 largest Chinese cities. Based on the above calculation and analysis, the research results can be summarized as follows. First, the scale dependence of Moran's \(I\) of urban form follows a power law. The power exponent changing with granularity can be linked to correlation dimension through Eq. 23. Second, in empirical study, the scale dependence of Moran's \(I\) of urban form only follows power law within a certain scale range, that is, there exists a scaling range, with granularity of about 1000\(\sim\)10m (Fig. 5). In empirical analysis of this paper, removing the coarse granularities where Moran's \(I\) of built-up area change irregularly, the scale dependence of Moran's \(I\) of most variables obeys power law. Third, the above power law of the scale dependence of Moran's \(I\) holds for all 19 Chinese cities and both variable types (p, t). In terms of cities, while the spatial pattern of built-up areas may differ across various cities, their Moran's \(I\) conform to the same power law after the turning points. In terms of variable types, since t variable is equivalent to raster images with the same resolution, Moran's \(I\) based on remote sensing images with different resolutions aggregated by majority will
also follow the same changing regularity. However, since t variable distorts the original pattern to some extent, its changing law is not as significant as that of p variable.
Scale-free spatial structure of cities is important for effective explaining urban evolution and proper predicting city development. The research has been ongoing for many years, but a large number of issues have not yet been resolved. In fact, scaling law is an interdisciplinary issue. Scale dependence in geography and ecosystem has attracted the attention of academia for long. Most studies remain in qualitative analysis of scale dependence of Moran's \(I\). It is found that at fine granularities, Moran's \(I\) decrease steadily with the increase of granularity (e.g., Qi & Wu, 1996; Bu et al., 2003; Xu et al., 2004; Tan et al., 2005); however, at coarser granularities, Moran's \(I\) fluctuate greatly with no obvious regularity (e.g., Overmars et al., 2003; Xie et al., 2006; Qiu et al., 2007; Yin et al., 2008). Therefore, the break point of Moran's \(I\) changing with granularity is called "intrinsic scale" of spatial autocorrelation in study area (Xu et al., 2007; Chen & Zheng, 2013; Feng et al., 2020; Yuan et al., 2020; Li et al., 2022), and the extreme point of Moran's \(I\) is called "optimum scale" (Feng et al., 2016; Wang et al., 2021). The concepts of "intrinsic scale" and "optimum scale" are enlightening, but still remain at a qualitative level and are different from "characteristic scale" in mathematical concepts. Chou (1991) quantitatively analyzed the scale dependence of Moran's \(I\). Linear, parabolic and logarithmic functions were fitted to the change of Moran's \(I\) with "resolution level", and logarithmic function was found to have the highest goodness of fit. However, the relationship between Moran's \(I\) and granularity is not directly established. Zhang et al. (2019) also observed a negative logarithmic relationship between Moran's \(I\) and resolution, with the
Figure 5: Schematic plot of the scale dependence of Moran’s \(I\). In the first scale range, Moran’s \(I\) fluctuate greatly without obvious regularity; in the second scale range that is actually the scaling range, Moran’s \(I\) follow a power law against granularity; in the third scale range, squeezed by the upper limit 1, Moran’s \(I\) tend to flatten.
characteristic scale of the logarithmic function exhibiting a positive correlation with box dimension. However, these findings have not been theoretically proven, nor have the eigenvalue of the Moran's \(I\) been given through the characteristic parameters of the logarithmic function. In fact, the characteristic scale of the logarithmic model exceeds the range of Moran's \(I\), implying that the scale dependence of Moran's \(I\) has no characteristic scale, and can instead be approximated by a power law. Moran's \(I\) is essentially a generalized spatial correlation equation. The author in this paper has derived a spatial autocorrelation function based on variable displacement parameters before, and established the mathematical relationship between spatial autocorrelation function and spatial correlation dimension. Indeed, the relative step function based on variable displacement parameters used to establish the spatial autocorrelation function in that research is similar to the localized contiguity matrix based on variable granularity in this study. Therefore, the results of the two studies can be connected approximately. Through these two different approaches, the link between fractal scaling and spatial autocorrelation has been established.
Compared with previous studies on fractal cities and urban spatial autocorrelation, the novelty of this paper is that it reveals the scale dependence of spatial autocorrelation in urban built-up area from the perspective of fractal scaling invariance. Based on fractal property and locality of spatial correlation, the mathematical expression of Moran's \(I\) changing with linear scale of spatial partition (that is, granularity) is derived, and the theoretical derivation is empirically analyzed with the help of observation data. According to previous studies and the calculation of this paper, the scale dependence of spatial autocorrelation is divided into two categories: one is the scale dependence with characteristic scales, which can be modeled by functions with characteristic scales (such as exponential function, logarithmic function, Gauss function), and effective spatial autocorrelation measure can be found by model parameters. The other is the scale dependence without characteristic scale, which can only be modeled by power law or its variants, and conventional spatial autocorrelation measure cannot be found in principle (Fig. 6). For the second case, there are two solutions now: one is to extend spatial autocorrelation index to spatial autocorrelation function, and the other is to replace spatial autocorrelation index with scaling exponent (Chen, 2021). Previous studies have found the scale dependence of spatial autocorrelation in urban landscape, or identified a link between fractal pattern and spatial autocorrelation process, but did not distinguish the geospatial properties behind scale dependence based on granularity analysis, nor reveal the specific
mathematical expression and physical basis of scale dependence.
Since empirical research requires lots of work, the scaling analysis for spatial autocorrelation of built-up area in this paper is limited. The main shortcomings of this study are as follows. First, spatial correlation in this work is only based on the postulate of locality, without considering action at a distance. Therefore, the spatial contiguity matrix is constructed based on nominal variables 0 and 1, whose essence is to use step function, which represents local action, to construct spatial weights matrix. Second, the mathematical derivation of this paper is based on the postulates of local spatial correlation and multifractal structure. If fractal properties are undeveloped in urban built-up area, the above derivation are invalid; if fractal properties are developed but the spatial relationship shows significant long-range effects, the results of this paper can only be used approximately. Third, there is no standard method to recognize turning points. The judgement of turning points in function fitting in this study is mainly based on experience and visual inspection, which is somewhat subjective. Indeed, some studies such as Zhang et al. (2019) and Li et al. (2022) have utilized semivariograms to identify the effective scale range of spatial autocorrelation. This approach is more objective and can be applied to recognize turning points in subsequent research.
## 5 Conclusions
The relationship between fractal scaling and Moran's \(I\) of urban form not only suggests new angle
Figure 6: Scale dependence of spatial autocorrelation in urban and classification of its solutions. Scale dependence of spatial autocorrelation can be divided into two categories: one is the scale dependence with characteristic scales, which can be modeled by functions with characteristic scales, and effective spatial autocorrelation measure can be found by model parameters. The other is the scale dependence without characteristic scale, which can only be modeled by power law or its variants, and conventional spatial autocorrelation measure cannot be found in principle.
of view of understanding spatial autocorrelation analysis, but also indicates new way of looking at cities. Urban spatial correlation and fractal structure represent different sides of the same coin. Fractal order of cities emerged from bottom-up self-organized evolution by ways of spatial correlation. Based on the above calculation, analysis and discussion, the main conclusions can be drawn as follows. First, the scale dependence of spatial autocorrelation measurement in urban space can be divided into two basic types: with and without characteristic scale. Particularly simple spatial patterns bear no scale dependence. If urban spatial pattern is a relatively simple system, the scale dependence has a characteristic scale, the relationship between Moran's \(I\) and measurement scale usually follows logarithmic function, and coefficient of the model indicates the characteristic value of Moran's \(I\). If urban spatial pattern is a complex system, the scale dependence has no characteristic scale, the relationship between Moran's \(I\) and measurement scale usually follows power law, and scaling exponent of the model may indicate the characteristic parameter to replace Moran's \(I\) of urban form. Second, if spatial effect of urban form shows power law, the urban structure follows scaling law. In this case, spatial autocorrelation analysis of cities needs to find another way. Urban patterns bears spatial complexity, and Moran's \(I\) calculated by conventional method is no longer comparable, even simple spatial autocorrelation index fails. There are two possible ways to solve the problem: one is to extend spatial autocorrelation index to spatial autocorrelation function, which can be used to reveal rich spatial dynamics; the second is to calculate scaling exponent based on the power-law relationship between measurement scale and calculation results, and replace conventional characteristic scale analysis with scaling analysis.
**Acknowledgments**: This research was sponsored by the National Natural Science Foundation of China (Grant No. 42171192). The supports are gratefully acknowledged.
|
2304.13813 | Robust Macroscopic Schrödinger's Cat on a Nucleus | We propose a scheme to generate spin cat states, i.e., superpositions of
maximally separated quasiclassical states on a single high-dimensional nuclear
spin in a solid-state device. We exploit a strong quadrupolar nonlinearity to
drive the nucleus significantly faster than usual gate sequences, achieving
collapses and revivals two orders of magnitude faster than the dephasing
timescale. Furthermore, these states are engineered without entanglement with
an ancilla, hence, are robust against error propagation. With our multitone
control, we can realize arbitrary high-spin rotations within an experimentally
feasible regime, as well as transform a spin coherent state to a spin cat state
using only phase modulation, opening the possibility of storing and
manipulating high-fidelity cat states. | Pragati Gupta, Arjen Vaartjes, Xi Yu, Andrea Morello, Barry C. Sanders | 2023-04-26T20:25:25Z | http://arxiv.org/abs/2304.13813v3 | # Robust Macroscopic Schrodinger's Cat on a Nucleus
###### Abstract
We propose an experimentally feasible scheme to create large Schrodinger cat states on a high-spin nucleus of a donor atom embedded in a solid-state system. The resulting cat state is robust against decoherence, macroscopic because its size scales linearly with nuclear spin, and tiny--at the femtometer scale. Our quantum-control scheme utilizes one-axis twisting caused by a non-linear quadrupole interaction and phase-modulated multi-tone radio-frequency pulses for universal high-dimensional rotations. Our scheme achieves fast generation and detection--within a fraction of the nuclear spin dephasing time--and can yield highly coherent cat states with a lifetime of tens of milliseconds on state-of-the-art hardware.
A Schrodinger cat state is an equal superposition of two macroscopically distinct quantum states [1; 2; 3], which can encode an error protected qubit, and can be a manifestation of many-body entanglement [4]. Cat states serve as an important resource for quantum sensing [5; 6; 7], information processing [8; 9; 10], fault-tolerant computation [11] and quantum communication [12; 13; 14]. Apart from technological applications, they are probes for fundamental aspects of quantum physics, including measurement problem [15], macro-realism [16; 1], quantum-classical transition [17; 18], and non-locality [19]. Though cat states can be realized on several platforms including photons [20; 21; 22; 23; 24], superconducting qubits [25; 26; 27], atoms [28; 29], ions [30; 31], their extreme fragility due to decoherence limits their usefulness.
We are motivated by recent experiments with high-spin nuclei [32; 33; 34; 35; 36; 37] that point to a new approach for utilising a coherent high-dimensional nuclear spin for scaling current technology. Here, we describe our method for generating spin Schrodinger cat states [3] on a nucleus, which are represented by a nuclear spin \(I\geq 1\) in a superposition of high clockwise and high anticlockwise spin, as shown in Fig. 1. Cat states on a high-spin nucleus are expected to be extremely robust, even for higher spins, unlike other spin cat states created via many-body interactions that decohere quickly--at least \(O(N)\) times fast as the number of particles \(N\) increases [38]. We devise a quantum-control strategy that exploits the electric quadrupole interaction for a high-spin nucleus and simultaneously driven transitions with phase-modulated multi-tone (MT) radio-frequency (RF) pulses to generate cat states. Our scheme overcomes fragility and can produce highly coherent cat states within a fraction of the decoherence time on state-of-the-art solid-state devices [32; 39].
Despite the femtometer scale of a nucleus, nuclear-spin cat states are macroscopic because their size, quantified by the relative quantum Fisher information for macroscopic spin systems [40], scales linearly with \(I\). The size or the degree of superposition \(N^{r}_{\text{eff}}\) of a nuclear-spin state \(\psi\) for measurement operator \(\hat{O}\) is connected to the variance \(\mathcal{V}(\psi,\hat{O}):=\bra{\psi}\hat{O}^{2}\ket{\psi}-\bra{\psi}\hat{O} \ket{\psi}^{2}\) as
\[N^{r}_{\text{eff}}:=\frac{2\mathcal{V}(\psi,\hat{O})}{I}. \tag{1}\]
For an appropriate \(\hat{O}\), the variance is \(I^{2}\) for bimodal cat states and we calculate the size of a nuclear-spin cat state to be \(2I\), which is equivalent to a Greenberger-Horne-Zeilinger state [4] made of \(N=2I\) entangled qubits, and denotes the maximum quantum Fisher information of the system [41]. The degree of superposition increases and decreases as we drive a coherent state to a cat state and back, and the periodic "death and revival" of a cat state is a signature of coherence.
We consider a high-spin donor (Group V element with a high-spin nucleus) implanted in a nanometer-scale solid-state chip made of \({}^{28}\)Si atoms (or any other spinless Group IV atoms), as described in [32]. The chip is affixed to a dilution refrigerator maintained at a low temperature (\(\sim\)14 mK) and placed inside a superconducting coil that produces a static magnetic field \(\mathbf{B}_{0}\). As shown in Fig. 2(a), the chip has electrostatic gates on its surface for changing the electric potential of the donor, a microwave antenna to supply electromagnetic pulses, and a single electron transistor (SET) for reading out the donor spin state. An implanted donor contains five electrons in its outermost shell, four of which form bonds with neighboring Si atoms and the remaining electron can be removed
Figure 1: Nuclear spin Schrödinger cat state: a superposition \(\ket{I,I}+\ket{I,-I}\) (ignoring the normalisation constant) of spin-up and spin-down orientations of a high-spin nucleus.
by ionizing the donor using electrostatic gates.
The nuclear spin of a donor atom is initialized, controlled and readout by passing DC and AC electric currents through the chip. In the above setup, the donor atom is subjected to a static potential, with the nuclear spin coupled to an electron spin, and initialization of the nuclear spin is done via flip-flop driving [42]. Then, the electrostatic potential is increased to ionize the donor atom by removing the electron and only the nuclear spin is manipulated using RF pulses [43]. When the nuclear spin reaches the desired final state, the donor atom is de-ionized such that the nuclear spin again couples to an electron, and is read out using pulsed electron spin resonance and spin dependent tunneling of the electron to the SET [44].
The electrostatic gates enhance the lattice strain in the chip, thereby distorting the bonds between the donor atom and neighboring Si atoms, as shown in Fig. 2(b). The electric field gradient (EFG) generated from distortion of the bonds results in a quadrupole interaction of the high-spin nucleus, which has a non-zero quadrupole moment due to its asymmetric nuclear charge distribution. For a spatially varying potential \(V(x,y,z)\), the EFG tensor is \(\mathcal{V}_{\alpha\beta}:=\frac{\partial^{2}V(x,y,z)}{\partial\alpha\partial\beta}\), where \(\alpha\), \(\beta\in\{x,y,z\}\) are the tensor indices. The tensor \(\mathcal{V}_{\alpha\beta}\) is real, traceless and symmetric and thus can be diagonalized in a principal axis system \(\{x^{\prime},\,y^{\prime},\,z^{\prime}\}\), such that \(|\mathcal{V}_{z^{\prime}z^{\prime}}|\geq|\mathcal{V}_{y^{\prime}y^{\prime}}| \geq|\mathcal{V}_{x^{\prime}x^{\prime}}|\), and the EFG is characterized by the asymmetry parameter \(\eta:=\frac{\mathcal{V}_{x^{\prime}x^{\prime}}-\mathcal{V}_{y^{\prime}y^{ \prime}}}{\mathcal{V}_{z^{\prime}z^{\prime}}}\) for \(0\leq\eta\leq 1\). For a nucleus with electric quadrupole moment \(q_{\rm n}\), the quadrupole coupling strength is \(f_{\rm q}=\frac{3q_{\rm n}\mathcal{V}_{x^{\prime}x^{\prime}}}{4(2I-1)\hbar}\), where e is elementary charge and h is Planck's constant. The quadrupole interaction is
\[\hat{H}_{\rm q}=f_{\rm q}\left[\hat{I}_{z^{\prime}}^{2}+\frac{\eta}{3}(\hat{I }_{x^{\prime}}^{2}-\hat{I}_{y^{\prime}}^{2}-I^{2})\right], \tag{2}\]
for a nucleus with spin operator vector \(\hat{\mathbf{I}}:=[\hat{I}_{x^{\prime}},\hat{I}_{y^{\prime}},\hat{I}_{z^{\prime}}]\) and \(I^{2}=\mathbf{I}\cdot\mathbf{I}\).
We manipulate the nuclear spin state using a static magnetic dipole interaction due to \(\mathbf{B}_{0}\), static electric quadrupole interaction due to EFG, and time dependent perturbation of amplitude \(A(t)\) driven by magnetic RF field \(\mathbf{B}_{1}\). For a nucleus with gyromagnetic ratio \(\gamma\), the nuclear spin dynamics are described by
\[\hat{H}=\gamma\hat{\mathbf{I}}\cdot\mathbf{B}_{0}+\hat{H}_{\rm q}+\gamma\hat{\mathbf{I}} \cdot\mathbf{B}_{1}A(t). \tag{3}\]
The \(2I+1\) degenerate levels of a nuclear spin undergo Zeeman splitting due to \(\gamma\hat{\mathbf{I}}\cdot\mathbf{B}_{0}\) which creates an equal spacing between the energy levels. The electric quadrupole interaction \(\hat{H}_{\rm q}\) causes an unequal shift of the levels such that the transition frequency \(\omega_{i,i-1}\) between two consecutive levels \(i\) and \(i-1\), for \(i\in[-I,I-1]\),is unique, as shown in Fig. 2(c). RF pulses drive transitions between spin states \(|I,m_{I}\rangle\), where \(m_{I}\in[-I,I]\), governed by the selection rule \(\Delta m_{I}=\pm 1\). For a spin-\(7\)\({}_{2}\)\({}^{123}\)Sb donor [32], the Zeeman splitting \(\gamma B_{0}=8.25\) MHz, \(f_{\rm q}=40\) kHz and perturbation strength \(\gamma B_{1}\approx 5\) kHz.
We consider the transformation of a nuclear-spin coherent state to a cat state due to non-linear terms like \(\hat{I}_{z^{\prime}}^{2}\) in the quadrupole interaction (Eq. (2)), similar to the dynamics of a non-linear rotator [3]. Though Eq. (2) is equivalent to one-axis twisting for \(\eta=0\) and two-axis counter twisting if \(\eta>0\)[45; 46], for high-spin donors that are under both electric quadrupole interaction (\(\hat{H}_{\rm q}\)) and magnetic dipole (\(\hat{H}_{\rm d}\)), only the component of the quadrupole interaction parallel to \(\mathbf{B}_{0}\) generates non-linear rotation. The perpendicular component does not affect the nuclear spin because the Zeeman splitting \(\gamma\mathbf{B}_{0}\) is much larger than quadrupole coupling \(f_{\rm q}\) and non-linear transitions between eigenstates are prevented by the large energy difference, which results in an effective one-axis twisting, regardless of the value of \(\eta\). In the lab frame of reference, we set \(\mathbf{B}_{0}=B_{0}\hat{z}\), \(\mathbf{B}_{1}=B_{1}\hat{y}\), \(\eta=0\), and EFG to be symmetric about \(z\) axis, i.e. \(\hat{z}^{\prime}=\hat{z}\), and our results can be generalized to other conditions of EFG.
The degree of superposition \(N^{r}_{\rm eff}\) (Eq. (1)) increases and decreases periodically during one-axis twisting, as shown in Fig. 3(a). \(N^{r}_{\rm eff}=1\) for a spin coherent state at \(t=0\) (Fig. 3(b)(i)), increases during squeezing as the spin state spreads over the equator of the
Figure 2: Solid-state device: a) High-spin donor implanted in a fabricated silicon-based chip with electrostatic gates, a single electron transistor (SET) and a microwave antenna. b) Electrostatic gates create a strain in the silicon lattice, which distorts the bond configurations and leads to an electric field gradient at the donor site.c) Energy level diagram of a high-spin nucleus under magnetic dipole and electric quadrupole interaction.
Bloch sphere(Fig. 3(b)(ii)), then reaches an inflection point where it interferes with itself (Fig. 3(b)(iii)) and finally increases again as two distinct components form (Fig. 3(b)(iv)) resulting in a cat state at \(t=\frac{\pi}{2\hat{I}_{q}}\) (Fig. 3(b)(v)). Under further evolution, the cat returns to a coherent state at \(t=\frac{\pi}{\hat{I}_{q}}\), which again squeezes to a cat state at \(t=\frac{3\pi}{\hat{I}_{q}}\), and so on. This periodic process of death and revival of the cat state has a time period of \(\nicefrac{{\pi}}{{f_{n}}}\) and is a signature of coherence [47]. From the plots for different values of nuclear spin in Fig. 3(a), we can note the macroscopicity of the cat states through the linear scaling of their size \((N^{r}_{\text{eff}})_{\text{max}}=2I\).
When the initial coherent state resides on the \(x\) axis of the Bloch sphere, the cat state produced by one-axis twisting lies on the \(y\) axis for a half-integer nuclear spin, as demonstrated by Fig. 3(b)(i) and Fig. 3(b)(v). The death and revival dynamics of one-axis twisting are thus revealed when \(\hat{O}=\hat{I}_{y}\). For donors in solid-state devices, however, nuclear spin measurement is limited to the eigenstates of \(\hat{I}_{z}\); as a result, we use global rotation to initialise and detect cat states along \(\hat{I}_{z}\) axis on the Bloch sphere.
We develop a multi-tone (MT) pulse that simultaneously drives all transitions resulting in high-dimensional rotations on a nuclear spin. The MT pulse is a superposition of \(2I\) tones [48], corresponding to the \(2I\) nearest neighbor sub-spaces with transition frequencies, represented by frequency vector \(\mathbf{\omega}=\{\omega_{i,i-1}\}_{i=I}^{-I-1}\). Each tone has a global phase \(\phi\) and a pulse switched on at a time \(t_{0}\) is described by the amplitude \(A^{\text{MT}}(t;\mathbf{\omega},\phi,t_{0})=\frac{1}{2I}\sum_{\omega_{j}\in\mathbf{ \omega}}\cos(\omega_{j}(t-t_{0})+\phi)\). Here, \(\phi\) tunes the axis of rotation on the Bloch sphere while preserving the relative phase between different energy levels. The frequency of global rotations \(\Omega\) is generalized for a high-spin nucleus, from the Rabi frequency \(\nicefrac{{\gamma B_{1}}}{{2}}\) for a two-level system divided by number of tones \(2I\), i.e. \(\Omega=\nicefrac{{\gamma B_{1}}}{{4I}}\). Applying this pulse for time \(\Delta t\) rotates the nuclear spin by angle \(\Theta\) given by \(\Theta=\Omega\Delta t\).
In our scheme to realize cat states, we initialize the nuclear spin to the lowest energy eigenstate \(|I,I\rangle\), and apply a \(\nicefrac{{\pi}}{{2}}\) global rotation to transform it to a coherent state on the equatorial plane. Then, we let the spin evolve freely to transform the coherent state to a cat state due to one-axis twisting. Finally, we apply another MT pulse such that the cat state generated on the equator is rotated by \(\nicefrac{{\pi}}{{2}}\) to lie along the z axis and is described by \(|I,I\rangle+|I,-I\rangle\), with unit norm implied. Here, the global rotation of \(\Theta=\nicefrac{{\pi}}{{2}}\) would take time \(\Delta t_{\nicefrac{{\pi}}{{2}}}=\frac{\pi}{2}\cdot\frac{4I}{\gamma B_{1}}\).
Figure 4 shows (a) our control pulse design to realize cat states and (b) the resulting dynamics illustrated using Husimi Q-functions. Thus, to implement the above steps, a square-shaped MT RF pulse is applied between initial time \(t_{0}=0\) and \(t_{1}=t_{\nicefrac{{\pi}}{{2}}}\), then switched off for time \(T\), and again applied between \(t_{2}=T+t_{\nicefrac{{\pi}}{{2}}}\) and \(t_{3}=T+2t_{\nicefrac{{\pi}}{{2}}}\), with the amplitude
\[A^{\text{cat}}(t;\mathbf{\omega},\Delta\phi,T)= A^{\text{MT}}(t;\mathbf{\omega},0,t_{0})\sqcap(t_{0},t_{1})\] \[+A^{\text{MT}}(t;\mathbf{\omega},\Delta\phi,t_{2})\sqcap(t_{2},t_{3})\,] (4)
for \(\sqcap\) the top-hat function. Setting the phase difference \(\Delta\phi=\nicefrac{{\pi}}{{2}}\) between the first and the second global rotations accounts for the angle between a coherent state and the resulting cat state on the equator. This pulse (4) transforms the squeezed spin states, which are generated on the equatorial (\(x\)-\(y\)) plane during one-axis twisting, to corresponding states along the \(z\) axis of the phase-space sphere.
For a spin-\(\nicefrac{{\pi}}{{2}}\)\(\nicefrac{{123}}{{\rm Sb}}\) nucleus, Fig. 4(c) shows the change in \(N^{r}_{\text{eff}}\), for \(\hat{O}=\hat{I}_{z}\), as \(T\) is varied. We observe the expected increase and decrease in \(N^{r}_{\text{eff}}\) due to squeezing of a coherent state to a cat state and back, similar to Fig. 3. In addition to the periodic death and revival of a cat state due to one-axis twisting, we observe a rapid variation of \(N^{r}_{\text{eff}}\) with a frequency of \(\nicefrac{{\gamma B_{0}}}{{\pi}}\), found by spectral analysis.
We introduce phase modulation to overcome these rapid oscillations of \(N^{r}_{\text{eff}}\). During free evolution, nuclear-spin states generated by one-axis twisting undergo Larmor precession at a rate of \(\omega=\gamma B_{0}\), matching the frequency above. If the axis of rotation, as specified by the phase difference \(\Delta\phi\), corresponds with the axis of the cat state, the second global rotation is unable to move the state from the equatorial plane to the \(z\) axis. To remedy this immobility, we assign a \(T\)-dependent shift as the
Figure 3: One-axis twisting of nuclear spin: a) Evolution of degree of superposition \(N^{r}_{\text{eff}}\) under periodic squeezing of a coherent spin state to a cat state at \(t=\frac{\pi}{\hat{I}_{q}}\). For different values of nuclear spin, the maximum value of \(N^{r}_{\text{eff}}\) scales as \(2I\), showing the macroscopicity of these cat states. b) Husimi Q-function graphs showing the transformation of a coherent state to a cat state on the equatorial plane (\(x-y\)) of the Bloch sphere.
phase difference: \(\Delta\phi=\nicefrac{{\pi}}{{2}}+\omega T\) in Eq. (4). As illustrated in Fig. 4(c), the phase-modulated pulses eliminate fast oscillations in \(N_{\text{eff}}^{r}\) and result in the periodic death and revival of a cat state solely due to one-axis twisting.
One-axis twisting achieves the quantum speed limit [49], taking minimum time possible for generating a cat state from a coherent state. The time-period of one-axis twisting scales inversely with the quadrupole coupling strength. For \(f_{\text{q}}=40\) kHz estimated in experiments [32], generating a cat state and observing the death followed by its revival (the second peak in Fig. 3) would take \(\sim\frac{3\pi}{2I_{\text{q}}}=0.12\) ms. When perturbation strength \(\gamma B_{1}\sim 4.5\) kHz, a \(\nicefrac{{\pi}}{{2}}\) global rotation would take \(\nicefrac{{\pi\cdot 2I}}{{\gamma B_{1}}}\sim 4.9\) ms. Thus, one-axis twisting and two MT pulses would take \(\sim\)10 ms for generating and validating the coherence of a cat state.
Our one-axis twisting approach works as we have shown, and we compare its performance to an intuitively appealing alternative approach for creating cat states, achieved by sequential Givens rotations [50]. Starting with \(\ket{I,I}\), we apply a \(\nicefrac{{\pi}}{{2}}\)-pulse of frequency \(\omega_{I,I-1}\) to form the superposition \(\ket{I,I}+\ket{I,I-1}\), with unit norm implied. Then, we apply a \(\pi\)-pulse of frequency \(\omega_{I-1,I-2}\) that drives the \(\ket{I,I-1}\leftrightarrow\ket{I,I-2}\) transition and leaves \(\ket{I,I}\) unaffected, thereby transforming the state to \(\ket{I,I}+\ket{I,I-2}\). Subsequently, we apply another \(\pi\)-pulse with frequency \(\omega_{I-2,I-3}\) to create the state \(\ket{I,I}+\ket{I,I-3}\) and so on; ultimately, we apply a sequence of \(2I-1\)\(\pi\)-pulses to finally create the cat state \(\ket{I,I}+\ket{I,-I}\).
A sequence of \(6I\) pulses are needed to observe the death and revival of cat states using Givens rotations. First, the above sequence of \(2I\) pulses transforms \(\ket{I,I}\rightarrow\ket{I,I}+\ket{I,-I}\). Then, repeating the same sequence evolves this cat state into the coherent state \(\ket{I,-I}\), opposite to the initial state \(\ket{I,I}\), signifying death of the cat state. In contrast to one-axis twisting, continuing the sequence will not lead to a revival. To achieve a revival, a different set of \(2I\) pulses--corresponding to descending the spin ladder--transforms this coherent state back into a cat state.
Realizing cat states using one-axis twisting and MT pulses beats the dephasing time of the nuclear spin on state-of-the-art devices and is significantly more efficient than Givens rotations. Silicon-based chips with an implanted spin-7/2 \({}^{123}\)Sb donor atom have a nuclear spin coherence time \(T_{2}^{*}\approx 100\) ms [32]. By exploiting one-axis twisting, we achieve the quantum speed limit to create cat states and for an experimentally observed quadrupole strength \(\sim 40\) kHz[32], our protocol is expected to have a time period \(\sim 0.12\) ms, two orders of magnitude faster than Givens rotations that require a sequence of \(2I=7\) Rabi pulses of frequency \(\sim 500\) kHz [32], taking \(\sim 7\) ms to make a cat state, and total \(\sim\)21 ms to verify its coherence. Even for a modest perturbation strength inferred from experiments, our approach can yield highly coherent cat states in \(\sim 10\) ms--within a fraction of the dephasing time. Faster Rabi frequencies are easily achieved for nuclear magnetic resonance (NMR) methods [51], that could reduce the time needed for MT pulses and lead to faster generation and detection of nuclear spin cat states.
In summary, we devised a powerful quantum control
Figure 4: a) Generalized Ramsey-like protocol for creating a cat state (Eq. (4)) comprises of a global \(\nicefrac{{\pi}}{{2}}\) rotation with a duration of \(t_{\nicefrac{{\pi}}{{2}}}\), followed by free evolution for time \(T\), and finally another \(\nicefrac{{\pi}}{{2}}\) global rotation. (Inset) The multi-tone (MT) pulse that generates global rotation is a superposition of \(2I\) components, corresponding to \(2I\) nearest neighbour transitions. b) Husimi Q-function plots showing the evolution of the nuclear spin under this control scheme. c) Varying \(T\) to observe the death and revival of a cat state, for a \(I=\nicefrac{{7}}{{2}}\) nucleus. When the phase difference between the two pulses is \(\Delta\phi=\nicefrac{{\pi}}{{2}}\) (purple), we observe fast oscillations with a frequency of \(\omega=\gamma B_{0}\). Phase-modulation using \(\Delta\phi=\frac{\pi}{2}+\omega T\) (black) eliminates the fast oscillations, resulting in death and revival solely due to one-axis twisting.
scheme to create Schrodinger cat states that overcome decoherence on a high-spin nucleus, using one-axis twisting and multi-tone pulses. We introduced a phase-modulated generalized Ramsey like protocol for initialization and detection of cat states on a high-spin donor, that could serve as a model for control of nuclear spins in NV centres [52], molecular spins [53; 54], and superconducting circuits [33], and paves the way towards optimal control of a high-spin nucleus using phase-modulated pulses [55]. Though we take \(I=7_{2}\)\({}^{123}\)Sb as an example, our scheme can produce macroscopic cat states with spin up to \({}^{9}\!/2\) using \({}^{209}\)Bi donors, i.e., equivalent to nine entangled qubits. The resulting states could serve as a resource for quantum information processing [8], error correction [56], parameter estimation [47; 57], and tests of quantum foundations [58; 59].
We acknowledge the traditional owners of the land on which this work was undertaken at the University of Calgary: the Treaty 7 First Nations (www.treaty7.org). We thank Dipankar Home for useful discussion on macroscopicity of nuclear spin cat states. P.G. and B.C.S. acknowledge support from NSERC. Work at UNSW was funded by an Australian Research Council Discovery Project (DP210103769). A.V. and X.Y. acknowledge support from the Sydney Quantum Academy.
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2307.15932 | Solution to the Uniformly Fully Inert Subgroups Problem for Abelian
Groups | A famous conjecture attributed to Dardano-Dikranjan-Rinauro-Salce states that
any uniformly fully inert subgroup of a given group is commensurable with a
fully invariant subgroup (see, respectively, [5] and [6]). In this short note,
we completely settle this problem in the affirmative for an arbitrary Abelian
group. | Andrey R. Chekhlov, Peter V. Danchev | 2023-07-29T08:56:54Z | http://arxiv.org/abs/2307.15932v3 | # Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups
###### Abstract
A famous conjecture states that _any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup_ (see [3] and [4]). In this short note, we settle this problem completely in the affirmative for an arbitrary Abelian group.
0
Footnote 0: 2010 AMS Subject Classification: Primary 20K10, Secondary 20K12. Key words and phrases: Abelian groups, (characteristically, fully) inert subgroups, uniformly (characteristically, fully) inert subgroups
## 1 Notations and Notions
Throughout this brief paper, all our groups are _additively_ written and _Abelian_. Our notation and terminology are mainly standard and follow those from [5].
Recall the standard concept that a subgroup \(F\) of an arbitrary group \(G\) is said to be _fully invariant_ if \(\phi(F)\subseteq F\) for any endomorphism \(\phi\) of \(G\), while if \(\phi\) is an invertible endomorphism (= an automorphism), then \(F\) is said to be a _characteristic_ subgroup. It is obvious that fully invariant subgroups are always characteristic, whereas the converse implication fails in general.
On the same vein, imitating [4] and [2], respectively, a subgroup \(S\) of a group \(G\) is called _fully (resp., characteristically) inert_, provided \((\phi(S)+S)/S\) is finite for all endomorphisms, respectively automorphisms, \(\phi\) of \(G\). The first of these two concepts is known to be refined in [3] and [4] to the so-called _uniformly fully inert_ subgroups by requiring the existence of a fixed positive integer \(m\) such that the cardinality of the quotient-group \((\phi(S)+S)/S\) is bounded by \(m\) (i.e., it has at most \(m\) elements, for each endomorphism \(\phi\) of \(G\)), while the second concept mentioned above was refined in [3] by defining a subgroup \(H\) of a group \(G\) to be _uniformly characteristically inert_, provided there is a positive integer
such that, for every automorphism \(\varphi\) of \(G\), the cardinality of the factor-group \((\varphi(H)+H)/H\) is bounded by \(m\) (that is, it has no more than \(k\) elements for each automorphism \(\varphi\) of \(G\)).
Remember also that two subgroups \(B\) and \(C\) of a group \(G\) are called _commensurable_ and write for short that \(B\sim C\) or, equivalently, that \(C\sim B\) since this is a symmetric relation, provided that both quotients \((B+C)/B\) and \((B+C)/C\) are finite. It is worthwhile noticing that in many special cases a fully inert subgroup is commensurable with a fully invariant subgroups as well as a characteristically inert subgroup is commensurable with a characteristic subgroup. However, it is not so hard to construct a fully inert subgroup that is _not_ commensurable with a fully invariant subgroup as well as a characteristically inert subgroup that is _not_ commensurable with a characteristic subgroup. Nevertheless, it was shown in [3, Corollary 1.9] the important fact that any uniformly characteristically inert subgroup is commensurable a characteristic subgroup and, moreover, in a way of similarity it was conjectured in [3, Conjecture 5.2] that _every uniformly fully inert subgroup of a given (not necessarily Abelian) group is commensurable with a fully invariant subgroup_ (see [4, Conjecture 1.6] too).
Our objective in the present short article is to give a complete positive solution of this difficult question in the next section.
## 2 The Solution
Our key tool, necessary to resolve the listed above conjecture, is the following straightforward but very useful statement. Note that the results from [1] and [2] used in it below are formulated only for \(p\)-groups, for some fixed prime \(p\), but actually they remain valid for arbitrary groups.
**Proposition 2.1**: _Let \(G\) be a group. Then \(G\) possesses the property that any its uniformly fully inert subgroup is commensurable with a fully invariant subgroup of \(G\) if, and only if, the square-group \(G\oplus G\) possesses the property that any its uniformly characteristically inert subgroup is commensurable with a characteristic subgroup of \(G\oplus G\)._
Proof. "\(\Rightarrow\)". Suppose \(K=G_{1}\oplus G_{2}\), where \(G_{1}\cong G\cong G_{2}\). An application of [5] (see also [2]) tells us that each endomorphism of \(K\) is a sum of three automorphisms of \(K\), so it then follows from [2, Proposition 4.2] that each uniformly characteristically inert subgroup \(C\) of \(K\) is uniformly fully inert in \(K\) (notice the important fact that, in our situation, every endomorphism of \(K\) is the sum of only three automorphisms of \(K\)). So, one can write that \(C\sim C_{1}\oplus C_{2}\), where \(C_{1}=C\cap G_{1}\), \(C_{2}=C\cap G_{2}\), and \(C_{1}\), \(C_{2}\) are uniformly fully inert in \(G_{1}\), \(G_{2}\), respectively. Furthermore, if \(\varphi:G_{1}\to G_{2}\) is an isomorphisms, then \(\varphi(C_{1})+C_{2}\sim C_{2}\) in accordance with [1, Lemma 2.1]. By assumption, the subgroup \(C_{1}\) is commensurable with a fully invariant subgroup, say \(H_{1}\). Set \(H_{2}=\varphi(H_{1})\). Since \(G_{1}\cong G_{2}\), it is easily seen that \(H=H_{1}\oplus H_{2}\) is fully invariant in \(K\) and, consequently, by what we have shown above it follows at once that \(H\sim C\), as needed.
"\(\Leftarrow\)". Suppose \(H\) is an uniformly fully inert subgroup in \(G\). Then, same as [2, Theorem 4.6], we derive that \(H\oplus H\) is an uniformly fully inert subgroup in \(G\oplus G\), whence \(H\oplus H\sim C\) for some characteristic subgroup \(C\) in \(G\oplus G\). But,
by what we noted above that each endomorphism in a square-group is a sum of finitely many automorphisms, it is readily observed that \(C\) is fully invariant in \(G\oplus G\) and also is of the form \(C=F\oplus F\), where \(F\) is fully invariant in \(G\) - indeed, as \(C\) is also fully invariant in \(G\oplus G\), we then can write \(C=(C\cap G)\oplus(C\cap G)\) and set \(F=C\cap G\). Thus, we infer that \(H\sim F\), as required.
We now come to our basic assertion that answers in the affirmative the aforementioned conjecture from [3, Conjecture 5.2] and [4, Conjecture 1.6], respectively. It is worth noticing that some recent work on that conjecture related to \(p\)-groups was done in [6] as well.
**Theorem 2.2**: _Any uniformly fully inert subgroup \(H\) of a group \(G\) is commensurable with some fully invariant subgroup of \(G\)._
Proof. We know that \(H\oplus H\) is uniformly fully inert in \(G\oplus G\) and so, in view of [3, Corollary 1.9], \(H\oplus H\) is commensurable with a characteristic subgroup \(K\) of \(G\oplus G\). But \(K\) is fully invariant in \(G\oplus G\), and hence \(K=(K\cap G)\oplus(K\cap G)\), where \(K\cap G\) is fully invariant in \(G\). However, \(H\) is commensurable with \(K\cap G\), as asked for.
**Acknowledgement.** The authors would like to thank Prof. Patrick W. Keef from Whitman College, Walla Walla, WA, for the professional checking of the truthfulness of the main result.
**Funding:** The scientific work of Andrey R. Chekhlov was supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2023-943). The scientific work of Peter V. Danchev was supported in part by the Bulgarian National Science Fund under Grant KP-06 No 32/1 of December 07, 2019, as well as by the Junta de Andalucia under Grant FQM 264, and by the BIDEB 2221 of TUBITAK.
|
2303.16714 | Cosmological models in $f(R,T)$-$Λ(φ)$ gravity | The Universe is currently in a phase of accelerated expansion, a fact that
was experimentally proven in the late 1990s. Cosmological models involving
scalar fields allow the description of this accelerated expansion regime in the
Cosmos and present themselves as a promising alternative in the study of the
inflationary eras, especially the actual one which is driven by the dark
energy. In this work we use the $f(R, T) - \Lambda(\phi)$ gravity to find
different cosmological scenarios for our Universe. We also introduce a new path
to derive analytic cosmological models which may have a non-trivial mapping
between $f$ and $T$. We show that the analytic cosmological models obtained
with this approach are compatible with a good description of the radiation era.
In addition, we investigated the inflationary scenario and obtained a good
agreement for the scalar spectral index $n_s$. Concerning the tensor-to-scalar
ratio $r$, we found promising scenarios compatible with current CMB data. | Joao R. L. Santos, S. Santos da Costa, Romario S. Santos | 2023-03-29T14:15:01Z | http://arxiv.org/abs/2303.16714v2 | # Cosmological models in \(f(R,T)-\Lambda(\phi)\) gravity
###### Abstract
The Universe is currently in a phase of accelerated expansion, a fact that was experimentally proven in the late 1990s. Cosmological models involving scalar fields allow the description of this accelerated expansion regime in the Cosmos and present themselves as a promising alternative in the study of the inflationary eras, specially the actual one which is driven by the dark energy. In this work we use the \(f(R,T)-\Lambda(\phi)\) gravity to find complete cosmological scenarios for our Universe. We show that the analytic cosmological parameters are compatible with a good description of several eras that the Universe passes through. We also introduce a new path to find analytic cosmological models which may have a non-trivial mapping between \(f\) and \(T\).
**Keywords**: Dark energy, accelerated expansion, cosmological parameters, scalar fields, \(f(R,T)\), \(\Lambda(\phi)\).
## I Introduction
The dark sector of our Universe is one of the main open problems in the actual science. The most recent data delivered by Planck Collaboration unveil that 69% of the content of our Universe is dark energy, which is responsible for the current accelerated expansion era [1]. This data set also established that the total amount of dark matter present in our Universe is 27% of its content. The standard model to describe the actual data and also the present evolution of our Universe is the \(\Lambda\)CDM model. However, such a model presents two fundamental problems: the huge discrepancy in the determination of the cosmological constant an the coincidence problem [2; 3]. In order to understand the nature of the dark sector and also to overcome the issues over the \(\Lambda\)CDM model, Harko et al. introduced the so-called \(f(R,T)\) gravity [4]. This new theory of gravity generalizes the so-called \(f(R)\) or Starobinsky gravity [5] by adding a function which can depend on the trace of the energy-momentum tensor. The function \(f(T)\) can be induced by the presence of exotic imperfect fluids or quantum effects due conformal anomaly in the theory [4]. This theory of gravity has been tested in several phenomenological and theoretical approaches so far as one can see in [6; 7; 8; 9].
Another route to amend the issues with the \(\Lambda\)CDM model is through a time-dependent cosmological constant or \(\Lambda(t)\), which is also known as the running vacuum model. This type of model has been studied in the literature since 1980 with the seminal paper published by Coleman et al. [10], and more recently by Polyakov [11], and Ranjantie et al. [12]. A running vacuum scenario can be used to explain the coincidence problem and also enables us to continuously describe two different acceleration regimes, one for early and other for late time values [13; 14]. Recently Santos and Moraes also studied how a complete description of the different eras of our Universe could raise in running vacuum model driven by a scalar field - \(\Lambda(\phi)\)[15]. There the authors also pointed out that the continuity equation is going to be satisfied for asymptotic values of field \(\phi\), since such a field imposes constant values for the Hubble parameter.
Time-dependent scalar fields have been applied to describe inflationary scenarios with great success. Since the beautiful works written by Kinney [16] and Bazeia et al. [17] we have seen different formulations to look for analytic inflationary models which continuously describe the different eras our Universe passes through. Despite this success, the inflationary one scalar field models faced a dilemma after the data delivered by Planck Collaboration in 2013 [19]. As pointed by Ellis et al. [18], inflationary one scalar field models are not able to reproduce the actual values of the spectral index and of the tensor to scalar ratio for Einstein-Hilbert gravity. However, multi fields inflation [18], Lorentz violating terms [20], new theories of gravity [21] or running vacuum models [15], are techniques which allow us to rescue inflation driven by scalar fields.
In this work, we intend to couple the \(f(R,T)\) with \(\Lambda(\phi)\) gravity. Along with our discussions, we are going to show how these two theories of gravity can work together to describe a complete history of the Universe. Moreover, such a theory of gravity can recover several well-known models, such as standard General Relativity, General Relativity plus
a cosmological constant, \(f(R)\) gravity plus a cosmological constant, etc. Besides, we also present a new method to derive analytic cosmological parameters using an inflaton field. Such an approach recovers and generalizes the results found by Moraes and Santos in [9]. This new method also enables us to generate analytic cosmological models for non-trivial forms of \(f(T)\).
For methodological reasons, the ideas presented in this paper are organized in the following nutshell: in section II we show the generalities of our theory of gravity, determining the Friedmann equations and the equation of motion for the inflaton field. After that, in section III we present our new method to derive analytic cosmological models. Examples of cosmological scenarios are derived and carefully analyzed in section IV. Our final remarks and perspectives are pointed in section V.
## II The \(f(R,T)\) - \(\Lambda(\phi)\) gravity
Let us start by introducing the generalities on our theory of gravity. Such a theory combines \(f(R,T)\) gravity with a cosmological constant depending on the inflaton field \(\phi\), i.e.
\[S=\frac{1}{2}\int d^{4}x\sqrt{-g}\left(f(R,T)-\Lambda(\phi)+\mathcal{L}\right)\,, \tag{1}\]
where \(\mathcal{L}\) is standard Lagrangian density for a real scalar field, whose specific form is
\[\mathcal{L}=\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi-V(\phi)\,. \tag{2}\]
Here \(f(R,T)\) gravity can exhibit non-linear geometric terms generalizing the Einstein-Hilbert sector and embeds the non-trivial contributions coming from a function that depends on the trace of the energy-momentum tensor \(T\). Such contributions refer to classical imprints due to quantum anisotropies after the end of primordial inflation. Moreover, the cosmological constant \(\Lambda(\phi)\) enables us to evade the coincidence problem, as pointed out by Coleman and Luccia [10] in their seminal paper. The explicit form of \(\Lambda(\phi)\) is such that
\[\Lambda(\phi)=c_{0}+c_{2}\,H^{\,2}(\phi)+c_{4}\,H^{4}(\phi)\,, \tag{3}\]
where \(H=H(\phi)\) is the Hubble parameter, which is going to depend on the evolution of the inflaton field [15].
In order to preserve the strong evidences which support General Relativity, we are going to work with
\[f(R,T)=f(R)+f(T)=-\frac{R}{2}+f(T)\,. \tag{4}\]
With this procedure we also intend to generalize the analytic models derived by Moraes and Santos in [9]. So, by taking that our action is constant in respect of metric variations we find
\[G_{\mu\nu}=2T_{\mu\nu}-2f\,g_{\mu\nu}-4f^{\,\prime}\partial_{\mu}\phi\partial _{\nu}\phi+2\rho_{\Lambda}g_{\mu\nu}\,, \tag{5}\]
for
\[T_{\mu\nu}=2\frac{\partial\mathcal{L}}{\partial g^{\mu\nu}}+\mathcal{L}g_{\mu \nu}\,;\qquad\rho_{\Lambda}=\frac{\Lambda}{2}\,, \tag{6}\]
where primes stand for derivatives in respect to \(T\). Once we would like to search for cosmological solutions for the field equations, we work with the standard Friedmann-Robertson-Walker metric, which is written as
\[d\,s^{2}=dt^{2}-a^{2}(t)\,d\vec{r}^{\,2}\,, \tag{7}\]
where \(a(t)\) is the scale factor. So, from (5) and working with a time-dependent field \(\phi=\phi(t)\), we can derive the Friedmann equations
\[\frac{3}{2}H^{2}=\left(\frac{1}{2}-2f^{\,\prime}\right)\dot{\phi}^{2}-f+V+\rho _{\Lambda}\,;\qquad H=\frac{\dot{a}}{a}\,, \tag{8}\]
and
\[\dot{H}=-\left(1-2f^{\,\prime}\right)\,\dot{\phi}^{2}\,, \tag{9}\]
where \(H\) is the so-called Hubble parameter.
Now, by taking our action constant in respect to the variation of field \(\phi\) we determine the equation of motion
\[(1-2f^{\,\prime})\,\left(\ddot{\phi}+3H\dot{\phi}\right)-2\dot{f}^{\,\prime}\dot {\phi}+(1-4f^{\,\prime})\,\tilde{V}_{\phi}=0\,, \tag{10}\]
where
\[\tilde{V}=V+\rho_{\Lambda}\,. \tag{11}\]
Here dots mean derivatives in respect to time, primes mean derivatives in respect to \(T\), and \(\tilde{V}_{\phi}=d\,\tilde{V}/d\,\phi\).
## III New method to derive cosmological models
In this section we are going to introduce a new path to find analytic cosmological scenarios in \(f(R,T)-\Lambda(\phi)\) gravity, generalizing the approach presented by Moraes and Santos in [9]. The main advantage of this method is that it can naturally accommodate contributions due to the time-dependent cosmological constant, and also it can be used to build analytic models for different forms of the Hubble parameter \(H\), and of \(f(T)\). Once we would like to search for analytic cosmological models, let us consider the following definition
\[H\equiv h(\phi)\,;\qquad\dot{H}=h_{\phi}\dot{\phi}\,, \tag{12}\]
which can be substituted in Eq. (9), yielding to the constraint
\[h_{\phi}=-\Big{(}1-2f^{\prime}\Big{)}\dot{\phi}\,. \tag{13}\]
Now, by defining that the field \(\phi\) obeys the first-order differential equation
\[\dot{\phi}=-W_{\phi}\,;\qquad W_{\phi}=\frac{dW(\phi)}{d\phi}\,, \tag{14}\]
we determine the following form for (13)
\[h_{\phi}=\Big{(}1-2f^{\prime}\Big{)}W_{\phi}\,. \tag{15}\]
Here \(W(\phi)\) is an arbitrary function of the field \(\phi\) which is going to establish an specific form for the cosmological potential. Thus, by taking Eqs. (12), (14), and (15) into (8) we yield to
\[\tilde{V}=\frac{3}{2}\,H^{\,2}+\frac{W_{\phi}^{2}}{2}-h_{\phi}\,W_{\phi}+f\,. \tag{16}\]
Therefore, once \(T=T(\phi)\), then \(f^{\prime}=\frac{df}{dT}=\frac{df/d\phi}{dT/d\phi}=\frac{f_{\phi}}{T_{\phi}}\), enabling us to write (15) as
\[f_{\phi}=\frac{1}{2}T_{\phi}\left(1-\frac{h_{\phi}}{W_{\phi}}\right)\,. \tag{17}\]
Moreover, if we are dealing with a perfect fluid, the trace of the energy-momentum tensor has the form
\[T=\rho-3p\,, \tag{18}\]
where
\[\rho=\frac{1}{2}\dot{\phi}^{2}+V\,;\qquad p=\frac{1}{2}\dot{\phi}^{2}-V\,, \tag{19}\]
which means that
\[T=-W_{\phi}^{2}+4V\,. \tag{20}\]
Consequently, by taking the derivative of (20) in respect to \(\phi\), we are able to rewrite (17) as
\[f_{\phi}=\left(\frac{h_{\phi}}{W_{\phi}}-1\right)\,(W_{\phi}W_{\phi\phi}-2V_{ \phi}). \tag{21}\]
However, by taking the derivative in respect fo \(\phi\) of (16) we obtain
\[f_{\phi}=W_{\phi\phi}h_{\phi}+W_{\phi}h_{\phi\phi}-W_{\phi}W_{\phi\phi}-3h\,h_{ \phi}+V_{\phi}+\rho_{\Lambda\,\phi}\,. \tag{22}\]
So, from Eqs. (21) and (22) we find the following constraint for the cosmological potential
\[V_{\phi}=\frac{\left(3hh_{\phi}-W_{\phi}h_{\phi\phi}-\rho_{\Lambda\,\phi} \right)W_{\phi}}{2h_{\phi}-W_{\phi}}\,. \tag{23}\]
Thus, by substituting Eq. (23) into (22) and (17) we determine that
\[f_{\phi}=(h_{\phi}-W_{\phi})\Big{(}\frac{W_{\phi\phi}(h_{\phi}-W_{\phi}/2)-3hh _{\phi}+W_{\phi}h_{\phi\phi}+\rho_{\Lambda\,\phi}}{h_{\phi}-W_{\phi}/2}\Big{)}\,,\] (24a) and \[T_{\phi}=-2W_{\phi}W_{\phi\phi}+\frac{4W_{\phi}\left(3hh_{\phi}-W_{\phi}W_{ \phi\phi}-\rho_{\Lambda\,\phi}\right)}{2h_{\phi}-W_{\phi}}\,, \tag{24b}\]
respectively.
Therefore, by choosing specific forms for \(h(\phi)\) and for \(W_{\phi}\) we can find different families of \(f(T)\) which analytic cosmological scenarios. The dependence between \(f\) and \(T\) can be derived analytically or numerically by depicting a parametric plot of Eqs. (24a) and (24b).
## IV Cosmological models
### First Model - \(f_{\phi}=\alpha\,T_{\phi}\)
As a first example we are going to work with the following definitions for \(W_{\phi}\) and \(h(\phi)\) are
\[W_{\phi}=b_{1}\left(\phi^{2}-1\right)\,;\qquad h(\phi)=(1-2\alpha)\;W\,, \tag{25}\]
where the superpotential \(W(\phi)\) is
\[W=b_{1}\left(-\phi+\frac{\phi^{3}}{3}\right)+\frac{b_{2}}{(1-2\alpha)}\,. \tag{26}\]
Here \(b_{1}\), \(b_{2}\), and \(b_{3}\) are free parameters, and this specific form of \(W(\phi)\) corresponds to the well-known \(\phi^{4}\) superpotential in classical field theory. Such superpotential is broadly applied in several subjects of investigations as one can see in [22] and references therein.
Thus, the correspondent first-order differential equation and its respective analytic solution are
\[\dot{\phi}=b_{1}\,\left(1-\phi^{2}\right)\,;\qquad\phi(t)=\tanh\left(b_{1}\,t +b_{3}\right)\,. \tag{27}\]
Now, taking (25), and (26) into Eqs. (23), (24a), and (24b) we yield to
\[V_{\phi}=-\frac{b_{1}\left(\phi^{2}-1\right)}{4\alpha-1}\bigg{(} (1-2\alpha)^{2}\,b_{1}\,\phi\left(\phi^{2}-3\right)+2(2\alpha-1)\,b_{1}\,\phi \tag{28}\] \[-2\,c_{4}\,\left(\frac{1}{3}\,b_{1}\,\phi\left(\phi^{2}-3\right)+ \frac{b_{2}}{1-2\alpha}\right)^{3}-\frac{1}{3}\,b_{1}\,c_{2}\,\phi\left(\phi^ {2}-3\right)+3(1-2\alpha)\,b_{2}+\frac{b_{2}\,c_{2}}{2\alpha-1}\bigg{)}\,,\]
\[f_{\phi}=\frac{1}{4\alpha-1}\bigg{[}4\,\alpha\,b_{1}\left(\phi^ {2}-1\right)\bigg{(}-(1-2\alpha)^{2}\,b_{1}\,\phi\left(\phi^{2}-3\right)+2(1- 2\alpha)\,b_{1}\,\phi+(1-4\alpha)\,b_{1}\,\phi\] \[+2\,c_{4}\,\left(\frac{1}{3}\,b_{1}\,\phi\left(\phi^{2}-3\right)+ \frac{b_{2}}{1-2\alpha}\right)^{3}+\frac{1}{3}\,b_{1}\,c_{2}\,\phi\left(\phi^{2 }-3\right)+3(2\alpha-1)\,b_{2}+\frac{b_{2}\,c_{2}}{1-2\alpha}\bigg{)}\bigg{]}\,, \tag{29}\]
\[T_{\phi}=4\,b_{1}\,\left(\phi^{2}-1\right)\bigg{[}-\frac{1}{4 \alpha-1}\bigg{(}(1-2\alpha)^{2}\,b_{1}\,\phi\,\left(\phi^{2}-3\right)+2(2\alpha- 1)\,b_{1}\,\phi\] \[-2\,c_{4}\,\left(\frac{1}{3}\,b_{1}\,\phi\left(\phi^{2}-3\right)+ \frac{b_{2}}{1-2\alpha}\right)^{3}-\frac{1}{3}b_{1}\,c_{2}\,\phi\left(\phi^{2} -3\right)+3(1-2\alpha)\,b_{2}+\frac{b_{2}\,c_{2}}{2\alpha-1}\bigg{)}-b_{1}\, \phi\bigg{]}\,. \tag{30}\]
From equations (29), and (30) we are able to check that \(f_{\phi}=\alpha\,T_{\phi}\), which means that \(f=\alpha\,T\), corroborating with the models studied by Moraes and Santos in [9]. Then, by integrating \(V_{\phi}\), \(f_{\phi}\) and \(T_{\phi}\) in respect to field \(\phi\) we determine
\[V=-\frac{1}{162(4\alpha-1)}\bigg{[}\,b_{1}\,\phi\bigg{(}\frac{6 \,b_{2}\,\left(\phi^{2}-3\right)\left(-108(\alpha-1)\alpha+2\,b_{1}^{2}c_{4}\, \phi^{2}\left(\phi^{2}-3\right)^{2}+9c_{2}-27\right)}{2\alpha-1}\] \[+b_{1}\,\phi\left(81\left(4\alpha(3\alpha-4)-3\phi^{2}+5\right)+ \phi^{2}\left(27\left((1-2\alpha)^{2}\phi^{2}+6\alpha(5-4\alpha)\right)-b_{1}^ {2}\,c_{4}\,\left(\phi^{2}-3\right)^{4}\right)-9\,c_{2}\,\left(\phi^{2}-3 \right)^{2}\bigg{)}\] \[-\frac{54\,b_{1}\,b_{2}^{2}\,c_{4}\,\phi\left(\phi^{2}-3\right)^ {2}}{(1-2\alpha)^{2}}+\frac{108\,b_{2}^{3}\,c_{4}\,\left(\phi^{2}-3\right)}{( 2\alpha-1)^{3}}\bigg{)}\bigg{]}-\frac{(2\alpha-1)b_{1}^{2}+3\,b_{2}^{2}}{8 \alpha-2}\,, \tag{31}\]
\[f=-\alpha\,b_{1}^{2}+\frac{\alpha}{81(2\alpha-1)^{3}(4\alpha-1)} \bigg{[}b_{1}\,\phi\bigg{(}12(1-2\alpha)^{2}b_{2}\left(\phi^{2}-3\right) \left(108(\alpha-1)\alpha-2b_{1}^{2}c_{4}\phi^{2}\left(\phi^{2}-3\right)^{2}-9 \,c_{2}+27\right)\] \[+(2\alpha-1)^{3}\,b_{1}\,\phi\bigg{(}2\left(\phi^{2}\left(-27(1-2 \alpha)^{2}\phi^{2}+324\alpha(2\alpha-3)+b_{1}^{2}\,c_{4}\,\left(\phi^{2}-3 \right)^{4}\right)+9\,c_{2}\,\left(\phi^{2}-3\right)^{2}\bigg{)} \tag{32}\]
\[T=\frac{1}{81(2\alpha-1)^{3}(4\alpha-1)}\bigg{[}b_{1}\,\phi \bigg{(}12(1-2\alpha)^{2}b_{2}\left(\phi^{2}-3\right)\left(108(\alpha-1) \alpha-2b_{1}^{2}c_{4}\phi^{2}\left(\phi^{2}-3\right)^{2}-9\,c_{2}+27\right)\] \[+(2\alpha-1)^{3}\,b_{1}\,\phi\bigg{(}2\left(\phi^{2}\left(-27(1-2 \alpha)^{2}\phi^{2}+324\alpha(2\alpha-3)+b_{1}^{2}\,c_{4}\,\left(\phi^{2}-3 \right)^{4}\right)+9\,c_{2}\,\left(\phi^{2}-3\right)^{2}\bigg{)} \tag{33}\]
respectively. Now let us move to the determination of the cosmological parameters. We firstly start with the Hubble parameter, whose form is
\[H=b_{2}-\frac{1}{3}(2\alpha-1)\,b_{1}\,\tanh(b_{1}\,t+b_{3})\left(\tanh^{2}(b_ {1}\,t+b_{3})-3\right)\,. \tag{34}\]
To find the last equation we took (12) together with the analytic solution presented in (27). The graphic of such a parameter is presented in Fig. 1, where we observe the presence of two inflationary eras continuously connected, where the Hubble parameter is approximately constant. We also realize that \(H\) is not affected by the presence of the cosmological constant \(\Lambda(\phi)\), since its contributions were embedded in the cosmological potential and in the trace of the energy-momentum tensor, and are going to be analyzed carefully below.
From the analytic form of the potential \(V\), together with the analytic solution \(\phi(t)\) and the definitions of the pressure and density related to the inflaton field, we can derive the Equation of State parameter \(\omega\). The analytic expression of \(\omega\) is going to be suppressed here for the sake of simplicity. However, its features are presented in Fig. 2. There the dashed curve stands for a simple \(f(R,T)\) model, while the blue curve represents the \(f(R,T)\) gravity plus non-trivial contributions from \(\Lambda(\phi)\). We can observe that the time-dependent cosmological constant can be used to make fine-tuning adjusts for \(\omega\), which can be also useful to constraint the theoretical model with other cosmological parameters, such as the tensor-to-scalar ratio and the spectral index, for instance. Moreover, both forms of \(\omega\) describe a continuous transition between two different inflationary eras (\(\omega\approx-1\)), one for remote and the other for big values of time. We also observe that the maximum value for the EoS parameter is \(\omega\approx 1/3\), corroborating with a description of the radiation era. Among these different eras, the EoS parameter presents a scenario of null pressure (\(\omega=0\)), standing for the matter era.
Finally, we take (33) and the inflaton field \(\phi(t)\) to depict the behavior of the trace of the energy-momentum tensor, which is plotted in Fig. 3. There we see that the cosmological constant \(\Lambda(\phi)\) may change the minimum value of \(T\) and also its asymptotic time evolution. It is relevant to point out that \(T\approx 0\) when \(\omega\approx 1/3\), probing the compatibility of our cosmological parameters. In order to see the functional dependence between \(f\) and \(T\), we build the parametric graphic presented in Fig. 4, where we used (32), (33), \(\phi(t)\), and the asymptotic values of the inflaton field. This graphic confirms that \(f=\alpha\,T\), corroborates with the discussions presented in [9].
Figure 3: Trace of the energy-momentum tensor for our first model of \(f(R,T)-\Lambda(t)\) gravity. The blue solid curve represents the case where \(\Lambda=0\), while the red dashed curve was depicted with \(c_{0}=1\), \(c_{2}=-0.05\), and \(c_{4}=-0.005\). In both curves we worked with \(\alpha=-4\), \(b_{1}=0.2\), \(b_{2}=1.31\), and \(b_{3}=-2.5\).
Figure 2: Equation of State parameter for our first model of \(f(R,T)-\Lambda(t)\) gravity. The blue solid curve represents the case where \(\Lambda=0\), while the red dashed curve was depicted with \(c_{0}=1\), \(c_{2}=-0.05\), and \(c_{4}=-0.005\). In both curves we worked with \(\alpha=-4\), \(b_{1}=0.2\), \(b_{2}=1.31\), and \(b_{3}=-2.5\).
### Second Model - \(f_{\phi}=(\alpha-\beta\,b_{1}\,\sin(\phi))\;T_{\phi}\)
As a second example, let us present a non-trivial mapping between \(f\) and \(T\), resulting in a new cosmological model. In order to do so, we work with the following forms for \(W(\phi)\), \(W_{\phi}\), and \(h(\phi)\)
\[W=b_{1}\,\sin(\phi)\,;\qquad W_{\phi}=b_{1}\,\cos(\phi)\,;\qquad h=b_{2}+(1-2\, \alpha)\,W+\beta\,W^{2}\,. \tag{35}\]
This model was also investigated by Moraes and Santos in [9], and its known as the sine-Gordon model which has several applications in different areas of physics, mainly due to its property of integrability, as we can see in [23; 24; 25] and references therein. The first-order differential equation and its analytic solution for this model are given by
\[\dot{\phi}=-b_{1}\,\cos(\phi)\,;\qquad\phi(t)=2\,\tan^{-1}\left(\tanh\left( \frac{1}{2}\left(b_{3}-b_{1}\,t\right)\right)\right)\,. \tag{36}\]
By repeating the procedure adopted in our first model, we substitute (36), and (36) into Eqs. (23), (24a), and (24b) to determine
\[V_{\phi}=\frac{b_{1}\,\cos(\phi)}{1-4\alpha+4\beta\,b_{1}\,\sin( \phi)}\bigg{(}-2\,\beta\,b_{1}^{2}\cos^{2}(\phi)+b_{1}\,\sin(\phi)\,(2\alpha( 6\,\alpha-7) \tag{37}\] \[+b_{1}\,\sin(\phi)\left((11-18\alpha)\beta-2\,b_{1}\,\left(c_{4} -3\beta^{2}\right)\sin(\phi)\right)+6\,\beta\,b_{2}-c_{2}+4\big{)}+(3-6\alpha) \,b_{2}\bigg{)}\,,\]
\[f_{\phi}=-\frac{1}{1-4\alpha+4\beta b_{1}\sin(\phi)}\bigg{[}2b_{1}\cos(\phi)( \beta b_{1}\sin(\phi)-\alpha)\bigg{(}b_{1}^{3}\left(c_{4}-3\beta^{2}\right) \sin(3\phi)+3(6\alpha-5)\beta b_{1}^{2}\cos(2\phi)\] \[+(11-18\alpha)\beta b_{1}^{2}+b_{1}\sin(\phi)\left(8\alpha(3 \alpha-4)-3b_{1}^{2}\left(c_{4}-3\beta^{2}\right)+12\beta b_{2}-2c_{2}+9 \right)+(6-12\alpha)b_{2}\bigg{)}\bigg{]}\,, \tag{38}\]
\[T_{\phi}=\frac{1}{1-4\alpha+4\beta b_{1}\sin(\phi)}\bigg{[}2b_{1}\cos(\phi) \bigg{(}-4\beta b_{1}^{2}\cos^{2}(\phi)+b_{1}\sin(\phi)\bigg{(}8\alpha(3 \alpha-4)\] \[+2b_{1}\sin(\phi)\left((13-18\alpha)\beta-2b_{1}\left(c_{4}-3 \beta^{2}\right)\sin(\phi)\right)+12\beta b_{2}-2c_{2}+9\bigg{)}+6(1-2\alpha) b_{2}\bigg{)}\bigg{]}\,. \tag{39}\]
After inspecting (38) and (39) we verify that \(f_{\phi}=(\alpha-\beta\,b_{1}\,\sin(\phi))\;T_{\phi}\), unveiling a non-trivial mapping between \(f\) and \(T\) for \(\beta\neq 0\). Such mapping generalizes the classes of analytic models for \(f(R,T)\) gravity. So, by integrating these previous equations in respect to \(\phi\) we find
\[V=-\frac{1}{384\beta^{4}}\,\bigg{[}b_{1}\bigg{(}\frac{3}{b_{1}} \log(-4\,\alpha+4\,\,\beta\,b_{1}\,\sin(\phi)+1)\bigg{(}\beta^{2}\left(-16\, \alpha^{2}-32\,\alpha+64\,\beta^{2}\,b_{1}^{2}-48\,\beta\,b_{2}\right. \tag{40}\] \[\left.+8(4\alpha-1)\,c_{2}+9)+(4\,\alpha-1)^{3}c_{4}\bigg{)}+64 \,\beta^{3}b_{1}^{2}\left(c_{4}-3\beta^{2}\right)\sin^{3}(\phi)-24\beta^{2}b_{ 1}\sin^{2}(\phi)\left((23-24\alpha)\beta^{2}-4\alpha c_{4}+c_{4}\right)\] \[+12\beta\sin(\phi)\left((1-4\alpha)^{2}c_{4}-\beta^{2}(4\alpha+48 \,\beta\,b_{2}-8\,c_{2}+9)\right)\bigg{)}\bigg{]}\,,\]
Figure 4: Parametric graph mapping \(f\) and \(T\) for our first model. The blue solid curve represents the case where \(\Lambda=0\), while the red dashed curve was depicted with \(c_{0}=1\), \(c_{2}=-0.05\), and \(c_{4}=-0.005\). In both curves we worked with \(\alpha=-4\), \(b_{1}=0.2\), \(b_{2}=1.31\), and \(b_{3}=-2.5\).
\[f=\frac{-b_{1}}{384\beta^{4}}\bigg{(}\frac{3}{b_{1}}\log(-4\alpha+4 \beta b_{1}\sin(\phi)+1)\left(\beta^{2}\left(-16\alpha^{2}-32\alpha+64\beta^{2}b_ {1}^{2}-48\beta b_{2}+8(4\alpha-1)c_{2}+9\right)\right.\] \[\left.+(4\alpha-1)^{3}c_{4}\right)+4\beta\bigg{(}-6\beta^{3}b_{1} ^{3}\left(c_{4}-3\beta^{2}\right)\cos(4\phi)+\sin(\phi)\left(c_{4}\left(48 \alpha^{2}-24\alpha+8\beta^{2}b_{1}^{2}+3\right)\right.\] \[\left.-3\beta^{2}\left(-8\beta^{2}b_{1}^{2}+4\alpha\left(24\beta^ {2}b_{1}^{2}+48\beta b_{2}+1\right)-48\beta b_{2}-8c_{2}+9\right)\right)+ \beta b_{1}\cos(2\phi)\left(-3\beta^{2}\left(96\alpha^{2}-104\alpha\right.\right.\] \[\left.\left.+24\beta^{2}b_{1}^{2}+48\beta b_{2}-8c_{2}+9\right) +3c_{4}\left(-4\alpha+8\beta^{2}b_{1}^{2}+1\right)-8\beta b_{1}\sin(\phi) \left(9(3-4\alpha)\beta^{2}+c_{4}\right)\right)\bigg{)}\bigg{]}\,, \tag{41}\]
\[T=\frac{1}{96\beta^{4}}\bigg{[}-96\beta^{4}b_{1}^{2}\cos^{2}( \phi)-3\log(-4\alpha+4\beta b_{1}\sin(\phi)+1)\left(\beta^{2}\left(64\beta^{2 }b_{1}^{2}-48\beta b_{2}\right.\right. \tag{42}\] \[\left.\left.+(4\alpha-1)(-4\alpha+8c_{2}-9))+(4\alpha-1)^{3}c_{4} \right)+4\beta b_{1}\sin(\phi)\bigg{(}2\beta b_{1}\sin(\phi)\left(8\beta b_{1 }\left(3\beta^{2}-c_{4}\right)\sin(\phi)\right.\right.\] \[\left.\left.+3\left((23-24\alpha)\beta^{2}-4\alpha c_{4}+c_{4} \right)\right)+3\beta^{2}(4\alpha+48\beta b_{2}-8c_{2}+9)-3(1-4\alpha)^{2}c_{ 4}\bigg{)}\bigg{]}\,.\]
All the previous ingredients yield us to compute our cosmological parameters. Firstly, using \(h(\phi)\) and the inflaton field \(\phi(t)\) we find the Hubble parameter
\[H=b_{1}\sin\left(2\tan^{-1}\left(\tanh\left(\frac{1}{2}(b_{3}-b_{1}t)\right) \right)\right)\left(-2\alpha+\beta b_{1}\sin\left(2\tan^{-1}\left(\tanh\left( \frac{1}{2}(b_{3}-b_{1}t)\right)\right)\right)+1\right)+b_{2}\,, \tag{43}\]
which is presented in detail in Fig. 5. One more time we derive a cosmological model describing a Universe which continuously passes through two different inflationary eras, where \(H\approx cte\). These results hold for both \(f(R,T)\) and \(f(R,T)-\Lambda(\phi)\), and also it recovers the features found by Moraes and Santos in [9] if we take the limit \(\beta\to 0\).
Using \(V\) from (40) together with \(\phi(t)\) and the definitions for \(\rho\) and \(p\) we were able to derive an analytic form for the EoS parameter \(\omega\), which is suppressed for the sake of simplicity. However, its features are illustrated in Fig. 6, where we can see an analogous behavior with our first model. This EoS parameter presents two inflationary eras, one for remote and another for future values of time, continuously connected with the radiation and the matter era, allowing the Universe to form complex objects such as hydrogen clouds and galaxies.
The time evolution of the trace of the energy-momentum tensor can be visualized in Fig. 7, where we see how \(\Lambda(\phi)\) can fine-tune the minimum and the asymptotic value of \(T\). We also observe that \(T\approx 0\) when \(\omega\approx 1/3\), confirming that our cosmological parameters are describing a solid history of the evolution of the Universe.
From Eqs. (41) - (42) and using the analytic solution \(\phi(t)\) which its asymptotic behavior we were able to derive the parametric plot shown in Fig. 8. There we are able to see the expected non-trivial functional mapping between \(T\) and \(f\). In order to find a functional analytic form connecting \(f\) and \(T\) we mapped its parametric curve with the function
\[T=\alpha_{4}-e^{-\alpha_{1}(f-\alpha_{2})}-\alpha_{3}(f-\alpha_{2})\,, \tag{44}\]
Figure 5: Time evolution of \(H\) for our second model of \(f(R,T)-\Lambda(t)\) gravity. The graphic was depicted for \(\alpha=0.01\), \(\beta=0.0412\), \(b_{1}=1\), \(b_{2}=1\), and \(b_{3}=4\).
characterising an asymmetric parabola, whose inverse function is
\[f=\frac{\alpha_{1}\alpha_{2}\alpha_{3}+\alpha_{1}\alpha_{4}+\alpha_{3}\mathcal{W} \left(-\frac{\alpha_{1}\epsilon^{\frac{\alpha_{1}\,T}{\alpha_{3}}-\frac{\alpha_ {1}\alpha_{4}}{\alpha_{3}}}}{\alpha_{3}}\right)-\alpha_{1}\,T}{\alpha_{1}\, \alpha_{3}}\,, \tag{45}\]
with \(\alpha_{1}=0.0648\), \(\alpha_{2}=1.1352\), \(\alpha_{3}=0.0435\), \(\alpha_{4}=2.5464\), and \(\mathcal{W}\) is the so-called Lambert or product logarithm function. This mapping is also presented in Fig. 8, where we show the optimal adjustment of the analytic function (44) with our parametric curves involving \(f\) and \(T\) (green dashed curve).
## V Final remarks
In this work, we show a new proposal for a theory of gravity, the \(f(R,T)-\Lambda(\phi)\). We were able to derive its modified Friedmann equations and also the equation of motion for the inflaton field. By imposing a constraint that the inflaton field satisfies a first-order differential equation, we were able to derive analytic cosmological models. The methodology used to derive such models generalizes the results obtained by Moares and Santos [9]. Moreover, such a new methodology also enables us to find analytic cosmological models for non-trivial mappings between \(f\) and \(T\).
Both examples covered in this paper proved to be well-behaved in the radiation era, where \(\omega=1/3\) and \(T=0\). Another interesting aspect about our models is that they present a continuous transition between different eras of our Universe. Besides, the extra terms coming from the running vacuum contributions can be used to fine-tune other cosmological parameters. These good behaviors of the theory open a new path to test the viability of these models of
Figure 6: Equation of State parameter for our second model of \(f(R,T)-\Lambda(t)\) gravity. The blue solid curve represents the case where \(\Lambda=0\), while the red dashed curve was depicted with \(c_{0}=1\), \(c_{2}=-0.0003\), and \(c_{4}=-0.00003\). In both curves we worked with \(\alpha=0.01\), \(\beta=0.0412\), \(b_{1}=1\), \(b_{2}=1\), and \(b_{3}=4\).
gravity under cosmological data, such as those from the Planck Collaboration. Moreover, the \(f(R,T)-\Lambda(\phi)\) gravity stands for a proposal that rescues the inflationary eras driven by one single scalar field, generalizing the results presented in the beautiful paper of Ellis et al. [18]. It also can be used to map non-trivial behavior of cosmological parameters in low redshift regimes which may appear in data coming from dedicated radio telescopes such as those from SKA [26] and from BINGO [27; 28].
The methodology here presented can be extended to several other modified theories of gravity, such as in \(f(Q)\)[29; 30], and \(f(Q,T)\)[31; 32]. Besides we can also add a term related to the presence of dust to verify its influence over the time evolution of our parameters [33]. These discussions can be applied in braneworld scenarios as well [34]. We hope to be able to report on some of these topics in the near future.
###### Acknowledgements.
JRLS would like to thank CNPq (Grant nos. 420479/2018-0, and 309494/2021-4), and PRONEX/CNPq/FAPESQ-PB (Grant nos. 165/2018, and 0015/2019) for financial support. RSS would like to thank CAPES for financial support. The authors are grateful to Jean Paulo Spinelli da Silva for reading the manuscript, offering suggestions, and improvements.
|
2303.01268 | Analyzing Effects of Fake Training Data on the Performance of Deep
Learning Systems | Deep learning models frequently suffer from various problems such as class
imbalance and lack of robustness to distribution shift. It is often difficult
to find data suitable for training beyond the available benchmarks. This is
especially the case for computer vision models. However, with the advent of
Generative Adversarial Networks (GANs), it is now possible to generate
high-quality synthetic data. This synthetic data can be used to alleviate some
of the challenges faced by deep learning models. In this work we present a
detailed analysis of the effect of training computer vision models using
different proportions of synthetic data along with real (organic) data. We
analyze the effect that various quantities of synthetic data, when mixed with
original data, can have on a model's robustness to out-of-distribution data and
the general quality of predictions. | Pratinav Seth, Akshat Bhandari, Kumud Lakara | 2023-03-02T13:53:22Z | http://arxiv.org/abs/2303.01268v1 | # Analyzing Effects of Fake Training Data on the Performance of Deep Learning Systems
###### Abstract
Deep learning models frequently suffer from various problems such as class imbalance and lack of robustness to distribution shift. It is often difficult to find data suitable for training beyond the available benchmarks. This is especially the case for computer vision models. However, with the advent of Generative Adversarial Networks (GANs), it is now possible to generate high-quality synthetic data. This synthetic data can be used to alleviate some of the challenges faced by deep learning models. In this work we present a detailed analysis of the effect of training computer vision models using different proportions of synthetic data along with real (organic) data. We analyze the effect that various quantities of synthetic data, when mixed with original data, can have on a model's robustness to out-of-distribution data and the general quality of predictions.
## 1 Introduction
Deep Neural Networks (DNNs) require large amounts of data to train efficiently and generalize. Computer vision models are always hungry for more data which they use in order to better understand the data distribution and become more accurate. Though data is produced in great volume, it is seldom usable to computer vision models. Large amounts of data in existence may need to be labeled and/or annotated in order to become usable by a computer vision model. All of this is cost intensive especially when the labeling and annotation need to be done by a human in the loop.
More recently, novel generative deep-learning techniques such as Generative Adversarial Networks (GANs) have demonstrated the ability to create high quality synthetic datasets [6]. With the advent of high quality synthetic datasets, various data-related challenges in deep learning and particularly computer vision can be combated. Challenges such as class imbalance and robustness to distribution shift can potentially be resolved through the use of synthetically generated datasets.
In this paper we present a study of the effect of synthetic data on a model's performance. By making use of Conditional Generative Adversarial Network (cGAN) [14], we create corresponding synthetic datasets for MNIST [9], Fahion-MNIST [23] and CIFAR10 [8]. We attempt to evaluate the model performance when trained on different compositions of the core dataset based on the ratio of synthetic and real images. Our results show adding only a few synthetic images to the training dataset not only helps to alleviate the problem of class imbalance but also improves model robustness to distribution shift.
## 2 Background
Computer vision tasks are made more difficult by the requirement for large amounts of annotated data. Using low-cost synthetically generated training images is one way to address this problem. This method, however, raises an important question: how should synthetic and real data be combined to optimise model training? With the growing popularity of GANs, numerous datasets have been introduced in recent years. Some of which include Flying Chairs[2], FlyingThings3D[11], UnrealStereo[25], SceneNet[5], SceneNet RGB-D [12], SYNTHIA [19], GTAV[18], Sim4CV[16], Virtual KITTI[3], SUNCG dataset by Princeton [21], MINOS [20], House3d [22] and MPI Sintel [1]. The use of synthetically generated data for training is one promising approach that addresses the issue of lack of data. However, models trained on synthetic images, frequently suffer from poor generalisation in the real world. Limitations in rendering quality, such as unrealistic texture, appearance, illumination, and scene layout, are common causes of domain gap between the synthetic and real images. As a result, networks are susceptible to overfitting to the synthetic domain, resulting in learned representations that differ from those obtained on real images.
Domain generalisation and adaptation techniques have been proposed to address these issues. [10; 17; 24] Domain adaptation, which aims to tailor the model to a particular target domain by jointly learning from the source synthetic data and the (often unlabeled) data of the target real domain, is frequently used to mitigate the domain mismatch between simulation and the real world. Domain generalisation, on the other hand, considers zero-shot generalisation without seeing the target data of real images, and is thus more difficult.
To the best of our knowledge no work so far has tried to combine the synthetic and real images as a single dataset for classification task and studied the effect that their relative ratio can have on model performance and robustness. In this work we investigate the effect of complementing a real dataset with synthetic data for classification tasks. We use conditional GAN (cGAN) to generate synthetic data and present the results for model accuracy when trained on various combinations of synthetic and original data.
## 3 Method
We design our experiments in order to evaluate model performance and ability to handle datasets of various domains and complexities to solve the multi-class image classification problem. Models were trained and tested on datasets of constant size throughout the experiments. However, the proportion
Figure 1: Visual Comparison between the Real and Synthetic Images of the datasets used for Experiments.
of original data and synthetically generated data was varied in different ratios. The accuracy of these trained models was calculated on the original organic test split and the synthetically generated test split, respectively.
### Data Generation
For our purpose of experimentation, we require real and synthetic data. We focus on the following datasets in this study:
The MNIST Dataset[9] dataset consists of handwritten digits having a training set of 60000 examples and a test set of 10000 examples of grayscale images of size 28x28. The digits have been size-normalized and centered in a fixed-size image.
The Fashion-MNIST Dataset[23]comprises of 28x28 grayscale images similar to MNIST. It consists of 70,000 fashion products from 10 categories, with 7,000 images per category. The training set has 60,000 images, and the test set has 10,000 images.
The CIFAR-10 Dataset[8] is a subset of the Tiny Images dataset and consists of 60000 32x32 color images. The images are labeled with one of 10 mutually exclusive classes. There are 6000 images per class with 5000 training and 1000 testing images per class.
OOD DataTo test the robustness of the models, we use the MNIST-C dataset[15].MNIST-C dataset has corruptions applied to the MNIST test set for benchmarking out-of-distribution robustness in computer vision[15]. These corruptions significantly degrade the performance of state-of-the-art computer vision models while preserving the semantic content of the test images[15]. We experimented with the Shot noise variant[15] for our experiments as it is one of the most common random corruptions that may occur during the imaging process.
We generate synthetic data with the help of conditional GAN (cGAN)[13]. cGANs are a type of Generative Adversarial Networks[4] (GANs). Vanilla GANs [4] do not provide control over the class or modes of the data being generated[13]. For our purpose here we require label specific images to be generated. Since we propose using synthetic data in place or in tandem with original data we need to keep the synthetic dataset structure aligned with that of the original data. cGANs provide a feasible solution by introducing a conditional version of GANs, which can be constructed by simply feeding the label we wish to condition into the generator and discriminator[13].
For the purpose of this study we have used StyleGAN2-ada for our experiment in addition to cGAN[13]. We trained cGAN for synthetic data generation to analyze the MNIST and Fashion MNIST datasets. For optimization of both Generator and the Discriminator, we use the Adam Optimizer with a learning rate of \(1\times 10^{-4}\) for the Discriminator and \(2\times 10^{-5}\) for the Generator.
Figure 2: (a) Test accuracy for different dataset combinations of original and synthetic data for the MNIST Dataset. The X-axis is labeled as training-test set where O: Original, S: Synthetic and OS: Mixed (b) Test accuracy for different dataset combinations of original and synthetic data for the Fashion-MNIST Dataset. The X-axis is labeled as training-test set where O: Original, S: Synthetic and OS: Mixed
We use Binary Cross Entropy to measure loss and kept a batch size of 100 images during training. We sample noise from Normal Distribution for noise as input for Generator. For CIFAR10 we use StyleGAN2 with adaptive discriminator augmentation(ADA)[7] to generate synthetic data. We used the official pre-trained class conditional model[7] trained on CIFAR-10 to produce 32x32 images for our experiments. We monitor image quality based on the FID score.
The different combinations of training and testing data generated and used for the experiments are delineated in table 2.
### Training Model
For our experiments, we use Deep CNN-based model architectures for the task of image classification. For MNIST[9] and F-MNIST[23], we have used a simple CNN model, while for CIFAR10[8] we used a slightly more complex CNN architecture comprising of 3 convolutional blocks followed by a dense layer and the output layer.
For each train-test set combination as described in table 2 we replicate the training procedure in its entirety. Once we perform the experiments with one train-test set combination we conduct the next set of experiments by initializing the model again using the same hyperparameters.
For MNIST and F-MNIST the models are trained for 10 epochs and for CIFAR10 the models are trained for 50 epochs.
## 4 Analysis
Based on the different combinations presented in table 2 we analyse our results on each dataset.
### MNIST and Fasion-MNIST
We observe that when the model was trained on just real data, it could not perform well on the synthesized test set. The model failed miserably on a purely synthetic test set. However the model trained solely on synthetic data and tested on the original data still showed relatively better performance.
We notice that on addition of just a small proportion of synthetic images to the original dataset, the model accuracy started improving. This however can be merely due to the fact that the model was now aware of the synthetic data distribution as well. However the model still shows an impressive revival of accuracy on adding just a few instances of the synthetic data to the training set. Table 1 shows the results for the various datasets.
We notice a significant increase in accuracy from the Original-Synthetic(1:1) to the Original-Synthetic(5:1) training-set category as shown in Table 2. We further tested the model performance on out of distribution data. For this we test a model trained on different combinations of the datasets on corrupted MNIST. Table 1 shows that model accuracy increases when it is trained on a hybrid dataset containing both synthetic and original images.
Hence models trained on synthetic datasets performed decently on a real dataset. We also observe that when a small amount of synthetic data is included in the training set of the model, it attains similar accuracy levels in both real and synthetic testing sets for both datasets (MNIST and Fashion-MNIST), making it robust to cases where it encounters out-of-distribution images.This is much more prevalent while testing the MNIST model on MNIST-C test set.
\begin{table}
\begin{tabular}{c c c c} \hline \hline Dataset & Model & Learning Rate & Epoch \\ \hline MNIST & Simple CNN & \(1\times 10^{-4}\) & 10 \\ Fashion-MNIST & Simple CNN & \(1\times 10^{-4}\) & 10 \\ CIFAR10 & Deep CNN & \(1\times 10^{-3}\) & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Different Models and hyperparameters used for different types of datasets for our analysis.
### Cifar10
We noticed that the model trained only on synthetic dataset gives comparable accuracy to the model trained only on original dataset when tested on the original testing split.
In addition to this a subtle yet similar pattern is seen for the cases when the model is trained on just original data and tested on synthetic data and when the model is trained on only synthetic data and tested on original data. The latter shows better performance.
Moreover, for the final combination of O:S (5:1) still shows a favourable trend as with MNIST and F-MNIST. Mixing a small amount of synthetic data improves the model and generally makes them more robust. We use the same number of total training images to train each variant of the model.
## 5 Results
We observe a general trend across models and datasets. Table 2 shows the effect that different kinds of synthetic datasets can have on a model's accuracy. We observe that a model trained on a dataset of only original images does not generalize well to out of distribution (and in this case synthetic images) whereas a model trained only on synthetic images does to some extent possess the ability to make correct predictions on original images.
We also make an important observation that adding only a small number of synthetic images to the original dataset (say in the 1:5 ratio) yields a dataset that is significantly more robust to out of distribution data. The test accuracy of a model trained with such datatsets remains very close to the original/baseline model however the accuracy of such a model on out of distribution images improves by a massive percentage, of the order of 700% for certain cases. This trend holds for all datasets.
\begin{table}
\begin{tabular}{l l l l} \hline Dataset & Train Data & Test Data & Test Accuracy \\ \hline \multirow{8}{*}{F-MNIST} & Original & Original & 0.882 \\ & Synthetic & Synthetic & 0.9771 \\ & Synthetic & Original & 0.5839 \\ & Original: Synthetic (1:1) & Original & 0.8699 \\ & Original: Synthetic (2:1) & Original & 0.8731 \\ & Original: Synthetic (1:2) & Original & 0.8658 \\ & Original: Synthetic (5:1) & Original & 0.8722 \\ & Original: Synthetic (5:1) & Synthetic & 0.8715 \\ \hline \multirow{8}{*}{MNIST} & Original & Original & 0.9484 \\ & Synthetic & Synthetic & 0.9926 \\ & Synthetic & Original & 0.8984 \\ & Original & Synthetic & 0.2365 \\ & Original: Synthetic (1:1) & Original & 0.9409 \\ & Original: Synthetic (2:1) & Original & 0.9427 \\ & Original: Synthetic (1:2) & Original & 0.9204 \\ & Original: Synthetic (5:1) & Original & 0.9322 \\ & Original: Synthetic (5:1) & Synthetic & 0.9528 \\ \hline \multirow{8}{*}{CIFAR10} & Original & Original & 0.6771 \\ & Synthetic & Synthetic & 0.7919 \\ \cline{1-1} & Synthetic & Original & 0.6523 \\ \cline{1-1} & Original & Synthetic & 0.6336 \\ \cline{1-1} & Original: Synthetic (1:1) & Original & 0.6728 \\ \cline{1-1} & Original: Synthetic (2:1) & Original & 0.684 \\ \cline{1-1} & Original: Synthetic (1:2) & Original & 0.6839 \\ \cline{1-1} & Original: Synthetic (5:1) & Original & 0.6843 \\ \cline{1-1} & Original: Synthetic (5:1) & Synthetic & 0.7363 \\ \hline \end{tabular}
\end{table}
Table 2: Test Accuracy results for Fashion-MNIST (F-MNIST),MNIST and CIFAR10 using synthetic datasets created by mixing different proportions of real and generated images.
Synthetic images are generated from random noise and do not possess a definite data distribution hence adding even a small number of such images to the original dataset serves to make it more robust in its prediction. We further verify our experiments on real world out of distribution data using corrupt MNIST. Hence we infer that synthetic data when added to the pure dataset not only serves to alleviate the problem of data shortage but also has secondary effects on the robustness of the model which it helps to improve.
We also observe that though our results hold on simple dataset such as Fashion-MNIST and MNIST, they are not as striking on more complex datasets such as CIFAR10, where classes do not possess many overlapping features and are drastically different from one another.
## 6 Conclusion and Future Work
In this paper, we performed a detailed analysis of using an amalgam of synthetic and original data for deep network training. This analysis led to several findings, of which we summarize the most important ones here: (1) A simple model trained only on original data is not as robust to OOD data as compared to a model trained with some synthetic images mixed in (2) while a combination of synthetic and real images benefits models trained for simpler datasets, it might not be so for more sophisticated datasets which possibly contain images with complex overlapping class-wise features.
Since real data is expensive to annotate, the impressive results of synthetic training are valuable. While we focused on datasets of similar complexity, for future analysis, we would like to expand it to different datasets of varying complexity while experimenting with more models and studying the effects of transfer learning. One area of focus will be evaluating various methods with respect to feature overlapping in the dataset.
We hope that our work provides insights on how synthetic images can impact a deep network, pointing the way for future research into developing cost-effective frameworks for training neural networks without needing large amounts of real data.
|
2310.01350 | A peridynamic-informed deep learning model for brittle damage prediction | In this study, a novel approach that combines the principles of peridynamic
(PD) theory with PINN is presented to predict quasi-static damage and crack
propagation in brittle materials. To achieve high prediction accuracy and
convergence rate, the linearized PD governing equation is enforced in the
PINN's residual-based loss function. The proposed PD-INN is able to learn and
capture intricate displacement patterns associated with different geometrical
parameters, such as pre-crack position and length. Several enhancements like
cyclical annealing schedule and deformation gradient aware optimization
technique are proposed to ensure the model would not get stuck in its trivial
solution. The model's performance assessment is conducted by monitoring the
behavior of loss function throughout the training process. The PD-INN
predictions are also validated through several benchmark cases with the results
obtained from high-fidelity techniques such as PD direct numerical method and
Extended-Finite Element Method. Our results show the ability of the nonlocal
PD-INN to predict damage and crack propagation accurately and efficiently. | Roozbeh Eghbalpoor, Azadeh Sheidaei | 2023-10-02T17:12:20Z | http://arxiv.org/abs/2310.01350v1 | # A peridynamic-informed deep learning model for brittle damage prediction
###### Abstract
In this study, a novel approach that combines the principles of peridynamic (PD) theory with PINN is presented to predict quasi-static damage and crack propagation in brittle materials. To achieve high prediction accuracy and convergence rate, the linearized PD governing equation is enforced in the PINN's residual-based loss function. The proposed PD-INN is able to learn and capture intricate displacement patterns associated with different geometrical parameters, such as pre-crack position and length. Several enhancements like cyclical annealing schedule and deformation gradient aware optimization technique are proposed to ensure the model would not get stuck in its trivial solution. The model's performance assessment is conducted by monitoring the behavior of loss function throughout the training process. The PD-INN predictions are also validated through several benchmark cases with the results obtained from high-fidelity techniques such as PD direct numerical method and Extended-Finite Element Method. Our results show the ability of the nonlocal PD-INN to predict damage and crack propagation accurately and efficiently.
Physics-informed neural networks, linearized peridynamics, damage prediction, residual-based loss function, brittle crack propagation, deep learning.
## 1 Introduction
High-fidelity numerical techniques such as Continuum Damage Mechanics (CDM), Extended Finite Element Method (XFEM), Phase-Field Method (PFM), and Cohesive Zone Model (CZM) [1-6] have long been a mainstay in the study of fracture mechanics, where intricate physical events are precisely studied to understand crack initiation and propagation. Later, meshless methods were introduced to eliminate mesh dependency and re-meshing strategies. Within these methods, Peridynamic (PD) theory has emerged as a prominent alternative to Classical Continuum Mechanics (CCM), specifically tailored to handle material discontinuities such as cracks or heterogeneities across micro- to macro-structures [7-10]. Notably, PD has found successful
applications in various domains, including brittle materials with elastic deformation [11, 12], rate-dependent inelastic deformations [13, 14], as well as heterogeneous and composite materials [15, 16]. It has proven effective in tackling challenges related to fatigue and impact loadings [17, 18], as well as multiphysics analysis [19, 20].
Despite the advantages of these high-fidelity techniques, computational demand arises with increasing structural and material complexity, especially when conducting extensive parametric studies or uncertainty quantification. To address these challenges, surrogate and machine learning (ML) models trained on high-fidelity simulations or experimental datasets, have emerged as effective solutions to maintain a balance between computational accuracy and efficiency [21-24]. Data-driven ML models can extract patterns and relationships directly from substantial datasets; however, their performance hinges on the quality and quantity of training data. Their susceptibility to overfitting the training data can hinder their effectiveness in extrapolating or handling scenarios beyond the training distribution.
Most recently, a new deep learning model known as Physics-Informed Neural Network (PINN) has emerged as a robust, low-to-no data-dependent ML tool to find a solution for various physical phenomena, including elasticity, fluid flow, heat, or sound propagation which are described and understood through solving the corresponding governing equations [25-30]. In PINN, governing equations, mainly Partial Differential Equations (PDEs), can be seamlessly integrated into the ML model loss function. This integration transforms the problem into an optimization task, effectively addressing the original physical problem through computational means. In the following, several studies including the application of PD in PINN will be reviewed. Niu et al. [31] enforced the governing equation of CCM for static equilibrium into the PINN's loss function consisting of individual losses for PDE, boundary conditions, and other constitutive relations to obtain finite-strain plasticity with isotropic hardening. Kamali et al. [32] utilized PINN to estimate the two-dimensional distribution of elastic modulus and Poisson's ratio from domain strain and normal stress boundary conditions. They validated the model with elasticity imaging experiments and numerical data from linear elastic materials under uniaxial loading. Haghighat et al. [33] introduced a PINN using the peridynamic differential operator (PDDO) as an alternative approach to the PD equations derivation in their nonlocal forms [34]. Their proposed PDDO-PINN method accurately captured elasto-plastic deformation of a square domain subjected to indentation by a
rigid body. Ning et al. [35] employed PINN with a PD approach to predict the elastic displacement of homogenous and heterogeneous plates. They showed that utilizing the gradual-refinement sampling technique instead of direct fine sampling leads to enhanced accuracy and time efficiency in the model performance.
While data-driven deep NNs have previously been employed to capture damage and crack propagation [36, 37, 38, 39, 40, 41], research focusing on PINN models in this context is limited. Dourado et al. [42] proposed a repeating recurrent NN to predict damage accumulation from corrosion-fatigue loading by incorporating a modified version of the Paris law, the Walker model, into the loss function to fine-tune network trainable parameters. Goswami et al. [43, 44] predicted crack propagation using phase-field theory integrated into the loss function of PINN. They achieved high accuracy with low computational cost by utilizing the transfer learning technique and re-training only the output layer of the network. Tu et al. [45] introduced the PointNet-based adaptive mesh refinement method to predict crack path in 2D plates with pre-crack by minimizing the variational energy of the system through the PINN optimization process. Zheng et al. [46] applied the principles of irreversible thermodynamics in CDM to the PINN loss function, where elastic residual energy of the damaged domain was minimized. They showed better convergence and robustness by decomposing the main domain into several subdomains and assigning separate networks to each subdomain. It is noteworthy to mention that despite the increasing utilization of PINNs in this field of study, a noticeable gap exists when exploiting the advantages of meshless and non-local methods such as PD.
This study explores a novel integration of the nonlocal PD method in conjunction with PINN to predict damage initiation and propagation in brittle materials. The main challenges in employing this method are the low convergence speed of training, low accuracy in interfaces and discontinuities, and the inefficacy of the PD-INN model in domains with high-density Material Points (MPs). Therefore, the proposed PD-INN is rigorously assessed through diverse case studies encompassing a spectrum of MP densities -- from low to high. A combination of the loss function with the linearized PD governing equation is proposed to increase accuracy and convergence speed. Since obtaining the PD forces introduces a considerable computing overhead, network architectures and hyperparameters are thoroughly assessed to reduce the required training epochs. Furthermore, deformation-gradient aware optimization technique is utilized while training high
density domains to prevent PD-INN from becoming trapped in its trivial solution. Eventually, the transfer learning technique is utilized to reduce the computational time needed for the subsequent time increments.
The remaining sections of this paper are structured as follows: In Section 2, an introduction to the theory of bond-based PD is provided, explaining the linearized form of its governing equation. This section also presents the PD-INN, covering a detailed explanation of the residual-based loss function and the details of the model hyperparameters. In section 3, detailed discussions results are provided and followed by comparisons between the predictions and true deformations. Finally, in section 4, concluding remarks are presented.
## 2 Methods and Implementation
In PD theory, the inherent non-local character of the method is a key feature. This arises from the fact that each material point (MP) is influenced by its neighboring points, incorporating length-scale parameters and long-range force interactions. Essentially, in PD, each MP interacts with points within a predefined radius, referred to as the horizon, forming a sphere (or circle in 2D). In this theory, a domain is discretized into particles with pairwise force interactions occurring through bonds inside the horizon. This discrete approach introduces an integral form of the equation of motion, distinguishing it from the differential equations found in CCM. This makes PD capable of dealing with problems including discontinuity and heterogeneity. By assuming small deformations and removing the time-derivative term in the equation of motion, linearized form of the PD governing equation can be implemented to find the quasi-static solution of the PD governing equation. The linearized PD governing equation is included in building the loss function in addition to other common residual-based loss terms such as boundary conditions and internal forces. In the subsequent subsections, a detailed description of the linearized PD as well as PD-informed NN is provided.
### Linearized bond-based peridynamic
The equation of motion in PD is achieved by considering the force balance of a MP **x**[7, 8]:
\[\left(\rho dV_{{}_{x}}\right)\ddot{\mathbf{u}}\left(\mathbf{x}\,;t\right)= \int_{\beta_{x}}\left[\ddot{\mathbf{f}}\left(\mathbf{q}\,;\mathbf{x}\,;t\right) dV_{{}_{\mathbf{q}}}\right]dV_{{}_{\mathbf{x}}}+\ddot{\mathbf{b}}\left( \mathbf{x}\,;t\right)dV_{{}_{x}} \tag{1}\]
where \(\beta_{x}\) is the domain in the horizon of \(\mathbf{x}\), \(\rho\) is density, \(\mathbf{b}\left(\mathbf{x}\text{',}t\right)\)is body force density applied, \(\mathbf{\bar{f}}\left(\mathbf{q}\text{',}\mathbf{x}\text{',}t\right)\) is pairwise bond force density exists in the horizon of \(\mathbf{x}\) and is governed by constitutive equations, \(dV_{\mathbf{x},\mathbf{q}}\) are volumes of the points in interaction, and \(t\) is time. Symbol (\(\mathbf{\bar{}}\)) stands for the deformed configuration. The integral term of the equation is to maintain interactions of all MPs within the horizon and is substituted by a summation of all bond forces that exist in \(\mathbf{x}\)'s neighborhood. MPs near the boundaries, where the horizon area is only partially defined, exhibit slightly softer mechanical properties compared to those within the interior of the domain. To address this issue, the commonly employed Geometry Modification method [47, 48, 49] is utilized. This method involves the addition of a layer of fictitious material points, typically with a thickness equal to the horizon, to the main domain where displacement boundary conditions are applied.
Bond-based PD is one of the most common and widely used material models in which two MPs exert the same amount of load in their bond direction [50, 51, 52]. The value of bond force is obtained as follows:
\[\mathbf{\bar{f}}\left(\mathbf{q}\text{,}\mathbf{x}\right)=\omega\,c\,\,s\left( 1-\mu\right)\mathbf{\bar{M}},\,\,c=\frac{9E}{\pi t\,\delta^{3}},\,\,s=\frac{e} {\xi},\,\,\,\omega=e^{-4\xi^{2}/\delta^{2}},\,\,\mu=\begin{cases}1&\,\,\,\,s \geq s_{cr}\\ 0&\,\,\text{otherwise}\end{cases} \tag{2}\]
where \(c\) is bond stiffness (or micromodules), \(E\) is Young's Modulus, \(t\) is thickness and \(\delta\) is horizon radius equal to 3 times grid space. The presented equation for micromodules is for 2D plane stress. 2D plane strain or 3D constitutive models of \(c\) can also be found in the literature. \(s\) is the bond stretch which is obtained by the division of bond elongation \(e\) to the initial length of the bond, \(\xi\). \(\omega\) is an influence function or length correction factor to reduce the interaction effects of MPs when \(\xi\) increases. \(\mu\) is brittle damage parameter which is equal to 1 when the stretch is beyond the critical stretch \(s_{cr}\) and 0 otherwise. In this study, \(s_{cr}\) is obtained by known fracture energy \(G_{0}\) as \(s_{cr}=\sqrt{4\pi G_{0}/9E\,\delta}\). Finally, \(\mathbf{\bar{M}}\) is unit vector of bond direction.
The analytical solution of the PD integro-differential motion equation (1) is not feasible, necessitating the utilization of numerical techniques for both time and space discretization. While explicit time integration schemes with controlled time steps are required for dynamic loadings [10, 53], quasi-static analyses are treated with different approach, mainly iterative or direct methods where dynamic effects like wave propagation are diminished and larger time steps are used.
Notably, Adaptive Dynamic Relaxation method (ADR) was introduced by Kilic and Madenci [54] to obtain steady-state solution of the dynamic PD equation, knowing the fact that the transient response will converge to its steady-state condition which is equal to static solution of the problem. In this method, an artificial damping term is added to the governing equation of PD, Eq. (1), and central-difference iterative method is utilized to find the steady-state solution after a number of iterations [11, 12]. However, this approach can be time-consuming when dealing with crack propagation problems since loads are applied incrementally and, in each increment, a certain number of iterations is required to converge to the steady-state solution. Moreover, identifying the most effective damping coefficient is not always straightforward, and improper numerical parameters may lead to a low convergence rate. To mitigate these challenges, direct solution is proposed by solving the system of equation \(\mathbf{K}\mathbf{u}=0\) where \(\mathbf{K}\) represents the global stiffness matrix and \(\mathbf{u}\) denotes the displacement vector of all MPs [55, 56]. In order to find stiffness matrix, the linearized form of the governing equation with respect to the displacements \(\mathbf{u}\) can be derived. This method will be reviewed here. According to Figure 1, elongation \(e\) can be obtained as \(\left|\vec{\xi}+\vec{\eta}\right|-\left|\vec{\xi}\right|\) where \(\vec{\eta}=\vec{\mathbf{u}}_{q}-\vec{\mathbf{u}}_{x}\) is relative displacement. Therefore, \(s=\left(\left|\vec{\xi}+\vec{\eta}\right|-\left|\vec{\xi}\right|\right)/ \left|\vec{\xi}\right|\) and \(\vec{\mathbf{M}}=\left(\left|\vec{\xi}+\vec{\eta}\right|\right)/\left|\vec{ \xi}+\vec{\eta}\right|\). The first step to linearize \(\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{}^{\prime}}\right)\) with respect to \(\vec{\eta}\) is to write the first order Taylor's series expansion by considering a small-near-to-zero \(\eta=\left|\vec{\eta}\right|\). Thus, we have:
\[\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{}^{\prime}}\right) =\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{}^{\prime}}\right) \bigg{|}_{\mathbf{q}=0}+\frac{\partial\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{ \prime}},\mathbf{x}^{{}^{\prime}}\right)}{\partial\mathbf{\eta}}\bigg{|}_{ \mathbf{q}=0}\eta \tag{3}\]
where \(\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{}^{\prime}}\right) \bigg{|}_{\mathbf{q}=0}=\vec{\mathbf{f}}\left(\mathbf{q},\mathbf{x}\right)=0\) states the undeformed configuration. Evaluating the second term of Taylor's expansion and substituting \(\eta=0\), Eq. (3) can be written as [57]:
\[\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{}^{\prime}}\right) =\frac{\partial\vec{\mathbf{f}}\left(\mathbf{q}^{{}^{\prime}},\mathbf{x}^{{} ^{\prime}}\right)}{\partial\mathbf{\eta}}\bigg{|}_{\mathbf{q}=0}\eta=\omega \left(1-\mu\right)\frac{\vec{\xi}\otimes\vec{\xi}}{\left|\vec{\xi}\right|^{3}}\, \mathbf{\eta} \tag{4}\]
where \(\otimes\) is dyadic product. Eq. (4) can be shown in vector representation as:
\[\bar{\mathbf{f}}\left(\mathbf{q}\,^{\prime},\mathbf{x}\,^{\prime}\right)=\frac{ \omega c\left(1-\mu\right)}{\left|\bar{\mathbf{\xi}}\right|^{3}}\Bigg{(}\left( \xi_{1}\quad\xi_{2}\right)\cdot\Bigg{(}\eta_{1}\Bigg{)}\Bigg{)}\Bigg{(}\xi_{1 }\Bigg{)} \tag{5}\]
in this equation, \((.)\) is vector dot product and subscripts 1,2 are directional components (in 2D). Final step is to rewrite the Eq. (5) in terms of the pairwise particle displacements \(\mathbf{\bar{u}}_{x}\) and \(\mathbf{\bar{u}}_{q}\) :
\[\bar{\mathbf{f}}\left(\mathbf{q}\,^{\prime},\mathbf{x}\,^{\prime}\right)= \frac{\omega c\left(1-\mu\right)}{\left|\bar{\mathbf{\xi}}\right|^{3}}\Bigg{(} \begin{matrix}\xi_{1}\xi_{1}&-\xi_{1}\xi_{1}&\xi_{1}\xi_{2}&-\xi_{1}\xi_{2}\\ \xi_{2}\xi_{1}&-\xi_{2}\xi_{1}&\xi_{2}\xi_{2}&-\xi_{2}\xi_{2}\end{matrix} \Bigg{)}\cdot\left(u_{q,1}\quad u_{x,1}\quad u_{q,2}\quad u_{x,2}\right)^{T} \tag{6}\]
where \(\left(\cdot\right)^{T}\) is vector transpose. By eliminating the acceleration term and substituting the integral term by the summation of all bond-forces inside the horizon of \(\mathbf{x}\), Eq. (1) is rewritten as \(\sum_{\beta_{x}}\bar{\mathbf{f}}\left(\mathbf{q}\,^{\prime},\mathbf{x}\,^{ \prime}\right)dV\)\({}^{2}=0\). Eventually, stiffness matrix can be easily derived by using the Eq. (6). Please note that it is assumed the domain is discretized uniformly, so \(dV_{x}=dV_{q}=dV\) and no body force is applied. A similar approach in FEM can be applied here to eliminate known boundary displacements from the system of equations and subsequently derive the right-hand-side (**rhs**) vector to solve the unknown deformations.
### Peridynamic-informed Neural Network
In this section, the details of the proposed PD-informed NN are discussed. PINNs can be embodied with thousands-to-millions of parameters, enabling them to provide more accurate results while capturing nonlinear patterns through multiple transformations from the input layer to the output layers. Therefore, fully connected deep NN (FCNN) is used to build our proposed PD-INN and
Figure 1: a schematic presentation of a PD bond, before and after a small deformation.
conduct feed forward process of the input variables, here the coordinates of all MPs, and train the network through backpropagation algorithm. Typically, PINNs consist of an input layer (L=0), \(l\) hidden layers (L=1:_l_) and an output layer (L=_l+1_). Each layer contributes its data as input to the subsequent layer, having processed the output of each unit through an activation function which can be represented as:
\[z^{L}=\varphi\Big{(}\mathbf{W}^{L-1}\mathbf{z}^{L-1}+\mathbf{b}^{L-1}\Big{)}, \ L=1:l \tag{7}\]
where \(z^{L}\) and \(z^{L-1}\) are the output of (_L_)th and (_L_-1)th layer, respectively, \(\mathbf{W}^{L-1}\) and \(\mathbf{b}^{L-1}\) are the concatenated weights and biases of previous layer, \(\varphi\) is activation function which is _tanh_ for all layers except it is linear for the output layer. The network's output should satisfy the PD constraints in order to predict a precise deformation. Consequently, the following total loss function, denoted as \(\mathcal{L}_{\text{total}}\), is constructed to provide FCNN with the knowledge of PD governing equation:
\[\mathcal{L}_{\text{total}}=\mathcal{L}_{\text{PD\_forces}}+\mathcal{L}_{\text {B.C.s}}+\mathcal{L}_{\text{linearized}}+\Big{(}\mathcal{L}_{\text{true\_data}} \Big{)} \tag{8}\]
where \(\mathcal{L}_{\text{PD\_forces}}\) is the loss corresponding to the forces of internal material points and free boundaries, \(\mathcal{L}_{\text{B.C.s}}\) is loss due to the applied boundary conditions like Dirichlet BCs, \(\mathcal{L}_{\text{linearized}}\) is the loss for the residual of the linearized PD, and finally, \(\mathcal{L}_{\text{true\_data}}\) is the (optional) data-driven loss according to the known data. A detail description of these losses is provided in the following equation:
\[\begin{split}\mathcal{L}_{\text{PD\_forces}}&= \mathcal{L}_{\text{int\_forces}}+\mathcal{L}_{\text{def. grad}}+\mathcal{L}_{\text{tmp}}\\ &=\Big{\|}\int_{\beta_{z}}\left[\overline{\mathbf{f}}\left( \mathbf{q}\,\text{'},\mathbf{x}\,\mathbf{i}\,\mathbf{f}\right)dV_{\mathbf{q}} \right]dV_{\mathbf{x}}\Big{\|},\ \mathbf{x}\in\Omega\,,\ \Omega_{2}\,,\ \Omega_{3}\\ \mathcal{L}_{\text{B.C.s}}&=\left\|\mathbf{u}_{\text{ BC\_pred.}}-\mathbf{u}_{\text{BC}}^{\ast}\right\|\\ \mathcal{L}_{\text{linearized}}&=\left\|\mathbf{K} \,\mathbf{u}_{\text{pred}}\right\|\ \text{or}\ \left\|\mathbf{K}^{-1}.\mathbf{rhs}-\mathbf{u}_{\text{pred}}\right\|\\ \mathcal{L}_{\text{true\_data}}&=\left\|\mathbf{u}_{ \text{pred.}}-\mathbf{u}^{\ast}\right\|\end{split} \tag{9}\]
where \(\Omega_{i}\) is forces of all internal MPs except those with boundary conditions applied, \(\Omega_{2}\) is the forces of MPs with higher deformation gradient and \(\Omega_{3}\) correspond to damaged MPs. (*) shows
the known displacements either from the applied boundary conditions or true numerical/experimental data. \(\ell^{2}\)-norm function \(\left\|\bullet\right\|\) is implemented to calculate each loss term's value. Two options exist for constructing the \(\mathcal{L}_{\text{linearized}}\): the first involves calculating \(\left\|\mathbf{K}\,\mathbf{u}_{\text{pred}}\right\|\) as the loss, which has the same order of magnitude as \(\mathcal{L}_{\text{PD\_forces}}\). The second option is to compute the residual of \(\left\|\mathbf{K}^{-1}\text{.}\mathbf{rhs}-\mathbf{u}_{\text{pred}}\right\|\) which shares a similar order of magnitude with losses \(\mathcal{L}_{\text{B.C.s}}\) and \(\mathcal{L}_{\text{true\_data}}\). The choice made here is important to build the weighted some of all losses. Here, we followed the second choice which found to enhance model performance better. It should also be noted that, while \(\mathcal{L}_{\text{true\_data}}\) is not a general requirement in PINNs, its presence would further accelerate model convergence rate. Once the weighted sum of all these losses is obtained, it is used to perform backpropagation algorithm. This is conducted using the automatic differentiation approach which calculates the gradient of the \(\mathcal{L}_{\text{total}}\) with respect to the model trainable parameters \(\mathbf{W}\) and \(\mathbf{b}\). To this end, one of the most used and open-source ML packages, called TensorFlow [58], is used to calculate such gradients. This library supports GPU accelerated operations via Nvidia CUDA which can increase training speed from several to a hundred times in comparison to CPU implementation.
As stated earlier, the order of magnitude of the loss \(\mathcal{L}_{\text{PD\_forces}}\) presented in Eq. (8) differs from that of either \(\mathcal{L}_{\text{B.C.s}}\) or \(\mathcal{L}_{\text{true\_data}}\) given their distinct units. Therefore, the loss minimization process is basically a multi-objective optimization process and is more likely to become trapped in the local minima of a non-convex loss function. To address this challenge, the balancing of different loss terms' contribution in \(\mathcal{L}_{\text{total}}\) is required. Therefore, the weighted sum of the different terms is recommended for controlling the magnitude of each term. Additionally, an alternative approach involves monitoring and adjusting the gradient directions corresponding to each term, which can enhance both time efficiency and accuracy. However, it's worth noting that this adjustment becomes less critical when \(\mathcal{L}_{\text{linearized}}\) or \(\mathcal{L}_{\text{true\_data}}\) are included.
Finally, an extension of stochastic gradient descent (STG) known as the Adam optimizer is employed to perform fine-tuning of the model trainable parameters. The Adam optimizer dynamically adjusts the learning rate for each trainable parameter based on the history of gradients.
This adaptive approach aids in achieving faster convergence and improved accuracy compared to using a fixed learning rate. Additionally, it is beneficial to reassign and reduce the initial learning rate throughout the training procedure by implementing a schedule decay rate. This strategy allows the algorithm to descend towards the global minimum, mitigating the risk of oscillations around this critical point.
Figure 2 provides a representation of the steps undertaken by the PD-INN to predict the deformation within an arbitrary domain subjected to loading conditions. The initial step is to discretize the domain into a series of material points, where their coordinates besides other geometrical parameters like pre-crack length and position (if applicable) are served as inputs to the first hidden layer. Subsequently, the deformation components of all material points in the domain are generated as outputs, which in turn are employed to calculate the overall loss. Identifying the MPs to which boundary conditions are applied, those MPs located within the domain, global stiffness matrix **[K],** and the neighbors inside the horizon of each MP would make it possible to compute \(\mathcal{L}_{\text{total}}\). Eventually, once the value of total loss reaches a threshold \(\mathcal{L}_{\text{thr}}\), the training is stopped, and the final deformation is obtained.
## 3 Results and Discussion
Figure 2: Schematic representation of the PD-INN model training process in this study
Here, the results obtained from PD-INN are validated through several benchmark problems and compared with the ground truth deformation from high-fidelity techniques such as PD direct numerical method, FEM and X-FEM. In section 3.1, several network architectures are investigated to identify those that offer superior accuracy and a fast convergence rate. In section 3.2, the PD-INN is trained to predict the deformation of a 2D plate with various pre-crack lengths and positions. The training loss curve is also presented to assess the effect of proposed loss function and hyperparameters on the model's accuracy and convergence rate. In section 3.3, the PD-INN is further evaluated to predict deformation of a domain consisting of stress localization around a circular cutout.
### PD-INN architecture
The objective of this analysis was to discover the network configuration that would meet the desired accuracy, \(\mathcal{L}_{\text{thr}}\), in the most time efficient way. Therefore, the PD-INN was trained to predict the deformation of a 2D plate with an arbitrary pre-crack as depicted in Figure 4. This analysis encompassed the choices over the network's input/output shape, number of hidden layers, and the units within each layer. Table 1 shows the details and specification of each network architecture, including the number of training parameters and the runtime corresponding to each network. DOF is the degree of freedom, equal to 2 for 2D analyses, and nMP is the number of material points within the domain.
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Network & \begin{tabular}{c} Input/Output \\ shape \\ \end{tabular} & \begin{tabular}{c} \# Hidden \\ layers \\ \end{tabular} & \begin{tabular}{c} \# \\ units \\ \end{tabular} &
\begin{tabular}{c} \# trainable \\ parameters \\ \end{tabular} & \(\mathcal{L}_{\text{total}}<\mathcal{L}_{\text{thr}}\) & Runtime\({}^{*}\) \\ \hline i & DOF & 6 & 60 & 18602 & No & \(>6\) hr \\ \hline ii & DOF & 6 & 100 & 51002 & No & \(>6\) hr \\ \hline iii & DOF & 10 & 60 & 33242 & No & \(>6\) hr \\ \hline iv & DOF & 10 & 100 & 91402 & No & \(>6\) hr \\ \hline v & DOF & 15 & 150 & 317852 & No & \(>6\) hr \\ \hline vi & DOF\(\times\)nMP & 6 & 60 & 1240460 & Yes & \(\sim 2\) hr, 50 min \\ \hline vii & DOF\(\times\)nMP & 6 & 100 & 2080700 & Yes & \(\sim 2\) hr, 55 min \\ \hline viii\({}^{**}\) & DOF\(\times\)nMP & 6 & 60 & 1240460 & Yes & \(\sim 15\) min \\ \hline \hline \end{tabular}
\end{table}
Table 1: Various networks of choice to build PD-INN and their statistics
In all architectural configurations detailed in Table 1, excluding network viii, the Glorot (Xavier) normal initializer and the hyperbolic tangent (_tanh_) activation function were adopted for constructing the FCNN. To fine-tune the network's trainable parameters, Adam optimizer was employed with an initial learning rate set at 5e-4 and a schedule decay rate 0.9 applied every 1000 epochs. The optimal learning rate for the networks i-v was observed to fall within the range of 5e-4 to 5e-3, although they failed to converge to the target threshold loss; values exceeding or falling below this range resulted in decreased network performance. In contrast, an examination of the training loss curve, as illustrated in Figure 3, reveals that this learning rate range would make the networks vi and vii struggle to minimize the loss and lead to considerable fluctuations during the training. For consistency in this comparative analysis, identical configurations are maintained. It is evident, both from Table 1 and Figure 3, that a reduction in the required number of epochs to attain the threshold loss has a noticeable effect on the overall computational time. This phenomenon can be attributed to the relatively more time-consuming nature of PD internal force calculations as opposed to gradient computations and network parameter updates. Observing almost similar performance in networks vi and vii, network vi architecture is selected to conduct analyses in the next subsections. Detailed information on the network viii performance, which exhibits a significant enhancement in convergence rate and training speed, is provided in section 3.2.
### Plates with pre-crack
As it was determined in section 3.1, network vii architecture with input/output shape of DOF\(\times\)nMP, and six hidden layers with 64 units in each layer is applied to build the FCNN. A training dataset consisting of the true deformations of a rectangular plate (W\(\times\)H) 0.1m\(\times\)0.05m under tension was used to train the network with hybrid loss function. The training dataset includes ground truth deformation associated with six different vertical pre-cracks of length 10mm and 30mm at positions x=0.02m, 0.05m and 0.08m. It should be noted that given the limited amount of training data in this study, the objective is not to create a surrogate model. Furthermore, a more complex ML model like parallel FCNNs or convolutional neural network (CNN) is required to build such a model. The purpose of this training step is to accelerate model training and create a sensitivity concerning the geometrical parameters of crack lengths (\(c_{len}\)) and positions (\(c_{pos}\)) as shown in Figure 4. Thus, these parameters are combined with the input layer of the FCNN. Once the network is trained, it is fine-tuned to meet the threshold training loss and predict an accurate deformation. Figure 4 presents the displacement contours with the mean absolute error (MAE) and the error contour of the relative final deformed configuration for three different cases. Each domain is under uniaxial tension of U1=0.2mm where it is ensured no further damage will happen. The following equation calculates the MAE:
Figure 3: Various PD-INN architectures and their performance on a plate with an arbitrary discontinuity
\[\text{MAE}=\frac{\sum_{i=1}^{n}\left|U_{\text{Predication}}^{\ i}-U_{\text{true\_ data}}^{\ i}\right|}{n} \tag{10}\]
where \(U\) is displacement and \(n\) is the total number of MPs. According to the results presented in Figure 4, the PD-INN prediction is in good agreement with true deformation where the MAE is as low as three orders of applied displacement boundary condition.
Upon close evaluation of the relative error contour presented in Figure 4, the area on the left side of the pre-cracks exhibits higher error in comparison to its other side. This observation stems from
Figure 4: PD-INN prediction of deformation U1 in presence of pre-crack b) ground truth U1 obtained from PD direct method implementation
the propagation hypothesis [59, 60] which states that in PINNs, the boundary conditions propagate from boundaries to the inner domain. It is also observed that the optimizer may stop this propagation and get stuck at its trivial solution \(u=\!0\) for the rest of the domain. As a result, a narrow region of a high deformation gradient will exist near the boundaries. This issue is more probable in domains with higher MP densities. While building the hybrid loss function with \(\mathcal{L}_{\text{true\_data}}\) would mitigate this issue, further regularization and consideration should be made in case of an unseen case without ground truth. Low speed of convergency is another concern regarding the cases where \(\mathcal{L}_{\text{true\_data}}\) is removed from the loss function. To address these issues, after extensive evaluation of the network's performances, the following enhancements was found to be effective for the framework of the PD-INN:
* Random Normal kernel initializer with 0.001 standard deviation was selected over the common choice of Glorot (or Xavier) initializer. This choice was made to specifically minimize the scattering of initial domain predictions.
* Initial learning rate equal to 1e-5 with an exponential decay learning rate of 0.9 is considered every 5e2 epochs.
* Forcing regions with higher deformation gradients by considering the \(\mathcal{L}_{\text{def\_grad}}\) to ensures smooth propagation from boundaries to the inside of the domain. Common masks for computing the components of gradient in a 2\(\times\)2 neighborhood [61] is M\({}_{\text{x}}\)=[-1 1;-1 1] or M\({}_{\text{y}}\)=[1 1;-1 -1]. In order to generalize this approach to various domains with or without crack, a cyclical annealing schedule [62] is found to be effective which considers the factor \(\beta\) in an increase-reset-to-zero pattern every 5e3 epochs as depicted in Figure 5.
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better indicate the influence of the proposed linearized PD loss, the training is performed with (\(\gamma=1\)) and without (\(\gamma=0\))\(\mathcal{L}_{\text{linearized}}\). The results indicate that attaining the desired accuracy with \(\gamma=0\) requires more than 50e3 epochs of training. In contrast, approximately 17e3 epochs were found to be sufficient when using \(\gamma=1\), although the potential for achieving even higher accuracy remains. It can also be seen from the Figure 3 and Figure 5 that having a PD-INN sensitive to the pre-crack parameters resulted in the considerably reduced loss value at the beginning of the training, where the displacement propagation from boundaries has already happened.
Once the PD-INN is trained, it is prepared to be implemented for the damage analysis. The trained model parameters are transferred to a new network for each following time increment using the transfer learning technique. This would decrease the training efforts considerably compared to a network without having a previous understanding of the physics of the problem. In addition, it is found that modifying the network so that only the parameters assigned to the output layer are trainable would be sufficient in converging to the final deformation. Considering all these methods as well as previously suggested enhancements would lead to a fast and accurate prediction in crack propagation and evaluation of the strength of the material. The crack propagation using PD-INN
Figure 5: Comparison of training loss for training with or without proposed enhancements
for the case \(c_{{}_{len}}=\,0.2\,\) and \(c_{{}_{pos}}=0.35\,\), shown in Figure 5, is investigated and compared to the true crack path obtained from direct method of calculation. Therefore, the plate is subjected to the uniaxial tension with the rate of 0.02 mm.s-1, material property \(E\)=192 GPa and fracture energy is \(G_{0}=83\,\)kJ/m\({}^{2}\), grid space 1 mm and horizon radius of 3 mm. Figure 6 presents the PD-INN crack path prediction and Force-Displacement curve. The PD-INN predictions are in good agreement with the results obtained from PD direct solution, with almost 3% error in stiffness calculation. It's worth noting that in this problem, the crack has not grown vertically due to the off-center position of the pre-crack and, as a result, existence of bending moments around the crack tip.
Next, a more complex case study is conducted to assess the PD-INN's predictive capabilities concerning crack propagation within a square plate containing ten randomly located pre-cracks. Thus, a square plate of width 2 in, grid space 0.015 in (with approx. 18600 MPs) is modeled as presented in Figure 7(a). The plate is stretched vertically until a complete failure happens and the final crack path is compared with the result obtained from Extended-FEM (XFEM) method [63] as depicted in Figure 7(b). In both methods, crack propagation originated from the central pre-crack, eventually merging with two adjacent pre-cracks situated on either side. Subsequently, the crack propagated from the tips of these two pre-cracks and extended towards the domain's boundary. Overall, our findings demonstrate a good agreement between the outcomes obtained through both methods.
Figure 6: The comparison of crack path (top) and force-displacement curves (bottom) obtained from PD-INN and direct PD solution
### A plate with circular cut-out
To further evaluate the effectiveness of the proposed PD-INN model in a domain with no initial cracks, a 50mm\(\times\)50mm square plate with the thickness of 0.5mm containing a central D=20mm circular hole was considered. Horizon radius of 0.6 mm, fracture energy of \(G_{0}=83\) kJ/m\({}^{2}\), and critical stretch \(s_{cr}=3.16\)e-2 was implemented. As illustrated in Figure 8-(a), a quarter of the domain is modeled where symmetry boundary condition was applied to both sides. The domain was stretched until complete failure occurred, and the results were compared with PD direct numerical solution.
In Figure 8-(b), the final crack path is depicted, demonstrating the expected crack propagation originating from regions with high stress concentrations. For visual purposes, the domain is
Figure 8: (a) schematic of the model and boundary conditions (b) crack path presentation (c) comparison of the PD-INN and PD numerical solution
Figure 7: A comparison of crack path predicted by PD-INN and XFEM [63] in a square plate with random pre-cracks (a) Initial configuration (b) Final crack path (c) sequences of crack growth
presented with a 90-degree rotational pattern. Figure 8-(c) compares the load-displacement curve obtained from PD-INN with from direct PD numerical solution. The small variations observed between these two results can be attributed to the value of threshold loss considered in this analysis. Despite these variations, our PD-INN demonstrates its capacity to accurately capture the correct crack growth, showcasing a satisfactory level of agreement when compared with the results obtained from direct numerical method.
## 5 Conclusion
This study investigates the potential of combining PD and PINN methods to study crack propagation in brittle structures. By incorporating the governing equations of peridynamics into the neural network's loss function, the proposed PD-INN has the capability to predict deformations in the presence of structural discontinuities. Several case studies involving plates with pre-existing cracks or circular cutout have been examined to evaluate the PD-INN's performance and validate its predictions. It has been shown that the conventional fully connected neural network (FCNN) architecture struggles to interpret geometric features such as pre-cracks within the domain, resulting in convergence failure. Therefore, different PD-INN architectures have been evaluated to discover the network with the potential of crack prediction.
The PD-INN model introduced in this study has maintained the significance of both accuracy and computational efficiency. To have a balance between these two aspects, a novel approach that integrates the residual of the linearized PD equation into the loss function has been introduced. This combination, enhanced by well-adjusted hyperparameters, has led to significant improvements in the effectiveness of the PD-INN. A substantial reduction in computational time has been achieved, by an approximately 12-fold acceleration of convergence rate, showing the practicality and feasibility of this approach. Moreover, a challenge associated with employing PD-INN for a high-density domain, wherein the risks arise converging towards the trivial solution, has been addressed through the implementation of a gradient-aware technique.
To further enhance efficiency in problems associated with damage evolution, the technique of transfer learning has been successfully applied. In addition, a partial training strategy for subsequent time increments has been implemented where only parameters associated with output layer are tuned. These combined advancements not only verify the effectiveness of the proposed
approach but also extend the horizons of PD and PINN, promising robust solutions in computational fracture mechanics applications.
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